Microscopic Black Holes and
Extra Dimensions
Olav Aursjø
Thesis submitted for the degree of
Candidatus Scientiarum
Department of Physics
University of Oslo
May 2005
Version 1.0: Submitted version (May 2005)
Version 1.1: Minor corrections (June 2005)
Acknowledgements
I would first of all like to thank my supervisor Finn Ravndal for all the help and
guidance he has given me. He suggested a topic that has proven interesting and
challenging to study.
I would also like to thank all my fellow students at the theory group for making the group a nice place to study. In addition, I would specially like to thank
Morad Amarzguioui and Torquil MacDonald Sørensen for useful discussions concerning both physics and computer problems.
Olav Aursjø
Oslo, April 2005
iii
Contents
1 Introduction
1
2 Gravity
2.1 Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Solid Angle . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Newtonian Gravity in D Dimensions . . . . . . . . . . . .
2.1.3 The Gravitational Potential . . . . . . . . . . . . . . . . .
2.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Schwarzschild Solution in D Dimensions . . . . . . . .
2.2.2 Isotropic Coordinates in D Dimensions . . . . . . . . . . .
2.2.3 Linearized Gravity . . . . . . . . . . . . . . . . . . . . . .
2.2.4 The Planck Scale . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 The Gravitational Constant in D Dimensions and the Fundamental Planck Mass . . . . . . . . . . . . . . . . . . . .
2.3 Extra Dimensions and the Hierarchy Problem . . . . . . . . . . .
2.3.1 Compact Spatial Dimensions . . . . . . . . . . . . . . . . .
2.3.2 The Retrieval of the Relations of Four Dimensions . . . . .
2.3.3 Higher Dimensional Black Holes in a World with Compact
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Black Hole Thermodynamics
3.1 The Black Hole Temperature . . . . . . . . . . . . . . . . .
3.1.1 Statistical Mechanics and Quantum Field Theory .
3.1.2 The Unruh Temperature . . . . . . . . . . . . . . .
3.1.3 The Temperature of a Black Hole in D Dimensions
3.2 The Black Hole Entropy . . . . . . . . . . . . . . . . . . .
3.2.1 The Black Hole Entropy in D Dimensions . . . . .
3.3 The Stefan-Boltzmann Law . . . . . . . . . . . . . . . . .
3.3.1 The Black Hole Luminosity . . . . . . . . . . . . .
3.3.2 The Black Hole Lifetime . . . . . . . . . . . . . . .
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vi
4 Pair Production and the Hawking Effect
4.1 Klein’s Paradox . . . . . . . . . . . . . . . . . . . . .
4.1.1 Introduction to Klein’s Paradox . . . . . . . .
4.1.2 The Paradox . . . . . . . . . . . . . . . . . .
4.1.3 The Resolution of the Paradox . . . . . . . . .
4.2 The Hawking Effect . . . . . . . . . . . . . . . . . . .
4.2.1 Quantum Field Theory in Curved Spacetime .
4.2.2 The Hawking Temperature . . . . . . . . . . .
4.2.3 The Energy-Momentum Tensor . . . . . . . .
4.2.4 A Heuristic Picture of the Hawking Radiation
CONTENTS
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5 Black Hole Production
5.1 The Black Hole Cross-Section . . . . . . . . . . . . . . . . .
5.2 Gravitational Capture . . . . . . . . . . . . . . . . . . . . .
5.2.1 The Classical Relativistic Approach . . . . . . . . . .
5.2.2 The Semi-Classical Approach . . . . . . . . . . . . .
5.3 Gravitational Shock-Waves . . . . . . . . . . . . . . . . . . .
5.3.1 Aichelburg-Sexl Shock-Waves . . . . . . . . . . . . .
5.3.2 Two Colliding Gravitational Shock-Waves . . . . . .
5.4 Other Calculations Concerning the Black Hole Cross-Section
5.4.1 The Sub-Relativistic Limit . . . . . . . . . . . . . . .
5.4.2 Eikonal-Approximation . . . . . . . . . . . . . . . . .
5.4.3 Voloshin Exponential Suppression . . . . . . . . . . .
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6 Conclusion
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A HEP-units (c = ~ = 1)
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B Alternative Definitions of the Gravitational Constant
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C The Surface Gravity of a Static and Diagonal Metric
121
D Calculations of Connection Coefficients
123
E Scalar Products for Bosons
127
F Scalar Products in the Schwarzschild Geometry
141
Bibliography
147
Chapter 1
Introduction
In physics, one of the ultimate goals is to unify the fundamental forces of nature.
Today physicists have been able to unify three of the four known fundamental
forces. The electromagnetic, the strong and the weak nuclear forces are described
in a single quantum field theory, the standard model. The fourth fundamental
force, gravity, on the other hand is described by the general theory of relativity.
Since the other fundamental interactions are quantized, it therefore seems natural
that in a grand unified theory, a theory of all the fundamental forces, gravity is
quantized as well (see Fig. 1.1).
String theories have by many in the recent years been regarded as the best
candidates for such a unified theory. These are theories where the fundamental
elements are one-dimensional strings and higher dimensional branes. A brane is a
higher dimensional generalization of a two dimensional membrane. The elementary particles we observe can then be described as different excitation modes of
the elementary strings. With these kind of theories it may be possible to unify the
known fundamental forces. However, to make the string theories mathematically
consistent, six extra spatial dimensions are needed. These theories all contain a
quantum theory of gravity. In addition, an eleven-dimensional theory of gravity
exists, which is connected to the ten-dimensional theories.
The idea of including extra dimensions, to achieve the goal of unifying physics,
is not a new one. Already the year before Einstein in 1915 introduced his theory
of general relativity, Gunnar Nordström suggested a unification of gravity and
electromagnetism with the introduction of a fifth dimension. These forces were
the two only forms of interaction known at that time. But this idea was forgotten for some time with the eruption of the First World War. But in April 1919
Theodor Kaluza introduced independently, in a letter to Einstein, a fifth dimension in an attempt to unify Einstein’s theory of gravity and Maxwell’s theory of
light. Oskar Klein (1926) contributed, in this quest, with his assumption that
the extra dimension was compactified. The Kaluza-Klein theory was a fact. This
theory includes an extra space dimension that is rolled up into a tiny circle, i.e.
compactified. And in this five dimensional theory, there is only one underlying
1
2
Chapter 1. Introduction
General
Relativity
Quantum
Gravity
Quantum
Field
Theory
Special
Relativity
Non-Relativistic
Quantum
Gravity
Newtonian
Mechanics
c−1
Galilean
Mechanics
G
Quantum
Mechanics
~
Figure 1.1: A visualization of the connections and evolution of
physics. Here the introduction of the gravitational constant G, the
light speed c and the Planck constant ~ signifies the introduction of
a new theory. Quantum gravity is here included as a possible next
step.
force, gravity. But in the four-dimensional spacetime observed at great distances,
it appears to be three kinds of forces, among these a gravitational and an electromagnetic force. This topic was initially a popular topic for research, but lost
much of its interest with the introduction of quantum mechanics.
In recent years the topic of extra dimensions has experienced a renewed interest. This renewed interest is also due to the exciting possibility of observing
new and spectacular physical phenomenas at far lower energy scales than otherwise. Even at energies available in the not so distant future, these phenomenas
could appear. Among these is the creation of higher dimensional semi-classical
microscopic black holes. The possibility of observing these objects, is viewed as
an opportunity to perhaps discover new intriguing physics.
This thesis is an attempt to give a consistent introduction to some of the theory behind this phenomenon of higher dimensional microscopic black holes. We
will investigate gravitational properties, as well as quantum mechanical properties
of such objects.
Chapter 2 is a general introduction to how gravity may be modified with
the inclusion of extra dimensions. Here both the Newtonian and Einsteinian
gravity are generalized. In the first section of this chapter we will give an introduction to Newtonian gravity and show how this can be changed into a theory
with more than three spatial dimensions. In the second section we will introduce
the theory of general relativity and show how the Newtonian theory is just a
3
limit scenario of this theory. We continue in this section to study how a generalization to spacetimes with extra dimensions modifies the theory. This part
of the chapter involves the derivation of the D-dimensional Schwarzschild spacetime. In connection with the retrieval of the Newtonian theory we express this
Schwarzschild spacetime in isotropic coordinates. This makes us able to find a
linearized isotropic D-dimensional Schwarzschild solution never before seen in
the literature. The linearized solution is used to find the Poisson equation in
D dimensions, i.e. the Newtonian limit. In the last section of this chapter we
review a possible solution to the hierarchy problem found in higher dimensional
theories. We introduce here compact dimensions to explain how a higher dimensional theory could be compatible with observations. In this section we show how
a relation between such a theory and the 4-dimensional theory may be found.
In Chapter 3 we consider thermodynamical properties of a higher dimensional
black hole. Here we first apply finite-temperature quantum field theory to the
example of an detector accelerated in a flat spacetime to produce an observed Unruh temperature. This is followed by use of the theory to obtain the temperature
of a D-dimensional black hole. With this temperature obtained we find the corresponding black hole entropy. In the last section of this chapter we generalize the
Stefan-Boltzmann law to calculate the lifetime of a semi-classical D-dimensional
black hole radiating onto a p-brane.
In Chapter 4 we describe the fundamental quantum field theoretical mechanism behind black hole thermodynamics. It will be shown that this may be
explained from spontaneous particle production in an exterior field. To introduce the frame work necessary to explain this particle production, we present
an improved resolution of Klein’s Paradox by means we find more consistent
with standard field theory than earlier work on the subject. In the next section
of this chapter we use this quantum field formalism to describe the observed
particle emission from a black hole. Here we generalize a procedure used on fourdimensional black holes. We are then also able to derive the temperature for a
D-dimensional black hole. In the end of this chapter we give a heuristic picture
of the emission process.
In Chapter 5 we consider the production of microscopic black holes at future
high energy colliders. Here we concentrate on discussing the cross-section of such
a process. We will then first introduce the simplest estimate of the black hole
cross-section for colliding ultra-relativistic particles. In the following sections we
will present and discuss several possible corrections to this cross-section. The
first of these is based on the general relativistic description of a photon gravitationally captured by a black hole. This scenario will also be analyzed quantum
mechanically. Another way to describe the process of black hole production is
to describe the colliding particles as colliding gravitational shock-waves. This
will involve finding the Aichelburg-Sexl shock-wave metric by boosting a static
Schwarzschild metric. Other corrections which will be discussed include the subrelativistic limit, an eikonal approximation and the Voloshin exponential suppres-
4
Chapter 1. Introduction
sion.
The concluding remarks are given in Chapter 6.
Chapter 2
Gravity
Gravity is the interaction between all massive bodies. This was one of the ideas
Sir Isaac Newton published in 1687, in his book “Philosophiae naturalis principia
mathematica”, or “Principia” as it is now known. According to his theory, gravity
was the force that causes both apples to fall down and planets to revolve around
the sun. This Newtonian theory was almost undisputed for over two centuries,
until Albert Einstein in 1907 started investigating how the theory agreed with
his special theory of relativity. In the following years he studied aspects such as
bending of light in a gravitational field and redshift of light escaping a gravitating
body. In an attempt to understand gravity better Einstein began around 1913
examining geometry aspects of the topic. This was a breakthrough. It could now
be argued that gravity was not a force, but a consequence of the curvature of the
four-dimensional spacetime. At this point Minkowski had already interpreted the
time as a dimension, and connected it to our otherwise three-dimensional reality,
in a spacetime continuum. It could now be explained how gravitation also could
affect massless particles such as photons. And in 1915 the general theory of
relativity was brought forth.
In the introduction we presented string theories, which are mathematical consistent only with the inclusion of extra dimensions, as possible important contributions to the objective of unifying gravity with the other known forms of
interaction in a single theory. Let us in this context study gravity in a spacetime
with one time dimension and more than three spatial dimensions. In the two
first sections of this chapter we assume the extra spatial dimensions to be equivalent to the three spatial dimensions we regard as infinitely large. Even though
questions to how such extra dimensions could be compatible with observations,
which correspond to a 4-dimensional world, do arise, this assumption will prove
useful. Let us now see how Newtonian and Einsteinian gravity is modified in D
spacetime dimensions.
5
6
Chapter 2. Gravity
2.1
Newtonian Gravity
We will in this section create a theory for Newtonian gravity in D = d + 1 infinite
spacetime dimensions. Here the spacetime has d spatial dimensions and one time
dimension.
But first let us remember the classical (3 + 1)-dimensional scenario. Newton
had in 1687 found that every massive object in the Universe attracts every other
massive object with a force, directed along the line of centers for the two objects,
that is proportional to the product of their masses and inversely proportional to
the square of the distance between the two objects. An object of mass m at a
distance r from another mass M experiences a gravitational force
F4 (r) = −GN
Mm
r,
r3
(2.1)
where GN is the proportionality constant, called the Newtonian gravitational
constant. We may here introduce the gravitational potential ΦN (r) ≡ V (r)/m,
where V (r) is the potential energy of the mass m. The force law may now be
expressed as F4 (r) = −m∇3 ΦN (r), where ∇3 is the usual gradient operator. This
leads to Newtonian gravitation in local form. Gravitational potential generates
motion according to
gN = −∇3 ΦN (r),
(2.2)
where gN is the field strength of the gravitational field. And mass generates
gravitational potential according to the Poisson equation
∇23 ΦN (r) = 4πGN ρ,
(2.3)
where ρ is the mass density and 4π is the total solid angle Ω2 for three spatial dimensions. In the pursuit of a higher dimensional Newtonian theory it is necessary
to introduce a generalization of this solid angle.
2.1.1
The Solid Angle
The total solid angle in D spacetime dimensions is the surface of a
(D − 2)-dimensional unit sphere.
First, in D = 4 dimensions the volume of a sphere is
Z
Z R
Z
Z R
4π 3
3
2
V3 = d x =
dr r
dΩ2 = 4π
dr r 2 =
R
(2.4)
3
0
0
R
R 2π R π
where dΩ2 = 0 dφ 0 dθ sin θ = 4π is the two-dimensional total solid angel.
We may now define the solid angle element for D = 4 (see Fig 2.1) as
dΩ2 ≡
dA2
.
r2
(2.5)
2.1 Newtonian Gravity
7
dΩ2
r=1
Figure 2.1: The solid angle element dΩ2 of a two-dimensional unit
sphere.
Generalized to D dimensions the solid angle element can be defined as
dΩD−2 ≡
dAD−2
.
r D−2
(2.6)
The generalized solid angle element may now be expressed in terms of the hyperspherical coordinates defined as
x1 = r cos χD−2 ,
x2 = r sin χD−2 cos χD−3 ,
..
.
xD−2 = r sin χD−2 sin χD−3 · · · sin χ2 cos χ1 ,
xD−1 = r sin χD−2 sin χD−3 · · · sin χ2 sin χ1 ,
(2.7)
where the D − 2 angular coordinates are defined so that 0 < χ1 < 2π, and
0 < χk < π for k = 2, . . . , D − 2. We now have that
∂(x1 , . . . , xD−1 ) D−1
drdχ1 · · · dχD−2 ,
d
x = (2.8)
∂(r, χ1 , . . . , χD−2 ) ∂(x1 ,...,xD−1 ) where ∂(r,χ
is the the absolute value of the Jacobian determinant
1 ,...,χD−2 )
∂x1
∂r
···
∂(x1 , . . . , xD−1 )
..
..
= .
.
∂(r, χ1 , . . . , χD−2 ) ∂xD−1
···
∂r
∂x1 ∂χD−2 .
∂xD−1 ..
.
(2.9)
∂χD−2
The solid angle element may now be written as
1 ∂(x1 , . . . , xD−1 ) dΩD−2 = D−2 dχ1 · · · dχD−2
r
∂(r, χ1 , . . . , χD−2 ) (2.10)
8
Chapter 2. Gravity
This reduces with use of Eq.(2.7) to
dΩD−2 = sinD−3 (χD−2 )dχD−2 dΩD−3
= [sinD−3 (χD−2 )dχD−2 ][sinD−4 (χD−3 )dχD−3 ] · · · sin(χ2 )dχ2 dχ1
=
D−2
Y
[sink−1 (χk )dχk ].
(2.11)
k=1
We may, in a straightforward manner, by integrating this recursion formula calculate the total solid angle. However, this result may also be found by another
method [1] which does not involve as much explicit calculations. We have that
the (D − 1)-dimensional volume becomes
Z
Z
Z R
RD−1
D−1
ΩD−2 .
(2.12)
dr r D−2 =
VD−1 = d
x = dΩD−2
D−1
0
From this, integrating a function f (r) over a (D − 1)-dimensional volume gives
R
RR
FD−1 = dD−1 x f (r) = ΩD−2 0 dr r D−2 f (r). Let us now consider the integral
R∞
2
ID−1 = −∞dD−1 x e−r . This integral may be solved by using the Γ-function
R ∞ z−1 −t
Γ(z) = 0 dt t e . This leads us to
Z ∞
Z ∞
D−3
1
D−2 −r 2
ID−1 = ΩD−2
dr r
e
= ΩD−2
dp p 2 e−p
2
0
0
1
D−1
= ΩD−2 Γ
.
(2.13)
2
2
But if we use that r 2 = x21 + x22 + · · · + x2D−1 , we may calculate the same integral
as follows,
Z ∞
Z ∞
Z ∞
√
2
−x22
−x21
···
dxD−1 e−xD−1 = ( π)D−1 .
dx2 e
ID−1 =
dx1 e
(2.14)
−∞
−∞
−∞
Comparing the two expressions for ID−1 we get [1]
D−1
ΩD−2
2π 2
= D−1 .
Γ( 2 )
(2.15)
In D = 4 dimensions the full solid angle becomes,
3
3
3
2π 2
2π 2
2π 2
Ω2 = 3 = 1 1 = 1 √ = 4π,
Γ( 2 )
π
Γ( 2 )
2
2
(2.16)
which is the result used in Eq.(2.3). This full solid angle is the area of a twodimensional unit sphere. Similarly, in two spatial dimensions we have Ω1 = 2π,
which is the circumference of a unit circle.
2.1 Newtonian Gravity
9
One may also notice that although Eq.(2.15) produces a result, Ω0 = 2, for
the solid angle in D = 2 spacetime dimensions, the definition of the solid angle
element in Eq.(2.6) may be a bit problematic to use in this scenario. In two
spacetime dimensions a volume is a distance on the number line. In this scenario
it may therefore be problematic to talk of a hypersurface element as it is done
in Eq.(2.6). It is here then hard to understand what the solid angle actually
describes. However, if we state that a 1-dimensional volume described in spherical
coordinates should be described solely by a radius r, a larger part of the problem
is revealed. In spherical coordinates the radius is ≥ 0. It is then possible to
describe only half of spacetime with such a coordinate. To compensate for this,
an integration over a 1-dimensional volume symmetric around the origin must
include the solid angle Ω0 = 2 to produce the correct result. This does of course
not solve how to describe the second half of the spacetime. And the conclusion
should be that in D = 2 dimensions spherical coordinates is more or less useless,
and one must be somewhat aware if the solid angle Ω0 should appear.
2.1.2
Newtonian Gravity in D Dimensions
We may now generalize Eq.(2.2) for the gravitational field ΦD (r). The field
strength in D dimensions is then defined to be
gD = −∇d ΦD (r),
(2.17)
where ∇d is the d-dimensional gradient operator. We may also generalize the
Poisson equation into
∇2d ΦD (r) ≡ ΩD−2 GD ρ,
(2.18)
which is our definition of the gravitational constant GD in D dimensions. Here ρ
is the mass density in the D-dimensional spacetime. Notice that there are other
ways to generalize the Poisson equation. With our definition, the force law in
D dimensions will be shown to have the same form as in the four-dimensional
theory. Likewise would other definitions of the gravitational constant correspond
to other quantities which are kept on the same form in the generalized theory as in
four dimensions. In Appendix B, some alternative generalizations are presented.
2.1.3
The Gravitational Potential
As a first step to find an expression for the gravitational potential ΦD (r) outside
a mass distribution ρ(r) = dM/dV in a spacetime with D − 1 spatial dimensions
we may combine the two equations, Eq.(2.17) and Eq.(2.18). And this gives
∇d · gD = −ΩD−2 GD ρ. Integrating this equation
over a volume
V and use of
R
H
Gauss’ divergence theorem, that states that dV ∇d · gD = dAD−2 · gD , on the
left hand side give us the generalized D-dimensional Gauss’ law
I
dAD−2 · gD = −ΩD−2 GD M.
(2.19)
10
Chapter 2. Gravity
This states that the gravitational flux out of a closed (D − 2)-dimensional surface
depends Honly of the mass Hinside the surface. Now calculating the left hand side
leads to dAD−2 ·gD = gD dAD−2 = (gD ·r)ΩD−2 r D−3 , where we integrate over a
(D −2)-dimensional sphere. And if we then combine the two previous expressions
we get (gD · r)ΩD−2 r D−3 = −ΩD−2 GD M . This produce the expression for the
gravitational field strength
GD M
gD = − D−1 r.
(2.20)
r
If we compare the (D = 4)-dimensional field strength to for instance the (D = 5)dimensional scenario, the field from a mass in five dimensions has to propagate in
one spatial dimension more than in the four-dimensional case. It therefore seems
natural that the field strength decreases faster, with increasing distance, in five
dimensions than it would in four. This is in agreement with the expression for
the gravitational field strength in Eq.(2.20).
Since we have that the gravitational force FD = mgD , the D-dimensional
gravitational force law becomes
FD (r) = −
GD M m
.
r D−2
(2.21)
To find the gravitational potential we use Eq.(2.20) and from Eq.(2.17) the
D r
fact that gD = − dΦ
, this gives us
dr r
ΦD (r) = GD M
Z
r 2−D dr = GD M
GD M
r 3−D
=−
.
3−D
(D − 3)r D−3
(2.22)
The gravitational potential in D = 3 + 1 dimensions becomes the familiar expression
GN M
,
(2.23)
Φ4 (r) = −
r
where we have defined the Newtonian Gravitational constant GN ≡ G4 . In
section 2.3 we shall see how the 4-dimensional force law may arise from the Ddimensional one.
As a curiosity we see that in D = 3 dimensions the general expression for the
gravitational potential in Eq.(2.22) does not hold. In this case the gravitational
force is inversely proportional to the radius. And by integrating this expression
we find that the potential becomes proportional to the logarithm of the radius.
2.2
General Relativity
After several years of working with the theory of general relativity, Einstein submitted his paper “The Field Equations of Gravitation” on the 25th of November
1915. But it has later been argued that the discovery of the gravitational field
2.2 General Relativity
11
equations was not solely Einstein’s. This argument is based on the misconception that the paper “The Foundations of Physics” submitted by Hilbert five days
earlier contained the correct field equations. Indeed, the final publication of the
Hilbert article contains the equations, but proofs dated before Einstein’s paper
was published do not contain these. On the other hand, Hilbert’s paper contained
other important contributions not found in Einstein’s publication.
The famous field equations can now be written as
Eµν = 8πGN Tµν ,
(2.24)
and are found in this form in all modern books written on the topic (e.g. Gravitation [2] or General Relativity [3]). These equations, now called the Einstein
field equations, connect the curvature of the spacetime, represented by the Einstein tensor components Eµν , to the energy in the spacetime, represented by the
energy-momentum tensor components Tµν .
Within a year, in 1916, Karl Schwarzschild had found a mathematical solution
to the field equations. This solution corresponds to the gravitational field of a
massive compact object. The Schwarzschild solution has later been important in
the study of black holes and will be derived in the following subsection. We will
there derive this solution for a D-dimensional spacetime. To do so we introduce
the Einstein field equations for D dimensions. We assume these to have the same
form as in four spacetime dimensions, i.e.
Eµν = κD Tµν .
(2.25)
Here is κD a constant depending on the number of dimensions D.
When the Schwarzschild solution is established, we will based on this derive
the isotropic coordinates in D dimensions and use these in the Newtonian limit
to produce the Poisson equation in D dimensions. From this equation we will be
able to express κD in terms of the gravitational constant GD . In the following
we will also by another method produce the Poisson equation. By assuming
the gravity to be linearized in the Newtonian limit we may find the desired
equation. Subsection 2.2.4 introduces the Planck scale which contributes to our
generalization of the Einstein-Hilbert action to D dimensions in Subsection 2.2.5,
and thereby introduces the fundamental Planck scale. In this last subsection we
express the gravitational constant GD by means of the fundamental Planck scale.
In the remainder of this paper all expressions will be written in HEP-units.
In these units the light speed c and the Dirac constant ~ are set to be equal 1.
A more thoroughly description of these units is to be found in Appendix A.
2.2.1
The Schwarzschild Solution in D Dimensions
As Schwarzschild did in 1916, we will derive the Schwarzschild solution to the
gravitational field equations. That is, we will derive the Schwarzschild metric.
12
Chapter 2. Gravity
But instead of working in a four-dimensional spacetime, we study the (D = d+1)dimensional scenario. The metric may then be described by the set of coordinates
{t, r, χ1 , . . . , χd−1 }. The solution to this higher dimensional scenario is known as
the Schwarzschild-Tangherlini solution [4] of the Einstein equations.
We want to find the solution to the field equations in empty space, Eµν = 0,
for a static spherically symmetric spacetime. These field equations originate
from the fact that the energy-momentum tensor components Tµν are vanishing
in empty space. One may then choose
ds2 = −e2α(r) dt2 + e2β(r) dr 2 + r 2 dΩ2d−1
(2.26)
as line element (using units so that c = 1), since this is the general form of a
metric describing a static spherically symmetric spacetime geometry. Here is
dΩ2d−1 = dχ2d−1 + sin2 (χd−1 )dΩ2d−2
=
dχ2d−1
+
d−2 Y
d−1
X
sin2 (χj )dχ2i
(2.27)
(2.28)
i=1 j=i+1
the squared solid angle element for d ≥ 3 dimensions. For d = 2, the square of
the solid angle element is dΩ21 = dχ21 . As previous, the angles χk are defined so
that 0 < χk < π for k = 2, . . . , d − 1, and 0 < χ1 < 2π. And eα(r) and eβ(r) are
functions we will determine.
In all of this subsection Greek indices run over all of the spacetime coordinates
and for these indices Einstein’s summation convention will be used. Latin indices
on the other hand, run over the angular coordinates and it will be expressed
explicitly when these indices are summed.
Here follows the stepwise algorithm we use to determine the components of
the Einstein tensor.
1. By introducing an orthonormal form basis {ω µ̂ } we find
ω t̂ = eα(r) dt,
ωr̂ = eβ(r) dr,
ˆ
ω d−1 = rdχd−1 ,
..
.
(2.29)
ω k̂ = r sin χd−1 sin χd−2 · · · sin χk+1 dχk ,
..
.
ω 1̂ = r sin χd−1 sin χd−2 · · · sin χ2 dχ1 ,
where k is the index corresponding to the arbitrary angular coordinate χk .
With use of these basis forms the line element may be expressed as
ds2 = ηµ̂ν̂ ωµ̂ ⊗ ω ν̂ ,
(2.30)
2.2 General Relativity
13
where
ηµ̂ν̂ = diag[−1, 1, ..., 1]
(2.31)
is the Minkowski metric in d + 1 dimensions.
2. Computing the connection forms by applying Cartan’s first structure equation
dω µ̂ = −Ωµ̂ν̂ ∧ ω ν̂ .
(2.32)
First we take the exterior derivative of ω t̂ ,
dω t̂ = d[eα(r) dt] = d[eα(r) ] ∧ dt + eα(r) d dt
= d[eα(r) ] ∧ dt = eα(r) α0 (r)dr ∧ dt
= eα(r) α0 (r)e−β(r) ω r̂ ∧ [e−α(r) ω t̂ ]
= −e−β(r) α0 (r)ω t̂ ∧ ω r̂ ,
(2.33)
where we in the third transition have used that d dt = 0. Using the final
result in Cartan’s first structure equation from Eq.(2.32) gives us
Ωt̂r̂ = e−β(r) α0 (r)ωt̂ + ftr (r)ω r̂ ,
(2.34)
where ftr (r) is an arbitrary function which arises from the fact that
dx ∧ dx = 0 for all x. And it leads to
Ωt̂k̂ = ftk (r)ω k̂ .
(2.35)
Calculating the exterior derivative of ω r̂ gives
dω r̂ = d[eβ(r) dr] = d[eβ(r) ] ∧ dr = eβ(r) β 0 (r)dr ∧ dr = 0.
(2.36)
Combined with Eq.(2.32) this leads to
Ωr̂α̂ = frα (r)ω α̂ .
(2.37)
Ωr̂t̂ = frt (r)ω t̂
(2.38)
Ωr̂k̂ = frk (r)ω k̂ .
(2.39)
And especially
and
To determine the f -functions we apply that the components of connection
forms in orthonormal basis are anti-symmetric in the indices, i.e.
Ωµ̂ν̂ = −Ων̂ µ̂ .
This gives us that ftr (r) = 0 and frt (r) = e−β(r) α0 (r).
(2.40)
14
Chapter 2. Gravity
Continuing in the same manner gives us all the non-zero connection forms
Ωt̂r̂ = Ωr̂t̂ = e−β(r) α0 (r)ω t̂ ,
1
Ωk̂r̂ = −Ωr̂k̂ = e−β(r) α0 (r)ω k̂ ,
r
cot χj
ĵ
Ωîĵ = −Ω î =
ω î
r sin χd−1 · · · sin χj+1
(2.41)
(2.42)
(i < j).
(2.43)
3. Determining the curvature forms by use of Cartan’s second structure equation
Rµ̂ν̂ = dΩµ̂ν̂ + Ωµ̂λ̂ ∧ Ωλ̂ν̂ .
(2.44)
This gives us
Rt̂r̂ =dΩt̂r̂ + Ωt̂λ̂ ∧ Ωλ̂r̂ = dΩt̂r̂ = d[e−β(r) α0 (r)ω t̂ ]
=d[e−β(r) α0 (r)] ∧ ω t̂ + e−β(r) α0 (r)dω t̂
=[−e−β(r) β 0 (r)α0 (r) + e−β(r) α00 (r)]dr ∧ ω t̂ + e−β(r) α0 (r)[−Ωt̂r̂ ∧ ω r̂ ]
=e−2β(r) [α00 (r) − β 0 (r)α0 (r) + α0 (r)2 ]ω r̂ ∧ ω t̂ ,
Rt̂k̂ =dΩt̂k̂ + Ωt̂λ̂ ∧ Ωλ̂k̂ = Ωt̂λ̂ ∧ Ωλ̂k̂ = Ωt̂r̂ ∧ Ωr̂k̂
1
= − e−β(r) α0 (r) e−β(r) ω t̂ ∧ ω k̂
r
1 0
= − α (r)e−2β(r) ω t̂ ∧ ω k̂ ,
r
X
d−1
1 −β(r) k̂
r̂
r̂
r̂
λ̂
ω +
Ωr̂î ∧ Ωîk̂
R k̂ =dΩ k̂ + Ω λ̂ ∧ Ω k̂ = d − e
r
i=1
d−1
X
1
1
=d − e−β(r) ∧ ω k̂ − e−β(r) dω k̂ +
Ωr̂î ∧ Ωîk̂
r
r
i=1
1
1
1
= 2 e−β(r) + β 0 (r)e−β(r) dr ∧ ω k̂ + 2 e−2β(r) ω k̂ ∧ ω r̂
r
r
r
1
= e−2β(r) β 0 (r)ω r̂ ∧ ω k̂ ,
r
(2.45)
(2.46)
(2.47)
and finally
Rîĵ = dΩîĵ + Ωîλ̂ ∧ Ωλ̂ĵ
1
= 2 [1 − e−2β(r) ]ω î ∧ ω ĵ .
r
(2.48)
4. By applying the the relation
1
Rµ̂ν̂ = Rµ̂ν̂ α̂β̂ ωα̂ ∧ ω β̂
2
(2.49)
2.2 General Relativity
15
and using the four symmetries of the Riemann curvature tensor
Rµ̂ν̂ α̂β̂ = −Rµ̂ν̂ β̂α̂ ,
(2.50)
Rµ̂[ν̂ α̂β̂] = 0,
(2.51)
Rµ̂ν̂ α̂β̂ = −Rν̂ µ̂α̂β̂ ,
(2.52)
Rµ̂ν̂ α̂β̂ = Rα̂β̂µ̂ν̂ ,
(2.53)
we get the components
Rt̂r̂t̂r̂ = −Rr̂t̂r̂t̂ = −e2β(r) [α00 (r) − β 0 (r)α0 (r) + α0 (r)2 ],
1
Rt̂k̂t̂k̂ = −Rk̂t̂k̂t̂ = − α0 (r)e−2β(r) ,
r
1
Rr̂k̂r̂k̂ = Rk̂r̂k̂r̂ = β 0 (r)e−2β(r) ,
r
1
Rîĵ îĵ = 2 [1 − e−2β(r) ].
r
(2.54)
(2.55)
(2.56)
(2.57)
5. Contraction of these components gives the components of the Ricci curvature tensor,
Rµ̂ν̂ ≡ Rα̂µ̂α̂ν̂ .
(2.58)
We get
1
Rt̂t̂ = e−2β(r) [α00 (r) − β 0 (r)α0 (r) + α0 (r)2 ] + α0 (r)e−2β(r) (d − 1), (2.59)
r
1
Rr̂r̂ = −e−2β(r) [α00 (r) − β 0 (r)α0 (r) + α0 (r)2 ] + β 0 (r)e−2β(r) (d − 1), (2.60)
r
1
1 −2β(r) 0
Rk̂k̂ = e
[β (r) − α0 (r)] + 2 [1 − e−2β(r) ](d − 2).
(2.61)
r
r
6. Another contraction gives us the Ricci curvature scalar,
R ≡Rµ̂µ̂ = η µ̂ν̂ Rν̂ µ̂ = −Rt̂t̂ + Rr̂r̂ +
= − Rt̂t̂ + Rr̂r̂ + Rk̂k̂ (d − 1)
d−1
X
Rîî
i=1
= − 2e−2β(r) [α00 (r) − β 0 (r)α0 (r) + α0 (r)2 ]
1
2
+ e−2β(r) [β 0 (r) − α0 (r)](d − 1) + 2 [1 − e−2β(r) ](d − 2)(d − 1).
r
r
(2.62)
7. In the end we may find the components of the Einstein tensor,
1
Eµ̂ν̂ = Rµ̂ν̂ − ηµ̂ν̂ R.
2
(2.63)
16
Chapter 2. Gravity
We then have
1
Et̂t̂ =Rt̂t̂ + R
2
1
1 −2β(r) 0
β (r)(d − 1) + 2 [1 − e−2β(r) ](d − 2)(d − 1),
= e
r
2r
1
Er̂r̂ =Rr̂r̂ − R
2
1 −2β(r) 0
1
= e
α (r)(d − 1) − 2 [1 − e−2β(r) ](d − 2)(d − 1),
r
2r
1
Ek̂k̂ =Rk̂k̂ − R
2
−2β(r) 00
=e
[α (r) − β 0 (r)α0 (r) + α0 (r)2 ]
1
− e−2β(r) [β 0 (r) − α0 (r)](d − 2)
r
1
− 2 [1 − e−2β(r) ](d − 3)(d − 2).
2r
(2.64)
(2.65)
(2.66)
Now we solve the equation Eµ̂ν̂ = 0. There are then only two linear independent
equations. From Et̂t̂ = 0 and Er̂r̂ = 0 we get
Et̂t̂ + Er̂r̂ = 0
α (r) + β 0 (r) = 0
α(r) + β(r) = C = constant.
0
(2.67)
We then have the line element
ds2 = −e−2β e2C dt2 + e2β dr 2 + r 2 dΩ2d−1 .
(2.68)
By choosing a suitable coordinate time t → eC t, we can achieve C = 0 and
α(r) = −β(r). The term eβ may then be determined by solving Et̂t̂ = 0. From
Eq.(2.64) we have
1
1 −2β(r) 0
e
β (r)(d − 1) − 2 [1 − e−2β(r) ](d − 2)(d − 1) = 0.
r
2r
(2.69)
If we examine this expression, we see that part of it is the derivative of r d−2 e−2β(r)
with respect to r. The equation may therefore be written as
[r d−2 e−2β(r) ]0 = (d − 2)r d−3 . Integrating this and rearranging it leads to
e−2β(r) = 1 +
where K is an arbitrary constant.
K
,
r d−2
(2.70)
2.2 General Relativity
17
To determine K we have to go to the Newtonian limit of the gravitational
acceleration. In the Newtonian limit we have
gD =
GD M
d2 r
= − D−2 .
2
dt
r
(2.71)
The D-acceleration, aα = d2 xα /dτ 2 , of a free particle in the D-dimensional spacetime is given by the geodesic equation
d 2 xα
+ Γαµν uµ uν = 0.
dτ 2
(2.72)
The connection coefficients Γαµν are established from
1
Γαµν = g αβ (gβµ,ν + gβν,µ − gµν,β ),
2
(2.73)
where gµν is the metric, and the comma notation is defined
,γ
≡
∂
.
∂xγ
(2.74)
For a particle instantaneously at rest, far from the mass distribution
(r
K) we have that
p time τ is approximately equal to the coordip the proper
nate time, since dτ = |gtt | dt = 1 + K/r d−2 dt. Using uµ = (1, 0, . . . , 0), and
dτ ≈ dt, we get
d−2
gD =
d2 r
= −Γrtt .
dt2
(2.75)
Γrtt then interprets as the gravitational acceleration. By using Eq.(2.73) we may
express the acceleration as
1 1
1
gtt,r ,
gD = −Γrtt = − g rβ (gβt,t + gβt,t − gtt,β ) =
2
2 grr
(2.76)
where we have used that g rβ = 1/grβ and gβt,t = 0. Combining this result with
gtt = −1/grr = −(1 + K/r d−2 ) we get
K
∂
K
1
gD = −
1 + d−2
1 + d−2
2
r
∂r
r
(d − 2)K
1
K
=−
1 + d−2
−
2
r
r d−1
(d − 2)K
=
+ O(r −(2d−3) )
2r d−1
(d − 2)K
(D − 3)K
≈
=
.
(2.77)
2r d−1
2r D−2
18
Chapter 2. Gravity
Comparing this with the Newtonian expression from Eq.(2.71) gives us
−
GD M
(D − 3)K
=
,
D−2
r
2r D−2
(2.78)
which determines the constant
K=−
2GD M
.
D−3
(2.79)
This leads to
e−2β(r) = e2α(r) = 1 −
2GD M
= 1 + 2ΦD (r),
(D − 3)r D−3
(2.80)
where we have used Eq.(2.22) in the last transition.
And finally the Schwarzschild metric may be expressed as
1
dr 2 + r 2 dΩ2D−2
1 + 2ΦD (r)
D−3
Y
X D−2
1
2
2
2
= −[1 + 2ΦD (r)]dt +
sin2 (χj )dχ2i + r 2 dχ2D−2 .
dr + r
1 + 2ΦD (r)
i=1 j=i+1
ds2 = −[1 + 2ΦD (r)]dt2 +
(2.81)
or, equivalently,

−[1 + 2ΦD (r)]
0
0
−1

0
[1
+
2Φ
(r)]
0
D

QD−2 2

2
0
0
r

k=2 sin (χk )
gµν = 
.
.
..
..
..

.


0
0
0
0
0
0

0
0

0

.
.. 
.

. . . r 2 sin2 (χD−2 ) 0 
...
0
r2
(2.82)
...
...
...
..
.
0
0
0
..
.
This general expression is valid only for D ≥ 4. In D = 3 dimensions we find
from Eq.(2.69) that e−2β = K, where K is still an arbitrary constant. In this case
the metric is on the form ds2 = −Kdt2 + (1/K)dr 2 + r 2 dφ2 . This metric describes
a locally flat spacetime outside the matter, which is evident since the Riemann
tensor is vanishing. One may also notice that if we in the Newtonian limit would
like to retrieve the Minkowski metric, the spacetime would always be described by
the Minkowski spacetime. More generally, in 3 dimensions the Riemann tensor
has only as many independent components as the Ricci tensor. The Riemann
tensor may then always be expressed in terms of the Ricci tensor and through
Eq.(2.63) in terms of the Einstein tensor. In empty space the Riemann tensor is
therefore vanishing and the spacetime is locally flat outside the matter. This was
2.2 General Relativity
19
pointed out by Myers and Perry [5] in their discussion concerning spinning black
holes in higher dimensional spacetimes. With empty space conditions in D = 3
dimensions, no horizons can exist and thus black holes are not possible.
In D = 4 dimensions, we get from Eq.(2.81) the standard Schwarzschild metric
1
ds2 = −[1 + 2Φ4 (r)]dt2 +
dr 2 + r 2 dΩ22
1 + 2Φ4 (r)
1
2GN M
dr 2 + r 2 dΩ22 ,
dt2 +
=− 1−
2GN M
r
1− r
(2.83)
where we have used Eq.(2.23) in the last transition.
The Schwarzschild radius in d = 3 spatial dimensions is defined as
RS4 ≡ 2GN M.
(2.84)
At this radius the metric has a singularity, gtt = 0. We will later see that
this singularity is a coordinate singularity, a singularity which is not connected
to the curvature of the spacetime. At RS4 the proper time τ is standing still
compared to the coordinate time t (proper time of the observer at infinity).
This corresponds to the fact that measured on a coordinate clock it will take
an infinite amount of time to reach the Schwarzschild radius, while measured
in proper time this will be done in finite time. For a radius r < RS4 we also
see from the metric in Eq.(2.83) that r becomes a timelike coordinate, while t
becomes spacelike. An in falling particle must move along a timelike worldline
which means for r < RS4 that r must constantly be changing, i.e. decreasing. A
message (photon) sent to the outside world must accordingly travel in direction of
decreasing r. It is therefore impossible for an observer outside the Schwarzschild
radius to receive information sent from inside the Schwarzschild radius RS4 . This
radius then describes an event horizon. For relatively small gravitating bodies like
the planets and the sun this radius is inside the surface of the mass distribution.
Since the empty space condition Tµν = 0 no longer holds inside the surface
of these bodies, the Schwarzschild solution may not be used to describe these
regions. In the description of such gravitating bodies the Schwarzschild solution
is only applicable outside the mass distribution, and the Schwarzschild radius
is not involved in the description of this region. For larger gravitating bodies
the Schwarzschild radius may exceed the radius of the massive body and the
spacetime will have an event horizon. Classically nothing may escape from inside
this horizon, not even light. Such bodies are therefore seemingly appropriately
called black holes. But as we shall see in the forthcoming chapters, quantum
mechanically a black hole does actually radiate.
In the general case, with D spacetime dimensions, we find the Schwarzschild
radius RSD by using that gtt = 0 when r = RSD . This gives us
1−
2GD M
=0
D−3
(D − 3)RSD
(2.85)
20
Chapter 2. Gravity
and the Schwarzschild radius of a massive body in D dimensions is determined
to be
RSD =
2GD M
D−3
1
D−3
.
Using this, the Schwarzschild metric in D dimensions becomes
D−3 RSD
1
2
dr 2 + r 2 dΩ2D−2 .
ds = − 1 − D−3 dt2 +
RD−3
r
1 − SD
(2.86)
(2.87)
r D−3
In addition to being singular at the Schwarzschild radius, we see that this metric
is singular at r = 0. This on the other hand is a curvature singularity, which
corresponds to the curvature of the spacetime.
In general a black hole is described by its charge, spin and mass. A Schwarzschild
black hole is solely specified by its mass, which is evident from the metric. The
Schwarzschild metric describes in other words a non-rotating, uncharged black
hole.
Our discussion concerning D-dimensional black holes will be based on the
derived D-dimensional Schwarzschild metric from this section, and hence be concentrated on the study of non-rotating, uncharged black holes.
2.2.2
Isotropic Coordinates in D Dimensions
It can often be favorable to express the Schwarzschild metric in isotropic coordinates. We will use such coordinates in an attempt to retrieve Newtonian gravity.
Isotropic coordinates are coordinates that give the same expression in front
of all the spatial components in the metric. Thus,
ds2 = −J(r̄)dt2 + F (r̄)[dr̄ 2 + r̄ 2 dΩ2D−2 ].
(2.88)
In the following we will determine these functions, J(r̄) and F (r̄), for our Ddimensional spacetime. If we in our metric from Eq.(2.81) let r be a function of
r̄, we have that dr = r 0 dr̄. This used in Eq.(2.81) and compared with Eq.(2.88)
give us the three equations
J(r̄) = 1 + 2ΦD (r),
(r 0 )2
F (r̄) =
1 + 2ΦD (r)
(2.89)
F (r̄)r̄ 2 = r 2 .
(2.91)
(2.90)
and
2.2 General Relativity
21
Combined, these give us a differential equation
(r 0 )2
= r̄ −2 ,
2
r (1 + 2ΦD (r))
(2.92)
which rearranged can be written as
Z
Z
dr
dr̄
p
.
=
r̄
r 1 + 2ΦD (r)
(2.93)
Using the expression for the gravitational potential ΦD (r) in Eq.(2.22) and then
integrating the previous expression we find that
s
!
√
1
D − 3 D−3
(D − 3)r 2D−6 2r D−3
1
+
ln − √
r
+
−
D−3
GD M
(GD M )2
GD M
D−3
r̄
= ln
+ C.
(GD M )1/(D−3)
(2.94)
Then by eliminating the logarithms this can be rewritten as
p
√
GD M
+ D − 3r D−3 + (D − 3)r 2D−6 − 2GD M r D−3 = C̃ r̄ D−3 .
−√
D−3
(2.95)
To find the constant C̃ we look at the situation were r → ∞. Here we assume
that r̄ → ∞ in the same manner, and by use of the last equation we find that
1=
r D−3
C̃
= √
,
D−3
r̄
2 D−3
(2.96)
√
C̃ = 2 D − 3.
(2.97)
and the constant becomes
Using this in Eq.(2.95) gives us the expression
r̄ D−3
GD M
1
=−
+ r D−3 +
2(D − 3) 2
s
1 2D−6
GD M D−3
r
−
r
.
4
2(D − 3)
(2.98)
We can then find that
GD M
r = r̄ 1 +
2(D − 3)r̄ D−3
2
D−3
.
(2.99)
22
Chapter 2. Gravity
Eq.(2.99) can now combined with the expressions in Eq.(2.91) and Eq.(2.89)
produce the relations
!2
2
1 + 12 ΦD (r̄)
=
J(r̄) =
,
1 − 12 ΦD (r̄)
1+
4
4
D−3
D−3
1
GD M
= 1 − ΦD (r̄)
.
F (r̄) = 1 +
2(D − 3)r̄ D−3
2
1−
GD M
2(D−3)r̄ D−3
GD M
2(D−3)r̄ D−3
(2.100)
(2.101)
Having determined these functions, the Schwarzschild metric given in isotropic
coordinates becomes
2
ds = −
1 + 12 ΦD (x)
1 − 12 ΦD (x)
2
1
dt + 1 − ΦD (x)
2
2
4
D−3
1 2
(dx ) + (dx2 )2 + · · · + (dxd )2 ,
(2.102)
where we have transformed the spatial coordinates into Cartesian coordinates.
Notice that as a function of the potential only the spatial components of the
metric are dependent of the number of dimensions. In D = 4 dimensions the line
element is reduced to the form on which it is found in e.g. Gravitation [6].
Let us now examine the above metric in the Newtonian limit, i.e. far from
the mass distributions. The potential ΦD (x) is then a small quantity and we may
expand the metric, and all other expressions to just the lowest order of ΦD (x).
This gives us the metric
2
2
2
ΦD (x) (dx1 )2 + (dx2 )2 + · · · + (dxd )2 ,
ds = −[1 + 2ΦD (x)]dt + 1 −
D−3
(2.103)
where we have used [1 − (1/2)ΦD (x)]4/(D−3) = 1 − 2ΦD (x)/(D − 3) to the lowest
order of ΦD (x). This metric is a generalization not yet seen in the literature.
This expression can now be used to find the Poisson equation in D dimensions.
We must then express the Einstein tensor component Ett . And we therefore start
by calculating the Riemann tensor components. We have in coordinate basis that
Rµναβ = Γµνβ,α − Γµνα,β + Γρνβ Γµρα − Γρνα Γµρβ ,
(2.104)
where Γαµν are the Christoffel symbols expressed in Eq.(2.73). We first calculate
Rktkt , where k is an arbitrary spatial coordinate which is not summed. That is
where ever it is summed over k (or any other latin indices) in this subsection, it
is done so explicitly.
Rktkt = Γktt,k − Γktk,t + Γρtt Γkρk − Γρtk Γkρt = ΦD (r),kk ,
(2.105)
2.2 General Relativity
23
where we have used that
Γktt,k = ΦD (x),kk ,
Γρtt Γkρk = 0
and
Γktk,t = 0,
Γρtk Γkρt = 0.
(2.106)
See Eq.(D.3)-(D.6) in Appendix D for the calculations of these coefficients. Because of the symmetries in the Riemann tensor we also have that
Rtktk = g tt Rtktk = g tt Rktkt = g tt gkk Rktkt
2
ΦD (x) ΦD (x),kk = −ΦD (x),kk .
= −[1 − 2ΦD (x)] 1 −
D−3
(2.107)
We then calculate
Rikik = Γikk,i − Γiki,k + Γρkk Γiρi − Γρki Γiρk
1
1
ΦD (x),ii +
ΦD (x),kk ,
=
D−3
D−3
where we have used that
1
1
Γikk,i =
ΦD (x),ii ,
Γiki,k = −
ΦD (x),kk ,
D−3
D−3
Γρkk Γiρi = 0
and
Γρki Γiρk = 0.
(2.108)
(2.109)
See Eq.(D.7)-(D.10) in Appendix D for the calculations of these coefficients. Using the expressions for Rµναβ we can calculate the components of the Ricci tensor
Rµµ . This gives us
Rtt = Rαtαt = ∇2d ΦD (x),
Rkk = Rαkαk = Rtktk +
D−1
X
(2.110)
Rikik
i=1
1 X
= −ΦD (x),kk +
[ΦD (x),ii + ΦD (x),kk ]
D − 3 i6=k
=
1
∇2 ΦD (x).
D−3 d
(2.111)
This enables us to calculate the Ricci scalar
X
X
R = Rαα = Rtt +
Rii = −Rtt +
Rii
i
i
1 X 2
∇d ΦD (x)
= −∇2d ΦD (x) +
D−3 i
D−1 2
D−3 2
∇d ΦD (x) +
∇ ΦD (x)
D−3
D−3 d
2
=
∇2 ΦD (x).
D−3 d
=−
(2.112)
24
Chapter 2. Gravity
We have now done all the calculation needed to express the Einstein tensor component
1
1
Ett = Rtt − gtt R = Rtt + R
2
2
1
∇2d ΦD (x)
= ∇2d ΦD (x) +
D
−
3
D−2
=
∇2d ΦD (x).
D−3
(2.113)
Now, in D dimensions we have from Eq.(2.25) that the Einstein equations is
generally Eµν = κD Tµν , And we have specially that Ett = κD Ttt . Tµν is the
energy-momentum tensor and Ttt is the mass density ρ. Generally Ttt is the
energy density, but since we have chosen c = 1, mass and energy has the same
dimension. This means that
D−2
Ett = κD ρ =
∇2d ΦD (x)
(2.114)
D−3
where we have used Eq.(2.113). This gives us
D−3
2
κD ρ.
∇d ΦD (x) =
D−2
(2.115)
Comparing this with what we have from Eq.(2.18), we find that the constant κD
can be related to GD by
D−2
ΩD−2 GD .
(2.116)
κD =
D−3
In D = 4 dimensions we have that κ4 = 8πGN and find that
∇23 Φ4 (x) = 4πGN ρ
(2.117)
which is the standard Poisson equation. And we may use this to find the expression for the potential Φ4 (x) as done in Subsection 2.1.3.
In Subsection 2.2.5 we will define the general constant κD in terms of a fundamental Planck mass. We will then be able to find a relation between the
gravitational constant GD and this Planck mass.
2.2.3
Linearized Gravity
The Newtonian limit may also be approached from another direction than done
in the previous subsection. In this subsection we will approximate the gravity to
be weak, as is the case in the Newtonian gravity. In general relativity this means
2.2 General Relativity
25
that the spacetime or the metric is almost flat. So our nearly flat metric can be
assumed to be [7, 8]
gµν = ηµν + 2f hµν ,
(2.118)
where ηµν is the Minkowski metric, 2f hµν is a small deviation from the otherwise
√
flat metric and f = κD . Here hµν (x) may be viewed as the graviton field.
The situation may then be viewed as gravitons propagating in a flat spacetime,
transmitting the gravitation in this spacetime.
Since the gravity is weak we will express Einstein’s equation to the lowest
order of hµν . This may be obtained by first to calculate the Christoffel symbols
from Eq.(2.73)
1
Γαµν = (η αβ + 2f hαβ )[(ηβµ + 2f hβµ ),ν + (ηβν + 2f hβν ),µ − (ηµν + 2f hµν ),β ]
2
= f η αβ (hβµ,ν + hβν,µ − hµν,β ).
(2.119)
Here we have in the last transition ignored all non-linear terms of hµν . With
the use of these Christoffel symbols we may express the components of the Ricci
tensor by contracting the components of the Riemann tensor in Eq.(2.104). We
then have
Rµν = Rαµαν = Γαµν,α − Γαµα,ν
= f [η αβ (hβµ,ν + hβν,µ − hµν,β )],α − f [η αρ (hρµ,α + hρα,µ − hµα,ρ )],ν
= f (hαµ,να + hαν,µα − hµν,αα − hαµ,αν − hαα,µν + hµα,αν )
= −f (hµν − ∂µ hν − ∂ν hµ ),
(2.120)
where = ∇2d − ∂ 2 /∂t2 is the d’Alembert operator, hµ = ∂λ hλµ − (1/2)∂µ h and
h = hλλ . These components of the curvature tensor Rµν is invariant under the
local gauge transformation hµν → hµν + ∂µ χν + ∂ν χµ and χµ is an arbitrary
vector function. This local invariance allows us to choose a convenient gauge.
The simplest choice of gauge is the Hilbert or harmonic gauge defined by the
condition hµ = 0 [7]. Using this we have that Rµν = −f hµν . Raising and
contracting the indices bring us to an expression for the Ricci scalar R = −f h.
Now all components necessary to calculate the Einstein tensor components from
Eq.(2.63) are available. These are now found to be
1
Eµν = −f (hµν − ηµν h).
2
(2.121)
Compared with the Einstein equations in Eq.(2.25) we have
[hµν − (1/2)ηµν h] = −f Tµν .
(2.122)
26
Chapter 2. Gravity
Raising and contracting the indices in this equation gives us h = 2f T /(D − 2).
Using this in Eq.(2.121) and comparing with Eq.(2.25) we get
1
hµν = −f Tµν + ηµν h
2
ηµν
= −f Tµν −
T .
D−2
(2.123)
This can now be used in evaluating the Newtonian limit. Our metric is time independent and therefore hµν = ∇2 hµν . The spatial diagonal components of the
energy-momentum tensor represents pressure p, and since we in the Newtonian
limit have no pressure the scalar T = T tt + T 11 + . . . + T dd = T tt . Eq.(2.123) then
gives
D−3
ηtt
1
2
t
∇ htt = −f Ttt −
T
Ttt = −f
Ttt .
= −f 1 −
D−2 t
D−2
D−2
(2.124)
Since Ttt is the mass density ρ, we have
D−3
2
ρ.
(2.125)
∇ htt = −f
D−2
Similarly ∇2 hkk = −f ρ/(D − 2). The motion of a free particle is given by the
geodesic equation
ν
α
d 2 xµ
µ dx dx
+
Γ
= 0.
να
dτ 2
dτ dτ
(2.126)
In the Newtonian limit dxk /dτ 1 and the proper time τ is approximately the
same as the coordinate time or the Newtonian time t. The geodesic equation in
Eq.(2.126) now gives us that Γttt = 0 since d2 xt /dτ 2 = 0 and dt/dτ = 1. It also
gives us that
d 2 xk
= −Γktt ,
dτ 2
(2.127)
where d2 xk /dτ 2 = ak is an arbitrary componentPof the acceleration of the free
particle. Having that a = gD = −∇ΦD (x) = − k ∂ k ΦD (x)ek , Eq.(2.127) combined with Eq.(2.119) give us −∂ k ΦD (x) = f ∂ k htt . Or ∇2 htt = −f −1 ∇2 ΦD (x).
Used in Eq.(2.125) this leads to
D−3
D−3
2
2
∇ ΦD (x) = f
ρ=
κD ρ.
(2.128)
D−2
D−2
Which is the same result derived in the isotropic coordinates. And the constant
κD relates therefore to GD in the same manner as in Eq.(2.116). In the Newtonian
limit we also have that gtt = ηtt + 2f htt = −(1 + 2ΦD (x)). In this limit we
then have the graviton field component htt ∼ ΦD (x). (For a more complete
introduction to the theory of linearized gravity see [8, 9].)
2.2 General Relativity
2.2.4
27
The Planck Scale
In the study of particles there is a scale in which the quantum effects starts to
play a dominant role in the description of the black hole. This scale is known
as the Planck scale. A particle with mass M has in a 4-dimensional spacetime
a Schwarzschild radius RS = 2M GN /c2 . This is the size of this particle if it
becomes a black hole. This same particle’s size may also be described by the
Compton wavelength λC = ~/(M c). Atpa certain mass, these two lengths are
the same, RS = λC . This mass is M = ~c/(2GN ). In this area quantum and
gravitational effects
p are of the same size. With use of units where c = ~ = 1 we
have that M = 1/(2GN ) ∼ 1019 GeV. We may from this define a rationalized
Planck mass
r
1
MP ≡
,
(2.129)
8πGN
where MP ∼ 1018pGeV. We may also define a length, the Planck length, as
RP = GN MP = GN /(8π) ∼ 10−34 cm. At this length the Schwarzschild radius
and the Compton wavelength are of the same magnitude. At distances smaller
than this, gravity is presumed to be quantized and must be described by possibly
some kind of string theory.
In our presentation and evaluation of black hole physics we take the Planck
mass for four dimensions to be a more fundamental constant than the Newtonian
gravitational constant. This will become evident in the following, where we make
a generalization based on this choice.
2.2.5
The Gravitational Constant in D Dimensions and
the Fundamental Planck Mass
We would now like to relate a fundamental Planck mass to the gravitational
constant GD defined by Eq.(2.18).
As we stated in the introduction to this section on general relativity, it has
wrongly been claimed that Hilbert found the correct field equations of gravitation
around the same time as Einstein did. But even tough Hilbert should not be
credited for having found the field equations, he should be credited for the other
important contributions to the theory. Perhaps the most important was to apply
the variational principle to gravitation. He showed that with the use of the
variational
R 4 √principle you can derive the Einstein field equations from the action
S = d x −g L , where L is the Lagrange density, or the Lagrangian, and g is
the determinant of the metric tensor. This action is called the Einstein-Hilbert
action. The Einstein-Hilbert action in 4 dimensions may be written as
Z
1
4 √
S = d x −g
R + LM ,
(2.130)
16πGN
28
Chapter 2. Gravity
where R is the Ricci scalar and LM is the Lagrangian√for matter. This expression
may be rewritten so that the Planck mass MP = 1/ 8πGN is incorporated,
Z
Z
√
√
1 1
1 2
4
4
R + LM = d x −g
M R + LM . (2.131)
S = d x −g
2 8πGN
2 P
To keep this simple form in (D = 4 + n)-dimensional theory as well, we define a
fundamental Planck mass so that the generalized Einstein-Hilbert action can be
expressed as
Z
1 2+n
4+n √
M R + LM .
(2.132)
S = d x −g
2 D
Here MD is the Planck mass for D dimensions, or the fundamental Planck scale.
This generalization of the Einstein-Hilbert action is different from the one used
by Myers and Perry [5], on which most of the later years publications are based.
We can now use this action to establish a relation between the D-dimensional
gravitational constant and the fundamental Planck mass. This can be done by
deriving the (D = 4+n)-dimensional Einstein equations from the Einstein-Hilbert
action. We begin then by varying the action in Eq(2.132). For all infinitesimal
variations gµν → gµν + δgµν , the variation of the action δS = 0. Varying the
Einstein-Hilbert action in 4 + n dimensions gives us
Z
√
1 2+n
4+n
δS = d x (δ −g)
M R + LM
2 D
Z
1 2+n
∂LM µν
4+n √
µν
µν
+ d x −g
, (2.133)
M (Rµν δg + g δRµν ) +
δg
2 D
∂g µν
where we have used that R = Rµµ = g µν Rµν . In the expression for δS we have a
term δRµν . This is a surface term that vanishes. This is non-trivial, but will not
be proven in this discussion. We now use an expression for the determinant,
g=
1
gµν γ µν ,
D
(2.134)
where γ µν is the cofactor of gµν . The cofactor is defined as γ µν ≡ (−1)µ+ν det[(gµν )r ],
and (gµν )r is the rest of matrix gµν without row µ and column ν. Now, by differentiation of Eq.(2.134), we find that
∂g
1
= γ µν .
∂gµν
D
(2.135)
This enables us to find the partial derivative of g with respect to xλ ,
g,λ = ∂λ g =
∂g
∂g ∂gµν
1
=
= γ µν ∂λ gµν = g g µν ∂λ gµν .
λ
λ
∂x
∂gµν ∂x
D
(2.136)
2.2 General Relativity
29
From this follows the expression for the variation of the determinant g,
δg = g g µν δgµν ,
(2.137)
√
1√
1 1
δ g = g − 2 g g µν δgµν = −
ggµν δg µν .
2
2
(2.138)
and also
In the last transition we have used that
δ(gµν g νλ ) = δgµν g νλ + gµν δg νλ = δ(δ λν ) = 0. This finally makes us able to simplify
the variation of the Einstein-Hilbert action,
Z
√
δS = d4+n x ( −g)
1 2+n
1 2+n
∂LM
1
M R + LM + MD Rµν +
δg µν . (2.139)
× − gµν
2
2 D
2
∂g µν
From this we have that
1
∂LM
1 2+n
1
MD Rµν − gµν R − gµν LM +
= 0,
2
2
2
∂g µν
(2.140)
where the expression inside the brackets is recognized to be the Einstein tensor
components Eµν . Rearranging this last equation we get
1
1
∂LM
Eµν = Rµν − gµν R = 2+n −2 µν + gµν LM .
(2.141)
2
∂g
MD
Since the Einstein field equation is Eµν = κ4 Tµν = (1/MP2 )Tµν in D = 4 dimensions, it is natural to generalize the equation in D = 4 + n dimensions as
Eµν = κD Tµν =
1
MD2+n
Tµν .
(2.142)
The constant κD introduced in the beginning of this section is now seen to directly
relate to the Planck mass MD . This gives us the energy-momentum tensor
Tµν ≡ −2
∂LM
+ gµν LM .
∂g µν
(2.143)
The energy-momentum tensor is to be symmetric in the indices, and from Eq.(2.143)
we see that this is automatically the case. So the Einstein field equations in
D = 4 + n dimensions are
1
1
Rµν − gµν R = 2+n Tµν .
2
MD
(2.144)
30
Chapter 2. Gravity
By raising one of the indices and contract we find that
1
1
R − g µµ R = 2+n T,
2
MD
(2.145)
where g µµ = δ µµ = D. Transforming this equation we get
R=
2T
,
(2 − D)MD2+n
(2.146)
which we may use in Eq.(2.144), so that
Rµν =
1
MD2+n
Tµν
T
1
+ gµν
2+n =
(2 − D)MD
MD2+n
Tµν
gµν T
+
2−D
.
(2.147)
The trace of the energy-momentum tensor T = T tt + T 11 + . . . + T dd = T tt , since
the spatial diagonal components represent pressure p and in our non-relativistic
case we have no pressure. We then have that
1
Ttt
1
D−3
Rtt = 2+n 1 −
Ttt = 2+n
.
(2.148)
D−2
D−2
MD
MD
Combining this with Eq.(2.110) and that Ttt is the mass density ρ, we have that
1
D−3
2
ρ.
(2.149)
∇d ΦD (x) =
D − 2 MD2+n
This is what we previously have found to be the D-dimensional Poisson equation,
where we now have introduced the fundamental Planck mass in the expression.
Using Eq.(2.18) we find that the gravitational constant in D = d + 1 dimensions
in terms of the Planck mass is
)
Γ( D−1
D−3
1
D−3
1
1
2
GD =
=
.
(2.150)
2+n
D−2
(D−1)/2
D − 2 MD ΩD−2
D − 2 2π
MD
In theories with D ≤ 11 the numerical factor involved is ∼ 10−2 . The factor could
however have been redused with other definitions of the gravitational constant
and the fundamental Planck mass. In D = 4 dimensions we then recover
1 1
G4 =
8πGN = GN .
(2.151)
2 4π
Eq.(2.150) now gives us the opportunity to rewrite the expression for the
Schwarzschild radius from Eq.(2.85) into
1
1 D−3
M D−3
1
2
(2.152)
RSD =
MD MD
(D − 2)ΩD−2
1
# D−3
1 "
)
Γ( D−1
1
M D−3
2
=
.
(2.153)
MD MD
(D − 2)π (D−1)/2
2.3 Extra Dimensions and the Hierarchy Problem
31
This expression reveals that the lower the fundamental Planck mass MD , the
larger the Schwarzschild radius. In the sequel we will explore the possibility that
the fundamental Planck mass may be several orders of magnitude smaller than
the observed Planck mass. A D-dimensional black hole will then be larger than
its 4-dimensional relative.
2.3
Extra Dimensions and the Hierarchy Problem
In this section we will examine a possible solution to the hierarchy problem. This
is the problem of the large difference between the two fundamental energy scales,
the Planck scale MP ∼ 1018 GeV and the electroweak scale mEW ∼ 103 GeV. But
while the electroweak interaction is probed at distances ∼ m−1
EW , gravitational
forces have only been accuratly mesured in the ∼ 1 mm range, not nearly down
to distances ∼ MP−1 . Since gravity is such a weak interaction several factors may
contribute to make the measurement difficult. Electromagnetic forces, seismic,
thermal and other background effects may cause problems while trying to measure the strength of the gravitational force. But when these effects are taken into
account no deviation from the Newtonian force law has been found. Most people would think that Newton’s inverse-square law should hold also for distances
smaller that those measured at, but another possibilty is that gravity is different
at these distances.
Arkani-Hamed, Dimopoulos and Dvali (ADD) [10] used this possibility to
propose a solution to the hierarchy problem. They supposed that mEW could be
the only fundamental energy scale. But how may the Newtonian gravitational
constant GN ∼ 1/MP2 arise with such an idea? The answer may be that there are
extra compact spatial dimensions that are undetectable at distances probed at.
2.3.1
Compact Spatial Dimensions
Until now we have threated the extra spatial dimensions in our D-dimensional
theory as if they were equivalent to the three spatial dimensions we observe in
every day life. But if this had been the case these dimensions would be possible
to observe, at least in some deviation in the gravitational force calculated from
Newton’s expression for three spatial dimensions in Eq.(2.1). Observations of
such deviations have not been found at the distances probed, which stretch from
astronomical distances down to approximately one millimeter. The possible extra
dimensions and our usual spatial dimensions may therefore not be similar in
nature.
Let us now assume that the extra dimensions are of a finite length Ln and
periodic, i.e. the dimensions are said to be compact. In such a Kaluza-Klein
compactified dimension the point x is equal to the point x + Ln . It is here
32
Chapter 2. Gravity
(a)
(b)
(c)
Figure 2.2: The figures show three possible compactifications of a
n = 2 dimensional surface. By identifying the edges on the opposite
side by letting the arrrows of same kind point in same direction, three
different flat manifolds are constructed. Fig. 2.2(a) gives a torus.
Fig. 2.2(b) gives a so-called Klein bottle. And Fig. 2.2(c) gives a
projective plane.
usual to introduce the radius of the compact dimension Rn . The length Ln is
then equal 2πRn . How may this compactification help us explain why we only
observe three spatial dimensions? To answer this let us look at a 2-dimensional
scenario with one infinite and one compact dimension. This can be described as
an infinitely long sylinder with the radius equal to that of the compact dimension.
At distances much larger than the length of the compact dimension the sylinder
would appear to be an 1-dimensional line and the compact dimension would be
unobservable. On the other hand, at distances much smaller than the length
of the compact dimension it would appear that the surface of the sylinder is an
uncompact infinitely large flat surface. If a generalization of this example is made
to our D-dimensional spacetime it would become apparent that the expressions
derived in the two previous sections is valid at distances much smaller than those
of the compact dimensions. And at much larger distances the 4-dimensional
expressions should be retrieved. Based on this assumption it should be possible
to find a connection between the 4-dimensional effective Planck mass MP and
the fundamental Planck mass MD .
When there are more than one extra dimension the compactification of these
is no longer done unambiguously. This is illustrated for n = 2 extra dimensions
in Fig. 2.2. In the following we will use the flat n-torus, the analogue to the
2-torus in Fig. 2.2(a), to describe the compact dimensions. We will assume a
symmetrical torodial compactification, i.e. all the compact dimensions have the
same radius R.
Even though we will not investigate this further, it should be pointed out that
compactification of the extra dimensions is not the only possible way to construct
a higher dimensional model compatible with the experimentally tested force law.
In 1999, an article by Randall and Sundrum [11] introduced an alternative. In
2.3 Extra Dimensions and the Hierarchy Problem
33
4 dimensional
spacetime
mirror
mass
L
M
extra
dimensions
Figure 2.3: A mass M placed in a (D = n+4)-dimensional spacetime
where the n extra compactified dimensions of length L are uncompactified by placing mirror masses periodically in the extra dimensions of infinite length. We have then a n-dimensional lattice. This
new uncompactified space is equal to Rn .
this Randall-Sundrum model, the metric describing our four familiar dimensions
is dependent of the coordinates of the extra dimensions. As a consequence we
may live in a (4 + n)-dimensional non-compact world, without going against
experimental gravity.
2.3.2
The Retrieval of the Relations of Four Dimensions
In the previous subsection we proposed that it should be possible to find a relationship between the 4-dimensional Planck mass and the fundamental Planck
mass, so that the 4-dimensional expressions are retrieved at large distances compared to the size of the compact dimensions. We will now present three ways
of estblishing this relationship. For each of the methods we will also discuss the
possible number of extra compact dimensions based on the desire to have only
one fundamental energy scale ∼ mEW .
Method 1: Gravitational Flux Conservation
Suppose that a mass M is placed at the origin. Because of the compact extra
dimensions this mass will not only contribute once to the gravitational force on a
test mass, but a infinite number of times. One may uncompactify this situation
by placing “mirror” masses M periodically, at intervalls L, in the uncompactified
extra dimensions, as in Fig. 2.3.
34
Chapter 2. Gravity
For a test mass m at distances r L from the mass M , the small contribution
to the force from the “mirror” masses is negligible and we have the D-dimensional
graviational force law
FD (r) = −
GD M m
GD M m
= − n+2
D−2
r
r
(r L).
(2.154)
But if the test mass is placed at a distance r L from the mass M , the test mass
will experience the usual gravitational force ∝ 1/r 2 , but now it would appear
that an infinitely long n-dimensional line with uniform mass density contributes
to the force. Now consider a “cylinder” C centered around the n-dimensional
lattice with side length l and end caps consisting of two-dimensional spheres of
radius r [12]. We now apply the D-dimensional Gauss’ law
I
gD dAD−2 = −
GD Min
ΩD−2 r D−2 = −ΩD−2 GD Min ,
D−2
r
where Min is the mass inside the surface C. We then have
Z
ln
(r L)
gD dAD−2 = −ΩD−2 GD M n
L
boundary C
(2.155)
(2.156)
and
Z
boundary C
g4 ln dA2 = −4πln GN M
(r L),
(2.157)
where ln dA2 is a D − 2-dimensional surface element. In Eq.(2.156), M l n /Ln is
the mass inside the surface C. Since the gravitational flux out of C should be
independent of r we may put Eq.(2.156) equal Eq.(2.157) and this gives us [12]
GN =
ΩD−2 GD
.
4π Ln
(2.158)
By using Eq.(2.150) and that GN = 1/(8πMP2 ) from Eq.(2.129), we have
1
ΩD−2
=
2
8πMP
4π
D−3
D−2
1
.
MD2+n ΩD−2 Ln
(2.159)
Rewriting this last equation we may express our standard Planck scale MP using
the higher-dimensional Planck scale MD ,
MP2
1
=
2
D−2
D−3
Ln MD2+n
(2.160)
2.3 Extra Dimensions and the Hierarchy Problem
The length L of the compact dimension is then
MP2
D−3
n
L =2
D − 2 MD2+n
1
36
2
D − 3 n 10 n (GeV) n
L∼ 2
2
+1
D−2
MDn
1
2 +1
30
D − 3 n 10 n −12 TeV n
=2 2
D−2
eV
MD
n1
2 +1
30
TeV n
D−3
−19
m
10 n
=2 2
.
D−2
MD
35
(2.161)
The hierarchy problem can now be solved by supposing that MD = mEW ∼
10 GeV. So we have that
1
30
D−3 n
(2.162)
× 10 n −19 m.
L∼2 2
D−2
3
Then for n = 1 extra dimension L ∼ 1011 m. This would imply deviations from
the Newtonian gravitation law at solar system distances. One extra dimension
is thus empirically impossible. For n = 2 we have L ∼ 10−2 cm. This is outside
the area of accurate measurement of gravity. For n > 2 and MD ∼ 103 GeV, the
modification of gravity only becomes noticeable at distances smaller than those
currently probed by experiments.
Method 2: An infinitely large (D − 4)-dimensional mass distribution
If we have the same situation as under Method 1, where we look at a infinitely
large (D − 4)-dimensional lattice of “mirror” point masses. At large distances
this lattice appears to be a hypersurface with homogeneous mass density
ρ = M/LD−4 . A mass element in this hypersurface may be described as
dM = ρΩD−5 aD−5 da, where ΩD−5 aD−5 is the surface of a (D − 5)-dimensional
sphere and a is the radius of the sphere. The mass element is then a spherical
shell with a infinitesimal thickness da. This also holds for D = 5, as long as
one remembers
that a ≥ 0. In a point P outside the hypersurface, at a distance
√
x = a2 + r 2 from such a mass element (see Fig. 2.4), the element will produce
a gravitational field strength
dgD = −GD
dM r
raD−5 da
=
−Ω
G
ρ
D−5
D
D−1 ,
xD−2 x
(a2 + r 2 ) 2
(2.163)
outward from the center of the sphere along r. Here r is the distance from the
center of the hypersurface to the point P . We have simplified the calculation by
arguing geometrically that the field strength is directed along r only, and thus
36
Chapter 2. Gravity
x
a
dM
θ
C
r
P
Figure 2.4: A point P at a distance r from the center C of a mass
distribution, will experience a gravitational field dg due to a mass
element
dM a distance a from C. The point P is then a distance x =
√
2
2
a + r from each point of the mass element. Each of these points
is source to a gravitational field strength in P along x. But because
of the symmetry of the mass element only the field strength along r
will contribute. The field strength along x may then be multiplied by
cos θ = r/x to find the contributing field strength along r.
the expression is multiplied with cos θ = r/x. Here θ is the angle between r and
x. We find that the total contribution a (D − 4)-dimensional hypersurface of
radius a makes to the gravitational field strenght at point P is
Z ∞
aD−5 da
gD (r) = −ΩD−5 GD rρ
D−1
(a2 + r 2 ) 2
0
"√
#
πΓ D−4
1
2 .
(2.164)
= −ΩD−5 GD ρ
4 r 2 Γ D−1
2
Here we have used that a → ∞ since the hypersurface is infinitely large. If we
use that the mass density ρ = M/LD−4 , we have that
"√
#
πΓ D−4
1
M
2 gD (r) = −ΩD−5 GD ρ
.
(2.165)
D−1
2
4 r Γ 2
LD−4
This gravitational field strength from the (D − 4)-dimensional hypersurface must
correspond to the field strength from a single point mass in four dimensions, i.e.
"√
#
πΓ D−4
GN M
1
GD M
2
=− 2 .
(2.166)
−ΩD−5
D−1
2
D−4
4r
L
r
Γ 2
This equation gives the connection between the D-dimensional and four-dimensional
gravitational constant,
"√
#
πΓ D−4
1
GD
ΩD−2 GD
2
GN = ΩD−5
=
.
(2.167)
D−3
D−4
4
L
4π LD−4
Γ 2
2.3 Extra Dimensions and the Hierarchy Problem
37
Expressing the two Γ-functions in terms of total solid angles in the last transition
simplifies the relation some. We now see that this is exactly what was found in
Eq.(2.158) with use of the previous method. This is perhaps no surprise when
we recall that the Newtonian force law was derived from Gauss’ law. These two
first methods then yield the same relation between the D-dimensional and the
four-dimensional Planck scales. Subsequently the discussion concerning possible
number of extra dimensions, if we assume to have only one energy scale
mEW ∼ 103 , will reveal the same result as the previous treatment of the subject.
The expression for GN could as easily have been derived by in stead using
the gravitational potential from the start. We would then have avoided the use
of the geometrical argument necessary in simplifying the calculation of the field
strength.
Method 3: Equal Force at Distance L
The third way to determine the relationship between the D-dimensional and the
four-dimensional gravitational constant, is to state that at one distance from the
mass at the origin the two gravitational forces from it must be the same. Since
FD (r) is valid r L and FN (r) is valid r L, these two gravitational forces are
said to be the same at the distance L. This gives us [13]
GD = GN LD−4 .
(2.168)
This result differ from the result of Eq.(2.158) and Eq.(2.167) only by the factor
ΩD−2 /4π. In theories with D ≤ 11 dimensions this factor is of order one. So the
results do not differ considerably.
By using Eq.(2.150) and that GN = 1/(8πMP2 ) from Eq.(2.129), we have
D−3
1
1
1
=
LD−4 .
(2.169)
2+n
2
D − 2 MD ΩD−2
8πMP
Rearranging this last equation we may express our standard Planck scale MP
using the higher-dimensional Planck scale MD ,
D−2 1
2
ΩD−2 Ln MD2+n
(2.170)
MP =
D − 3 8π
The length L of the compact dimension is then
D−3
8π MP2
n
L =
D − 2 ΩD−2 MD2+n
n1
2 +1
30
D−3
8π
TeV n
−19
L∼2
10 n
.
m
D−2
ΩD−2
MD
(2.171)
38
Chapter 2. Gravity
Using MD = mEW ∼ 103 GeV we have that
L∼2
D−3
D−2
8π
ΩD−2
n1
30
10 n −19 m.
(2.172)
Then for n = 1 extra dimension L ∼ 1011 m and for n = 2 we have L ∼ 10−2 cm.
Again for n > 2 extra dimensions the gravity becomes modified only at distances
smaller than currently probed.
We see that the lowest possible number of extra dimensions, found from these
three methods, do not differ. And the calculated lengths of the extra dimensions
are of the same order, independently of what method used. Because the expressions for GN found in Eq.(2.158), Eq.(2.167) and Eq.(2.168) only differ in factors
more or less of order 1, we have that MP2 ∼ LD−4 MDD−2 for all the methods.
The method used to find the exact connection between the D-dimensional and
four-dimensional gravitational constant is therefore not of great importance.
2.3.3
Higher Dimensional Black Holes in a World with
Compact Dimensions
As long as the Schwarzschild radius of a black hole is much larger than the
Planck length, gravitational effects dominate the description of the black hole.
In this region the classical relativistic approach we have used to describe this
graviational phenomenon is sufficient. And is still the most accurate theory of
gravity we have got at our disposal. The black holes are then restricted to have
a mass MBH MD , or i.e. the radius RSD 1/MD .
With the proposal that a higher dimensional fundamental Planck mass could
be much smaller than the 4-dimensional one, the door to the possible production
of observable black holes, at relatively low energies, described semi-classically,
could be opened. But for the black holes to be affected by such a fundamental
Planck mass, it has to be a higher dimensional black hole. For a black hole
to be accurately described by the higher dimensional Schwarzschild metric, in a
world with extra compact spatial dimensions, we have argued that the size of
the black hole must be much smaller than the length of the compact dimensions.
This restricts the Schwarzschild radius of the black hole to be RSD L. As the
size of a black hole horizon approaches the size of the compact dimensions, the
horizon becomes more and more distorted and our description of the black hole
as higher dimensional object in infinitely large extra dimensions does not apply
anymore.
For our description of a higher dimensional black hole to be accurate we have
2.3 Extra Dimensions and the Hierarchy Problem
39
to restrict its radius to be
1
RSD L.
MD
(2.173)
If we assume that a difference of two orders of magnitude is sufficiently large to
obey these restrictions, a situation of a black hole with mass MBH ∼ 100 TeV in
a spacetime with a fundamental Planck mass MD ∼ 1 TeV, would obey the lower
limit restriction and could be described semiclassically. In, say a 10-dimensional
spacetime, the size of such a black hole would be RSD ∼ 10−18 m. Compared to
the length of the compact dimensions L ∼ 10−14 m, we see that also the restriction
on the upper limit would be satisfied.
We have in the end of Section 2.2 already seen that a lower valued fundamental
Planck mass results in a larger Schwarzschild radius. The possibility that a
black hole is higher dimensional, rather than 4-dimensional, means that its radius
is governed by a possibly much smaller fundamental Planck mass, rather than
the conventional 4-dimensional Planck mass, and therefore may be much larger
than expected from a 4-dimensional theory. This may be seen directly from the
relation RSD ∼ (L/RSD )D−4 RS4 , originating from MP2 ∼ LD−4 MDD−2 found in
the previous subsection.
One should emphasize that in a D-dimensional spacetime with symmetrical
torodial compactification, where all the compact dimensions are of same length, a
black hole is restricted to either being 4-dimensional or D-dimensional, depending
its size. To have, say a (D−2)-dimensional black hole in a theory of D dimensions,
two of the compact dimensions must be much smaller than the rest and the black
hole radius must be larger than the two but smaller than the rest of the compact
dimensions.
Chapter 3
Black Hole Thermodynamics
When Stephen Hawking in his article “Particle Creation by Black Holes” [14]
from 1975 described how a black hole radiates thermally, it had for some years
already been proposed that black holes could be assigned certain thermodynamical properties.
It seems that Wheeler was the first to point out that the very existence of
black holes violates the second law of thermodynamics, which states that the
entropy is non-decreasing. A black hole could swallow a hot body with a certain
entropy. And for an observer outside the black hole, it would appear that after the
relaxation processes were complete the entropy in the part of universe accessible
to him had decreased.
Later, in articles by Floyd and Penrose [15] and Christodoulou [16], it was
found that the area of a black hole increases irreversibly under almost any transformation. Hawking showed in 1971 that the area actually never decreases classically. Bekenstein [17] suggested that this implied that the area of the black
hole in some sense was the entropy of the black hole, due to the similar behaviors of the two quantities. Bekenstein also suggested a generalized second law
of thermodynamics: The classical entropy of matter + some multiple of the area
of the black hole never decreases. This would solve Wheeler’s entropy-problem.
In this work Bekenstein expressed a quantity Tbh with the same dimension as
temperature, but he emphasize that this should be regarded as merely a mathematical expression, not the actual temperature of the black hole. On the other
hand, Hawking found that the generalized second law could be violated unless the
black hole could emit particles as well as absorb them. From this idea of particle
emission he could show that a black hole actually radiates like a blackbody of a
certain temperature. This effect has later been known as the Hawking radiation
and the temperature as the Hawking temperature. With the discovery of this
black hole temperature, the connection between black hole physics and classical
thermodynamics was strengthened.
The connection between black hole physics and classical thermodynamics has
been quite fertile. As well as the general laws of thermodynamics, gedanken
41
42
Chapter 3. Black Hole Thermodynamics
experiments with specific thermodynamic devices, like the heat engine, has its
analogue in black hole dynamics. Some of these laws of black hole thermodynamics will be briefly reviewed in the forthcoming sections.
In Section 3.1, the Hawking temperature is investigated and the temperature
of a D-dimensional black hole is derived.
In Section 3.2, the corresponding black hole entropy is discussed and derived.
And in Section 3.3, a generalized Stefan-Boltzmann law is derived, to produce
an estimate of the black hole lifetime.
3.1
The Black Hole Temperature
We will in this section describe how Hawking derived a temperature for a black
hole and how he explained this astonishing effect. Then we will examine the problem another way, which also provides the same result. This method of assigning
a temperature will first be illustrated with the Unruh effect. And in the end of
the section we will use it to derive the Hawking temperature in D dimensions.
Hawking [14] had no consistent quantum theory of gravity, as is still the situation to day. And as Hawking stated, classical general relativity still provides the
most accurate description of gravity. In general relativity the Einstein equations
describe how matter curves the spacetime. These equations consist of a classical
metric combined with the stress-energy-momentum tensor of the classical matter
field. But we know that quantum mechanics plays an important role in the behavior of the matter field. In lack of a satisfactory quantum theory, Hawking used in
his paper an approximation to a presumably deeper theory. This semi-classical
approximation consisted of applying ordinary quantum field theory to matter
fields in a fixed curved background. That means that the matter fields, such as
electromagnetic or neutrino fields, are treated quantum mechanically and obey
the usual wave equations where the Minkowski metric is replaced by a classical
metric which describes a curved spacetime. This approximation works as long
as the gravitating mass MBH is much larger than the Planck mass and quantum
gravity effects may be disregarded.
The idea behind quantum particle production in curved spacetime is that a
particle is depending on the observer’s reference frame. There is no invariant way
to decompose the field into positive and negative frequency. An observer with
proper time t1 define particles as positive frequency components with respect
to this time t1 , but another observer with proper time t2 define particles with
respect to this time. In general the number of positive frequency components
will be different in the two reference frames. This means that these two observers
will observe a different number of particles. Even in flat spacetime this effect
may occur. If the two frames differ with only a Lorentz transformation, they
will observe the same amount of particles (because quantum field theory in flat
spacetime is Lorentz invariant). But if the two reference frames have a relative
3.1 The Black Hole Temperature
43
acceleration, they will measure different particle numbers. This is described by
Unruh [18].
To derive a temperature associated with a black hole, Hawking now studied
the gravitational collapse of a star. He started with a spacetime containing a
star. Far from the star the the spacetime is flat. He supposed that the star
collapses into a black hole. A wave packet originating far from the collapsing
star propagates through the star, just escaping being captured behind the event
horizon of the newly formed black hole, and propagates back to the flat region.
Let the wave start out with only positive frequency waves with respect to the
time of the flat region t. When the wave escapes the forming horizon, it is in a
area of great curvature. Back in the flat region again, the wave will consist of
both positive and negative frequency components. These new negative frequency
components correspond to particle production. Hawking found that the expected
number of particles spontaneously created at the horizon is precisely the same as
that emitted from a perfect blackbody at a temperature [14]
T4 =
κ
,
2π
(3.1)
where κ is the surface gravity on the horizon and the Boltzmann constant kB = 1.
For the 4-dimensional Schwarzschild metric κ = 1/(2RS4 ). (See Appendix C.)
So the temperature is inversely proportional to the Schwarzschild radius of the
black hole. So when the black hole gains mass, the temperature decreases. A
black hole that does not obtain mass, will lose all its mass because of radiation. The expression of the lifetime of a Schwarzschild black hole is found in
Subsection 3.3.2.
Although Hawking derived the temperature in connection with the gravitational collapse of a star, the resulting particle creation does not depend on the
details of the collapse at late times. And the black hole temperature may therefore also be derived from a static black hole.
The Hawking effect is usually explained by the creation of virtual particle
pairs at the horizon. One particle has negative energy, the other has positive.
The particle with negative energy will travel into the black hole and cause its
mass to decrease. The other particle of the pair, having positive energy, will
travel outward to infinity where it will constitute a part of the thermal specter
emitted from the black hole. The exact process will be described and explained
by a more direct approach in Chapter 4, by use of quantum field theory in a
curved spacetime.
In the following sections we derive the black hole temperature in another way.
By use of statistical mechanics and the connection to quantum field theory, it is
possible to assign a temperature to a quantum system. As an illustration we will
first use this to derive the Unruh temperature, which is observed by accelerated
detectors. And in the end, we will derive the Hawking temperature for a black
hole in D dimensions.
44
Chapter 3. Black Hole Thermodynamics
3.1.1
Statistical Mechanics and Quantum Field Theory
We will in this subsection express the partition function of statistical mechanics
by use of a Euclidean path integral and by doing so, showing the connection
between a statistical mechanical and a quantum mechanical system.
Now, in statistical mechanics the partition function of a system in thermodynamic equilibrium with discrete states is defined as
X
X
e−Eq /(kB TD ) ,
(3.2)
Z=
e−βEq =
q
q
where kB is the Boltzmann constant, TD is the temperature of the system and Eq
is the energy of the state | qi. If one uses that the Hamilton operator Ĥ acting
on an eigenstate | qi gives Eq | qi, the partition function may be expressed as
X
X
Z=
e−βEq =
hq | e−β Ĥ | qi = Tr(e−β Ĥ ).
(3.3)
q
q
If the system is a system of continuous states, the sum is replaced with an integral
over the states so that the partition function becomes
Z
Z = dqhq | e−β Ĥ | qi.
(3.4)
This integrand may be expressed using complete sets of states which gives
Z
Z
Z
−β Ĥ
hqB = q | e
| qA = qi = dqN −1 dqN −2 · · · dq1 hqN = q | e−εĤ | qN −1 i
× hqN −1 | e−εĤ | qN −2 i · · · hq1 | e−εĤ | q0 = qi, (3.5)
where β = N ε. Here, by expanding to the lowest order of εĤ and using that
hqk | qk−1 i = δ(qk − qk−1 ), an arbitrary element of this product may be written as
hqk | e−εĤ | qk−1 i = δ(qk − qk−1 ) − εhqk | Ĥ | qk−1 i,
(3.6)
where we have that
hqk | Ĥ | qk−1 i =
Here is
2
hqk | p̂ | qk−1 i =
Z
1
hqk | p̂2 | qk−1 i + hqk | V (q̂) | qk−1 i.
2m
2
dp hqk | pihp | p̂ | qk−1 i =
and
hqk | V (q̂) | qk−1 i = V (qk )δ(qk − qk−1 ) = V (qk )
Z
Z
dp ip(qk −qk−1 )
e
2π
dp ip(qk −qk−1 )
e
.
2π
(3.7)
(3.8)
(3.9)
3.1 The Black Hole Temperature
45
This gives us
hqk | e
−εĤ
2
dp ip(qk −qk−1 )
p
| qk−1 i =
1−ε
e
+ V (qk )
2π
2m
Z
dp ip(qk −qk−1 ) −ε p2 −εV (qk )
e
e 2m e
=
2π
Z
(3.10)
We may now Rcalculate this √
integral by completing the square in the exponent
∞
−n2
= π. We then have
and use that −∞ dn e
Z
√m
2
ε
m
dp −[√ 2m
2
p−i 2ε
(qk −qk−1 )] − 2ε
−εĤ
e
hqk | e
| qk−1 i =
e (qk −qk−1 ) e−εV (qk )
2π
r
qk −qk−1 2
1 2m √ − mε
πe 2 ( ε2 ) e−εV (qk )
=
2π
ε
r
h
i
m −ε m2 ( qk −qεk−1 )2 +V (qk )
=
e
.
(3.11)
2πε
Finally we get
hqB = q | e
−β Ĥ
| qA = qi =
Z NY
−1
j=1
∀ {qA =qB =q}
dqj
r
m − Pk ε
e
2πε
h
m qk −qk−1 2
(
) +V
2
ε
(qk )
i
. (3.12)
If we now let ε → 0 and N → ∞, we get that (qk − qk−1 )/ε → q̇k (the derivative
of qk ), and we may define the Euclidean Lagrangian
m 2
q̇ + V (qk ).
(3.13)
2 k
R
P
→ Dq and
→ . This all combined gives
LEk =
We also have that
us
QN −1
j=1
dqj
hqB = q | e
p
−β Ĥ
m
2πε
| qA = qi =
Z
Dq
∀ {qA =qB =q}
e−
Rβ
0
dτ LE
.
(3.14)
If we now recall the Eq.(3.4), we may now express the partition function as
Z
Z
Z
R
Rβ
− 0β dτ LE
=
Dq e− 0 dτ LE .
(3.15)
Z = dq
Dq e
∀ {qA =qB =q}
∀ {closed paths}
This is very similar to the Schrödinger time-evolution operator, expressed with
path integral,
Z
R tB
(3.16)
U (qA , qB ) = Dq ei tA dtL ,
46
Chapter 3. Black Hole Thermodynamics
where qA = q(tA ) and qB = q(tB ) is the start- and end-position of the path.
With use of a Wick rotation t → −iτ , one may go from the time-evolution
operator to our partition function. This means that β is an imaginary time
difference. But it also may be regarded as our time coordinate t has been transformed into a spatial coordinate τ . Now, since in Eq.(3.15) our start- and endposition is to be the same, the coordinate q must be periodic with period β. Thus
q(τ ) = q(τ + β), where τ is the Euclidean time. This property of periodicity in
imaginary time is characteristic of a thermal state. So if we introduce a set of
coordinates that are the same for a time t and a time t + iβ, we may associate a
temperature with this system.
3.1.2
The Unruh Temperature
After Hawking published his paper on particle creation by black holes, many
publications on the subject followed. In one of these, Unruh [18] examined the
behavior of accelerated particle detectors in flat Minkowski spacetime, where all
relevant quantum fields is in the vacuum state. He found that a detector that
experiences a uniform acceleration detects particles, while a detector at rest in the
flat spacetime does not. And this particle radiation corresponds to a blackbody
radiation with a specific temperature determined by the uniform acceleration
experienced by the detector. This temperature is the Unruh temperature. With
use of what we found in the previous section we are able to derive this temperature
in a simple manner.
First we have to find the Rindler coordinates which give the proper time
and proper distance as measured by the accelerated detector. In this reference
frame the detector is uniformly accelerated. As first step toward the Rindler
coordinates, we may parametrize the worldline xµ (τ ) of the detector by the proper
time τ . Suppose the detector has a 4-velocity uµ ≡ dxµ /dτ at some time. The comoving reference frame is at that moment an inertial frame that moves with the
4-velocity uµ . In this co-moving frame the velocity has the components (1, 0, 0, 0)
at that time. We then have the Lorentz-invariant 4-velocity identity
uµ uµ = −1,
(3.17)
The derivative of this identity with respect to the proper time gives another
Lorentz-invariant property,
aµ uµ = 0,
(3.18)
where aµ ≡ d2 xµ /dτ 2 = duµ /dτ is the 4-acceleration. From this property we find
that the 4-acceleration in the co-moving frame has the components (0, a1 , a2 , a3 ).
Here (a1 , a2 , a3 ) ≡ a is the 3-acceleration , a = d2 x/dτ 2 , observed in the comoving frame. We now have
aµ aµ = |a|2
(3.19)
3.1 The Black Hole Temperature
47
For simplicity we look at an observer accelerating along the x-axis, a =
(a, 0, 0). We need then only obtain the functions x(τ ) and t(τ ) for the Minkowski
coordinates, since y and z remain constant throughout the motion. From now
on we therefore look at the (1 + 1)-dimensional spacetime. Then from Eq.(3.17)
and Eq.(3.19) we have that
0 2 1 2
du
du
−
+
= a2 ,
(3.20)
dτ
dτ
−(u0 )2 + (u1 )2 = −1.
(3.21)
If we assume that u0 > 0 (t grows with τ ) and du1p
/dτ > 0 (the acceleration points
0
in the positive x direction), we find that u = 1 + (u1 )2 and its derivative is
du0 /dτ = [1 + (u1 )2 ]−1/2 u1 (du1 /dτ ). With the initial condition u1 (0) = 0 this
derivative inserted into Eq.(3.20) and integration leads to
dx
dt
= u1 (τ ) = sinh(aτ )
and
= u0 (τ ) = cosh(aτ ).
(3.22)
dτ
dτ
By choosing the initial conditions x(0) = a−1 and t(0) = 0, integration of the
previous expressions gives the world line
1
1
and
t(τ ) = sinh(aτ ).
(3.23)
x(τ ) = cosh(aτ )
a
a
The world line of the detector is then the hyperbola x2 − t2 = a−2 .
We now use this point (t, x) as origin of a co-moving reference frame were the
detector is at rest, but observes a uniform acceleration a. The detector trajectory
(t, x) then corresponds to a world line in the co-moving reference frame where
the spatial coordinate xR = 0. The coordinate time at an arbitrary point in
the co-moving frame is defined by tR ≡ τ0 , where τ0 is the proper time of the
detector. At each proper time τ0 the co-moving reference frame is an inertial
frame moving with 4-velocity uµ = dxµ /dτ0 . A measuring stick of proper length
xR placed by the detector is accelerated together with the detector. The stick
has the endpoints (tR , 0) and (tR , xR ). The stick at time tR may be represented
by the 4-vector sµ = (0, xR ), which connect the events (tR , 0) and (tR , xR ). Since
the co-moving frame is an inertial frame of reference the coordinates sµ and sµlab
(measured in the Minkowski frame), are connected by a Lorentz transformation,
s1lab = γ(s1 + us0 ) = u0 s1 + u1 s0 ,
s0lab = γ(s0 + us1 ) = u0 s0 + u1 s1 ,
(3.24)
(3.25)
where uµ = γ(1, u) and γ = [1 − u2 ]−1/2 . The stick in the Minkowski frame,
represented by the sµlab , is then
dtM
= xR cosh(aτ0 ),
dτ0
dxM
= u 1 xR = x R
= xR sinh(aτ0 ).
dτ0
s1lab = u0 xR = xR
(3.26)
s0lab
(3.27)
48
Chapter 3. Black Hole Thermodynamics
In the Minkowski frame the coordinates for the far end of the stick is given by
1
(3.28)
tM = t + s0lab = (1 + axR ) sinh(atR ),
a
1
xM = x + s1lab = (1 + axR ) cosh(atR ),
(3.29)
a
where we have used the expressions for t and x from Eq.(3.23). The coordinates
tR and xR of the co-moving frame is the Rindler coordinates
1
xM + t M
tR =
,
(3.30)
ln
2a xM − tM
q
−1
xR = −a + x2M − t2M
(3.31)
And the Rindler metric is
ds2 = −(1 + axR )2 dt2R + dx2R ,
(3.32)
This metric has a singularity at xR = −a−1 and for xR < −a−1 we notice that
dtR /dτ < 0, which means that an accelerated observer cannot measure proper
distances longer than a−1 in the direction opposite to the acceleration. The line
xR = −a−1 may therefore be regarded as an event horizon. So the coordinates
in this co-moving frame are defined for −∞ < tR < +∞ and −a−1 < xR < +∞,
and cover only a part of the Minkowski space, xM > |tM |.
Since sinh(atR +2πi) = sinh(atR ) we see from Eq.(3.28) and Eq.(3.29) that the
Minkowski coordinates are periodic. And in the previous subsection we showed
that coordinates with a periodicity β in imaginary time, could be associated with
a temperature. To obtain this, 2π have to be equal to aβ. The temperature
measured in the uniformly accelerated reference frame, the Unruh temperature,
is then found to be
a
TU =
,
(3.33)
2πkB
since β = 1/kB T from the previous subsection.
If a conformal flat metric is preferred, we may introduce the conformal Rindler
coordinate x̃R so that the metric is expressed as
ds2 = (1 + axR )2 (−dt2R + dx̃2R ).
(3.34)
When we compare this metric with the one of Eq.(3.32) we see that dx̃R =
dxR /(1 + axR ), which when integrated reveals that x̃R = (1/a) ln(1 + axR ) or
that eax̃R = (1 + axR ). If we use this last relation in Eq.(3.28) and Eq.(3.29) we
find that the conformal Rindler coordinate is related to Minkowski coordinates
by
1
(3.35)
tM = eax̃R sinh(atR ),
a
1
xM = eax̃R cosh(atR ).
(3.36)
a
3.1 The Black Hole Temperature
49
The conformal coordinate is defined in the interval −∞ < x̃R < +∞. And the
conformal Rindler metric becomes
ds2 = e2ax̃R (−dt2R + dx̃2R ),
(3.37)
which is finite for finite values of x̃R . We notice that these coordinates show
the same periodic behavior as the previous used coordinates. And therefore the
temperature observed is the same as found in Eq.(3.33).
3.1.3
The Temperature of a Black Hole in D Dimensions
We will now use the same argument as in the preceeding subsection to deduce
the temperature of D-dimensional black hole.
Consider a metric of the form
ds2 = −f (r)dt2 +
dr 2
+ r 2 dΩ2D−2 ,
f (r)
(3.38)
where f (r) is an arbitrary function of r. We see that this metric appears to
have a couple of singularities, one at f (r) = 0, one at r = 0 and the others at
f (r) = ±∞. The determinant of gµν vanishes at r = 0, so the inverse metric
components g µν are singular at this point. By definition, at the horizon r+ , is
f (r = r+ ) = 0. We are interested in observations in the vicinity of the horizon
and we would like to have a well behaved space in this region. The singularity
at the horizon is therefore a problem. But if this singularity is just a coordinate
singularity, that is if it is merely a product of a bad choice of coordinates, it is
possible to get rid of it with a change of coordinates. A goal would be to have
the metric as similar to the Minkowski metric as possible in the area close to the
horizon or to get rid of the singularity at r = r+ .
To achieve this, we may expand the function f (r) around the horizon, to the
lowest order of r [19],
df (r) (r − r+ ) = f (r+ ) + f 0 (r+ )(r − r+ ),
(3.39)
f (r) = f (r+ ) +
dr r=r+
wherepf (r+ ) is by definition zero. And by introducing a new radial coordinate
ρ = 2 (r − r+ )/f 0 (r+ ), we may write the metric as
ds2 = −f 0 (r+ )(r − r+ )dt2 + dρ2 + r 2 dΩ2D−2
ρ
2
= − f 0 (r+ ) dt2 + dρ2 + r 2 dΩ2D−2 .
2
(3.40)
ds2 = −ρ2 dφ2 + dρ2 + r 2 dΩ2D−2 .
(3.41)
Now we introduce a time coordinate φ = 12 f 0 (r+ )t, which gives us the metric
50
Chapter 3. Black Hole Thermodynamics
It appears that we still have a singularity at ρ = 0 (r = r+ ). In this case we may
define some coordinates [20]
r
r − r+ −f 0 (r+ )t/2
−(φ−ln ρ)
−φ
e
,
(3.42)
P = −e
= −ρe = −2
f 0 (r)
r
r − r+ f 0 (r+ )t/2
φ+ln ρ
φ
Q=e
= ρe = 2
e
,
(3.43)
f 0 (r)
which gives us the metric
ds2 = −dP dQ + r 2 dΩ2D−2 .
(3.44)
We have with these coordinate transformations gotten rid of all the initial singularities, except the one at r = 0 which is a curvature singularity. Since we have
gotten rid of the singularity at r = r+ and introduced coordinates (P, Q) that are
the same at the time t as at the time t + 2πi, we are near our aim to associate a
temperature to our system. Now, since we from Subsection 3.1.1 have that P (t)
0
should be equal P (t + iβ) = P e−if (r+ )β/2 we have to have (β/2)f 0 (r+ ) = 2π.
Since β = 1/(kB TD ), the temperature may be calculated to
TD =
f 0 (r+ )
.
4πkB
(3.45)
This is the same as the Hawking expression in Eq.(3.1) if we use the expression for
the surface gravity from Eq.(C.9) in Appendix C. Notice that a new coordinate
transformation P̃ ≡ (Q + P )/2, Q̃ ≡ (Q − P )/2, converts the metric into a
Minkowski-like metric ds2 = −dP̃ 2 + dQ̃2 + r 2 dΩ2D−2 , which is as we wanted. But
this is an uninteresting transformation in our case.
The temperature found is not the one physically observed close to the black
hole horizon. It is the coordinate temperature and is the temperature near the
horizon as it is observed from infinity where the metric is flat. This is where the
metric components are equal to 1. To find the observed temperature at a radius
r, one has to divide the previously found
p temperature expression by a redshift
√
√
factor Vredshift = −η µ ηµ = −gtt = f (r) which may be found from Eq.(C.3)
and [21]. Far from the black hole, when r → ∞, the redshift factor Vredshift → 1
and we retrieve the Hawking temperature. Near the black hole (r → r+ ) we have
κ/Vredshift → a, where a is the proper acceleration of the detector, and the effect
may be interpreted as an effect originating from the acceleration of the detector.
The temperature observed is then the one found in the previous Subsection 3.1.2.
If we now remember our Schwarzschild metric in Eq.(2.87) we recognize the
form of our metric in Eq.(3.38), which gives us
f 0 (r+ ) =
(D − 3)
.
RSD
(3.46)
3.1 The Black Hole Temperature
51
The temperature for the black hole is therefore
(D − 3)
4πkB RSD
1
1
MD D−3 (D − 3) (D − 2)ΩD−2 D−3
= MD
,
M
4πkB
2
TD =
(3.47)
(3.48)
where we have used the expression for the D-dimensional Schwarzschild radius
in Eq.(2.152). This expression for the black hole temperature is valid only for
D ≥ 4. But that is no surprise when we from our derivation of the generalized
Schwarzschild metric in Subsection 2.2.1 remember that this only holds for D ≥ 4
dimensions as well. The temperature may also be written as
TD = M D
MD
M
"
1
D−3
D−3
(D − 3)
(D − 2)
2D−6
(D−5)/2
2
π
Γ( D−1
)
2
1
# D−3
1
.
kB
(3.49)
The temperature expression from Eq.(3.47), which is independent of the definitions of the gravitational constant and the fundamental Planck mass, is well
known in the literature [22].
In D = 4, this simplifies to the standard expression
T4 =
1
.
8πkB GN M
(3.50)
We see from Eq.(3.47) that the temperature is inversely proportional to the
Schwarzschild radius. That means that a small black hole has a higher temperature than a larger one. Previously we have seen that a higher dimensional black
hole is larger than a 4-dimensional one. It then follows that a higher dimensional
black hole also is colder than its 4-dimensional relatives.
Kruskal-Szekeres Coordinates
When deriving the Hawking temperature from the 4-dimensional Schwarzschild
spacetime the usual approach for elimination of the coordinate singularity in
r = RSD is to introduce ingoing and outgoing null Kruskal-Szekeres coordinates.
In D dimensions this approach will not make this singularity vanish. This will
become apparent from Eq.(3.55) where the D-dimensional line element will be
expressed in terms of the Kruskal-Szekeres coordinates. On the other hand we
will also see that use of these coordinates may still prove fertile.
In D dimensions Kruskal-Szekeres coordinates are defined as [23]
∗
U ≡ −e−(D−3)(t−rD )/(2RSD ) ,
V ≡e
∗ )/(2R
(D−3)(t+rD
SD )
,
(3.51)
(3.52)
52
Chapter 3. Black Hole Thermodynamics
where
∗
rD
=
Z
∗
drD
≡
Z
1
1−
RD−3
SD
r D−3
dr,
(3.53)
∗
is the “tortoise coordinate” in D dimensions. And drD
is the radial part of our
line element. By rational fraction decomposition of the integrand in the first
transition, we have [23]
∗
rD
=
Z
1
RD−3
SD
1 − rD−3
D−3
X
=r+
=r+
n=1
D−3
X
n=1
dr =
1
D−3
D−3
X
n=1
Z
1
D−3
Z
dr
1−
ei2πn/(D−3) R
SD r
−1
dr
−1
e−i2πn/(D−3) RSD
r−1
ei2πn/(D−3)
−1
RSD ln e−i2πn/(D−3) RSD
r−1 .
D−3
(3.54)
We get a metric
D−3 2
4RSD
RSD
ds =
1 − D−3 dU dV + r 2 dΩ2D−2
(D − 3)2 U V
r
D−3 2
RSD
4RSD
∗
1 − D−3 e−(D−3)rD /RSD dU dV + r 2 dΩ2D−2 .
=−
2
(D − 3)
r
2
(3.55)
In our D-dimensional spacetime we notice that we have not eliminated the singularity in r = RSD . As discussed earlier this is presumably a problem for observing
a temperature near the horizon. But if we ignore this fact and notice that our
Kruskal coordinates are periodic, it is possible to calculate a temperature. Since
∗
U (t) = −e−(D−3)(t−rD )/(2RSD ) should be equal U (t + iβ) = U e−iβ(D−3)/(2RSD ) this
gives
β(D − 3)
= 2π,
2RSD
(3.56)
which is exactly the same expression as in Eq.(3.47).
In addition to have been proven useful in the derivation of the Hawking temperature, the Kruskal-Szekeres coordinates will be useful in the quantum field
theoretical explanation and derivation of the Hawking effect in Chapter 4.
3.2
The Black Hole Entropy
Of the many similarities between black hole physics and thermodynamics, Bekenstein found the behaviors of the black hole area and of the entropy the most
3.2 The Black Hole Entropy
53
obvious. Classically both these quantities increase irreversibly. He discussed
black hole physics in a information theoretical context. For an observer outside
the black hole horizon all information sent into a black hole is presumably lost.
Bekenstein showed [17] that it is natural to postulate a black hole entropy as a
measure of the information about the black hole interior, which is not accessible
to an observer outside the horizon. Generally, in a spacetime with a horizon
information may be lost and therefore it is possible to view this as a growth of
entropy. This suggests that a horizon may be described thermodynamically with
quantities as entropy and temperature.
The ordinary second law of thermodynamics says that the total entropy S, of
matter in the universe never decreases. But in a universe with black holes matter
get lost inside the black hole, not measurable by an external observer, and it
would seem that the entropy of matter does decrease. But Bekenstein suggested
that this problem could be solved if one make a total entropy by adding a black
hole entropy to the classical entropy of matter. This generalized entropy may be
defined as
S 0 = S + S4bh .
(3.57)
Bekenstein argued that this black hole entropy S4bh , the Bekenstein-Hawking entropy, could be expressed as
k B c3
1
ln 2
A3 .
(3.58)
2π
4~GN
Hawking later showed that the factor (ln 2)/2, which came from a information
theoretical argument, along with the factor 1/π should be dropped. This followed
directly from his discovery of the particle creation near the horizon. Before
this discovery of the quantum mechanical particle creation, there were situations
where the generalized entropy decreases. Hawking pointed out, as an example,
that without a particle emission, a black hole immersed in black body radiation
at a lower temperature than that formally assigned to the black hole, would
violate a generalized second law. Because this would produce a heat flow from
a cold body to a hotter body, i.e. the classical entropy of matter decreases as
well as the black hole entropy decreases. But taking the quantum effects into
account, a reduction in black hole entropy, δS4bh < 0, would be accompanied by
an increase in the classical entropy of matter, S > 0, and vice versa. This leads
to a generalized second law,
S4bh =
that is valid.
3.2.1
δS 0 ≥ 0,
(3.59)
The Black Hole Entropy in D Dimensions
The discovery of the Hawking temperature made it possible to easily derive the
black hole entropy with use of the first law of thermodynamics.
54
Chapter 3. Black Hole Thermodynamics
Consider the area AD−2 of a spherical surface with the radius RSD ,
D−2
AD−2 = ΩD−2 RSD
,
(3.60)
where ΩD−2 is the solid angle element and RSD is the Schwarzschild radius. This
area may be differentiated into
D−2
2GD
dAD−2 = ΩD−2
RSD
dM,
(3.61)
D−3
D−3
with respect to M . Rearranging the expression, gives us
dM = TD 2πkB MDD−2 dAD−2 .
(3.62)
Remembering the first law of thermodynamics dE = T dS, which we also assume
is valid in extra dimensions, we see that our expression in Eq.(3.62) has the same
form. Since we are using units with c = ~ = 1, energy and mass have the same
bh
dimensions. This gives us the black hole entropy SD
of the D-dimensional black
hole given as
2π
D−3
1
D−2
bh
SD = 2πkB MD AD−2 =
kB AD−2
(3.63)
ΩD−2 D − 2 GD
1
D−2
D−3
2
M D−3 4πkB
.
(3.64)
=
MD
D − 2 (D − 2)ΩD−2
It also gives us the first law of black hole thermodynamics for a Schwarzschild
black hole where no matter enters the black hole,
bh
dM = TD dSD
.
(3.65)
In D = 4 dimensions the black hole entropy is
S4bh =
kB
A2 ,
4GN
(3.66)
which we recognize from Bekenstein’s expression. With use of a different definition of the gravitational constant the D-dimensional expression would also reduce
to this form. This may be seen in Appendix B in Table B.1.
3.3
The Stefan-Boltzmann Law
We have earlier in this chapter seen that a black hole may be assigned a temperature. It would in other words radiate thermally and is therefore reasonable to
regard as a blackbody. We should then be able to derive the Stefan-Boltzmann
law for a D-dimensional black hole. So let us examine an object in thermal
3.3 The Stefan-Boltzmann Law
55
equilibrium radiating particles. We assume that standard model particles are
confined to a hypersurface with 4 spacetime dimensions, a 3-brane. And only
non-standard model particles, such as the graviton, may propagate in all the
spacetime dimensions, the bulk. Let us as a generalization instead study particles confined to a p-brane. We have then, more or less, the Stefan-Boltzmann law
for particles propagating in the extra dimensions as well. A non-rotating black
hole such as the Schwarzschild black hole does however not radiate gravitons since
it has no quadrupole moment [24].
For each particle species i inside the blackbody, the average particle number
in a state u is
hnu i =
eε/(kB TD )
1
,
− (−1)2si
(3.67)
where ε is this state’s energy and si is the spin of the particle. For fermions
this gives the usual Fermi-Dirac distribution function and for bosons the BoseEinstein distribution function.
Since the mass of these particles mi kB TD , the particles are viewed as ultra
relativistic (mi /ε 1) and the mass of the particles may be neglected in this
calculation. Because of the particle-wave duality of matter the particles have the
and the energy ε = ω. For ultra relativistic
De Broglie wavelength λ = hq = 2π
k
particles we have that ε = q, which gives us that k = ω. The average energy of
this state is then
hεi = hnu iω =
ω
.
eω/(kB TD ) − (−1)2si
Summing over all states of one kind of particle we find the total energy
X
X
ω
,
Ei = ci (p)
hεi = ci (p)
ω/(k
T
)
B
D
e
− (−1)2si
u
u
(3.68)
(3.69)
where ci (p) is the degrees of freedom of that kind of particle propagating
R inVp spaP
tial dimensions. By using the statistical mechanical transition u → (2π)p dp k,
we have
Z
ci (p)V ∞ p
ω
Ei =
d k ω/(k T )
p
(2π) 0
e B D − (−1)2si
Z ∞
ci (p)V Ωp−1
kp
=
dk
(2π)p
ek/(kB TD ) − (−1)2si
0
ci (p)V Ωp−1 (kB TD )p+1
fi (p + 1)Γ(p + 1)ζ(p + 1),
(3.70)
=
(2π)p
P
1
where ζ(b) = ∞
n=1 nb is the Riemann zeta function, fi (p + 1) = 1 for bosons and
−p
fi (p + 1) = (1 − 2 ) for fermions. This fi (p + 1) comes from the integration [25].
56
Chapter 3. Black Hole Thermodynamics
c
θ
z
dAp−1
ds
Figure 3.1: Model of a blackbody radiator. A cavity sphere with
radius R. Here dAp−1 = Ωp−2 (R sin θ)p−2 ds is an infinitesimal surface
area and ds = Rdθ. The speed of the rays associated with the standing waves inside the cavity is denoted c. These rays may only escape
through the hole to the right.
The total energy density ρ for all species of particles in the black hole is now
p+1
E X ci (p)fi (p + 1)Ωp−1 kB
=
Γ(p + 1)ζ(p + 1)TDp+1 .
ρ=
p
V
(2π)
i
(3.71)
The relation between the total energy density ρ and the energy flux out of it, i.e.
the total energy emitted Eem at temperature T per unit time per unit area, is
dEem 1
1
= hcz iρ,
dt Ap−1
2
(3.72)
where hcz i is the average velocity of the particles radiated from the blackbody
and Ap−1 = Ωp−2 Rp−1 . This relation may be explained by looking at particles
that transports some energy dE through a volume element Ap−1 ds. This energy
is equal to the energy density of the particles multiplied with this volume element.
The energy flux through the area Ap−1 may then be written as the energy density
multiplied with the average velocity of the particles through the surface. To
explain how this factor 1/2 appears in Eq.(3.72), let us look at Fig. 3.1. This
figure illustrates a blackbody radiator. For all the radiation in the cavity only
one half of the particles (rays) will have positive velocity components in the z
direction and transports energy through the Ap−1 . The rays have then through
the hole of the sphere an average velocity given by the z component of the velocity,
cz = c cos θ = cos θ, averaged over the right hemisphere. In other words,
R
cz dAp−1
hcz i = R
.
(3.73)
dAp−1
3.3 The Stefan-Boltzmann Law
57
For a (p − 1)-dimensional sphere with arbitrary radius R we have
dAp−1 = Ωp−2 Rp−1 (sin θ)p−2 dθ. This gives us
hcz i =
R π/2
0
cos θΩp−2 Rp−1 (sin θ)p−2 dθ
R π/2
=
R π/2
0
cos θ(sin θ)p−2 dθ
R π/2
(sin θ)p−2 dθ
0
Ωp−2 Rp−1 (sin θ)p−2 dθ
h
iπ/2
1
p−1
1
(sin
θ)
p−1
2Ωp−2
p−1
0
=
.
=
=
√
p−1
1/2
(p−1)/2
π 2π
Ωp−1
πΓ( 2 )
(p
−
1)Ω
p−1
4π p/2 Ωp−2
2Γ( p2 )
0
(3.74)
Combining this with Eq.(3.72) and the expression for the energy density ρ in
the blackbody in Eq.(3.71), the radiant emittance (the energy flux) may now be
calculated to be
X 1 2Ωp−2 ci (p)fi (p + 1)Ωp−1 k p+1
dEem 1
B
Γ(p + 1)ζ(p + 1)TDp+1
=
p
dt Ap−1
2 (p − 1)Ωp−1
(2π)
i
X
σi (p),
(3.75)
= TDp+1
i
where
p+1
Ωp−2 kB
σi (p) =
Γ(p + 1)ζ(p + 1)ci (p)fi (p + 1)
(p − 1)(2π)p
(3.76)
is the Stefan-Boltzmann constant for a particle species i propagating in p spatial
dimensions. The expression in Eq.(3.75) is the Stefan-Boltzmann law for a blackbody radiating onto a p-brane. Since the temperature of a black hole is inversely
proportional to the Schwarzschild radius, we see from the Stefan-Boltzmann law
that a small black hole has a higher outward energy flux than a larger black hole.
If we now look at a blackbody of D = 4 dimensions that just radiate photons
onto a 3-brane we have that the degree of freedom for a photon is
cphoton (3) = 3 − 1 = 2. This and Eq.(3.75) gives the standard Stefan-Boltzmann
law,
dEem 1
= σphoton (3)T44 ,
dt A2
(3.77)
where we have retrieved the standard Stefan-Boltzmann constant
σphoton (3) =
4
4
Ω1 k B
π 2 kB
Γ(4)ζ(4)c
(3)
=
.
photon
2(2π)3
60
(3.78)
58
3.3.1
Chapter 3. Black Hole Thermodynamics
The Black Hole Luminosity
The luminosity, or the energy emitted by a blackbody per unit time, may now
be found to be
X
dEem
= TDp+1 Rp−1 Ωp−1
σi (p)
dt
i
X
Ωp−1 Ωp−2 p−1
p+1
R
(k
T
)
Γ(p
+
1)ζ(p
+
1)
ci (p)fi (p + 1)
=
B
D
(p − 1)(2π)p
i
X
1
=
Rp−1 (kB TD )p+1 Γ(p + 1)ζ(p + 1)
ci (p)fi (p + 1)
(3.79)
π(p − 1)!
i
√
where we have used Eq.(2.15) combined with Γ(n/2)Γ(n/2 + 1/2) = ( π/2)Γ(n).
The natural choice of radius for the black hole emitter is seemingly the corresponding Schwarzschild radius. However, we have in this section assumed that
all particles emitted from the black hole horizon reach infinity where we measure
the luminosity. But as we will see in Chapter 4, because of the spacetime curvature, some of the particles will be scattered back and never reach infinity. It
will in that chapter be shown that particles propagating toward infinity, tunnel
through to infinity with a probability Γωl (p). And accordingly the black hole
do not radiate as a perfect black body. Because of this filtering of the original
blackbody spectrum emitted from the horizon, the factors are called graybody
factors. We will come back to these tunneling factors in the next chapter.
To simplify the calculation of the graybody factors De Witt [26] proposed to
use a geometrical optics approximation. In this approximation he showed that
a black hole emitter acts as a perfect absorber (Γωl (p) = 1) with a somewhat
larger radius than the Schwarzschild radius. It has been suggested [13] that this
should also be the case in higher dimensions. In Section 4.2.2 we will show, in
the same manner as De Witt, that this is correct in D dimensions. The radius
for the black hole absorber is then found to be
1 r
D − 1 D−3 D − 1
R = bcr '
RSD .
(3.80)
2
D−3
The derivation of this radius is given in Chapter 5. It is important to emphasize
that this geometrical optics approximation is usually used in the high energy limit
(RSD ω 1) where it becomes a good approximation. However, in D. N. Page’s
calculations of the graybody factors [27] and subsequently the power spectra
from a black hole in 4 dimensions, he found for low energies (RSD ω . 1) that
as RSD ω becomes larger the black hole radiation approaches that of a thermal
body of radius R = bcr . So apparently, at least in 4 dimensions, a geometrical
approximation should prove sufficient for a broader range of energies.
We notice from Eq.(3.79) that, with or without our approximation, the luminosity is inversely proportional to the square of the black hole radius, independently of the number of brane dimensions.
3.3 The Stefan-Boltzmann Law
3.3.2
59
The Black Hole Lifetime
We may now from our previously found Stefan-Boltzmann law calculate the lifetime of the black hole. Since dM/dt = −dEem /dt, the lifetime τD will be a
function of it’s mass M by rearranging and integrating the luminosity expression
in Eq.(3.79).
Let us again use the geometrical optics approximation, and assume that this
is an adequate approximation. The lifetime of a D-dimensional black hole that
radiate onto an p-brane, may then be written as
Z τD
Z 0
dM
1
τD =
dt = − P
p+1
0
i σi (p) MBH Ap−1 TD
p−1 D−3
p−1 Z MBH
2
1
D−3 2
dM
=P
p−1 p+1
D−1
D−1
RSD TD
0
i σi (p)Ωp−1
p−1 D−3
p−1
p+1 Z MBH
2
1
D−3 2
4πkB
2
RSD
dM
=P
D−1
D−1
D−3
0
i σi (p)Ωp−1
D−1
1
MBH D−3
= µ(p, D)
,
(3.81)
MD MD
where the dimensionless constant
µ(p, D) ≡
[(D − 1)(D − 3)]−(p+1)/2
p−1
(D − 1) D−3
1
(D − 2)ΩD−2
2
D−3
p+1
2 D−3 (4πkB )p+1
P
. (3.82)
i σi (p)Ωp−1
The black hole lifetime is then τD ∼ (1/MD )(MBH /MD )(D−1)/(D−3) . A four dimensional black hole would have a lifetime τ4 ∼ (1/MP )(M/MP )3 . From Section
2.3 we have that the ordinary Planck mass relates to the fundamental Planck
mass as MP2 ∼ LD−4 MDD−2 . The relationship between the lifetime of a Ddimensional black hole and that of a 4-dimensional one, of same mass, then
goes as τD ∼ (L/RSD )2(D−4) τ4 . The lifetime is longer and possibly extremely
much longer than would have been expected from the 4-dimensional theory.
It has been argued [13] that a black hole radiates mainly onto the brane, into
standard model particles. In a scenario with D = 10 spacetime dimensions and
MD ∼ 1 TeV, the lifetime of a black hole of mass MBH ∼ 100 TeV which radiate
onto a 3-brane would be τ10 ∼ 10−25 s.
However, when the mass of the black hole is ∼ MD quantum effects becomes
dominant and our semi-classical description of the black hole may become inadequate. This could then lead to observations of unknown physical properties,
perhaps described by string theory.
Chapter 4
Pair Production and the
Hawking Effect
As mentioned in the introduction to Section 3.1 the Hawking effect is usually
explained by the spontaneous pair production at the horizon of the black hole. To
explain how this pair production works, it is useful to study the related situation
of pair production in an external electromagnetic field. To do so we look at
the case of Klein’s Paradox and show how this paradox can be resolved by use
of quantum field theory. This is done in a way we find more consistent with
standard definitions in field theory than earlier work.
When we have resolved this paradox, we will use the same formalism to explain
how the spontaneous pair production at the horizon of a black hole is possible.
We will here generalize, to D dimensions, a method which in the past has been
used to describe the particle emission from four-dimensional black holes.
4.1
Klein’s Paradox
Oskar Klein found in 1928 [28] that in certain situations one may experience
difficulties within the relativistic quantum theory introduced by Dirac earlier that
same year. Klein had studied the problem of electron scattering on an electrostatic potential step, and found that if the strength of this potential was greater
than a certain value the results would appear to yield a paradox. Electrons are
sent toward a potential barrier and reflected or transmitted. This problem of
electron scattering is a typical problem in non-relativistic quantum mechanics.
The probabilities of reflection or transmission may in the relativistic case be
found in exactly the same manner, by calculating the reflection and transmission
coefficients. But if the electro-static potential is strong enough, Klein found that
the reflection coefficient is greater than one and the transmission coefficient is
less than zero. That is, there would apparently be more electrons reflected than
coming in.
61
62
Chapter 4. Pair Production and the Hawking Effect
In the forthcoming subsections the potential problem will be introduced, followed by the exact calculations leading up to the paradox. And in the last
subsection it will all be resolved by means of quantum field theory.
4.1.1
Introduction to Klein’s Paradox
In this subsection and the next we will look at the potential problem not only
in the fermionic case, but also in the bosonic case for spin-0 particles. It will
become clear later on why we also review the bosonic situation. Let us remember
the one-particle theories for these two types of particles and eventually see how
these are modified with the introduction of an external electromagnetic field.
For a free fermion the Dirac equation is
[iγ µ ∂µ − m]ψ = 0,
(4.1)
where γ µ is defined by the Clifford algebra
[γ µ , γ ν ]+ = γ µ γ ν + γ ν γ µ = −2η µν .
We use the standard Dirac-representation
1 0
0
γ =
and
0 −1
i
γ =
0 σi
,
−σ i 0
where 1 is the 2 × 2 unit matrix and σ i are the Pauli matrices,
0 1
0 −i
1 0
1
2
3
σ =
,
σ =
,
σ =
.
1 0
−i 0
0 −1
(4.2)
(4.3)
(4.4)
The Dirac equation then admits plane wave solutions of the form
ψ(x) = w(E, p)eip
µx
µ
,
(4.5)
where w(E, p) is a four-component spinor. By inserting this solution into the
Dirac equation and multiplying with (iγ µ ∂µ + m) we find that
p
(4.6)
E = ± p2 + m2 .
In other words, we have two classes of wave solutions corresponding to E ≥ m and
E ≤ −m, respectively. A physical particle has positive energy and the negative
energy solutions appear to make no sense. The wave solution corresponding to a
free positive energy particle is of the form
s
E+m
µ
χs
ψ(x) =
eip xµ ,
(4.7)
σ·p
χ
|p|
E+m s
4.1 Klein’s Paradox
63
p
where (E + m)/|p| is a normalizing constant convenient in potential barrier
problems. With quantization of the spin along the z-axis, χs are the eigenspinors
1
0
χ+ =
and χ− =
(4.8)
0
1
of σ 3 with eigenvalues +1 and −1, respectively.
For a free spin-0 boson the Klein-Gordon equation is
[∂µ2 − m2 ]φ(x) = 0,
(4.9)
where the negative sign in front of the mass originates from the chosen signature
of the metric (−, +, . . . ) and φ(x) is a scalar wave function. The Klein-Gordon
equation admits plane wave solutions of the form
1
µ
(4.10)
φ(x) = p eip xµ ,
|p|
p
where pµ = (E, p) is the 4-momentum and 1/ |p| is a normalizing constant
convenient in potential barrier problems. This solution inserted into the KleinGordon equation yields the expression found in Eq.(4.6) and, like the Dirac equation, has both positive and negative energy solutions.
In the presence of an electromagnetic field Aµ the wave equations are modified
with the minimal coupling
i∂µ → i∂µ + eAµ ,
(4.11)
where e is the electrical charge of the particles and Aµ = (V, A). While the
Klein-Gordon equation takes the form
[(∂µ − ieAµ )2 − m2 ]φ = 0,
(4.12)
the Dirac equation becomes
[γ µ (i∂µ + eAµ ) − m]ψ = 0,
(4.13)
where γ µ Aµ = −γ 0 V + γ k Ak . Both the modified Klein-Gordon equation and the
modified Dirac equation implies that
−(pµ − eAµ )2 = m2
i.e.
(E − eV )2 = (p − eA)2 + m2 ,
(4.14)
which represent the mass-shell condition of a free particle in the external field.
Now consider the static potential step Klein studied
(
0,
z<0
V (z) =
,
(4.15)
V,
z>0
64
Chapter 4. Pair Production and the Hawking Effect
eV (z)
eV
region I
region II
z
Figure 4.1: A particle wave of energy E in region I travels toward a
potential step. The wave will be transmitted into region II and partly
reflected back into region I.
which gives
Aµ = (V (z), 0).
(4.16)
The corresponding modified Klein-Gordon equation and the modified Dirac equation now give that
E 2 = p 2 + m2
(E − eV )2 = q 2 + m2
(z < 0),
(z > 0).
(4.17)
We may define
E≡
√
k 2 + m2 ,
where k is the momentum of the particle, so that
(
E,
z<0
±E =
.
E − eV,
z>0
(4.18)
(4.19)
A wave solution where the negative sign in front of E is chosen, is said to be a
negative frequency solution, while one with a positive sign, is a positive frequency
solution. So we have the possibility to have negative frequency solutions in both
regions. These solutions make seemingly no sense physically.
Let us now consider a free particle with positive energy E > m, i.e. positive
frequency, and electric charge e moving along the z-axis in positive z-direction.
Such a particle will to the left of the step (region I) have momentum
√
p = E 2 − m2
(4.20)
and to the right (region II) it will have momentum
p
q = (E − eV )2 − m2 ,
(4.21)
4.1 Klein’s Paradox
65
where we have chosen the positive square roots. The situation is visualized in
Fig. 4.1. The particle may at the barrier either be transmitted or reflected and
we may find the respective probabilities by calculating the respective currents.
In the following we will calculate the reflection and transmission coefficients for
both fermions and spin-0 bosons.
Fermions
For incident fermions we have to match each component of the one-particle spinor
across the step at z = 0 in order to calculate the reflected and transmitted waves.
Let us assume that the incident fermion has spin up, i.e. χs = χ+ . The general
wave solution in region I may then be expressed as
s
E+m
χ+
χ+
ipz
−ipz
ψI (t, z) =
C
e + D −p
e
e−iEt ,
(4.22)
p
χ
χ
|p|
E+m +
E+m +
where the first term represents an incident wave, the second term represents a
reflected wave from the potential barrier and p is the momentum from Eq.(4.20).
In region II the general wave solution will be
s
E − eV + m
χ+
eiqz−iEt ,
(4.23)
F
ψII (t, z) =
q
χ
|q|
+
E−eV +m
which describes only a transmitted wave and q is the momentum from Eq.(4.21).
There is no spin-flip at the potential boundary, and neither the reflected nor the
transmitted wave have terms corresponding to a spin down fermion. We demand
that the total wave solution across the potential step at z = 0 is continuous,
i.e. ψI (t, 0) = ψII (t, 0). This gives us two equations, one for the upper part and
one for the lower part of the spinors, respectively. These equations establish the
connection between the arbitrary coefficients C, D and F . We find directly that
√
(C + D) α = F
where
q
α≡
p
and
1
(C − D) √ = F,
α
E+m
E − eV + m
.
Combining the two equations in Eq.(4.24), we find that
√
D
1−α
F
2 α
=
and
=
.
C
1+α
C
1+α
(4.24)
(4.25)
(4.26)
To calculate the probabilities of reflection and transmission, we need the 4-vector
J µ = (ρ0 , J), where ρ0 is the probability density and J is the current density. This
66
Chapter 4. Pair Production and the Hawking Effect
4-vector is for fermions given by J µ = ψ̄γ µ ψ, where ψ̄ = ψ † γ 0 . In our situation
the current density becomes
J = J z = ψ † γ 0 γ 3 ψ.
(4.27)
The current density of the incoming wave is therefore
†
Jinc = ψinc
γ 0 γ 3 ψinc
E+m †
p
χ+
1 0
0 σ3
†
†
C χ+ ,
χ
C
=
p
χ
−σ 3 0
0 −1
p
E+m +
E+m +
p
p
E+m 2
χ
E+m +
|C| χ†+ ,
χ†+
=
= 2|C|2 ,
(4.28)
χ+
p
E+m
where we have used that χ†+ χ+ = 1. In the same manner the current density of
the reflected wave is
−p
−p
E +m
χ
+
†
†
2
E+m
= −2|D|2
(4.29)
|D| χ+ ,
χ+
Jref =
χ
p
E+m
+
while the current density of the transmitted wave is
q
E − eV + m 2
q
χ+
†
†
E−eV
+m
Jtr =
= 2|F |2 .
|F | χ+ ,
χ
χ+
q
E − eV + m +
(4.30)
Since we initially have only an incident wave, the reflection coefficient is simply
2
|D|2
1−α
|Jref |
=
=
ρ≡
.
(4.31)
|Jinc |
|C|2
1+α
And the transmission coefficient is
τ≡
|F |2
4α
|Jtr |
=
=
.
2
|Jinc |
|C|
(1 + α)2
(4.32)
If we add these two coefficients we find that
τ +ρ=
(1 − α)2 + 4α
(1 + 2α + α2
=
= 1.
(1 + α)2
(1 + α)2
(4.33)
This is in accordance with current conservation, i.e. transmission and reflection
are the only two possibilities in our problem.
Bosons
For an incident boson we demand that the wave function and its first derivative
with respect to the z-coordinate are continuous across the potential step at z = 0,
i.e.
∂
∂
φI (t, 0) = φII (t, 0)
and
φI (t, z)
=
φII (t, z) .
(4.34)
∂z
∂z
z=0
z=0
4.1 Klein’s Paradox
67
In region I the wave solution may be expressed as
1
φI (t, z) = p [Ceipz + De−ipz ]e−iEt ,
|p|
(4.35)
where the first term again represents the incident wave and the second represents
the reflected wave. And in region II the wave solution may be written as
1
φII (t, z) = p F ei(qz−Et) ,
|q|
(4.36)
which describes a transmitted wave. The demands of continuity from Eq.(4.34)
then give
√
(C + D) α = F
and
1
(C − D) √ = F,
α
(4.37)
where now
q
α≡ .
p
(4.38)
Combining the two equations in Eq.(4.37) we find the connection
√
D
1−α
F
2 α
=
and
=
.
C
1+α
C
1+α
(4.39)
We notice that this is the same result as we got for the fermion case, except for
a different definition of α. In the same manner as for fermions, to calculate the
reflection and transmission coefficients, respectively, we use the 4-current density
Jµ which for bosons is
↔
↔
Jµ = −iφ† ∇µ φ = −iφ† [ ∂ µ − 2ieAµ ]φ,
(4.40)
and
↔
A ∂ µ B ≡ A∂µ B − (∂µ A)B.
(4.41)
The current density in our case is therefore
J = Jz = −iφ† ∂z φ + i(∂z φ† )φ,
(4.42)
since A = 0. The current density of the incident wave is then
1
1
Jinc = −i |C|2 e−ipz (ip)eipz + i |C|2 (−ip)e−ipz eipz = 2|C|2 .
p
p
(4.43)
68
Chapter 4. Pair Production and the Hawking Effect
The current densities of the reflected and transmitted waves are
1
1
Jref = −i |D|2 eipz (−ip)e−ipz + i |D|2 (ip)eipz e−ipz = −2|D|2
p
p
(4.44)
1
1
Jtr = −i |F |2 e−iqz (iq)eiqz + i |F |2 (−iq)e−iqz eiqz = 2|F |2 ,
q
q
(4.45)
and
respectively. We now find the reflection coefficient to be
|Jref |
|D|2
ρ≡
=
=
|Jinc |
|C|2
1−α
1+α
2
,
(4.46)
while the transmission coefficient is
τ≡
|Jtr |
|F |2
4α
=
=
.
2
|Jinc |
|C|
(1 + α)2
(4.47)
We see that this is the same results as for the fermion case, though with another
constant α. It follows directly that we have current conservation also in the boson
situation.
One problem with interpreting the Klein-Gordon equation as a one-particle
wave equations is that it is not possible to construct a positive definite probability
density. And since people insisted on building a one-particle theory in analogy
to non-relativistic quantum mechanics, the Klein-Gordon equation was rejected
for a long time as a true candidate to a relativistic quantum theory. But it was
revived in 1934, with the quantization of the scalar field.
4.1.2
The Paradox
We have seen that for both fermions and bosons the reflection coefficient is
2
|Jref |
1−α
ρ≡
=
.
(4.48)
|Jinc |
1+α
and the transmission coefficient is
τ ≡
|Jtr |
4α
=
.
|Jinc |
(1 + α)2
where α for fermions and bosons are given as respectively
E+m
q
q
and
α≡ .
α≡
p E − eV + m
p
(4.49)
(4.50)
4.1 Klein’s Paradox
69
eV (z)
Strong Potential Region
E+m
Intermediate Potential Region
E
Intermediate Potential Region
E−m
Weak Potential Region
0
Figure 4.2: Diagram of the different potential regions
With regards to the forthcoming discussion it is convenient to introduce the
reflection and transmission amplitudes,
1−α
√
R= ρ=
1+α
and
T =
√
√
2 α
.
τ =
1+α
(4.51)
In the further discussion we distinguish between three kind of potentials.
These three types of potentials are classified according to the diagram shown in
Fig. 4.2.
Weak Potentials
The potential may be described as weak if eV < E − m is satisfied. A particle
subjected to such a potential will in region II have a momentum that is real and
q > 0. This means that the solutions in Eq.(4.23) and Eq.(4.36) are oscillatory
plane waves that travel in positive z direction. This situation is illustrated in
Fig. 4.3. The shaded areas in this and the forthcoming figures are energy intervals
which are classically forbidden for the particles. From Eq.(4.50) we find that for
both fermions and bosons α > 0. Consequently are the transmission coefficient
τ > 0 and the reflection coefficient ρ < 1. This result is as expected and do not
pose any problem.
70
Chapter 4. Pair Production and the Hawking Effect
eV (z)
ρ
1
τ
eV region
I
m
−m
region II
eV + m
eV − m
z
Figure 4.3: The particle is transmitted with probability τ > 0 or
reflected with a probability ρ < 1 in a weak potential. The classically
forbidden regions are shaded.
Intermediate Potentials
The potential may be described as intermediate if it satisfies E−m < eV < E+m.
A particle subjected to a potential of this group will in region II have a momentum
that is totally imaginary. The momentum may then be expressed as q = ia, where
a = [m2 − (E − eV )2 ]1/2 > 0. The wave solutions in Eq.(4.23) and Eq.(4.36) is
then respectively ψII ∼ e−az and φII ∼ e−az . The wave solution is exponential
damped and there is no current present in region II. The wave is reflected totally
(see Fig. 4.4). This means that ρ = 1 and τ = 0. This result is also in accordance
with what would be expected.
eV (z)
ρ=1
1
eV
## m $
## $
## $## $## region
## $## I $##
$
$
$$$$$$$$
−m
"!"! "!"! "!"! "!"! "!"! "!"! "!"! "!"!
region II
eV + m
eV − m
z
Figure 4.4: The particle is transmitted with probability τ = 0 or
reflected with a probability ρ = 1 in an intermediate potential.
4.1 Klein’s Paradox
71
eV (z)
region
II&
%
%
%
%
%
%
%
%
&
&
&
&
&
&
&
eV &
% &% &% &% &% &% &% &%
region I
ρ
m
−m
'' (
'' (
'' (
'' ('' ('' (''
(
(((((((
1
τ
eV + m
eV − m
z
Figure 4.5: Klein’s paradox. The reflection coefficient ρ > 1 and
transmission coefficient τ < 0 in a strong potential.
Strong Potentials
The potential may be described as strong if it satisfies eV > E + m. Since
E > m, a strong potential also satisfies eV > 2m. A particle subjected to such
a potential will in region II have a momentum that is real and q > 0. And again
the solutions in Eq.(4.23) and Eq.(4.36) describes oscillatory plane waves. For
fermions αF = (q/p)[(E +m)/(E −eV +m)] < 0, since we have E < eV −m. The
reflection and transmission coefficients then become ρ > 1 and τ < 0, respectively.
This indicates a negative transmitted current and a reflected current greater than
the incident one. This poses a paradox and is the core of Klein’s paradox. It
could seem as the particle wave is moving in negative z-direction in region II,
which is in disagreement with our initial conditions. The energy part E − eV is
negative in this region, more specifically E − eV < −m. The wave solutions in
this region are in other words negative frequency solutions, with −E = E − eV .
The paradox shows that also the negative frequency solutions must be interpreted
physically.
The paradox may seemingly be solved by considering the group velocity of
the particle in region II, given by
v≡
q
dE
=
.
dq
E − eV
(4.52)
From E < eV − m we then find that q/v < −m, which means that the direction
of the group velocity is opposite to that of the momentum. A wave in region II
traveling in positive z-direction is then obtained if q < 0. At the same time the
parameter α becomes positive and sensible reflection and transmission coefficients
are retrieved. Bosons should for the same reasons have q < 0. This makes α < 0
and we see that the paradox is merely transfered to the boson scenario. So let
us disregard this cul-de-sac and as originally use only positive momentum values.
And we have still not attempted to interpret physically the negative frequency.
72
Chapter 4. Pair Production and the Hawking Effect
P2
t
z
P1
N1
Figure 4.6: This figure is a schematic description of the process
analyzed in the previous subsections. We see that in the infinite
future this wave solution is described purely as a positive frequency
solution P2 , while it in the infinite past is described as combination of
positive and negative frequency solutions. Because of this behavior
the solution is specified as an outgoing particle solution.
4.1.3
The Resolution of the Paradox
The resolution of Klein’s paradox is found in the possibility for creation of
particle-antiparticle pairs which kinematically can take place in strong potentials, since eV > 2m. The negative transmission coefficient in the fermion case
may then be explained by a current of anti-particles in positive z-direction. And
|τ | will therefore relate to the probability for spontaneous production of a single
particle-anti-particle pair at the potential step. Both Sommerfeld [29] and Hund
[30] tried to explain this pair production within the physical framework of the
time, without revealing completely satisfactory explanations. The best modern
discussion is given by Hansen and Ravndal [31], who used quantum field theory
to fully explain the pair creation occurring in a strong electric potential. Within
the framework of the second quantization we shall see that the creation of matter takes place even if there is no incoming matter to stimulate the process. To
describe this scattering process it is convenient to define so-called incoming and
outgoing solution sets. We will here choose a definition of these sets we find
more consistent than that of earlier work. This will then involve explicit new
calculations in order to resolve the paradox.
From the foregoing discussion we will only consider those one-particle solutions for which the energy lies in the interval m < E < eV − m. Let us now introduce the asymptotic particle solutions pk (t, z) and the asymptotic anti-particle
solutions nk (t, z). These are specified by their behavior at t → ±∞. If a solution
in the infinite past or future is completely constituted of a positive frequency
solution, i.e. a plane wave particle solution, it is labeled pk (t, z). The index k
denotes the conserved quantum numbers of the corresponding modes which here
in general will be the total particle energy E and its transverse momentum pT .
As earlier remarked, in scattering problems it is usual to find two complete sets
of solutions to the wave equations. Conventionally these two sets are classified
4.1 Klein’s Paradox
73
by their behavior in the infinite past or future. A wave is defined as an incoming
particle wave if it in the infinite past (t = −∞) displays the behavior of a particle.
While an outgoing anti-particle wave is defined as a wave that in the infinite future
(t = ∞) displays the behavior of an anti-particle wave, i.e. is a negative frequency
solution. This is the definition we will use here. And this is a different definition
of incoming and outgoing modes than those used in previous modern discussions
of the paradox, e.g. Hansen and Ravndal [31] or Holstein [32]. Fig. 4.6 illustrates
the wave solution studied in the previous subsections which now can be defined
as an outgoing particle solution. A field operator may now be expanded in either
of these two sets as
X
ψ̂(t, z) =
[a1k p1k (t, z) + b†1k n1k (t, z)]
(4.53)
k
or
ψ̂(t, z) =
X
[a2k p2k (t, z) + b†2k n2k (t, z)],
(4.54)
k
where index 1 denotes incoming waves and 2 denotes outgoing waves. The operator a1k will annihilate an incoming particle with wave function p1k (t, z) while b†2k
will create an outgoing anti-particle with wave function n2k (t, z). The physical
transitions we will consider will be between states with the same conserved quantum numbers. The equations therefore hold for each separate k and the index
will for convenience be dropped. We then have the solutions
(
1−α
√ P + 1+α
√ P ,
z<0
2 α 1
2 α 2
p1 (t, z) =
,
(4.55)
N2 ,
z>0
(
1+α
√ P + 1−α
√ P ,
z<0
2 α 1
2 α 2
p2 (t, z) =
,
(4.56)
N1 ,
z>0
(
P2 ,
z<0
,
(4.57)
n1 (t, z) =
√ N1 + 1+α
√ N ,
− 21−α
z>0
α
2 α 2
(
P1 ,
z<0
n2 (t, z) = 1+α
,
(4.58)
√ N − 1−α
√ N ,
z>0
2 α 1
2 α 2
where
1
1
P1 = p ei(pz−Et) = p ei(pz−Et) ,
|p|
|p|
1
1
N1 = p ei(qz−Et) = p ei(qz+Et) e−ieV t
|q|
|q|
(4.59)
(4.60)
74
Chapter 4. Pair Production and the Hawking Effect
for bosons and

1

E+m
 0p  ei(pz−Et) ,
P1 =
|p|  E+m 
0


1
s
 i(qz−Et)
E − eV + m 
e
 0q
N1 =


|q|
E−eV +m
0

s
(4.61)
(4.62)
for fermions. The solutions P2 and N2 are obtained from these expressions by
letting p → −p and q → −q. Remember that the parameter α is positive for
bosons and negative for fermions, because of the decision to use only momenta of
positive value. These solutions look superficially like those of Hansen and Ravndal
[31] or Holstein [32]. But their solutions are not specified by the behavior in the
infinite past and infinite future, but rather by the behavior at z → −∞ and
z → +∞. The process in Fig. 4.6 would then have been classified as an incoming
anti-particle solution.
We see that the solution in Eq.(4.56) describes the situation studied in the
previous two subsections and is illustrated in Fig. 4.6. The other three solutions
may be found in the same way, but with different initial conditions. Notice that
the solutions of the wave equation in the region z < 0 correspond to particles,
and solutions in the region z > 0 correspond to anti-particles. We only have
anti-particles in the region exposed to the potential.
A general quantization of the field would also include negative frequency solutions corresponding to one-particle solutions with negative total energy. These
solutions correspond to anti-particles in the free field theory.
From Eq.(4.53) and Eq.(4.54) we have that
a1 p1 + b†1 n1 = a2 p2 + b†2 n2 .
(4.63)
If we now for both fermions and bosons define a scalar product of two solutions
u and v such that
(u, v) = (v, u)∗ ,
(4.64)
we may combine this with the previous Eq.(4.63) to give us the operator relations
a1 = Aa2 + Bb†2 ,
b†1
=
a2 =
b†2
=
B̃a2 + Ãb†2 ,
A∗ a1 + B̃ ∗ b†1 ,
B ∗ a1 + Ã∗ b∗1 .
(4.65)
(4.66)
(4.67)
(4.68)
4.1 Klein’s Paradox
75
Here we have introduced the Bogoliubov coefficients
(p1 , p2 )
(p1 , p1 )
(n1 , n2 )
à =
(p1 , p1 )
A=
(p1 , n2 )
,
(p1 , p1 )
(n1 , p2 )
B̃ =
(p1 , p1 )
,
B=
,
(4.69)
and their complex conjugates. Additionally we have used that (n1 , n1 ) = (n2 , n2 ) =
(p1 , p1 ) = (p2 , p2 ), where
(
+1
=
−1
for fermions
for bosons
.
(4.70)
For bosons this inner product is given as
(u, v) ≡ i
↔
Z
∞
↔
dz u∗ ∇0 v,
(4.71)
−∞
↔
where ∇0 ≡ (∂0 + 2ieV (z)). And for fermions it is defined as
(u, v) ≡
Z
∞
dz u† v.
(4.72)
−∞
In the boson case we find that the inner products are computed to be
(p − q)2
δ(E − E 0 ),
pq
(p − q)2
(n1 , n01 ) = (n2 , n02 ) = − π
δ(E − E 0 ),
pq
p2 − q 2
δ(E − E 0 ),
(p1 , p02 ) =π
pq
p2 − q 2
(n1 , n02 ) =π
δ(E − E 0 ),
pq
p−q
(p1 , n02 ) =2π √ δ(E − E 0 ),
pq
p−q
(n1 , p02 ) =2π √ δ(E − E 0 ),
pq
(p1 , p01 ) = (p2 , p02 ) =π
(4.73)
(p1 , n01 ) =0,
(n2 , p02 ) =0,
where the complete calculations of these inner products are given in Appendix E.
76
Chapter 4. Pair Production and the Hawking Effect
Canonical quantization requires bosons to be quantized by commutators and
fermions to be quantized by anti-commutators. Thus in our case
[a1 , a†1 ] = a1 a†1 + a†1 a1 = 1,
(4.74)
[b1 , b†1 ] = b1 b†1 + b†1 b1 = 1,
[a2 , b2 ] = a2 b2 + b2 a2 = 0.
(4.75)
(4.76)
We may then immediately, from Eq.(4.65) used in Eq.(4.74), find the relation
1 = [a1 , a†1 ]
= (Aa2 + Bb†2 )(A∗ a†2 + B ∗ b2 ) + (A∗ a†2 + B ∗ b2 )(Aa2 + Bb†2 )
= A∗ A + B ∗ B
= |A|2 + |B|2 .
(4.77)
In the same manner we may use Eq.(4.67) and Eq.(4.68) in Eq.(4.76) to derive
the constraint
AB ∗ = −Ã∗ B̃.
(4.78)
Using the relations between the two sets of field operators in Eq.(4.65)Eq.(4.68) with Eq.(4.63) we may express the incoming wave solutions by use
of the outgoing wave solutions alone. Thus we have
a1 p1 + b†1 n1 = (Aa2 + Bb†2 )p1 + (B̃a2 + Ãb†2 )n1
= a2 (Ap1 + B̃n1 ) + b†2 (Bp1 + Ãn1 ) = a2 p2 + b†2 n2
(4.79)
and
a2 p2 + b†2 n2 = (A∗ a1 + B̃ ∗ b†1 )p2 + (B ∗ a1 + Ã∗ b†1 )n2
= a1 (A∗ p2 + B ∗ n2 ) + b†1 (B̃ ∗ p2 + Ã∗ n2 ) = a1 p1 + b†1 n1 .
(4.80)
Separately these two equations give us
p1 = A∗ p2 + B ∗ n2 ,
∗
(4.81)
∗
n1 = B̃ p2 + Ã n2 ,
(4.82)
p2 = Ap1 + B̃n1 ,
(4.83)
n2 = Bp1 + Ãn1 .
(4.84)
At z = 0 we find that for example Eq.(4.83) may be written as
N2 = (1/A)N1 − (B̃/A)P2 . If we use the expressions for the reflection and transmission amplitudes, Eq.(4.51), in Eq.(4.57) and compare it with this previous
result, we obtain the relations
R=
1
A
and
T = −
B̃
.
A
(4.85)
4.1 Klein’s Paradox
77
And when we do the same with Eq.(4.55) and Eq.(4.84) it follows that
R=−
1
Ã
and
T =−
B
.
Ã
(4.86)
This shows that there are only two independent Bogoliubov coefficients,
à = −A
and
B̃ = −B.
(4.87)
We may therefore also express the transmission coefficient as
T =
B
.
A
(4.88)
Alternatively, these relations between the Bogoliubov coefficients and the reflection and transmission coefficients, may be shown by using the inner products of
which the Bogoliubov coefficients consist. If we again consider the boson inner
products from Eq.(4.73) we then see that this will give the same relations between
the different coefficients as found previously.
These expressions for the reflection and transmission amplitudes may be used
in Eq.(4.77) to derive the relation
|R|2 − |T |2 = 1.
(4.89)
In the same way they will transform Eq.(4.78) into
T = −T ∗
(4.90)
which shows that the transmission amplitude T is purely imaginary for fermions.
As a consequence this implies that ρ > 1 in this case. We now see that this result
is a direct consequence of the fact that fermions obey the Pauli principle and
hence is quantized by anti-commutators.
The incoming and outgoing operators may be used to construct two complete
sets of incoming and outgoing states. These are related by the unitary S-matrix
operator S,
| Ψ1 i = S | Ψ2 i.
(4.91)
The incoming state | Ψ1 i is obtained by applying incoming creation operators on
the incoming vacuum | 01 i. An incoming one-particle state with quantum number
p is then
| p; 1i = a†1p | 01 i = a†1p S | 02 i,
(4.92)
| p; 1i = S | p; 2i = Sa†2p | 02 i.
(4.93)
but also
78
Chapter 4. Pair Production and the Hawking Effect
Combined, these give us
a†1 = Sa†2 S † .
(4.94)
The two vacua are both annihilated by their corresponding particle destruction
operators,
a1 | 01 i = a2 | 02 i = 0.
(4.95)
And both vacuum states are required to be normalized to unity,
h01 | 01 i = h02 | 02 i = 1.
(4.96)
The number operator for particles described by p1 is N̂1 = a†1 a1 . The expectation
value of this, with respect to the in-vacuum state, is then h01 | N̂1 | 01 i = 0.
Let us look at the probability amplitude for spontaneous pair production
Spair = hΨout | Ψin i, where in this case | Ψout i = a†2 b†2 | 02 i and | Ψin i =| 01 i. We
then get
B †
1
a 1 − b2 | 0 1 i
Spair = hΨout | Ψin i = h02 | b2 a2 | 01 i = h02 | b2
A
A
B
B
= h02 | b2 − b†2 | 01 i = − h02 | (1 − b†2 b2 ) | 01 i
A
A
B
B
= − h02 | 01 i = eiW −
= −T eiW ,
(4.97)
A
A
where we have used Eq.(4.65), Eq.(4.75) and Eq.(4.95). In the last transition
we have introduced the probability amplitude eiW ≡ h02 | 01 i for the vacuum
to remain a vacuum under the influence of the external potential. The absolute
probability for pair production is therefore
|Spair |2 =
|B|2 −2ImW
e
= |T |2 Cvac = |τ |Cvac ,
|A|2
(4.98)
where Cvac is then the probability for the vacuum to remain vacuum. We see that
the absolute probability of pair production is the probability for the vacuum not
to become something else, combined with the absolute value of the transmission
coefficient. This is in accordance with what was claimed in the introduction to
this subsection. In the same manner we may calculate the absolute probability
amplitude for elastic scattering of a particle on the barrier,
Sscatt = h02 | a2 a†1 | 01 i = h02 | a2 (A∗ a†2 + B ∗ b2 ) | 01 i
|B|2 iW
|A|2 iW
e +
e
= A∗ h02 | 01 i + B ∗ h02 | a2 b2 | 01 i =
A
A
1
1
= (|A|2 + |B|2 )eiW = eiW = −ReiW ,
A
A
(4.99)
4.1 Klein’s Paradox
79
where we have used Eq.(4.65), Eq.(4.76), Eq.(4.97) and Eq.(4.77) to obtain the
result. The absolute probability for elastic scattering then becomes
|Sscatt |2 =
1 −2ImW
e
= |R|2 Cvac = |ρ|Cvac .
|A|2
(4.100)
The average rate of spontaneous produced pairs of particles in a certain mode
is given as
hni = h01 | a†2 a2 | 01 i,
(4.101)
where a†2 a2 is the number operator for outgoing particles. With a2 from Eq.(4.67)
we find that
|T |2 τ †
2
2
2
.
(4.102)
=
hni = h01 | a2 a2 | 01 i = |B̃| = (−) |B| =
|R|2 ρ Klein’s paradox arose from the Dirac equation which was treated as a oneparticle equation and thus did not obey the Pauli principle. But as we will see
here this principle is exactly what resolves the paradox. If we scatter a fermion
with a certain energy from a strong potential, it will be totally reflected. Since an
outgoing particle then already is present, there is according to the Pauli principle
no possibility for creating particles in that state. Because of the existence of
only this possible process, the absolute probability for scattering fermions must
−1
be |Sscatt |2 = |ρ|Cvac = 1. This means that Cvac
= |ρ|. We see that this is
in accordance with Eq.(4.102) since the number of fermion pairs spontaneously
produced is just
hni =
1
X
n=0
2
nPn = 0 · Cvac + 1 · |Spair | = |τ |Cvac
τ = .
ρ
(4.103)
Here Pn is the probability to produce n pairs.
In the situation of an incident boson in the same kind of potential, the boson
will be totally reflected, but may now also stimulate the emission of pairs of the
same energy mode. In accordance to the Pauli principle any number of pairs of
the same quantum state may appear. The probability amplitude to have n pairs
produced in addition to the scattering, is
1
(n)
Sscatt = p
h02 | bn2 an+1
a†1 | 01 i,
2
n!(n + 1)!
(4.104)
where we have included the normalization factor of the final state. With use of
the expression for a1 in Eq.(4.65) and use of the commutator relations for bosons,
80
Chapter 4. Pair Production and the Hawking Effect
we find that
1
(n)
Sscatt = p
h02 | bn2 an+1
(A∗ a†2 + B ∗ b2 ) | 01 i
2
n!(n + 1)!
h
i
1
† n+1
∗
n
n
∗
n+1
=p
A h02 | b2 [(n + 1)a2 + a2 a2 ] | 01 i + B h02 | (b2 a2 )
| 01 i
n!(n + 1)!
"
n
n+1 #
1
B
B
=p
(n + 1)A∗ n! −
eiW
+ B ∗ (n + 1)! −
A
A
n!(n + 1)!
n
√
√
B
1
−
(4.105)
eiW = n + 1(−R)(−T )n eiW ,
= n+1
A
A
after using Eq.(4.77) and the expressions for Spair and Sscatt . The corresponding
probability is now
(n)
|Sscatt |2 = (n + 1)|R|2 |T |2n Cvac = (n + 1)|ρ||τ |n Cvac .
The sum of the scattering probabilities must then be
∞
X
Cvac |ρ|
(n + 1)|τ |n =
Cvac |ρ|
= 1,
(1 − |τ |)2
n=0
(4.106)
(4.107)
because of probability conservation. The probability Cvac is then equal to |ρ|.
This is also consistent with the result in Eq.(4.102) for the average rate of produced pairs from vacuum, since
X
τ Cvac |τ |
2
3
. (4.108)
=
hni =
nPn = Cvac |τ | + 2|τ | + 3|τ | + . . . =
ρ
2
(1
−
|τ
|)
n
The results derived in this subsection is exactly those found by Hansen and
Ravndal [31]. So use of either two definitions of incoming and outgoing solution
sets will give the same interesting results. This resolution of Klein’s paradox has
shown that in an external field, pairs may be spontaneous created from a vacuum
state.
4.2
The Hawking Effect
In the following we will use the framework, introduced in the previous section, in
a curved spacetime to explain the Hawking effect and derive the Hawking temperature in D dimensions. We shall consider our D-dimensional Schwarzschild
spacetime and apply the quantum field formalism introduced in the previous
section on this curved spacetime background.
Hawking showed, by studying the gravitational collapse of a star, that a black
hole continuously emits particles thermally. But as noted in the introduction to
Section 3.1 it should also be possible to derive the same result by studying a
static black hole. This would reveal that the particle production is a result of the
curved spacetime geometry itself. This was first noticed by Unruh [18] in 1976.
4.2 The Hawking Effect
4.2.1
81
Quantum Field Theory in Curved Spacetime
We now consider the possibility for spontaneous produced boson pairs near the
black hole horizon in a static spacetime. For simplicity let us examine the behavior of massive scalar fields in this curved spacetime. Since we now are dealing
with bosons, the Lagrangian for matter LM is the Klein-Gordon Lagrangian
LKG = −[(∂α φ)† (∂ α φ) + µ2 φ† φ],
(4.109)
where µ is the mass of the field. The sign of the Lagrangian is due to our choice
of signature of the metric, and is consistent with our definitions of the EinsteinHilbert action and the energy-momentum tensor in Subsection 2.2.5. To obtain a
generalization of the Klein-Gordon equation to curved spacetime we may variate
this action from Eq.(2.132). One obtains then first an Euler-Lagrange equation,
where a square root of minus the determinant is included. And by using the
Lagrangian LKG in this equation we find the desired Klein-Gordon equation for
curved spacetime,
√
1
αβ
2
√ ∂α −gg ∂β − µ φ(t, x) = 0,
(4.110)
−g
where g ≡ det(gαβ ). In the past this wave equation has been applied to the fourdimensional Schwarzschild geometry to produce the correct wave equation in that
case. We will now instead use it on our D-dimensional spacetime geometry.
Considering the D-dimensional Schwarzschild metric from Eq.(2.81), the square
root of minus the determinant may be calculated to give
√
−g = r D−2 sinD−3 (χD−2 ) · · · sink−1 (χk ) · · · sin(χ2 ).
(4.111)
The above wave equation may then be written as
" 1
r D−1
2
D−3
∂
+
−
∂r r(r D−3 − RSD
)∂r
t
D−3
D−4
r
r D−3 − RSD
+ r2
D−3
X
k=1
+
sin
g kk
∂k sink−1 (χk )∂k
k−1
sin (χk )
1
∂D−2 sinD−3 (χD−2 )∂D−2
(χD−2 )
#
D−3
− r 2 µ2 × φ(t, r, χD−2 , . . . , χ1 ) = 0, (4.112)
where g kk = [r 2
QD−2
j=k+1 sin
2
(χj )]−1 for k ≤ D − 3. Separating the variables we let
φ(t, r, χD−2 , . . . , χ1 ) = R(r)Ylm (χD−2 , . . . , χ1 )e−iωt ,
(4.113)
82
Chapter 4. Pair Production and the Hawking Effect
where Ylm (χD−2 , . . . , χ1 ) are the hyperspherical harmonics of degree l [33]. Here
the index m represents the D − 3 quantum numbers which are restricted by
l ≥ m1 ≥ m2 ≥ · · · ≥ |mD−3 | ≥ 0. The hyperspherical harmonics Ylm (χD−2 , . . . , χ1 )
are eigenfunctions of the operator
" D−3
X
g kk
L2 = − r 2
∂k sink−1 (χk )∂k
k−1
sin
(χ
)
k
k=1
#
1
∂D−2 sinD−3 (χD−2 )∂D−2
(4.114)
+
sinD−3 (χD−2 )
" D−3
X
1
∂k sink−1 (χk )∂k
=−
Q
D−2
k−1
2
sin (χk ) j=k+1 sin (χj )
k=1
#
1
+
∂D−2 sinD−3 (χD−2 )∂D−2 ,
(4.115)
D−3
sin
(χD−2 )
so that L2 Ylm = l(l + D − 3)Ylm . For each l, the m-quantum numbers may
take on ds (D, l) combinations, i.e. for each l there is a degeneracy. With use of
binomial coefficients, this degeneracy may be expressed as [34]
D−2+l
D−4+l
ds (D, l) =
−
.
(4.116)
D−2
D−2
For four spacetime dimensions, we then have the usual degeneracy ds (4, l) = 2l+1.
We use the Darwin definition of spherical harmonics [33, 35] for which
Y∗lm1 ...mD−3 = Ylm1 ...−mD−3 ≡ Yl,−m . The hyperspherical harmonics form an orthonormal set and satisfy
Z
dΩD−2 Y∗lm (χD−2 , . . . , χ1 )Yl0 m0 (χD−2 , . . . , χ1 ) = δll0 δmm0 .
(4.117)
The radial function R(r) must now obey
1
r D−4
∂r r(r
D−3
−
D−3
RSD
)∂r
ωr D−1
2 2
+ D−3
− l(l + D − 3) − µ r R(r) = 0.
D−3
r
− RSD
(4.118)
∗
Using the “tortoise coordinate” rD
from Eq.(3.54) we are able to solve this radial
∗
∗
equation. This coordinate rD → −∞ as r → RSD and rD
→ ∞ as r → ∞. We
now find that
dr ∗ d
d
r D−3
d
= D ∗ = D−3
.
∗
D−3
dr
dr drD
r
− RSD drD
(4.119)
4.2 The Hawking Effect
83
This leads to
2
D−3
r D−3
d2
−RSD
d
d
D−4
+
=(D − 3)r
.
∗
D−3 2
D−3
2
∗2
D−3
D−3
dr
drD
(r
− RSD ) drD
r
− RSD
(4.120)
Assuming that
R(r) = √
1
r D−2
f (r),
we find with a little algebra that f (r) satisfies the equation
2
d
2
+ ω − Ul (r) f (r) = 0.
∗2
drD
(4.121)
(4.122)
where the effective potential Ul (r) is given as
D−3 RSD
1
1
Ul (r) = 1 − D−3
(D − 2)(D − 4) 2
r
4
r
D−3
1
l(l + D − 3)
2 RSD
2
+ (D − 2) D−1 +
+ µ . (4.123)
4
r
r2
We notice that our generalization to D dimensions introduces a part in the effective potential which includes the factor (D − 4), i.e. a part which is non-existing
in the 4-dimensional potential.
Just outside the black hole horizon (r & RSD ), the differential equation in
Eq.(4.122) changes asymptotically into
2
d
2
(4.124)
+ ω f (r) = 0,
∗2
drD
∗
which has the solutions f (r) ∼ e±iωrD , where we take ω > 0 in the following. In
the asymptotic region r → ∞, the equation becomes
2
d
2
2
+ ω − µ f (r) = 0,
(4.125)
∗2
drD
∗
which has the solutions f (r) ∼ e±iprD , with p = (ω 2 − µ2 )1/2 as the wave’s radial
momentum. We see that in these regions the new part of Eq.(4.123) due to our
generalization does not contribute to any changes in the form of the asymptotic
equations compared to the usual 4-dimensional scenario. In the following, we
need the radial function in the asymptotic regions only. The wave equation in
Eq.(4.112) has two linearly independent solutions
1 →
Rωl (r)Ylm (χD−2 , . . . , χ1 )e−iωt ,
4πω
←
1 ←
φ ωlm (t, r, χD−2 , . . . , χ1 ) = √
Rωl (r)Ylm (χD−2 , . . . , χ1 )e−iωt ,
4πω
→
φ ωlm (t, r, χD−2 , . . . , χ1 ) = √
(4.126)
(4.127)
84
Chapter 4. Pair Production and the Hawking Effect
U
FUTURE SINGULARITY
V
r = RSD
t=∞
I
III
→
←
φ ωlm
φ ωlm
PAST SINGULARITY
r = RSD
t = −∞
→
←
Figure 4.7: Kruskal-Szekeres diagram with the solutions φ and φ
in the exterior region I of the Schwarzschild spacetime. Here the
solutions are taken to be massless and therefore the radial geodesics
are described as lines at 45 degrees in the diagram.
where the radial solutions are

→
∗
 √ 1 eiωrD∗ + √ 1 A
−iωrD
→
,
ωl e
D−2
D−2
r
r
Rωl (r) =
→
 √ 1 B ωl eiprD∗ ,
D−2
 r
←
∗
√ 1 B
−iωrD
←
e
,
ωl
D−2
r
Rωl (r) =
←
∗
 √ 1 e−iprD + √ 1 Aωl eiprD∗ ,
r D−2
∗
rD
→ −∞
,
(4.128)
∗
→ −∞
rD
.
(4.129)
∗
rD
→∞
∗
rD
→∞
r D−2
→
The φ ωlm (t, r, χD−2 , . . . , χ1 ) solution describes a wave originating from the past
horizon (t = −∞, r = RSD ) in the exterior region I (see Fig. 4.7) of the extended
→
Schwarzschild geometry. It is propagating toward infinity with an amplitude B ωl ,
or is scattered back to the future horizon (t = ∞, r = RSD ) with an amplitude
→
←
Aωl . The φ ωlm (t, r, χD−2 , . . . , χ1 ) describes a wave from past infinity. This will
←
either reach the future horizon with an amplitude B ωl , or be scattered back to
←
future infinity with an amplitude Aωl . In Fig. 4.7 these solutions are drawn
schematically into a Kruskal-Szekeres diagram.
By using that for any two linearly independent solutions R1 and R2 , the
Wronskian [26]
R1 R2 (4.130)
W ≡ dR1 dR2 ∗
drD
∗
drD
is constant, we may find how the asymptotic solutions (and their complex con∗
∗
jugates) for rD
→ ∞ relate to those for rD
→ −∞. We obtain by using different
4.2 The Hawking Effect
85
combinations of solutions that
ω ← 2
|B ωl | ,
p
→
p →
1 − |Aωl |2 = |B ωl |2 ,
ω
←
1 − |Aωl |2 =
→
←∗
←∗ →
→∗
←
−pB ωl Aωl = ω B ωl Aωl ,
pB ωl = ω B ωl ,
(4.131)
(4.132)
(4.133)
(4.134)
which also give that
→
←
|Aωl | = |Aωl |.
(4.135)
←
We notice that the absorption probability |B ωl |2 is equal to the tunneling proba→
bility |B ωl |2 only if the particle is massless. In this case we define the probability
→
←
amplitudes to be Bωl ≡ B ωl = B ωl .
→
←
The solutions φ ωlm and φ ωlm together with their complex conjugates form a
complete set of normalized solutions in region I. Unruh [18] called this way of
defining particles the η-definition. This refers to the Killing vector η = ∂/∂t,
here used to specify the modes with positive frequency (ω > 0). The modes of
positive frequency are then eigenfunctions of the Lie derivative £η , i.e.
£η φωlm =
→
∂
φωlm = −iωφωlm .
∂t
(4.136)
←
The solutions φ ωlm and φ ωlm behave asymptotically as plane waves in the regions
→
→
√
∗
rD
→ ±∞ only. Hence, φ ωlm behaves as φ ωlm ∼ (1/ r D−2 )e−iωu right outside
the past horizon (see Fig. 4.7). Here u is the outgoing Eddington-Finkelstein
coordinate
∗
u ≡ t − rD
.
←
(4.137)
←
Similarly, φ ωlm behaves as φ ωlm ∼ e−iωv in the vicinity outside the future horizon,
where v is the ingoing Eddington-Finkelstein coordinate
∗
v ≡ t + rD
.
(4.138)
These solutions are, however, not well-behaved near the past horizon. In the
Kruskal-Szekeres coordinates, defined in Eq.(3.51) and Eq.(3.52), the past horizon
is given by V = 0 which gives v → −∞ along this surface. In other words, the
←
φ ωlm solutions oscillate infinitely rapid when r → RSD in this region.
86
Chapter 4. Pair Production and the Hawking Effect
U
V
FUTURE SINGULARITY
r = RSD
t=∞
wωlm
III t
uωlm
PAST SINGULARITY
t
I
r = RSD
t = −∞
Figure 4.8: Kruskal-Szekeres diagram with the solutions uωlm and
wωlm propagating from the past horizon V = 0.
When studying the process of particle production in the Schwarzschild spacetime we assume that no particles are coming in from past infinity. And thereby
only the modes originating from the past horizon, propagating toward future infinity, are of special interest. Since only the past horizon is of special interest, we
use V = 0 as the surface on which we expand our quantum field. In region I we
consider then the particle modes originating from the past horizon. The modes
on this surface become
(
√ 1
e−iωu Ylm (χD−2 , . . . , χ1 ),
U <0
4πωr D−2
.
(4.139)
uωlm =
0,
U >0
Additionally, we need to take into account the modes defined on the surface V = 0
as
(
0,
U <0
.
(4.140)
wωlm =
1
iωu
√
e
Y
(χ
,
.
.
.
,
χ
),
U
>
0
lm
D−2
1
D−2
4πωr
These particle modes are originating from the past horizon at U > 0 and are unobservable to an observer at infinity, as they are always inside the future horizon
U = 0. The modes wωlm have a time dependence of the form eiωu . This difference
between the uωlm and wωlm modes is due to the the reversal of the coordinate time
direction in region III of the extended Schwarzschild spacetime. The dependence
of eiωu does also contribute to a positive scalar product norm for the wωlm modes.
In Fig. 4.8 the behavior of the two kinds of modes are schematically drawn into
a Kruskal-Szekeres diagram.
In our D-dimensional curved spacetime the inner product for bosons may be
4.2 The Hawking Effect
defined as
(φ1 , φ2 ) = −i
87
Z
↔
(φ∗1 ∂ U φ2 )g U V
√
−gdU dχD−2 · · · dχ1 ,
(4.141)
↔
using Kruskal-Szekeres coordinates and that A ∂ U B ≡ A∂U B − [∂U B]A. An
explanation of how this definition come to rise is given in [26, 36]. In KruskalSzekeres coordinates the metric, from Eq.(3.55), is
D−3 2
RSD
4RSD
∗
2
1 − D−3 e−(D−3)r /RSD dU dV + r 2 dΩ2 .
(4.142)
ds = −
2
(D − 3)
r
And we have that
√
−g =
2RSD
(D − 3)
s
1−
D−3
1
RSD
∗
e− 2 (D−3)r /RSD r D−2 sinD−3 (χD−2 ) · · · sin2 (χ2 ).
D−3
r
(4.143)
The contravariant component g U V is
1
D−3 − 2
1
D−3
RSD
∗
UV
g =−
1 − D−3
e− 2 (D−3)r /RSD .
2RSD
r
(4.144)
Combined with Eq.(4.141) this gives
Z
↔
(φ1 , φ2 ) = i (φ∗1 ∂ U φ2 )r D−2 sinD−3 (χD−2 ) · · · sin2 (χ2 )dU dχD−2 · · · dχ1
Z
↔
= i (φ∗1 ∂ U φ2 )r D−2 dU dΩD−2 .
(4.145)
If we now use the transformations
U = −e−(D−3)u/(2RSD ) ,
U =e
−(D−3)u/(2RSD )
,
U <0
(4.146)
U >0
(4.147)
to rewrite our two kinds of modes, uωlm and wωlm , these become
(
√ 1
(−U )iω2RSD /(D−3) Ylm (χD−2 , . . . , χ1 ),
U
D−2
4πωr
uωlm =
0,
U
(
0,
U
wωlm =
−iω2RSD /(D−3)
√ 1
U
Ylm (χD−2 , . . . , χ1 ),
U
4πωr D−2
<0
,
(4.148)
<0
.
>0
(4.149)
>0
We are now able to calculate the inner products and show that the uωlm modes
satisfies
(uωlm , uω0 l0 m0 ) = −(u∗ωlm , u∗ω0 l0 m0 ) = δ(ω − ω 0 )δll0 δmm0 ,
(uωlm , u∗ω0 l0 m0 ) = 0,
(4.150)
88
Chapter 4. Pair Production and the Hawking Effect
and the wωlm modes satisfies
∗
, wω∗ 0 l0 m0 ) = δ(ω − ω 0 )δll0 δmm0 ,
(wωlm , wω0 l0 m0 ) = −(wωlm
(wωlm , wω∗ 0 l0 m0 ) = 0.
(4.151)
The explicit calculations are done in Appendix F. The products between the
uωlm modes and the wωlm modes are zero, i.e. these form an orthonormal set of
functions on the past horizon. The quantum field Φ̂ may then be expanded in
terms of these modes and written as
i
XZ ∞ h
∗
dω aωlm uωlm + a†ωlm u∗ωlm + cωlm wωlm + c†ωlm wωlm
Φ̂ =
.
(4.152)
l,m
0
Here aωlm annihilates particles originating from the half of the past horizon where
U < 0. And c†ωlm creates particles emanating from the half of the past horizon
where U > 0. The creation and annihilation operators follow the commutation
relations
[aωlm , a†ω0 l0 m0 ] = δ(ω − ω 0 )δll0 δmm0 ,
[cωlm , c†ω0 l0 m0 ] = δ(ω − ω 0 )δll0 δmm0
(4.153)
and all other commutators vanish. We may now define the vacuum state for this
expansion as
aωlm | 0η i = cωlm | 0η i = 0.
(4.154)
And this is normalized to unity,
h0η | 0η i = 1.
(4.155)
The η refers to the earlier mentioned definition given by Unruh. This vacuum
state is also known as Boulware vacuum and corresponds to the state of minimum energy as measured from infinity and yield no particle production in the
Schwarzschild spacetime.
Unruh [18] argued that one may define an alternative vacuum state on the
past horizon of the Schwarzschild black hole. This is the ξ-vacuum | 0ξ i where ξ
referes to the Killing vector ξ = ∂/∂U . We define the positive frequency solutions
(ω̃ > 0) such that they on the past horizon are eigenfunctions of the Lie derivative
with respect to ξ,
£ξ φ̃ω̃lm =
∂
φ̃ω̃lm = −iω̃ φ̃ω̃lm .
∂U
(4.156)
We may by integrating this equation, obtain a complete set of normalized modes
in the form
φ̃ω̃lm = √
1
4π ω̃r D−2
e−iω̃U Ylm (χD−2 , . . . , χ1 ).
(4.157)
4.2 The Hawking Effect
89
The Lie derivative £ξ of the modes φ̃ω̃lm yields
1 ∂r
1
φ̃ω̃lm − iω̃ φ̃ω̃lm .
£ξ φ̃ω̃lm = − (D − 2)
2
r ∂U
(4.158)
But since (∂r/∂U ) vanishes on the past horizon, V = 0, the condition from
Eq.(4.156) is satisfied. And the modes in Eq.(4.157) do in fact define positive
frequency with respect to the Killing vector ξ = ∂/∂U . The inner products
on this surface is given by Eq.(4.145) and we find that these modes satisfy the
orthonormality relations
∗
, φ̃ω̃∗ 0 l0 m0 ) = δ(ω̃ − ω̃ 0 )δll0 δmm0 ,
(φ̃ω̃lm , φ̃ω̃0 l0 m0 ) = −(φ̃ω̃lm
(φ̃ω̃lm , φ̃ω̃∗ 0 l0 m0 ) = 0.
(4.159)
The explicit calculations are done in Appendix F. It can be shown [36] that
these modes actually are solutions to the wave equation. Let us now expand the
quantum field operator on the past horizon as
i
XZ ∞ h
†
∗
Φ̂ =
dω̃ aω̃lm φ̃ω̃lm + aω̃lm
φ̃ω̃lm
.
(4.160)
l,m
0
†
Here aω̃lm is the annihilation operator for particles described by φ̃ω̃lm and aω̃lm
is the creation operator for that same mode. The commutation relations these
operators obey are given by
[aω̃lm , aω̃† 0 l0 m0 ] = δ(ω̃ − ω̃ 0 )δll0 δmm0 ,
[aω̃lm , aω̃0 l0 m0 ] =
†
[aω̃lm
, aω̃† 0 l0 m0 ]
= 0.
(4.161)
(4.162)
The vacuum state associated with this field expansion is the ξ-vacuum, or the
Unruh vacuum, | 0ξ i and is defined by
aω̃lm | 0ξ i = 0.
(4.163)
h0ξ | 0ξ i = 1.
(4.164)
This state is normalized to unity,
This vacuum is different from the η-vacuum. Unruh [18] favored | 0ξ i to be the
real vacuum state on the past horizon of the Schwarzschild manifold and backed
this up with a number of reasons. A particle detector falling freely near the
future horizon will respond to the presence of ξ, not η, positive frequency modes.
And physically the ξ-vacuum corresponds to the absence of particles originating
from the past horizon, as measured by such a detector. Hawking’s gravitational
collapse scenario may be said to have been replaced by Unruh’s choice of the
ξ-vacuum to be the real vacuum.
90
Chapter 4. Pair Production and the Hawking Effect
In the same manner as for the Klein paradox we may compare the field operator Φ̂ expanded in the two sets to find the relations between the two sets creation
and annihilation operators. Two of these relations are
i
XZ ∞ h
∗
0
0
0
0
0
0
(4.165)
dω̃ αωω̃ll mm φ̃ω̃l m + βωω̃ll mm φ̃ω̃l0 m0 ,
uωlm =
l 0 m0
aωlm =
0
XZ
l 0 m0
∞
h
dω̃ α
0
ωω̃ll0 mm0
a
ω̃l0 m0
−β
ωω̃ll0 mm0
†
aω̃l
0 m0
i
,
(4.166)
where we have introduced the Bogoliubov coefficients
αωω̃ll0 mm0 = (φ̃ω̃l0 m0 , uωlm ),
(4.167)
∗
βωω̃ll0 mm0 = −(φ̃ω̃l
0 m0 , uωlm ).
We may now find a relation between these two coefficients by combining Eq.(4.166)
with the commutator relation in Eq.(4.161),
†
1 = [aω̃lm , aω̃lm
]
Z
X ∞ h
†
dω̃ (αωω̃ll0 mm0 aω̃l0 m0 − βωω̃ll0 mm0 aω̃l
=
0 m0 ),
l 0 m0
0
†
∗
(αω∗ ω̃ll0 mm0 aω̃l
0 m0 − βω ω̃ll0 mm0 aω̃l0 m0 )
XZ ∞ 2
2
0
0
0
0
=
dω̃ |αωω̃ll mm | − |βωω̃ll mm |
l 0 m0
i
(4.168)
0
These scalar products are calculated in Appendix F and we find that
αωω̃ll0 mm0 = αωω̃ δll0 δmm0 ,
βωω̃ll0 mm0 = βωω̃ δll0 δm,−m0 ,
with
αωω̃
βωω̃
(4.169)
r
RSD
ω πω RSD −iω 2RSD
2RSD
D−3
D−3
=
e
ω̃
Γ i
ω ,
π(D − 3) ω̃
D−3
r
ω −πω RSD −iω 2RSD
2RSD
RSD
D−3 ω̃
D−3 Γ
e
ω .
=−
i
π(D − 3) ω̃
D−3
(4.170)
We see from this that
αωω̃ = −e2πωRSD /(D−3) βωω̃ .
And used in Eq.(4.168) we find that
Z ∞
dω̃ |βωω̃ |2 =
0
1
e4πωRSD /(D−3)
−1
(4.171)
.
(4.172)
4.2 The Hawking Effect
91
We have now, in the same manner as for Klein’s step potensial, established a
consistent quantum field formalism for a curved spacetime. Therefore we should
now be able to calculate the vacuum expectation value of particles spontaneously
produced in the Schwarzschild spacetime or the vacuum expectation value of the
energy-momentum tensor.
4.2.2
The Hawking Temperature
We have already chosen the ξ-vacuum | 0ξ i to be the real vacuum state. So let
us now calculate the mean number of particles of the η-definition spontaneously
produced from the ξ-vacuum | 0ξ i near the horizon, as it is observed from infinity.
We have that
hnωlm i = h0ξ | a†ωlm aωlm | 0ξ i
XZ ∞
†
∗
=
dω̃ h0ξ | (αω∗ ω̃ll0 mm0 aω̃l
0 m0 − βω ω̃ll0 mm0 aω̃l0 m0 )
l 0 m0
0
†
× (αωω̃ll0 mm0 aω̃l0 m0 − βωω̃ll0 mm0 aω̃l
0 m0 ) | 0 ξ i
Z ∞
XZ ∞
1
.
=
dω̃ |βωω̃ |2 = 4πωR /(D−3)
dω̃ |βωω̃ll0 mm0 |2 =
SD
e
−1
0
0
0 0
(4.173)
lm
This is a Planck spectrum with a temperature of
TD =
(D − 3)
,
4πkB RSD
(4.174)
and is the same result as found in Eq.(3.47), in the chapter concerning black hole
thermodynamic. However, due to the spacetime curvature some of the particles
will be scattered back and not reach infinity. From Eq.(4.139) we have that
uωlm = √
1
∗
4πωr D−2
eiωrD eiωt Ylm (χD−2 , . . . , χ1 )
(4.175)
just outside past horizon V = 0. When these waves propagate toward infinity
→
∗
(rD
→ ∞), they follow the behavior of the φ ωlm solutions emanating from the
past horizon given in Eq.(4.126) and Eq.(4.128). We then have that
r ∗ →∞
uωlm D−→ √
→
1
4πωr D−2
→
∗
B ωl eiprD eiωt Ylm (χD−2 , . . . , χ1 ).
(4.176)
Here B ωl is the probability amplitude for the wave to reach infinity and p is the
radial momentum given earlier.
92
Chapter 4. Pair Production and the Hawking Effect
For ultra relativistic particles the Bogoliubov coefficients will then simply be
modified in such fashion that at infinity the mean number of observed particles
is
Γωl
,
(4.177)
hnωlm i = 4πωR /(D−3)
SD
e
−1
and may be regarded as a filtered Planck spectrum. Here Γωl ≡ |Bωl |2 is the
probability for a particle, specified by ω and l, to tunnel through to infinity. This
tunneling factor filters away a part of the radiation from the black hole. This
Γωl is therefore often refered to as a graybody factor, and the filtered Planck
spectrum as a graybody spectrum.
For massive particles the modification of the Bogoliubov coefficients is not as
simple. A more extensive review of the problem is necessary.
4.2.3
The Energy-Momentum Tensor
Let us now calculate the vacuum expectation value of the energy-momentum
tensor with respect to the ξ-vacuum.
From the definition in Eq.(2.143), the quantized operator for a neutral scalar
field becomes
i
1 µν h
α
β
2 2
µν
µ
ν
gαβ ∂ Φ̂∂ Φ̂ + µ Φ̂ ,
(4.178)
T̂ = ∂ Φ̂∂ Φ̂ − g
2
where we have used that LM = −(1/2)[∂α Φ∂ α Φ + µ2 Φ2 ] for this kind of field.
The vacuum expectation value is then hT̂ µν i = h0ξ | T̂ µν | 0ξ i. But in its current
form hT̂ µν i is formally divergent [26]. The naive way out, is by use of normal
ordering [26]. We then have
hT̂ µν i ≡ h0ξ |: T̂ µν :| 0ξ i.
(4.179)
Let us now examine the component
hT̂ tr i = h0ξ |: T̂ tr :| 0ξ i = h0ξ |: ∂ t Φ̂∂ r Φ̂ :| 0ξ i,
(4.180)
which at infinity represents the energy flux out of the black hole. By lowering
the indices, the expectation value hT̂ tr i = −h0ξ |: ∂t Φ̂∂r Φ̂ :| 0ξ i at infinity. Using
the expansion of Eq.(4.152), only the uωlm modes contribute to the particle flux
since the wωlm modes are hidden behind the future horizon. And we may expand
the field operators in terms of uωlm modes only. In order to find the energy flux
at infinity, we normal order with respect to the operators aωlm and a†ωlm . By
expanding the field operators, the energy flux then becomes
i
XZ ∞ h
tr
dω aωlm ∂t uωlm + a†ωlm ∂t u∗ωlm
hT̂ i = −h0ξ |:
l,m
×
0
XZ
l0 ,m0
h
i
dω 0 aω0 l0 m0 ∂r uω0 l0 m0 + a†ω0 l0 m0 ∂r u∗ω0 l0 m0 :| 0ξ i.
∞
0
(4.181)
4.2 The Hawking Effect
93
By normal ordering the operators and using the Bogoliubov transformation in
Eq.(4.166), the expression is transformed to
XZ ∞
tr
dω dω 0 dω̃ αω∗ ω̃ βω∗ 0 ω̃ ∂t uωlm ∂r uω0 l,−m − βωω̃ βω∗ 0 ω̃ ∂t u∗ωlm ∂r uω0 lm
hT̂ i =
l,m
0
− βω∗ ω̃ βω0 ω̃ ∂t uωlm ∂r u∗ω0 lm + βωω̃ αω0 ω̃ ∂t u∗ωlm ∂r u∗ω0 l,−m ,
(4.182)
where we also have used the commutation relation of Eq.(4.162) and the definition
of the Unruh vacuum in Eq.(4.163). With use of Eq.(4.170) and the expression
for the Hawking temperature, the integrals over ω̃ are found to be
2
Z ∞
RSD √ 0 π RSD (ω−ω0 ) 2RSD
∗
∗
δ(ω + ω 0 )
dω̃ αωω̃ βω0 ω̃ = −
ω
ωω e D−3
Γ
i
π(D − 3)
D−3 0
= 0,
(4.183)
2
Z ∞
RSD √ 0 π RSD (ω−ω0 ) 2RSD
δ(ω + ω 0 )
dω̃ βωω̃ αω0 ω̃ = −
ωω e D−3
Γ
i
ω
π(D − 3)
D−3 0
Z
Z
= 0,
∞
0
dω̃ βωω̃ βω∗ 0 ω̃ =
∞
0
dω̃ βω∗ ω̃ βω0 ω̃ =
(4.184)
eω/(2kB TD )
δ(ω − ω 0 ),
1
2 sinh π 2kB TD ω
eω/(2kB TD )
δ(ω − ω 0 ),
2 sinh π 2kB1TD ω
where we also have used that
Z
∞
0
dω̃ ±iCρ 2π
ω̃
=
δ(ρ)
ω̃
C
(4.185)
(4.186)
(4.187)
and that |Γ(ix)|2 = Γ(−ix)Γ(ix) = π/[x sinh(πx)]. Additionally, ω > 0 gives us
that δ(ω + ω 0 ) = 0. Thus, the energy flux takes the form
XZ ∞
1
tr
dω ω/(k T )
hT̂ i = −
[∂t u∗ωlm ∂r uωlm + ∂t uωlm ∂r u∗ωlm ].
(4.188)
B D − 1
e
0
l,m
To find the behavior of the energy flux at infinity, we insert Eq.(4.176), which
expresses the behavior of the uωlm wave solutions at infinity, into Eq.(4.188). This
reveals the energy flux at infinity to be
XZ ∞
pdω
1
tr r→∞
hT̂ i −→
Γ Y Y∗ ,
(4.189)
ω/(kB TD ) − 1 ωl lm lm
2πr D−2
e
0
l,m
where Γωl is the graybody factors introduced earlier.
94
Chapter 4. Pair Production and the Hawking Effect
The total luminosity at infinity is then
Z
dEem
= lim
dΩD−2 r D−2 hT̂ tr i
r→∞
dt
Z
pdω
1 X ∞
Γ ,
=
ω/(k
TD ) − 1 ωl
B
2π l,m 0 e
(4.190)
where we have used the normalization criterion in Eq.(4.117). Since the graybody
factors are only dependent of ω and l we may express the luminosity as
∞ Z
pdω
1 X ∞
dEem
ds (D, l) ω/(k T )
=
Γωl ,
(4.191)
dt
2π l=0 0
e B D −1
where ds (D, l) is the degeneracy expressed in Eq.(4.116).
The luminosity derived here is for a black hole radiating uniformly in D dimensions. We now assume, as in Section 3.3, that it instead radiates onto a
brane. Remembering the expression for luminosity found in Eq.(3.79), let us
transform our newly found expression so it describes ultra relativistic scalar particles confined to a q-brane. The wave modes will then be normalized on this
hypersurface and the degenaracy of the hyperspherical harmonics here will accordingly be ds (q + 1, l). Additionally, we have that the momentum p = ω for
ultra relativistic particles. Let us now use the De Witt approximation [26] where
the graybody factors are determined by geometrical optics. De Witt used this
approximation to find the luminosity for a four dimensional black hole, and from
this show that the black hole acts as a perfect blackbody with a larger radius
than the Schwarzschild radius. It has been suggested [13] that the same should be
the case in higher dimensions. We will now show that this is indeed correct for a
D-dimensional black hole radiating onto a q-brane. In the geometrical optics approximation, the outgoing wave is totally reflected if the effective potensial Ul (r)
from Eq.(4.123) is larger than the energy ω of the particle and totally transmitted
if Ul (r) is less than ω. The outgoing wave is then totally reflected if l is larger
than ωbcr or totally transmitted if l is less than ωbcr . Here bcr is the absorption
radius which will be derived in Section 5.2. This approximation corresponds to
a high energy limit approximation, where ωRSD 1. The luminosity of scalar
particles propagating on the brane is then
ωbcr Z ∞
1 X
dEem
ωdω
=
ds (q + 1, l) ω/(k T )
dt
2π l=0 0
e B D −1
Z ∞
1
ωdω
q + ωbcr
q − 2 + ωbcr
,
(4.192)
=
−
q
q
2π 0
eω/(kB TD ) − 1
where we have used that [37]
m X
k=0
n+k
n
=
n+m+1
.
n+1
(4.193)
4.2 The Hawking Effect
95
In the high energy limit, where we have that ωbcr 1, the binomial coefficients
is then somewhat reduced and the final expression of the luminosity of scalar
particles, confined to the b-dimensional brane, is
q−1 Z ∞
ω q dω
dEem
1 2bcr
=
dt
2π (q − 1)! 0 eω/(kB TD ) − 1
1
q−1
bcr
(kB TD )q+1 Γ(q + 1)ζ(q + 1),
(4.194)
=
π(q − 1)!
where Γ(q + 1) and ζ(q + 1), which arose from the integration [25], is the gamma
function and the Riemann zeta function, respectively. De Witt used this approximation of the luminosity even for small ωRSD . Even though this may seem as a
crude approximation, it could be supported by results published by D. N. Page
[27] the year after.
For scalar particles the degree of freedom in Eq.(3.79) is cscalar (q) = 1. We
then see that Eq.(4.194) is equal to Eq.(3.79) if the blackbody radius R = bcr .
In the geometrical optics approximation a Schwarzschild black hole acts as a
blackbody with radius bcr .
Numerical calculations of the graybody factors in 4 dimensions have been
made as early as 1976 [27]. More recently calculations have been made in higher
dimensions for particles with spin s = 0, 1/2, 1 in the low energy limit [38, 39].
It is here important to notice that there is a different definition of the graybody
factor used in these two latest articles. Here the graybody factor has dimensions
of area and is dependent of the absorption probability, i.e. our graybody factor.
This alternative definition of the graybody factor is widely used.
4.2.4
A Heuristic Picture of the Hawking Radiation
The thermal particle emission from the black hole horizon may be viewed as
emerging from spontaneous produced particle pairs at the black hole horizon. One
particle will travel outward, toward infinity and is a part of the thermal radiation
observed from infinity. The other particle will tunnel through the horizon and
travel in the black hole’s interior. Viewed from infinity this particle traveling
inside the horizon has negative energy and will contribute to a decrease in the
black hole mass. (Viewed by a local observer the particle has of course positive
energy.) To see how this is possible let us first define the energy of a particle with
D-momentum pµ measured by an observer with D-velocity uµ to be
E = −uµ pµ .
(4.195)
A static observer at infinity that has in his own referance frame a D-velocity uµ∞ ,
with ut∞ = 1 as its only non-zero component, measures the energy of the particle
to be E∞ . We then have that E∞ = −pt . Let us imagine an observer inside
the event horizon. For simplicity let him move along a trajectory for which the
96
Chapter 4. Pair Production and the Hawking Effect
only non-zero component of the D-velocity is ur . We then have from the identity
uα uα = gαβ uα uβ = −1, that
ur = −
D−3
RSD
−1
r D−3
21
,
(r < RSD )
(4.196)
where we have chosen negative sign in front of the square root since the observer
is going inward. For this local observer, the measured energy of a particle must be
positive, which gives that −pµ uµ > 0. The energy of a zero angular-momentum
massless particle moving radially inside the horizon will by the local observer be
measured to be
µ
r r
E = −pµ u = −grr p u = −
D−3
RSD
−1
r D−3
− 21
pr .
(4.197)
This energy is positive only if pr < 0. By expressing the Lagrange function in
terms of its constants of motion, for such a particle trajectory, it may be shown
that pr = ±E∞ . This will be shown explicitly for a massless particle in the
next chapter, where ω denotes the energy measured at infinity. But still this sets
no restriction on E∞ . In other words, a massless particle may travel inside the
horizon with negative E∞ , as long as pr < 0.
Equivalently, we could have defined the energy of a particle relative to infinity
as [40]
E∞ ≡ −pµ · ηµ ,
(4.198)
where η is the Killing vector which we earlier used to define positiv frequency
modes as observered from infinity. Inside the Schwarzschild radius the Killing vector becomes spacelike and thus a particle inside the horizon may have a timelike
D-momentum pµ and negative energy E∞ . The modes uωlm from our quantum
field discussion then represents particles propagating outward from the black hole,
and the modes wωlm represents particles falling into the black hole singularity.
We remember from Eq.(4.139) and Eq.(4.140) that uωlm ∼ e−iωu and wωlm ∼ eiωu .
And we may in some sence view these modes as having opposite signs of energy
and the particles inside the horizon therefore cause the mass of the black hole to
decrease.
This picture of the mechanism responsible for the thermal emission and the
area decrease is heuristic only and should, as Hawking noted, not be taken too
literally. But the result is inevitably the same, we have a net flux of positive
energy particles emitted toward infinity which constitute a thermal specter. This
is accompanied by a decrease in the black hole mass.
Chapter 5
Black Hole Production
The possibility of producing black holes at high energy colliders operational in the
close future, became evident, as we have seen in Section 2.3, with the proposal
of large extra dimensions. There we showed how the possibility of having a
world with compact dimensions of size < 1 mm also means that a fundamental
Planck scale MD ∼ 103 GeV is possible. This would then be the solution to the
hierarchy problem, but it would also imply the possibility of producing semiclassical, microscopic black holes at far lower energies than otherwise possible.
This production of black holes could potentially be the dominant process at future
high energy colliders. Even collisions between cosmic neutrinos and atmospheric
nucleons could yield production of microscopic black holes. A black hole created
in such a process will then subsequently decay via Hawking radiation and the
traces of the black hole would be detected as standard model particles radiating
thermally.
We will here confine our discussion to a process of colliding hadrons, specifically protons. This process will be realized at CERN when the LHC (Large
Hadron Collider) starts up in a couple of years. This collider will produce a
center of mass energy of 14 TeV and may possibly become a black hole factory.
We will concentrate on the cross-section of such a potential black hole producing process. We introduce an initial approximation, before discussing possible
corrections to this.
5.1
The Black Hole Cross-Section
In a collision process, the cross-section is a very useful quantity. The cross-section
contains all the physics in the process. The cross-section may intuitively [41] be
defined as the effective size of the target as seen by an incoming beam. (And
is often measured in terms of barns. One barn is 10−26 m2 .) A particle beam
is sent toward the target with an effective area σ. This beam has a hit area
A. The probability for particles constituting the beam to hit the target is then
97
98
Chapter 5. Black Hole Production
4
MD = 1 TeV
σ̂I × (TeV)2
3
PSfrag replacements
2
1
D =7
D =9
0
10
20
30
√ 40 50
s/TeV
D
D
D
D
=
=
=
=
60
70
80
6
8
10
11
Figure 5.1: The black hole cross-section
σ̂I from Eq.(5.2) as a func√
tion of the center of mass energy s in a scenario where the fundamental Planck mass MD = 1 TeV, plotted for D = 6, 8, 10, 11 dimensions.
P (hit) = σ/A. If the number of particles in the beam is NB , the number of hits,
events, is then NE = NB σ/A. We then have that the cross-section is
σ=
transition rate
NE /t
=
,
JB
JB
(5.1)
where the transition rate is the number of events per second and JB = NB /(At) is
the incoming current density. This is a more formal definition of a cross-section.
The first to describe the process in which a microscopic black hole is formed
were Banks and Fischer [42] in 1999. They argued that a collision between two
particles, with center of mass energy much larger than the relevant Planck scale,
would be dominated by the formation of a black hole if the impact parameter
of the process was smaller that the Schwarzschild radius corresponding to the
center of mass energy. The impact parameter describes the distance at which the
particles would pass if they did not interact. Now consider a collision between two
particles. These are sent toward each other with an impact parameter b. Then if
this parameter is smaller
√ than the Schwarzschild radius RSD associated with the
center of mass energy s, the relativistic particles pass within the corresponding
event horizon. The event horizon is of course entirely fictitious, until the particles
pass it. The particles will then be trapped inside the horizon and a black hole is
formed.
5.1 The Black Hole Cross-Section
99
Figure 5.2: This plot, taken from an article by Giudice, Rattazzi
and Wells√[44], shows possible the black hole production cross-section
at LHC ( s = 14 TeV) as a function of minimum black hole mass.
The cross-section is plotted for n = 2, 6 extra dimensions dimensions
and fundamental Planck mass MD = 1.5 TeV, 3 TeV.
From the intuitive formulation of the cross-section, the black hole cross-section
should simply be approximated by the geometrical cross-section of an absorptive
black disk [43] with radius RSD ,
2
# D−3
"
2
√ D−3
D−1
Γ(
)
s
1
2
2
.
(5.2)
σ̂I = πRSD
=π 2
MD MD
(D − 2)π (D−1)/2
The cross-section grows like s1/(D−3) , which is more rapid than any other standard
model cross-section [22]. This would mean that at center of mass energies MD ,
black hole production becomes the dominant process. In Fig.
√ 5.1 this crosssection is displayed as a function of the center of mass energy
s of the system.
√
We have here plotted the cross-section even for values of s for which our semiclassical approximation is not reliable.
This description of the collision is adequate in a process involving elementary
particles, but a process of colliding non-elementary particles, such as protons,
should more appropriately be described as parton collisions. In the parton description, particles consists of partons which each carries a different fraction x
of the theP
original
particle’s momentum and energy. These fractions satisfies the
R
property i0 dx xfi0 (x, p) = 1, where fi (x, p) is a parton distribution function
which describes the probability that the parton i carries a fraction x of the original particles momentum p. The resulting cross-section of such a process is given
100
Chapter 5. Black Hole Production
by the expression [45]
Z
σBH =
1
2
min /s
MBH
dx
Z
1
x
√
dy
fi (y, p)fj (x/y, p)σ̂ij (x s),
y
(5.3)
√
where σ̂ij ( s) is the cross-section between parton i and j and could be described
min
by the geometrical cross-section in Eq.(5.2). MBH
is the minimal black hole
min
mass for which the semi-classical cross-section is valid (MBH
MD ), and the
summation over i and j include all possible pairs of the initial parton states. In
Fig. 5.2 [44] the cross-section σBH is plotted as a function of minimum black hole
min
mass. In this plot the lines are extended down to MBH
= MD , beyond the region
where our semi-classical model is reliable.
However, many have argued that the initial approximation for the crosssection σ̂ may be somewhat crude. And there have been suggested several corrections to this absorptive black disk cross-section [46, 47, 48, 44]. Others again
[49, 50, 51] have tried to explain and motivate this geometric value of the crosssection more thoroughly. We will in the following review a couple of these models
and corrections.
5.2
Gravitational Capture
In the description of the process leading to our initial cross-section, the particles
are propagating in a flat spacetime. A more correct description should perhaps
be to have the particles propagate in a spacetime curved by the energy of the
two particles. However, in general relativity the classical two-body problem,
such as the process of two colliding particles, is not generally solved. In order
to study the black hole formation from a collision between two particles general
relativistically, we must therefore somehow overcome this. It was suggested in an
article from 2002 by Solodukhin [46] that one may study a process approximately
equivalent to the two particle problem, which in general relativity is described in
a straight forward manner. The process in mind is a particle falling freely in a
Schwarzschild spacetime.
Let us now see how this approximation may come to be. In Newtonian mechanics two particles sent toward each other with energy E1 and E2 respectively,
could be described in terms of a particle with reduced√
energy ω = E1 E2 /(E1 +E2 )
moving in a gravitational field produced by the mass s = E1 +E2 located at the
system’s center of mass. The particle is then moving toward the center of mass
with an impact parameter b. This parameter may now be associated with the
shortest distance between the particle of energy ω and the center of mass. An
approximative general relativistic description of our two-body problem, would
then be a particle with energy ω
√ moving in a Schwarzschild spacetime with a
black hole horizon at r = RSD ( s). And if the test particle crosses this fictitious horizon, a real one is formed. In the original process this corresponds to
5.2 Gravitational Capture
101
black hole production. The advantage of this approximation, is that it is a well
formulated and well known general relativistic process.
In the sequel we consider ultra-relativistic particles with velocity close to the
speed of light and assume that the motion of the system as a whole is in one
plane.
5.2.1
The Classical Relativistic Approach
In this general relativistic approximation it is easy to derive the geodesic equation.
This may be done by evaluating the Lagrange function
1
L(xµ , ẋµ ) = gµν ẋµ ẋν ,
2
(5.4)
where ẋµ now is the derivation of xµ with respect to a non-vanishing affine parameter λ. For a ultra-relativistic particle, confined to a 3-brane, moving in
the equatorial plane in the D-dimensional Schwarzschild spacetime the Lagrange
function becomes
D−3 1
1 2 2
1
1
RSD
2
µ
µ
r φ̇ .
(5.5)
L(x , ẋ ) = −
1 − D−3 ṫ2 +
D−3 ṙ +
2
r
2 1 − RSD
2
r D−3
The canonical momentums conjugated to the time t and the angle φ are constants
of motion, i.e. the energy
D−3 ∂L(xµ , ẋµ )
RSD
ω ≡ −pt = −
= 1 − D−3 ṫ
(5.6)
r
∂ ṫ
and the angular momentum
Lφ ≡ p φ =
∂L(xµ , ẋµ )
= r 2 φ̇
∂ φ̇
(5.7)
are constants. We may then transform the respective derivatives into
ω
ṫ =
1−
RD−3
SD
r D−3
and
φ̇ =
Lφ
.
r2
In addition we have that the canonical momentum conjugated to r is
D−3 −1
RSD
pr = 1 −
ṙ.
r
(5.8)
(5.9)
And then it follows that pr = ṙ. For a lightlike particle, the 4-velocity identity
becomes ẋµ ẋµ = 0, and thus we set Eq.(5.5) to be equal zero. We then have that
D−3 L2φ
RSD
2
2
ṙ = ω − 2 1 − D−3 .
(5.10)
r
r
102
Chapter 5. Black Hole Production
This corresponds to an energy equation with an effective potential V (r) given by
L2φ
V (r) = 2
r
2
D−3
RSD
1 − D−3
r
.
(5.11)
(A zero angular momentum particle will have V 2 (r) = 0 and pr = ±ω.) From
Eq.(5.8) we find that
D−3 1
RSD
dλ =
1 − D−3 dt
(5.12)
ω
r
and our equation of motion may be written as
dr
dt
2
=
D−3
RSD
1 − D−3
r
2 1 2
1 − 2 V (r) .
ω
(5.13)
The impact parameter b now describes the shortest distance from the particle to
the point r = 0 if the particle had been traveling in a flat spacetime. At infinity
our Schwarzschild spacetime is asymptotically flat. We expect here the particle
movement to be along a straight line with a constant velocity v. In this region
of spacetime we have that
b ' r sin φ ' rφ
−v '
and
d
dr
(r cos φ) ' .
dt
dt
(5.14)
If we in Eq.(5.13) let r become large and use the two approximations for this
region, we find that v 2 = 1 and b = Lφ /ω. And we may write the equation of
motion as
"
#
2 D−3 2
dr
RSD
1
1
= 1 − D−3
b2 2 − 2 V 2 (r) .
(5.15)
dt
r
b
Lφ
The effective potential takes its maximal value at
rmax =
D−1
2
1
D−3
RSD
(5.16)
and
2
V (rmax ) =
D−1
2
2
− D−3
D−3
D−1
L2φ
.
2
RSD
(5.17)
We may now find the critical impact parameter bcr for which the particle is
captured by the gravitating center and eventually falls into the horizon. By
evaluating the equation of motion when the radial velocity dr/dt become zero
5.2 Gravitational Capture
103
and the effective potential takes it maximal value, we find that the critical impact
parameter for gravitational capture is
bcr =
D−1
2
r
1
D−3
D−1
RSD .
D−3
(5.18)
When D = 4 we retrieve the result,
derived in almost all books concerning general
√
relativity (e.g. [2, 3]), bcr = 3 3RS4 /2. For impact parameters b < bcr capture of
the particle is inevitable.
Having in mind the original picture of colliding particles, these particles form
a black hole with a Schwarzschild radius RSD if b < bcr . If b > bcr , the particles
just scatter of the center of mass and the gravitational capture is not achieved.
This corresponds to the particles passing each other without forming a black hole.
Classically, the cross-section for a black hole formation is then given by
σ̂cl =
πb2cr
=
D−1
2
2
D−3
D−1 2
πR .
D − 3 SD
(5.19)
Thus in this classical relativistic model two colliding particles form a black hole if
the shortest distance between them is b < bcr . We see that the absorption crosssection is [(D − 1)/2]2/(D−3) (D − 1)/(D − 3) times the geometrical cross-section.
With this approach, the cross-section is, ranging from 4 (with D = 5) to 1,87
(with D = 11) times larger than the initial geometrical absorption cross-section.
5.2.2
The Semi-Classical Approach
The picture described in the beginning of this section is also possible to analyze
more accurately in a quantum mechanical picture. This analysis is the one done
in Section 4.2 and the quantum mechanical analogue to the classical equation
from Eq.(5.15) is the wave equation found in Eq.(4.122). In the analysis of ultrarelativistic particles, we set µ = 0. This equation may then be written as
d2
2
+ ω − Ul (r) Rωl (r) = 0,
∗2
drD
(5.20)
where
Ul (r) =
D−3
RSD
1 − D−3
r
D−3
1
1
1
l(l + 1)
2 RSD
(D − 2)(D − 4) 2 + (D − 2) D−1 +
4
r
4
r
r2
(5.21)
is the new effective potential. We remember that in the regions r → ∞ and
←
r & RSD this equation had a solution Rωl found in Eq.(4.127). This solution
104
Chapter 5. Black Hole Production
described the radial part of a wave originating from past infinity, propagating
toward the future horizon. And, as we remember, it will either reach future
←
horizon with an amplitude B ωl or be scattered to future infinity with an amplitude
←
←
←
Aωl . We have that |B ωl |2 + |Aωl |2 = 1, for ultra-relativistic particles.
Let us now look at the scattering of a plane wave moving toward the black
hole. This scattering process has been thoroughly studied in four dimensions by
applying the method of partial waves [43]. Using this method in D dimensions we
construct a solution of the wave equation outside the region of interaction, where
∗
the particles are free, i.e. where rD
→ ∞. If we define the z-axis to be parallel
to the incoming beam, the wave function ψ has in this region the asymptotic
solution
∗
eiωrD
1
iωz
.
(5.22)
e + f (θ, φ) √
ψ=√
2ω
r D−2
This solution represents an plane wave ∼ eiωz traveling in the direction of the
black hole and an scattered wave moving outward, away from it. The term f (θ, φ)
↔
is the scattering amplitude. The current density Jz = ψz∗ ∂ z√
ψz of the plane wave
is normalized to unity with the inclusion of the term 1/ 2ω. A plane wave
undisturbed by interaction with the black hole’s gravitational field may now be
expressed as a sum of spherical waves [43],
∞
1 iωz
1 Xp
∗
ψz = √ e = √
4π(2l + 1)il jl (ωrD
)Yl0 (θ, φ)
2ω
2ω l=0
r
∞
i
∗ →∞
rD
lπ
1
2π X √
il+1 h −i(ωrD∗ − lπ )
∗
2
2l + 1
− ei(ωrD − 2 ) Yl0 (θ, φ),
e
=
ωr ω l=0
2
(5.23)
∗
where Yl0 (θ, φ) is a spherical harmonic and jl (ωrD
) is a spherical Bessel function
which is
∗
lπ
∗ rD →∞ 1
∗
jl (ωrD
) =
sin(ωrD
− ).
(5.24)
ωr
2
∗
We have here used that the “tortoise coordinate” rD
→ ∞ in the same manner
as the coordinate r. The first exponential in the spherical wave expansion in
Eq.(5.23) describes a wave traveling from infinity toward the coordinate position
z = 0, and the second term describes a wave propagating outward to infinity.
With the black hole present at z = 0, the outgoing part of the wave is disturbed.
←
To describe this disturbance we may introduce the scattering coefficient Aωl from
Eq(4.129), which enable us to express ψ as
r
∞
2π X √
il+1 h −i(ωrD∗ − lπ ) ← i(ωrD∗ − lπ ) i
1
2
2
2l + 1
e
− Aωl e
Yl0 (θ, φ)
(5.25)
ψ=
ωr ω l=0
2
= ψin + ψout .
(5.26)
5.2 Gravitational Capture
105
In the process of black hole formation we are not interested in the scattering, but
rather the absorption of waves. It can be shown that the differential absorption
cross-section may be expressed as
Jab 2
dσ̂quant
=
r ,
dΩ2
Jz
(5.27)
where the absorption current density is Jab = Jin −Jout . The two involved current
densities are calculated to be
∞
↔
π X
∗
∂ r ψin = 2 2
Jin = ψin
(2l + 1)|Yl0 (θ, φ)|2
(5.28)
ω r l=0
and
↔
∗
Jout = ψout
∂ r ψout =
When we use that
section becomes
R
(5.29)
∗
dΩ2 Ylm
(θ, φ)Yl0 m0 (θ, φ) = δll0 δmm0 , the total absorption cross-
σ̂quant =
←
∞
←
π X
2
2
(2l
+
1)|Y
(θ,
φ)|
|
A
l0
ωl | .
ω 2 r 2 l=0
Z
dΩ2 Jab r 2 =
∞
←
π X
2
(2l
+
1)(1
−
|
A
ωl | )
ω2
(5.30)
l=0
∞
π X
(2l + 1)Γωl ,
= 2
ω l=0
(5.31)
←
where Γωl = |B ωl |2 = 1 − |Aωl |2 is the absorption probability. Here the impact
parameter is
1 1
b= l+
.
(5.32)
2 ω
In the high energy limit, geometrical optics becomes a good approximation,
since ωRBH 1. In this approximation, the incoming wave is totally absorbed
[26] if ω 2 is less than the maximum value of Ul (r). For l = 1, this effective
potential has its maximum value close to the rmax found in Eq.(5.16). For higher
values of l this radius is further approached. By putting ω 2 equal the maximal
value of the effective potential we may find the critical impact parameter to be
2
D − 1 D − 1 D−3 2
1
(D − 2)2 1
2
bcr =
RSD −
(D − 2)(D − 4) +
, (5.33)
D−3
2
2
D − 1 ω2
where we have used that l(l + 1) ' ω 2 b2 . In our approximation we have that the
graybody factor, or the absorption probability may be approximated to be [52]
(
1,
l + 21 < ωbcr
Γωl =
.
(5.34)
0,
l + 12 > ωbcr
106
Chapter 5. Black Hole Production
P cr ω
If we use that bl=0
(l + 1/2) ' (1/2)b2cr ω 2 we find that, in the high energy limit,
the absorption cross-section is
σ̂quant ' πb2cr
π
(D − 2)2 1
= σ̂cl −
(D − 2)(D − 4) +
,
2
D − 1 ω2
(5.35)
where σ̂cl is the cross-section found in Eq.(5.19) by use of the classical relativistic
approach. In Section 4.2 we noticed that the D-dimensional wave equation in
Eq.(4.122) had a part which vanishes when D = 4. In our discussion on the
D-dimensional Hawking effect this part made no contribution, but as we now
notice here it contributes to our absorption cross-section. In D = 4 dimensions
2
we then get the result [46] σ̂quant ' (27/4)πRS4
− (2/3)πω −2 .
5.3
Gravitational Shock-Waves
Later in the year 2002, as a motivation for the geometrical cross-section, an alternative to the previous approximation was proposed in an article by Eardley and
Giddings [49]. It proposed that the collision process should more appropriately
be described as two particles each which at rest curve the spacetime according
to the Schwarzschild solution. The geometry around a particle moving with a
particular speed can then be obtained simply by doing an appropriate Lorentz
boost√transformation. In the case of two colliding particles, each has an energy
µ = s/2 in the center of mass frame. In the limit of approaching the speed of
light and zero mass, with fixed total energy µ, the Aichelburg-Sexl solution [53]
of the Einstein equations is found. In our collision process where the center of
mass energy is much larger than the rest mass contribution, this solution can be
considered a good approximation. We will in the following show that the solution
for a particle moving in positive z̄ ≡ x̄1 direction is
2
ds = −dūdv̄ +
D−1
X
(dx̄i )2 + Θ(x̄i )δ(ū)dū2 ,
(5.36)
i=2
where x̄i is one of the D − 2 transverse coordinates,
ū = t̄ − z̄
and
v̄ = t̄ + z̄
(5.37)
are the light-cone coordinates, and Θ(x̄i ) may be found to be
Θ(x̄i ) = −8GN µ ln(ρ̄),
D − 2 ΩD−2
GD µ
i
,
Θ(x̄ ) = 2
D − 3 ΩD−3 (D − 4)ρ̄D−4
q
where ρ̄ = x22 + · · · + x2D−1 is the transverse radius.
D = 4,
(5.38)
D ≥ 4,
(5.39)
5.3 Gravitational Shock-Waves
5.3.1
107
Aichelburg-Sexl Shock-Waves
To see how the form of the presented metric may come about, let us in the same
manner as Aichelburg and Sexl [53] consider a particle with mass m, at rest in a
coordinate system (t0 , x0 , y 0 , z 0 ), which curves the four-dimensional spacetime continuum according to the four-dimensional isotropic Schwarzschild solution which
may be found from Eq.(2.102). This metric is then revealed to be
2
4
1
0
ds = −
dt + 1 − Φ4 (r ) (dz 02 + dx02 + dy 02 ),
(5.40)
2
p
where Φ4 (r 0 ) is given in Eq.(2.23) and r 0 = z 02 + x02 + y 02 . If this particle now
is boosted with a speed v in positive z direction, an observer at rest will see the
metric deformed by a Lorentz transformation
2
1 + 21 Φ4 (r 0 )
1 − 12 Φ4 (r 0 )
02
1
t = (1 − v 2 )− 2 (t0 + vz 0 ),
1
z = (1 − v 2 )− 2 (z 0 + vt0 ),
x = x0 ,
y = y 0.
(5.41)
The line element changes into
2
ds =
1
1 − Φ4 (r 0 )
2
4
+
(−dt2 + dz 2 + dx2 + dy 2 )
"
1
1 − Φ4 (r 0 )
2
4
1
− 1 − Φ4 (r 0 )
2
4 #
(dt − vdz)2
, (5.42)
(1 − v 2 )
with
Φ4 (r 0 ) = −
GN µ(1 − v 2 )
GN m
=
−
1 .
r0
[(z − vt)2 + (1 − v 2 )(x02 + y 02 )] 2
(5.43)
We have then introduced the energy µ = m(1 − v 2 )−1/2 which in the limit v → 1
is kept constant, while the particle’s rest mass goes to zero. This ensures that
limv→1 Φ4 (r 0 ) = 0, while
(
GN µ
,
z 6= t
GN µ
|z−t|
=
.
(5.44)
lim
1
v→1 [(z − vt)2 + (1 − v 2 )(x02 + y 02 )] 2
divergent, z = t
In the case of z 6= t we have in the limit v → 1 that
ds2 = −dt2 + dz 2 + dx2 + dy 2 +
4GN µ
(dt − vdz)2 ,
|z − t|
(5.45)
108
Chapter 5. Black Hole Production
for which the Riemann tensor is zero [53], i.e. the spacetime is flat. A suitable
coordinate transformation [53]
x̄ = x,
ȳ = y,
z̄ − v t̄ = z − vt,
(5.46)
z̄ + v t̄ = z + vt − 4µ ln[
p
(z − vt)2 + (1 − v 2 ) − (z − t)],
may be introduced which enables the evaluation of the spacetime in z = t as well.
We may then find that the line element from Eq.(5.40), in the limit v → 1, is
transformed after some calculations into
ds2 = −dt̄2 + dz̄ 2 + dx̄2 + dȳ 2 − 8GN µδ(z̄ − t̄) ln ρ̄(dt̄ − dz̄)2 .
(5.47)
We see that this is in agreement with the form presented in Eq.(5.36) as well as
the function Θ(x̄i ) in Eq.(5.38). It has here been used that
n
− 1 o
− 1 lim fv (t̄, z̄, x̄, ȳ) = lim (z̄ − v t̄)2 + ρ̄(1 − v 2 ) 2 − (z̄ − v t̄)2 + (1 − v 2 ) 2
v→1
v→1
= −2δ(z̄ − t̄) ln ρ̄.
(5.48)
This result may be found by integrating fv (t̄, z̄ 0 , x̄, ȳ) over z̄ 0 from −∞ to z̄, and
then taking the limit. The uncovered result involves a step function. Differentiating the step function gives the delta function displayed above. This result of
Eq.(5.47) is in accordance with that found by Dray and
’t Hooft [54], when they presented another derivation of the shock-wave solution
of a massless particle some years later.
The Aichelburg-Sexl solution is a solution of a point particle moving with the
speed of light. With use of light-cone coordinates the metric has found the form
of Eq.(5.36). This is the form of plane-fronted gravitational waves with parallel
rays (pp-waves) [55, 56]. This spacetime solution is flat except in the null plane
ū = 0 of the shock-wave. The only non-zero component of the Ricci tensor is
1
Rūū = − ∇2 Θ(x̄i )δ(z̄ − t̄),
2
(5.49)
where ∇ is the flat-space derivative in the D − 2 transverse coordinates. We also
have that the only non-zero Einstein tensor component is Eūū = Rūū .
Another, simpler way [50] to find Θ(x̄i ) is to boost the energy momentum tensor and then solve the Einstein equations for the resulting pp-wave, i.e. we have
to assume that the resulting Aichelburg-Sexl shock wave is a pp-wave. However,
in this way we do not see from where the form of the line element originates. A
point particle of mass m at rest has
i
Tt0 t0 = mδ D−2 (x0 )δ(z 0 )
(5.50)
5.3 Gravitational Shock-Waves
109
as its only non-zero energy momentum tensor component. Boosted in positive z̄
direction we have a Lorentz transformation
1
t̄ = (1 − v 2 )− 2 (t0 + vz 0 ),
1
z̄ = (1 − v 2 )− 2 (z 0 + vt0 ).
(5.51)
The energy momentum tensor transforms according to the tensor transformation
0
0
M µµ = ∂xµ /∂xµ . We then have that
0 2
m
∂t
Tt̄t̄ =
Tt 0 t 0 = √
δ D−2 (x̄i )δ(z̄ − v t̄),
(5.52)
∂ t̄
1 − v2
0 0
−v
∂t
∂t
Tt 0 t 0 = √
Tt̄z̄ = Tz̄ t̄ =
δ D−2 (x̄i )δ(z̄ − v t̄),
(5.53)
∂ z̄
∂ t̄
1 − v2
0 2
∂t
v2
Tz̄z̄ =
δ D−2 (x̄i )δ(z̄ − v t̄)
(5.54)
Tt 0 t 0 = √
2
∂ z̄
1−v
and the rest is zero. By letting v → 1, introducing the non-vanishing energy µ
and using the light-cone coordinates from Eq.(5.37) we are able to find
µ ν
∂x
∂x
Tµν
Tūū =
∂ ū
∂ ū
2
2
∂ z̄
∂ t̄
∂ z̄
∂ t̄
Tt̄t̄ + 2
Tz̄z̄
Tz̄ t̄ +
=
∂ ū
∂ ū
∂ ū
∂ ū
= µδ D−2 (x̄i )δ(ū).
(5.55)
Since Rūū = Eūū we have from Eq.(5.49) combined with Eq.(2.142) that
1
∇2 Θ(x̄i ) = −2µδ D−2 (x̄i ) D−2 .
(5.56)
MD
By integrating this Poisson equation over the transverse coordinates, the delta
function vanishes.H So for D ≥ 4, by use
H of Gauss’ divergence theorem this may be
transformed into dAD−3 ·∇Θ(x̄i ) = dΩD−3 ρ̄D−4 ρ̄·∇Θ(x̄i ) = −2µ/MDD−2 , where
we have used that dAD−3 = dΩD−3 ρ̄D−3 . Assuming that Θ(x̄i ) is symmetric and
only dependent of ρ̄, the Laplace operator ∇ reduces to (ρ̄/ρ̄)∂/∂ ρ̄. We then
have that ρ̄[∂Θ(x̄i )/∂ ρ̄] = −2µ/[ΩD−3 ρ̄−4 MDD−2 ], which with integration over ρ
reduces to
2µ
1
(5.57)
Θ(x̄i ) =
D−2
D−4
(D − 4)ΩD−3 ρ̄
MD
D − 2 ΩD−2
GD µ
=2
,
(5.58)
D − 3 ΩD−3 (D − 4)ρ̄D−4
where we in the last transition have used the relation between the fundamental
Planck mass and the D-dimensional gravitational constant from Eq.(2.150). This
again is in agreement with what was displayed in Eq.(5.39). This method of
obtaining Θ(x̄i ) works just as well in D = 4 dimensions and the result from
Eq.(5.38) would then be confirmed.
110
5.3.2
Chapter 5. Black Hole Production
Two Colliding Gravitational Shock-Waves
We have now established how the line element of a single massless particle is
constituted. Let us see what happens when two such particles are sent toward
each other, one particle in the v̄ direction, the other in the ū direction. The
particles are sent toward each other with an impact parameter b, i.e. the initial
trajectories may be chosen to be along x̄i = (±b/2, 0, . . . , 0). Even though the
metric in the future of the collision cannot be calculated exactly, it is possible
by extending a method [49] due to Penrose, to calculate a lower bound on the
area of the horizon that will form in the collision, and thus a lower bound on the
black hole cross-section. With this method Eardley and Giddings [49] were able to
show the existence of a marginally trapped surface S [57, 49, 50]. This implies the
existence of an apparent horizon, since this is the outermost marginally trapped
horizon. Then there either exists an apparent horizon outside the marginally
trapped surface found, or this is the apparent horizon. This apparent horizon
corresponds to that of a black hole in the future of the collision. They could then
use this marginally trapped surface S as a lower bound on the area of this black
hole horizon.
For an impact parameter b = 0 it could then be calculated that the area
[49, 50]
1
1 (D − 2)ΩD−2 D−3
D−2
ΩD−2 RSD
Atrap (S) =
2
4ΩD−3
D−2
,
(5.59)
≡ ΩD−2 RSD
2
where Atrap (S)
while
√ is the total area of the marginally trapped surface, √
RSD = RSD ( s) is the radius of a Schwarzschild black hole of mass s. The area
Atrap (S) may also be expressed as the area of a sphere, Atrap (S) = ΩD−2 r D−2 ,
where r ≤ RSD (MBH ). This leads to the restriction [49, 50]
1
1 (D − 2)ΩD−2 D−2 √
MBH ≥
s.
(5.60)
2
2ΩD−3
√
The lower bound on
√ the black hole mass [49] then ranges from 0.707 11 s for
D = 4 to 0.581 05 s for D = 11. That means that at least this much of the
center of mass energy of the process contributes to the black hole mass produced
in the collision.
For an impact parameter b 6= 0, Eardley and Giddings [49] could in D = 4
dimensions construct a marginally trapped surface and found the maximal impact
parameter for which the surface
would form. This maximal value
√
bmax,1 ≈ 3.219GN µ < RS4 ( s) could then be used to construct a lower limit on
the cross-section so that [49]
σ̂ ≥ πb2max,1 ≈ 32.552(GN µ)2 .
(5.61)
5.4 Other Calculations Concerning the Black Hole Cross-Section
111
With another approximative construction Kang and Nastase could later calculate
the maximal √
impact
in D = 4 dimensions to be
√ parameter √
bmax,2 = RS4 ( s)/ 2 = 4GN µ/ 2, which would give that [50]
σ̂ ≥ πb2max,2 = 8π(GN µ)2 ≈ 25.133(GN µ)2 .
(5.62)
Even though this alternative construction gives a more conservative estimate, it
enabled them to examine scenarios where D > 4 as well. The maximal impact
parameter for D > 4 could then be found to be [50]
b2max,2 = 2
(RSD )2
D−2
(D − 2) D−3
(D − 3)
(5.63)
and the cross-section
σ̂ ≥ πb2max,2 = 2π
(RSD )2
D−2
(D − 2) D−3
(D − 3).
(5.64)
In D = 5 dimensions this would give [50] bmax,2 = 0.952 3RSD and
σ̂ ≥ 11.40(G5 µ)2 .
These lower bounds on the black hole cross-section indicates that it may be
somewhat smaller than the geometric cross-section due to a black hole of mass
√
s. This reduction in the cross-section is often referred to as a mass ejection
correction [22].
Publications following the article by Eardley and Giddings [49] questioned
this classical description of the black hole production process. In analyses of the
process Rychkov [58, 59] found that the classical spacetime has a large curvature
in the region relevant to the horizon formation and concluded that quantum
effects must be taken into account. However, in an later article by Giddings and
Rychkov [51] a semi-classical analysis of the process was used to argue that the
geometric cross-section still remains a robust approximation.
5.4
Other Calculations Concerning the Black Hole
Cross-Section
In addition to the corrections already described, there are some other calculations
worth reviewing and commenting [22]. We will therefore end this section with a
brief description and discussion of these.
5.4.1
The Sub-Relativistic Limit
An argument for having a larger cross-section than our initial geometric crosssection, may be found by studying a sub-relativistic collision of two particles
112
Chapter 5. Black Hole Production
with rest mass about E = MBH /2. Such a correction to the cross-section is in
the literature called a sub-relativistic limit correction. The particles may then
be regarded as black holes with that rest mass. If now the impact parameter of
the process b ≤ 2RSD (MBH /2), the horizons of the particles will merge on impact
and a black hole of mass MBH is formed. In this picture the cross-section should
be [22]
D−4
2
(MBH ).
σ̂ & π [2RSD (MBH /2)]2 = 4 D−3 πRSD
(5.65)
For large D this lower bound will approach a value of four times the initial crosssection in Eq.(5.2). However, the situation on which this argument is based may
be greatly changed in the ultra-relativistic limit.
5.4.2
Eikonal-Approximation
One may alternatively use an eikonal approximation to calculate the cross-section.
One then calculate the eikonal probability amplitude [44] for scattering of any two
partons. In our transplanckian collision, the eikonal approximation
may be used
√
with small scattering angles. In this approximation, MD / s 1 and −t/s 1,
where t is the Lorentz invariant Mandelstam variable and −t/s is a measure of
the scattering angle in the center of mass frame.
The eikonal scattering amplitude may be found to be [44]
2
2
)+...
) + A1-loop (q⊥
Aeik = ABorn (q⊥
Z
= −2is d2 b⊥ eiq⊥ b⊥ eiχ − 1 ,
where the eikonal scattering phase
Z 2
1
d q⊥ iq⊥ b⊥
χ(b⊥ ) =
ABorn (q⊥ ),
e
2s (2π)2
(5.66)
(5.67)
q⊥ is the transverse momentum transfer and b⊥ is the impact parameter.
One may then directly express the elastic cross-section as [44]
Z
1
σ̂el =
d2 q⊥ |Aeik |2
16π 2 s2
Z
1
2
2 (2)
0
2
−Imχ
−Imχ
=
4s
(2π)
δ
(b
−
b
)
d
b
1
+
e
−
2e
cos(Reχ)
⊥
⊥
⊥
16π 2 s2
Z
= d2 b⊥ 1 + e−Imχ − 2e−Imχ cos(Reχ) ,
(5.68)
while the total cross-section for such a parton-parton scattering may from the
optical theorem be found as [44]
Z
ImAeik (0)
= 2 d2 b⊥ 1 − e−Imχ cos(Reχ) .
(5.69)
σ̂tot =
s
5.4 Other Calculations Concerning the Black Hole Cross-Section
113
Here Aeik (0) is the eikonal scattering amplitude in the forward direction and
Imχ = 0 indicates no absorption in the process. The absorption cross-section is
from Eq.(5.68) and Eq.(5.69) directly
Z
Z
2
−Imχ
σ̂abs = d b⊥ 1 − e
= 2π db b⊥ 1 − e−Imχ .
(5.70)
An absorption cross-section of a black hole producing process would have
Imχ 1 at impact parameters smaller than the critical impact parameter. And
the result would be an absorption cross-section similar to that of the previous
considered models.
5.4.3
Voloshin Exponential Suppression
The most questioned and controversial correction to a geometrical cross-section
is the Voloshin exponential suppression, proposed in two articles by Voloshin
[47, 48]. He argues there that in a collision process where a few particles interact
and the outcome is a black hole with a mass MBH MD , the cross-section should
be suppressed by an exponential factor dependent of the system’s entropy. From a
statistical argument, he viewed the black hole as an object realized by the number
bh
bh
of micro-states N = exp(SD
/kB ), determined by the entropy SD
. He did this
for a four-dimensional black hole, but it can easily be done in D-dimensions
where this number of micro-states would be determined by the D-dimensional
bh
entropy SD
= 2πkB MDD−2 AD−2 from Eq.(3.63). The probability to create a
black hole from a few colliding particles could then be found by summing the
probabilities to create a black hole in a given micro-state. The total probability
is then proportional to N . By CPT symmetry, each such probability is related
to the probability of the reverse process of a black hole in that state decaying
thermally into a few particles. The probability of such a decay is proportional
to exp(Ei /(kB TD )), where Ei is the energy of the produced particle and TD of
course the black hole temperature. The total probability of producing a black
hole from a few colliding particles is [47]
P (few particles → black hole) ∼ N P (black hole → few particles)
X
bh
∼ exp[SD
/kB −
Ei /(kB TD )].
(5.71)
i
The black hole decays until it disappears, thus Voloshin stated that
P
bh
i Ei /(kB TD ) = MBH /(kB TD ) = [(D − 2)/(D − 3)]SD /kB . And the probability
bh
becomes proportional to exp{−SD
/[kB (D − 3)]}. The black hole cross-section
would then be
2
bh
σ̂ ∼ πRSD
exp{−SD
/[kB (D − 3)]}.
(5.72)
114
Chapter 5. Black Hole Production
This is in agreement with what obtained in [22]. In a counterargument, Solodukhin [46] considered the fact that the decay of the black hole causes the temperature to change since the mass of the black hole decreases. In that instant
when the black hole of initial mass MBH has radiated particles with total energy
ω, the black hole temperature is
TD (ω) = (D − 3)/(4πkB RSD ). The probability can then be calculated to be [46]
bh Z MBH
dω
SD
−
P (few particles → black hole) ∼ exp
,
(5.73)
kB
kB TD (ω)
0
RM
bh
where 0 BH dω/(kB TD (ω)) = SD
/kB , and the exponent vanishes. It could then
be argued that this description of the process would not be correct when the
black hole decays into a few particles at once, and Voloshin’s method is then
correct. Solodukhin’s response to this was that such a decay could hardly be
called thermal and the thermal form of the probabilities should not be used at
all. This exponential suppression was also be discussed by use of path integrals,
but Voloshin’s results could also here be questioned.
Chapter 6
Conclusion
This thesis has been an introduction to black hole physics in D spacetime dimensions. This led us to a D-dimensional generalization of Einstein’s theory
of general relativity where the higher dimensional Schwarzschild solution of the
Einstein field equations, or the Schwarzschild-Tangherlini solution, was derived.
This became the basis for our study of higher dimensional black holes. In the
following, we derived the higher dimensional Schwarzschild metric in isotropic
coordinates and used this to show that Newtonian gravity is retrieved in the
weak field limit. Later on, we introduced the Planck mass which helped us in
the generalization of the Einstein-Hilbert action, where a fundamental Planck
mass entered. From this action we were able to retrieve the Einstein field equations and find the relation between the D-dimensional gravitational constant and
the fundamental Planck mass. Since the value of this fundamental Planck mass
is unknown, there is a possibility that it may be several orders of magnitude
smaller than the observed Planck mass. Because of the specific dependence of
the Schwarzschild radius on this fundamental Planck mass, a higher dimensional
black hole could be much larger than its four-dimensional relative. It was then
discussed how such a higher dimensional theory was compatible with results consistent with a four-dimensional one. This could be solved by having compact
extra dimensions. We then discussed three different methods to find how the
four-dimensional gravitational constant relates to the D-dimensional constant.
At the end our treatment of gravity and extra dimensions, it was evident that for
a black hole to be higher dimensional, its Schwarzschild radius is restricted to be
much smaller than the length of the extra dimensions. Additionally, for such a
black hole to be described semi-classically its mass is restricted to be much larger
than the fundamental Planck mass.
With this restriction on the higher dimensional black hole, we could find
its thermodynamical properties. The temperature and entropy of the higher
dimensional black hole were found. The results showed that a small black hole
is hotter than a larger one, which means that a higher dimensional black hole
is colder than a black hole in four dimensions of the same mass. This thermal
115
116
Chapter 6. Conclusion
property of the black hole suggested that it could be described as a blackbody
and a generalized Stefan-Boltzmann law was derived. We could then calculate
the lifetime of a radiating black hole, and found that with a fundamental Planck
mass of ∼ 1 TeV a black hole of mass ∼ 100 TeV which radiates onto a 3-brane
would have a lifespan of 10−25 s.
To understand the thermal behavior of such an object better, we studied
spontaneous pair production in an exterior field. This led to an improvement in
the resolution of Klein’s paradox, found in quantum field theory, by introducing
a more consistent definition of incoming and outgoing modes than those used in
previous discussions of the paradox. The framework used here could then be applied to our Schwarzschild spacetime and show that a D-dimensional black hole
evidently displays the thermodynamical properties found earlier. We also found
that the Planck spectrum observed at infinity is not a perfect blackbody spectrum, but a filtered, a so-called graybody spectrum. However, we were able to
show that with a geometric optics approximation, a black hole would act as a perfect blackbody with a radius somewhat larger than the black hole’s Schwarzschild
radius.
In the last part of this thesis, one property of a black hole producing process, the cross-section, was investigated. A first approximation of the black hole
2
cross-section was introduced, given by the geometrical value σ̂I = πRSD
. This
was followed by a discussion of possible corrections to this. The cross-section
2
could then be approximated to be σ̂ ∼ πRSD
. This cross-section indicates that
black hole production may become the dominant process at future high energy
accelerators.
In the future, an analysis of decay products from the produced black holes
would be appropriate. Additionally, a string theoretical analysis of black hole
production could be interesting. String corrections to the semi-classical black
hole cross-section could then be investigated. We may conclude by saying that
this is still an area where much work remains.
Appendix A
HEP-units (c = ~ = 1)
It is in physics practical to set certain quantities to be 1. But this also means
that other quantities get other dimensions than what we are used to. Let us see
what happens when we put c = ~ = 1. This is usual in High Energy Physics
(and the units we get are called HEP-units).
We have that c = 1 leads to
1 s = 299 792 458 m,
(A.1)
since c = 299 792 458 m/s in ordinary units. We now see that time and length
have the same dimension. This means that acceleration, as an example, no longer
has the dimension [length] · [time]−2 , but the dimension [time]−1 or [length]−1 .
Another quantity that changes dimension is mass,
[energy] = [mass]
[length]2
= [mass].
[time]2
(A.2)
We also have that ~ = 1 leads to
1 s−1 = 6.582 122 0 · 10−16 eV,
(A.3)
since ~ = 6.582 122 0 · 10−16 eVs in ordinary units. In other words, combined
with c = 1, we have
[energy] = [mass] = [length]−1 = [time]−1 .
(A.4)
So in HEP-units one may express the dimensions of all these quantities in
some power of [energy].
To obtain an expression where the correct powers of c and ~ are included, one
simply multiply the HEP-unit expression with a factor ~a cb , where the exponents
a and b may be determined by the correct dimension of the expression.
117
Appendix B
Alternative Definitions of the
Gravitational Constant
The Poisson equation which defines the gravitational constant may be generalized
in a number of ways. One example is [49]
D−3
2
∇d ΦD (r) ≡
8πG∗ ρ,
(B.1)
D−2
another is
∇2d ΦD (r) ≡ (D − 3)ΩD−2 G̃D ρ.
(B.2)
We see that also these generalizations of the equation is reduced to the usual
Poisson equation in four dimensions.
While our definition of the gravitational constant keeps the force law on a
familiar form in D dimensions, the generalization displayed in Eq.(B.1) keeps the
black hole entropy,
1
bh
SD
=
kB AD−2 ,
(B.3)
4G∗
on the same form in D dimensions as in four. And Eq.(B.2) keeps the expression
for the gravitational potential as
ΦD (r) = −
G̃D M
.
r D−3
(B.4)
In Table B.1, some of the quantities presented in this thesis is given with
use of different definitions of the gravitational constant. There also the relations
between each of the definitions are shown.
119
120 Appendix B. Alternative Definitions of the Gravitational Constant
Alt. 1
Alt. 2
D−3
D−2
Alt. 3
∇2d ΦD (r)
ΩD−2 GD ρ
ΦD (r)
GD M
− (D−3)r
D−3
8πG∗ M
− (D−2)Ω
D−3
D−2 r
DM
− G̃rD−3
FD (r)
DM
− GrD−2
(D−3)8πG∗ M
− (D−2)Ω
D−2
D−2 r
G̃D M
− (D−3)
r D−2
h
(2G̃D M ) D−3
2GD M
D−3
RSD
bh
SD
2π
ΩD−2
D−3
D−2
GD
G∗
G̃D
1
D−3
1
k A
GD B D−2
D−2
D−3
ΩD−2
8π
1
G
D−3 D
16πG∗ M
(D−2)ΩD−2
1
i D−3
1
k A
4G∗ B D−2
D−3
D−2
GD
8πG∗ ρ
GD
8π
G
ΩD−2 ∗
G∗
1
D−2
8π
G
ΩD−2 ∗
(D − 3)ΩD−2 G̃D ρ
1
2π
1
k A
ΩD−2 (D−2) GD B D−2
(D − 3)G̃D
(D − 2) ΩD−2
G̃D
8π
G̃D
Table B.1: Relations between some important quantities and the
different definitions of the gravitational constant. Alt. 1 corresponds
to our definition from Eq.(2.18), while Alt. 2 and Alt. 3 correspond
to Eq.(B.1) and Eq.(B.2), respectively.
Appendix C
The Surface Gravity of a Static
and Diagonal Metric
The surface gravity is an expression of the acceleration of gravity at the horizon of
a black hole. It may be calculated by use of the Killing vector which is orthogonal
to the horizon [60]. The horizon of a black hole is a null surface. That means
that any vector normal to the surface is a null vector. This again is a vector that
dotted with itself is zero. The Killing vector η = η µ eµ is in a static and diagonal
metric simply η = et = ∂/∂t. Since the Killing vector is normal to the horizon,
ηµ η µ = 0 and thus ηµ η µ is constant. So the gradient ∇α (ηµ η µ ) is also normal
to the horizon. In other words there exist a function κ, the surface gravity, that
fulfills
∇α (ηµ η µ ) = −2κη α .
(C.1)
It is now possible to express the surface gravity as [21]
1
κ2 = − (∇µ ην )(∇µ η ν ).
2
(C.2)
Since our metric is diagonal we have
η µ = δtµ
ηµ = gµα η α = gµα δtα .
and
(C.3)
The definition of covariant derivative is
∇µ ην ≡ ην,µ − Γανµ ηα .
(C.4)
Since ηt,r = gtt,r is the only nonzero ηµ,ν , the Christoffel symbols are symmetric
in the lower indices and by use of the Killing equation that states that
∇µ ην = −∇ν ηµ ,
121
(C.5)
122 Appendix C. The Surface Gravity of a Static and Diagonal Metric
we have
∇t ηr = ηr,t − Γtrt ηt = −(ηt,r − Γttr ηt ) = −∇r ηt .
(C.6)
Since ηr,t = 0 and Γtrt = 12 g tt gtt,r we get that ∇t ηr = 12 gtt,r . This makes us able to
calculate the surface gravity
r
r
1
1
(C.7)
κ = − (∇µ ην )(∇µ η ν ) = − g rr g tt (gtt,r )2 .
2
4
If we use the metric
ds2 = −f (r)dt2 +
dr 2
+ r 2 dΩ2D−2 ,
f (r)
(C.8)
at the horizon r = r+ we find that
1
1
= f 0 (r+ ).
κ = gtt,r 2
2
r=r+
(C.9)
Or for a Schwarzschild black hole in D dimensions the surface gravity becomes
κ=
D−3
.
2RSD
(C.10)
In four dimensions, the surface gravity then becomes κ = 1/(4GM ).
The surface gravity may also be calculated in terms of the D-acceleration and
D-velocity of a free particle. The surface gravity is then defined by
a
,
r→r+ ut
κ = lim
where a =
√
aµ aµ .
(C.11)
Appendix D
Calculations of Connection
Coefficients
Here follows the calculations of the connection coefficients in Eq.(2.105) and
Eq.(2.108), in Subsection 2.2.2, to the lowest order of ΦD (x). The connection
coefficients are calculated from
1
Γαµν = g αβ (gβµ,ν + gβν,µ − gµν,β ),
2
(D.1)
and
2
ΦD (x) (dx1 )2 + (dx2 )2 + · · · + (dxd )2 ,
ds = −[1 + 2ΦD (x)]dt + 1 −
D−3
(D.2)
2
2
Calculations Used in Eq.(2.105)
Γktt,k
!
=0
=0
1 kβ z}|{ z}|{
1 kβ
=
g ( gβt,t + gβt,t −gtt,β )
= − g gtt,β
2
2
,k
,k
1
2
=
ΦD (x) [1 + 2ΦD (x)],k
1+
2
D−3
,k
1
= 2ΦD (x),k = ΦD (x),kk ,
2
,k
Γktk,t =0,
123
(D.3)
(D.4)
124
Appendix D. Calculations of Connection Coefficients
Γρtt Γkρk
!
=0
=0
1 kγ
1 ρβ z}|{ z}|{
g ( gβt,t + gβt,t −gtt,β )
g (gγρ,k + gγk,ρ − gρk,γ )
=
2
2
1 ρβ
1 kk
= − g gtt,β
g (gkρ,k + gkk,ρ − gρk,k )
2
2
D−1
1 X jj
1 ρβ
kk
g gtt,j g kk gkk,j
= − g gtt,β g gkk,ρ = −
4
4 j=1
2
1
2
=−
1+
ΦD (x)
4
D−3
D−1
X
2
ΦD (x)
(−1)[1 + 2ΦD (x)],j 1 −
×
D
−
3
,j
j=1
D−1
1 X
=
[ΦD (x),j ]2 = 0
D − 3 j=1
=0
Γρtk Γkρt =
=0
z}|{ z}|{
1 ρβ
g (gβt,k + gβk,t − gtk,β )
2
=0
(D.5)
!
=0
1 kγ z}|{
g ( gγρ,t +gγt,ρ − gρt,γ )
2
z}|{
1
1
= [g ρβ gβt,k ][g kk ( gkt,k −gρt,k )] = − g tt g kk gtt,k gtt,k
4
4
2
1
= [1 − 2ΦD (x)] 1 +
ΦD (x) [−(1 + 2ΦD (x)),k ]2
4
D−3
= [ΦD (x),k ]2 = 0
!
(D.6)
Calculations Used in Eq.(2.108)
Γikk,i
1 iβ
=
g (gβk,k + gβk,k − gkk,β )
2
,i
!
=0
=0
1 ii z}|{ z}|{
g ( gik,k + gik,k −gkk,i )
=
2
,i
"
#
1
2
2
= −
ΦD (x)
1−
ΦD (x)
1+
2
D−3
D−3
,i
1
=
ΦD (x),ii
D−3
,i
(D.7)
125
Γiki,k
!
=0
1 iβ
z}|{
=
g (gβki + gβik − gki,β )
2
,k
!
=0
1 ii z}|{
g ( gik,i +gii,k )
=
2
,k
" #
2
1
2
=
ΦD (x)
1−
ΦD (x)
1+
2
D−3
D−3
,k
,k
1
ΦD (x),kk
=−
D−3
Γρkk Γiρi
=
(D.8)
1 ρβ
g (gβk,k + gβk,k − gkk,β )
2
1 iγ
g (gγρ,i + gγi,ρ − gρi,γ )
2
1
= [g ρβ (2gβk,k − gkk,β )][g ii (giρ,i + gii,ρ − gρi,i )]
4
D−1
1 X ii
[g gii,j ][g jj (2gjk,k − gkk,j )]
=
4 j=1
X
1
= g ii [g kk gii,k gkk,k −
g jj gkk,j gii,j ]
4
j6=k
2
2
1
1+
ΦD (x)
=
4
D−3
"
2 X 2 #
2
2
×
ΦD (x),k −
ΦD (x),j
=0
D−3
D−3
j6=k
=0
Γρki Γiρk
=
z}|{
1 ρβ
g (gβk,i + gβi,k − gki,β )
2
!
1 iγ
g (gγρ,k + gγk,ρ − gρk,γ )
2
(D.9)
=0
1
z}|{
= [g ρβ (gβk,i + gβi,k )][g ii (giρ,k + gik,ρ −gρk,i )]
4
1
= [−g kk gkk,ig ii gkk,i + g ii gii,k g ii gii,k ]
4
2 2
1
2
2
=−
1+
ΦD (x)
−
ΦD (x),i
4
D−3
D−3
2
1
2
+
1+
ΦD (x) [−2ΦD (x),k ]2 = 0
4
D−3
(D.10)
Appendix E
Scalar Products for Bosons
In the following sections we have done the complete calculations of the inner
products listed i Eq.(4.73). We will use the definition of the inner product for
bosons given in Eq.(4.71). This equation define the scalar product between u and
v as
(u, v) ≡ i
↔
Z
∞
↔
dz u∗ ∇0 v,
(E.1)
−∞
↔
where ∇0 ≡ (∂0 + 2ieV (z)). In our case the potential is a step-function defined
in Eq.(4.15).
In the calculations that follow we will need to use mainly two known results.
The first comes from complex analysis and states that
Z
∞
dxeikx =
0
i
1
= iP( ) + πδ(k),
k + iε
k
(E.2)
where P indicates the principal branch of 1/k and ε is a positive infinitesimal
number. This principal branch may be chosen freely and is in our calculations
chosen to be zero. The second result states that a δ-function
∂g(x) δ(g(x)),
δ(x) = ∂x where g(x) is some function of x.
127
(E.3)
128
Appendix E. Scalar Products for Bosons
The Inner Product (p1, p01)
To calculate the scalar product (p1 , p01 ) we use the expression for p1 from Eq.(4.55).
With use of the definition from Eq.(E.1) this then becomes
(p1 , p01 )
Z
∞
Z
↔
p∗1 ∇0 p01
=i
−∞
Z 0
Z
↔
∗
0
dz p1 ∂0 p1 + i
=i
=i
dz
−∞
0
∞
↔
dz p∗1 (∂0 + 2ieV (z))p01
−∞
∞
↔
(E.4)
dz p∗1 (∂0 + 2ieV )p01
=I1 + I2
For simplicity we have divided the inner product into two separate integrals, I1
and I2 . Let us calculate the first integral,
I1 =i
Z
Z
0
dz
−∞
0
↔
p∗1 ∂0 p01
=i
Z
0
dz
−∞
↔
1
(K1 P1∗ + K2 P2∗ ) ∂0 (K10 P10 + K20 P20 )
4
1 h
0
0
0
dz √ 0 (K1 e−ipz + K2 eipz )eiEt ∂0 (K10 eip z + K20 e−ip z )e−iE t
=i
4 pp
−∞
i
0 ip0 z
0 −ip0 z −iE 0 t
−ipz
ipz iEt
− (K1 e + K2 e
)e
∂0 (K1 e
+ K2 e )e
Z 0
1 h
0
0
0
dz √ 0 (K1 e−ipz + K2 eipz )eiEt (−iE 0 )(K10 eip z + K20 e−ip z )e−iE t
=i
4 pp
−∞
i
0 ip0 z
0 −ip0 z −iE 0 t
−ipz
ipz iEt
− (K1 e + K2 e
)e
iE(K1 e
+ K2 e )e
Z
0
E0 + E
0
0
0
= √ 0 ei(E−E )t
dz (K1 e−ipz + K2 eipz )(K10 eip z + K20 e−ip z )
4 pp
−∞
Z 0
Z 0
0
E + E i(E−E 0 )t
0
0
i(p0 −p)z
0
= √ 0e
K1 K1
dz e
+ K 1 K2
dz e−i(p +p)z
4 pp
−∞
−∞
Z 0
Z 0
0
0
+K2 K10
dz ei(p +p)z + K2 K20
dz ei(p−p )z
−∞
−∞
Z ∞
Z ∞
E 0 + E i(E−E 0 )t
0
0
i(p −p)z
0
0
dz e−i(p +p)z
dz e
+ K 1 K2
K1 K1
= √ 0e
4 pp
0
Z ∞
Z ∞
0
0
i(p +p)z
0
i(p−p0 )z
+K2 K1
dz e
+ K 2 K2
dz e
,
(E.5)
0
0
where we have introduced
1−α
K1 = √ =
α
r
p
−
q
r
q
p
and
1+α
K2 = √ =
α
r
p
+
q
r
q
.
p
(E.6)
129
We may now use the result from complex analysis shown in Eq.(E.2) to simplify
this expression for I1 . Use of this result leads to
i
E 0 + E i(E−E 0 )t
i
+ K1 K20
I1 = √ 0 e
K1 K10 0
p − p + iε
−p − p0
4 pp
i
i
+ K2 K10
+ K2 K20 0
0
p+p
p − p + iε
E = π K12 δ(p0 − p) + iK1 K2 δ(p + p0 ) − iK2 K1 δ(p + p0 ) + K22 δ(p − p0 )
2p
E = π K12 + K22 δ(p − p0 ).
(E.7)
2p
With use of the expressions for K1 and K2 from Eq.(E.6) and we get
" r
r 2 r
r 2 #
p
q
p
q
E
−
+
δ(p − p0 )
+
I1 = π
2p
q
p
q
p
E
p q
p q
0
= π
δ(p − p ) = π
δ(E − E 0 ),
(E.8)
+
+
p
q p
q p
p
where we have used the result from Eq.(E.3) and E = p2 + m2 in the last
transition.
Calculation of the second integral from Eq.(E.4) does not involve the same
amount of work, since it does not contain linear combinations of different wave
solutions. The second integral is calculated to be
Z ∞
Z ∞
↔
↔
∗
0
I2 =i
dz p1 (∂0 + 2ieV )p1 = i
dz N2∗ (∂0 + 2ieV )N20
0
Z0 ∞
i
1 h iqz+iEt ↔ −iq0 z−iE 0 t
0
0
=i
dz √ 0 e
∂0 e
+ 2ieV eiqz+iEt e−iq z−iE t
qq
0
Z ∞
1
0
0
=√ 0 i
dz [(−iE 0 ) − iE + 2ieV ] ei(q−q )z ei(E−E )t
qq 0
Z
(E − eV ) + (E 0 − eV ) i(E−E 0 )t ∞ i(q−q0 )z
√ 0
e
dz e
=
qq
0
2(E − eV )
πδ(q − q 0 ).
(E.9)
=
q
Here we have used Eq.(E.2) in the last line, in a similar manner as done in the
calculationp
of the first integral I1 . If we now use the result from Eq.(E.3) and
that E = q 2 + m2 + eV , we find that δ(q − q 0 ) = [−q/(E − eV )]δ(E − E 0 ).
The minus sign in front of the momentum q originates from the fact that in our
situation (E − eV ) < 0. This gives us that the second integral in the end becomes
I2 = −2πδ(E − E 0 ).
(E.10)
130
Appendix E. Scalar Products for Bosons
Now we only need to add the two integrals I1 and I2 to have the final inner
product (p1 , p01 ). This becomes
(p1 , p01 )
=I1 + I2 = π
=π
p q
+
q p
δ(E − E 0 ) − 2πδ(E − E 0 )
(p − q)2
δ(E − E 0 ).
pq
(E.11)
We see that the norm of p1 is unnormalized.
The Inner Product (p2, p02)
To calculate the scalar product (p2 , p02 ) we use the expression for p2 from Eq.(4.56).
With use of the definition from Eq.(E.1), this then becomes
(p2 , p02 )
=i
Z
∞
dz
↔
p∗2 ∇0 p02
Z
=i
−∞
Z 0
Z
↔
∗
0
dz p2 ∂0 p2 + i
=i
−∞
0
∞
↔
dz p∗2 (∂0 + 2ieV (z))p02
−∞
∞
↔
dz p∗2 (∂0 + 2ieV )p02
(E.12)
=I1 + I2
Let us calculate the first integral,
I1 =i
Z
Z
0
↔
dz p∗2 ∂0 p02
−∞
0
↔
1
dz (K2 P1∗ + K1 P2∗ )∂0 (K20 P10 + K10 P20 )
4
−∞
Z
h
0
E + E i(E−E 0 )t 0
0
0
= √ 0e
dz K2 K20 ei(p −p)z + K2 K10 e−i(p+p )z
4 pp
−∞
i
0
0 i(p+p0 )z
+K1 K2 e
+ K1 K10 ei(p−p )z
E 0 + E i(E−E 0 )t
i
i
= √ 0e
K2 K20 0
+ K2 K10
p − p + iε
−p − p0
4 pp
i
i
+ K1 K10
+K1 K20
0
p+p
p − p0 + iε
E
π
=π [K22 + K12 ]δ(p − p0 ) = [K22 + K12 ]δ(E − E 0 )
2p
2
p q
δ(E − E 0 ),
=π
+
q p
=i
(E.13)
131
where we have used Eq.(E.2), Eq.(E.3) and the expressions for K1 and K2 from
Eq.(E.6). We calculate the second integral in the same manner,
I2 =i
Z
∞
0
dz
Z
↔
p∗2 (∂0
+ 2ieV
)p02
=i
Z
∞
↔
dz N1∗ (∂0 + 2ieV )N10
0
∞
i
0
0
=√ 0
dz e−iqz+iEt [(−iE 0 ) − iE + 2ieV ]e−ip z−iE t
qq 0
Z
(E − eV ) + (E 0 − eV ) i(E−E 0 )t ∞ i(q0 −q)z 2(E − eV )
√ 0
dz e
=
πδ(q − q 0 )
=
e
q
qq
0
0
= − 2πδ(E − E ),
(E.14)
where we have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two last
transitions. The total inner product then becomes
(p2 , p02 ) =I1 + I2 = π
(p − q)2
δ(E − E 0 ).
pq
(E.15)
We see that this inner product is (p2 , p02 ) = (p1 , p01 ), as would be expected.
The Inner Product (n1, n01)
To calculate the scalar product (n1 , n01 ) we use the expression for n1 from Eq.(4.57).
With use of the definition from Eq.(E.1), this then becomes
(n1 , n01 )
=i
Z
∞
dz
Z
↔
n∗1 ∇0 n01
=i
−∞
Z 0
Z
↔
∗
0
=i
dz n1 ∂0 n1 + i
−∞
0
∞
↔
dz n∗1 (∂0 + 2ieV (z))n01
−∞
∞
↔
dz n∗1 (∂0 + 2ieV )n01
(E.16)
=I1 + I2
Let us calculate the first integral,
Z
0
↔
n∗1 ∂0 n01
Z
0
↔
dz P2∗ ∂0 P20
dz
=i
−∞
−∞
Z 0
i
0
0
=√ 0
dz eipz+iEt [(−iE 0 ) − iE]e−ip z−iE t
pp −∞
Z
0
E + E i(E−E 0 )t 0
2E
0
πδ(p − p0 )
= √ 0 e
dz ei(p−p )z =
p
pp
−∞
=2πδ(E − E 0 ),
I1 =i
(E.17)
132
Appendix E. Scalar Products for Bosons
where we have used Eq.(E.2) and Eq.(E.3) in the two last transitions. We calculate the second integral in the same manner,
I2 =i
Z
∞
↔
dz n∗1 (∂0 + 2ieV )n01
Z0 ∞
↔
1
dz (−K1 N1∗ + K2 N2∗ )(∂0 + 2ieV )(−K10 N10 + K20 N20 )
=i
4
0
Z
(E − eV ) + (E 0 − eV ) i(E−E 0 )t ∞ h
0
0
√ 0
=
e
dz K1 K10 ei(q −q)z − K1 K20 e−i(q+q )z
4 qq
i0
0
0
−K2 K10 ei(q+q )z + K2 K20 ei(q−q )z
i
(E − eV ) + (E 0 − eV ) i(E−E 0 )t
i
√ 0
K1 K10 0
e
− K1 K20
=
q − q + iε
−q − q 0
4 qq
i
i
+ K2 K20
−K2 K10
0
q+q
q − q 0 + iε
E − eV 2
π
=π
[K1 + K22 ]δ(q − q 0 ) = − [K12 + K22 ]δ(E − E 0 )
2q
2
p q
δ(E − E 0 ),
(E.18)
=−π
+
q p
where we have used Eq.(E.2), Eq.(E.3), the expressions for K1 and K2 from
Eq.(E.6) and that (E − eV ) < 0. The total inner product then becomes
(n1 , n01 ) =I1 + I2 = −π
(p − q)2
δ(E − E 0 ).
pq
(E.19)
We see that this inner product is (n1 , n01 ) = −(p1 , p01 ). And that the norm of the
unnormalized n1 is negative.
The Inner Product (n2, n02)
To calculate the scalar product (n2 , n02 ) we use the expression for n2 from Eq.(4.58).
With use of the definition from Eq.(E.1), this then becomes
(n2 , n02 )
Z
∞
Z
=i
−∞
Z 0
Z
↔
∗
0
=i
dz n2 ∂0 n2 + i
=i
dz
↔
n∗2 ∇0 n02
−∞
=I1 + I2
0
∞
↔
dz n∗2 (∂0 + 2ieV (z))n02
−∞
∞
↔
dz n∗2 (∂0 + 2ieV )n02
(E.20)
133
Let us calculate the first integral,
Z 0
Z 0
↔
↔
∗
0
I1 =i
dz n2 ∂0 n2 = i
dz P1∗ ∂0 P10
−∞
−∞
Z 0
i
0
0
dz e−ipz+iEt [(−iE 0 ) − iE]eip z−iE t
=√ 0
pp −∞
Z
0
E + E i(E−E 0 )t 0
2E
0
= √ 0 e
πδ(p − p0 )
dz ei(p −p)z =
p
pp
−∞
=2πδ(E − E 0 ),
(E.21)
where we have used Eq.(E.2) and Eq.(E.3) in the two last transitions. We calculate the second integral in the same manner,
Z ∞
↔
I2 =i
dz n∗2 (∂0 + 2ieV )n02
Z0 ∞
↔
1
=i
dz (K2 N1∗ − K1 N2∗ )(∂0 + 2ieV )(K20 N10 − K10 N20 )
4
0
Z
(E − eV ) + (E 0 − eV ) i(E−E 0 )t ∞ h
0
0
√ 0
dz K2 K20 ei(q −q)z − K2 K10 e−i(q+q )z
e
=
4 qq
i0
0 i(q+q 0 )z
0 i(q−q 0 )z
−K1 K2 e
+ K 1 K1 e
(E − eV ) + (E 0 − eV ) i(E−E 0 )t
i
i
√ 0
=
− K2 K10
e
K2 K20 0
q − q + iε
−q − q 0
4 qq
i
i
+ K1 K10
−K1 K20
0
q+q
q − q 0 + iε
E − eV 2
π
=π
[K2 + K12 ]δ(q − q 0 ) = − [K22 + K12 ]δ(E − E 0 )
2q
2
p q
+
δ(E − E 0 ),
(E.22)
=−π
q p
where we have used Eq.(E.2), Eq.(E.3), the expressions for K1 and K2 from
Eq.(E.6) and that (E − eV ) < 0. The total inner product then becomes
(n2 , n02 ) =I1 + I2 = −π
(p − q)2
δ(E − E 0 ).
pq
(E.23)
We see that this inner product is (n2 , n02 ) = (n1 , n01 ) = −(p1 , p01 ) = −(p2 , p02 ).
The Inner Product (p1, n01)
To calculate the scalar product (p1 , n01 ) we use the expressions for p1 and n1 from
Eq.(4.55) and Eq.(4.57). With use of the definition from Eq.(E.1), this then
134
Appendix E. Scalar Products for Bosons
becomes
(p1 , n01 )
=i
Z
∞
dz
↔
p∗1 ∇0 n01
Z
=i
−∞
Z
Z 0
↔
0
∗
dz p1 ∂0 n1 + i
=i
−∞
0
∞
↔
dz p∗1 (∂0 + 2ieV (z))n01
−∞
∞
↔
dz p∗1 (∂0 + 2ieV )n01
(E.24)
=I1 + I2
Let us calculate the first integral,
I1 =i
Z
Z
0
dz
−∞
0
↔
p∗1 ∂0 n01
Z
0
↔
1
(K1 P1∗ + K2 P2∗ ) ∂0 P20
2
−∞
↔
0
0
K1 e−ipz + K2 eipz eiEt ∂0 e−ip z−iE t
=i
dz
1
dz √ 0
2 pp
−∞
Z 0
1
0
0
=i
dz √ 0 K1 e−ipz + K2 eipz eiEt [(−iE 0 ) − iE]e−ip z−iE t
2 pp
−∞
Z 0
Z 0
0
E + E i(E−E 0 )t
i(p−p0 )z
−i(p+p0 )z
dz e
= √ 0e
dz e
+ K2
K1
2 pp
−∞
−∞
Z ∞
Z ∞
E + E 0 i(E−E 0 )t
i(p−p0 )z
−i(p+p0 )z
= √ 0e
dz e
dz e
+ K2
K1
2 pp
0
0
i
E + E 0 i(E−E 0 )t
i
+ K2
= √ 0e
K1
−p − p0
p − p0 + iε
2 pp
E
= π[iK1 δ(p + p0 ) + K2 δ(p − p0 )]
p
=π[iKδ(E + E 0 ) + K2 δ(E − E 0 )],
(E.25)
=i
where we have used Eq.(E.2) and Eq.(E.3) in the two last transitions. We calculate the second integral in the same manner,
Z ∞
Z ∞
↔
↔
∗
0
dz N2∗ (∂0 + 2ieV )(−K10 N10 + K20 N20 )
dz p1 (∂0 + 2ieV )n1 = i
I2 =i
0
Z0 ∞
i
1 h iqz+iEt ↔
0
0
0
dz √ 0 e
=i
(∂0 + 2ieV )(−K10 eiq z + K20 e−iq z )e−iE t
2 qq
Z0 ∞
1
0
0
0
=i
dz √ 0 eiqz+iEt [(−iE 0 ) − iE + 2ieV ](−K10 eiq z + K20 e−iq z )e−iE t
2 qq
0
Z ∞
Z ∞
(E − eV ) + (E 0 − eV ) i(E−E 0 )t
0
i(q+q 0 )z
0
i(q−q 0 )z
√
−K1
dz e
+ K2
dz e
=
e
2 qq 0
0
0
E − eV
=
π[iK1 δ(q + q 0 ) + K2 δ(q − q 0 )]
q
=π[−iK1 δ(E + E 0 ) − K2 δ(E − E 0 )],
(E.26)
135
where we again have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two
last transitions.
The total inner product then becomes
(p1 , n01 ) =I1 + I2 = 0.
(E.27)
The Inner Product (p1, p02)
To calculate the scalar product (p1 , p02 ) we use the expressions for p1 and p2 from
Eq.(4.55) and Eq.(4.56). With use of the definition from Eq.(E.1), this then
becomes
(p1 , p02 )
=i
Z
∞
dz
↔
p∗1 ∇0 p02
Z
=i
−∞
Z
Z 0
↔
0
∗
=i
dz p1 ∂0 p2 + i
−∞
0
∞
↔
dz p∗1 (∂0 + 2ieV (z))p02
−∞
∞
↔
dz p∗1 (∂0 + 2ieV )p02
(E.28)
=I1 + I2
Let us calculate the first integral,
Z
Z
0
↔
1
=i
dz (K1 P1∗ + K2 P2∗ )∂0 (K20 P10 + K10 P20 )
4
−∞
−∞
Z 0
0
E +E
0
0
0
= √ 0 ei(E−E )t
dz (K1 e−ipz + K2 eipz )(K20 eip z + K10 e−ip z )
4 pp
−∞
Z 0
Z 0
0
E + E i(E−E 0 )t
0
0
i(p−p0 )z
0
K1 K2
dz e
+ K 1 K1
dz e−i(p+p )z
= √ 0e
4 pp
−∞
−∞
Z 0
Z 0
0
0
+ K2 K20
dz ei(p+p )z + K2 K10
dz ei(p−p )z
−∞
−∞
i
i
E 0 + E i(E−E 0 )t
+ K1 K10 0
K1 K20 0
= √ 0e
p − p + iε
−p − p
4 pp
i
i
+ K2 K20 0
+ K2 K10
p +p
p − p0 + iε
E = π 2K1 K2 δ(p − p0 ) + (iK12 − iK22 )δ(p + p0 )
2p
E
= π [K1 K2 δ(p − p0 ) − 2iδ(p + p0 )]
p
=π[K1 K2 δ(E − E 0 ) − 2iδ(E + E 0 )],
(E.29)
I1 =i
0
dz
↔
p∗1 ∂0 p02
136
Appendix E. Scalar Products for Bosons
where we have used Eq.(E.2), Eq.(E.3) and that K12 − K22 = 2, in the two last
transitions. We calculate the second integral in the same manner,
I2 =i
Z
∞
0
dz
Z
↔
p∗1 (∂0
+ 2ieV
)p02
=i
Z
∞
0
↔
dz N2∗ (∂0 + 2ieV )N10
∞
↔
i
0
dz eiqz+iEt (∂0 + 2ieV )eiq z−iEt
=√ 0
qq 0
Z ∞
i
0
=√ 0
dz eiqz+iEt ((−iE 0 ) − iE + 2ieV )eiq z−iEt
qq 0
Z
(E − eV ) + (E 0 + eV ) i(E−E 0 )t ∞ i(q+q0 )z
√ 0
dz e
e
=
qq
0
2(E − eV )
πδ(q + q 0 ) = i2πδ(E + E 0 ),
=−i
q
(E.30)
where we have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two last
transitions. With use of the expressions for K1 and K2 from Eq.(E.6), the total
inner product then becomes
(p1 , p02 ) =I1 + I2 = πK1 K2 δ(E − E 0 )
r
r r
r p
q
p
q
=π
δ(E − E 0 )
−
+
q
p
q
p
p2 − q 2
p q
−
δ(E − E 0 ).
δ(E − E 0 ) = π
=π
q p
pq
(E.31)
The Inner Product (p1, n02)
To calculate the scalar product (p1 , n02 ) we use the expressions for p1 and n2 from
Eq.(4.55) and Eq.(4.58). With use of the definition from Eq.(E.1), this then
becomes
(p1 , n02 )
Z
∞
Z
=i
−∞
Z 0
Z
↔
∗
0
=i
dz p1 ∂0 n2 + i
=i
dz
↔
p∗1 ∇0 n02
−∞
=I1 + I2
0
∞
↔
dz p∗1 (∂0 + 2ieV (z))n02
−∞
∞
↔
dz p∗1 (∂0 + 2ieV )n02
(E.32)
137
Let us calculate the first integral,
Z 0
Z 0
↔
↔
1
∗
0
I1 =i
dz p1 ∂0 n2 = i
dz (K1 P1∗ + K2 P2∗ )∂0 P10
2
−∞
−∞
Z 0
i
0
0
= √ 0
dz (K1 e−ipz + K2 eipz )eiEt [(−iE 0 ) − iE]eip z−iE t
2 pp −∞
Z 0
Z 0
E 0 + E i(E−E 0 )t
i(p+p0 )z
i(p0 −p)z
dz e
= √ 0e
dz e
+ K2
K1
2 pp
−∞
−∞
E
=π [K1 δ(p − p0 ) − iK2 δ(p + p0 )]
p
=π[K1 δ(E − E 0 ) − iK2 δ(E + E 0 )],
(E.33)
where we have used Eq.(E.2) and Eq.(E.3) in the two last transitions. We calculate the second integral in the same manner,
Z ∞
Z ∞
↔
↔
∗
0
I2 =i
dz p1 (∂0 + 2ieV )n2 = i
dz N2∗ (∂0 + 2ieV )(K20 N10 − K10 N20 )
0
0
Z
E + E 0 − 2eV i(E−E 0 )t ∞
0
0
√ 0
dz (K20 ei(q+q )z − K10 ei(q−q )z )
e
=
2 qq
0
E − eV
π[−iK2 δ(q + q 0 ) − K1 δ(q − q 0 )]
=
q
=π[iK2 δ(E + E 0 ) + K1 δ(E − E 0 )],
(E.34)
where we have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two last
transitions.With use of the expressions for K1 and K2 from Eq.(E.6), the total
inner product then becomes
r
r p
q
0
0
−
(p1 , n2 ) =I1 + I2 = 2πK1 δ(E − E ) = 2π
δ(E − E 0 )
q
p
p−q
=2π √ δ(E − E 0 ).
(E.35)
pq
The Inner Product (n1, p02)
To calculate the scalar product (n1 , p02 ) we use the expression for n1 and p2 from
Eq.(4.57) and Eq.(4.56). With use of the definition from Eq.(E.1), this then
becomes
Z ∞
Z ∞
↔
↔
∗
0
0
dz n∗1 (∂0 + 2ieV (z))p02
dz n1 ∇0 p2 = i
(n1 , p2 ) =i
−∞
−∞
Z 0
Z ∞
↔
↔
(E.36)
=i
dz n∗1 ∂0 p02 + i
dz n∗1 (∂0 + 2ieV )p02
−∞
=I1 + I2
0
138
Appendix E. Scalar Products for Bosons
Let us calculate the first integral,
Z 0
Z 0
↔1
↔
∗
0
dz P2∗ ∂0 (K20 P10 + K10 P20 )
dz n1 ∂0 p2 = i
I1 =i
2
−∞
−∞
Z 0
0
E +E
0
0
0
dz [K20 ei(p+p )z + K10 ei(p−p )z ]
= √ 0 ei(E−E )t
2 pp
−∞
E
=π [−iK2 δ(p + p0 ) + K1 δ(p − p0 )]
p
=π[−iK2 δ(E + E 0 ) + K1 δ(E − E 0 )],
(E.37)
where we have used Eq.(E.2) and Eq.(E.3) in the two last transitions. We calculate the second integral in the same manner,
Z ∞
↔
I2 =i
dz n∗1 (∂0 + 2ieV )p02
Z0 ∞
↔
1
=i
dz (−K1 N1∗ + K2 N2∗ )(∂0 + 2ieV )N10
2
0
Z
(E − eV ) + (E 0 − eV ) i(E−E 0 )t ∞
0
0
√ 0
e
dz [−K1 ei(q −q)z + K1 ei(q+q )z ]
=
2 qq
0
E − eV
=π
[−K1 δ(q − q 0 ) − iK2 δ(q + q 0 )]
q
=π[K1 δ(E − E 0 ) + iK2 δ(E + E 0 )],
(E.38)
where we have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two last
transitions. With use of the expressions for K1 and K2 from Eq.(E.6), the total
inner product then becomes
(n1 , p02 ) =I1 + I2 = 2πK1 δ(E − E 0 )
r
r p−q
p
q
δ(E − E 0 ) = 2π √ δ(E − E 0 ).
−
=2π
q
p
pq
(E.39)
The Inner Product (n1, n02)
To calculate the scalar product (n1 , n02 ) we use the expression for n1 and n2 from
Eq.(4.57) and Eq.(4.58). With use of the definition from Eq.(E.1), this then
becomes
Z ∞
Z ∞
↔
↔
∗
0
0
dz n∗1 (∂0 + 2ieV (z))n02
dz n1 ∇0 n2 = i
(n1 , n2 ) =i
−∞
−∞
Z 0
Z ∞
↔
↔
(E.40)
=i
dz n∗1 ∂0 n02 + i
dz n∗1 (∂0 + 2ieV )n02
−∞
=I1 + I2
0
139
Let us calculate the first integral,
I1 =i
Z
0
dz
−∞
0
↔
n∗1 ∂0 n02
Z
=i
Z 0
0
↔
dz P2∗ ∂0 P10
−∞
E +E
0
0
= √ 0 ei(E−E )t
dz ei(p+p )z
pp
−∞
2E
= π(−i)δ(p + p0 ) = −i2πδ(E + E 0 ),
p
(E.41)
where we have used Eq.(E.2) and Eq.(E.3) in the two last transitions. We calculate the second integral in the same manner,
Z ∞
↔
dz n∗1 (∂0 + 2ieV )n02
I2 =i
Z0 ∞
↔
1
=i
dz (−K1 N1∗ + K2 N2∗ )(∂0 + 2ieV )(K20 N10 − K10 N20 )
4
0
Z ∞
(E − eV ) + (E 0 − eV ) i(E−E 0 )t
0
0
√ 0
=
e
i
dz [−K1 K20 ei(q −q)z + K1 K10 e−i(q+q )z
4 qq
0
0 i(q+q 0 )z
0 i(q−q 0 )z
+ K 2 K2 e
− K 2 K1 e
]
E − eV
π[−2K1 K2 δ(q − q 0 ) + iK12 δ(q + q 0 ) − iK22 δ(q + q 0 )]
=
2q
=π[K1 K2 δ(E − E 0 ) + 2iδ(E + E 0 )],
(E.42)
where we have used Eq.(E.2), Eq.(E.3), that (E −eV ) < 0 and that K12 −K22 = 2,
in the two last transitions. With use of the expressions for K1 and K2 from
Eq.(E.6), the total inner product then becomes
(n1 , n02 ) =πK1 K2 δ(E − E 0 ) = π
p2 − q 2
δ(E − E 0 ).
pq
(E.43)
The Inner Product (n2, p02)
To calculate the scalar product (n2 , p02 ) we use the expression for n2 and p2 from
Eq.(4.58) and Eq.(4.56). With use of the definition from Eq.(E.1), it then becomes
Z ∞
Z ∞
↔
↔
∗
0
0
dz n∗2 (∂0 + 2ieV (z))p02
dz n2 ∇0 p2 = i
(n2 , p2 ) =i
−∞
−∞
Z 0
Z ∞
↔
↔
(E.44)
=i
dz n∗2 ∂0 p02 + i
dz n∗2 (∂0 + 2ieV )p02
−∞
=I1 + I2
0
140
Appendix E. Scalar Products for Bosons
Let us calculate the first integral,
Z 0
Z 0
↔
1 ↔
∗
0
dz P1∗ ∂0 (K20 N10 + K10 N20 )
dz n2 ∂0 p2 = i
I1 =i
2
−∞
−∞
Z
0
E0 + E
0
0
0
= √ 0 ei(E−E )t
dz [K20 ei(p −p)z + K10 e−i(p+p )z ]
2 pp
−∞
0
i
i
E + E i(E−E 0 )t
0
0
+ K1
= √ 0e
K2 0
p − p + iε
−p − p0
2 pp
E
= π[K2 δ(p − p0 ) + iK1 δ(p + p0 )]
p
=π[K2 δ(E − E 0 ) + iK1 δ(E + E 0 )],
(E.45)
where we have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two last
transitions. We calculate the second integral in the same manner,
Z ∞
Z ∞
↔
↔
1
∗
0
dz (K2 N1∗ − K1 N2∗ )(∂0 + 2ieV )N10
dz n2 (∂0 + 2ieV )p2 = i
I2 =i
2
0
Z0 ∞
1
0
0
=i
dz √ 0 K2 e−iqz − K1 eiqz eiEt [(−iE 0 ) − iE + 2ieV ]eiq z−iE t
2 qq
0
Z ∞
Z ∞
(E − eV ) + (E 0 − eV ) i(E−E 0 )t
i(q 0 −q)z
i(q+q 0 )z
√
=
K2
e
dz e
− K1
dz e
2 qq 0
0
0
(E − eV ) + (E 0 − eV ) i(E−E 0 )t
i
i
√
=
− K1
K2
e
q − q 0 + iε
q + q0
2 qq 0
E − eV
π[K2 δ(q − q 0 ) + iK1 δ(q + q 0 )]
=
q
=π[−K2 δ(E − E 0 ) − K1 δ(E + E 0 )],
(E.46)
where we again have used Eq.(E.2), Eq.(E.3) and that (E − eV ) < 0 in the two
last transitions.
The total inner product then becomes
(n2 , p02 ) =I1 + I2 = 0.
(E.47)
Appendix F
Scalar Products in the
Schwarzschild Geometry
In the following sections we have calculated the inner products needed in Section
4.2. In curved spacetime we have from Eq.(4.141) defined the inner product as
(φ1 , φ2 ) = i
Z
↔
(φ∗1 ∂ U φ2 )r D−2 dU dΩD−2 ,
(F.1)
↔
where A ∂ U B ≡ A∂U B − [∂U B]A. In the following calculations we will make use
of the particle modes
uωlm =
wωlm =
√ 1
(−U )iω2RSD /(D−3) Ylm (χD−2 , . . . , χ1 ),
4πωr D−2
0,
0,
√ 1
U −iω2RSD /(D−3) Ylm (χD−2 , . . . , χ1 ),
4πωr D−2
U <0
,
U >0
(F.2)
U <0
U >0
(F.3)
and
φ̃ω̃lm = √
1
4π ω̃r D−2
e−iω̃U Ylm (χD−2 , . . . , χ1 ),
from Eq.(4.148), Eq.(4.149) and Eq.(4.157), respectively.
141
(F.4)
142
Appendix F. Scalar Products in the Schwarzschild Geometry
Scalar Products between uωlm modes
In this section we will do the calculations which lead to the results in Eq.(4.150).
From the definition of the scalar product we have
Z
↔
(uωlm , uω0 l0 m0 ) = i (u∗ωlm ∂ U uω0 l0 m0 )r D−2 dU dΩD−2
Z
2RSD ↔
i
0 2RSD
= √
Yl0 m0 Y∗lm (−U )−iω D−3 ∂ U (−U )iω D−3 dU dΩD−2
4π ωω 0
(F.5)
And according to Eq.(4.117) we have that
Z
dΩD−2 Y∗lm (χD−2 , . . . , χ1 )Yl0 m0 (χD−2 , . . . , χ1 ) = δll0 δmm0 .
(F.6)
Thus the product becomes
Z 0
2RSD ↔
0 2RSD
i
(uωlm , uω0 l0 m0 ) = √
dU (−U )−iω D−3 ∂ U (−U )iω D−3 δll0 δmm0
4π ωω 0 −∞
Z 0
2RSD
dU
ω + ω0
2RSD
0
(−U )−i D−3 (ω−ω ) δll0 δmm0 .
=− √
4π ωω 0 D − 3
−∞ U
(F.7)
By using the transformation U = −e−(D−3)u/(2RSD ) we see the integral is a δfunction and finally the inner product becomes
Z ∞
ω + ω0
0
(uωlm , uω0 l0 m0 ) = √
du ei(ω−ω )u δll0 δmm0
0
4π ωω −∞
= δ(ω − ω 0 )δll0 δmm0 .
(F.8)
Similarly, we have
(u∗ωlm , u∗ω0 l0 m0 )
↔
(uωlm ∂ U u∗ω0 l0 m0 )r D−2 dU dΩD−2
Z 0
2RSD
0
dU −i2RSD
i
0
(ω + ω )
(−U )i D−3 (ω−ω )
= √
0
D−3
4π ωω
−∞ U
0 Z ∞
ω+ω
0
=− √
du e−i(ω−ω )u δll0 δmm0
4π ωω 0 −∞
= −δ(ω − ω 0 )δll0 δmm0
=i
and
(uωlm , u∗ω0 l0 m0 )
Z
Z
↔
(F.9)
(u∗ωlm ∂ U u∗ω0 l0 m0 )r D−2 dU dΩD−2
Z 0
2RSD
2RSD
dU
0
ω − ω0
(−U )−i D−3 (ω+ω ) δll0 δm,−m0
= √
0
4π ωω D − 3
−∞ U
0
ω−ω
δ(ω + ω 0 )δll0 δm,−m0 .
= √
(F.10)
0
2 ωω
=i
143
In the calculation of the last inner product, we have used that Y∗lm = Yl,−m . But
since the frequency ω > 0, the δ-function is always zero, and
(uωlm , u∗ω0 l0 m0 ) = 0.
(F.11)
Scalar Products between wωlm modes
In this section we will do the calculations which lead to the results in Eq.(4.151).
From the definition of the scalar product we have
Z
↔
∗
∂ U wω0 l0 m0 )r D−2 dU dΩD−2
(wωlm , wω0 l0 m0 ) = i (wωlm
Z ∞
2RSD ↔
i
0 2RSD
= √
dU U iω D−3 ∂ U U −iω D−3 δll0 δm,m0
4π ωω 0 0
Z ∞
2RSD
ω + ω0
dU i 2R
0
SD
= √
U D−3 (ω−ω ) δll0 δmm0 .
(F.12)
U
4π ωω 0 D − 3
0
By using that U = e−(D−3)u/(2RSD ) we find that
Z ∞
ω + ω0
0
du e−i(ω−ω )u δll0 δmm0
(wωlm , wω0 l0 m0 ) = √
0
4π ωω −∞
= δ(ω − ω 0 )δll0 δm,m0 .
(F.13)
Similarly, we have
∗
(wωlm
, wω∗ 0 l0 m0 )
Z
↔
(wωlm ∂ U wω∗ 0 l0 m0 )r D−2 dU dΩD−2
Z ∞
2RSD
0
dU i2RSD
i
0
U −i D−3 (ω−ω )
(ω + ω )
= √
0
U
D−3
4π ωω
0
0 Z ∞
ω+ω
0
=− √
du ei(ω−ω )u δll0 δmm0
0
4π ωω −∞
= −δ(ω − ω 0 )δll0 δmm0
(F.14)
=i
and
(wωlm , wω∗ 0 l0 m0 )
Z
↔
∗
(wωlm
∂ U wω∗ 0 l0 m0 )r D−2 dU dΩD−2
Z
0
ω − ω 0 2RSD ∞dU i 2R
SD
U D−3 (ω+ω ) δll0 δm,−m0
= √
4π ωω 0 D − 3 0 U
ω − ω0
= √
δ(ω + ω 0 )δll0 δm,−m0 = 0.
0
2 ωω
=i
(F.15)
In the calculation of the last inner product we have used that Y∗lm = Yl,−m , and
since the frequency ω > 0 the δ-function is always zero.
144
Appendix F. Scalar Products in the Schwarzschild Geometry
Scalar Products between φ̃ω̃lm modes
In this section we will do the calculations which lead to the results in Eq.(4.159).
From the definition of the scalar product we have
Z
↔
∗
∂ U φ̃ω̃0 l0 m0 )r D−2 dU dΩD−2
(φ̃ω̃lm , φ̃ω̃0 l0 m0 ) = i (φ̃ω̃lm
Z ∞
↔
i
0
dU eiω̃U ∂ U e−iω̃ U δll0 δmm0
= √
4π ω̃ ω̃ 0 −∞
Z ∞
i
0
0
= √
dU ei(ω̃−ω̃ )U δll0 δmm0
(−i)(ω̃ + ω̃ )
4π ω̃ ω̃ 0
−∞
0
0
0
= δ(ω̃ − ω̃ )δll δmm .
(F.16)
Similarly, we have
∗
(φ̃ω̃lm
, φ̃ω̃∗ 0 l0 m0 )
Z
↔
(φ̃ω̃lm ∂ U φ̃ω̃∗ 0 l0 m0 )r D−2 dU dΩD−2
Z ∞
↔
i
0
= √
dU e−iω̃U ∂ U eiω̃ U δll0 δmm0
0
4π ω̃ ω̃ −∞
= −δ(ω̃ − ω̃ 0 )δll0 δmm0
=i
and
(φ̃ω̃lm , φ̃ω̃∗ 0 l0 m0 )
Z
↔
∗
(φ̃ω̃lm
∂ U φ̃ω̃∗ 0 l0 m0 )r D−2 dU dΩD−2
Z ∞
↔
i
0
dU eiω̃U ∂ U eiω̃ U δll0 δmm0
= √
0
4π ω̃ω̃ −∞
ω̃ − ω̃ 0
δ(ω̃ + ω̃ 0 )δll0 δm,−m0 = 0.
= √
4π ω̃ω̃ 0
=i
(F.17)
(F.18)
Where have used that the frequency ω̃ > 0 and therefore the δ-function is always
zero.
The Bogoliubov Coefficients αωω̃ll0 mm0 and βωω̃ll0mm0
In this section we will do the calculations which lead to the results in Eq.(4.169)
and Eq.(4.170). From the definition of the scalar product we have
Z
↔
∗
αωω̃ll0 mm0 = (φ̃ω̃l0 m0 , uωlm ) = i (φ̃ω̃lm
∂ U uωlm ))r D−2 dU dΩD−2
Z 0
↔
i
0 2RSD
= √
dU eiω̃U ∂ U (−U )iω D−3 δll0 δmm0
4π ω ω̃ −∞
= αωω̃ δll0 δmm0 ,
(F.19)
145
where
αωω̃
i
√
"
i2RSD ω
=
−
D−3
4π ω ω̃
Z
0
dU eiω̃U (−U )
−∞
These integrals may be evaluated by use of
Z ∞
1
dx xν−1 e−µx = ν Γ(ν)
µ
0
iω 2RSD
−1
D−3
− iω̃
Z
0
dU eiω̃U (−U )
−∞
[Reµ > 0, Reν > 0]
iω 2RSD
D−3
(F.20)
(F.21)
given by formula 3.381.4 in Table of Integrals, Series, and Products. By letting
x = −U we have
#
"
Z ∞
Z
2RSD
2RSD
1
2RSD ω ∞
αωω̃ = √
dx e−iω̃x xiω D−3 . (F.22)
dx e−iω̃x xiω D−3 −1 + ω̃
4π ω ω̃ D − 3 0
0
By letting µ = ε + iω̃ and ν = δ + iω2RSD /(D − 3) with ε and δ as small positive
constants. In the limit ε → 0, δ → 0 the first integral becomes
Z ∞
2RSD
2RSD
−1
−iω
−iω̃x iω 2RSD
D−3 Γ
ω .
(F.23)
= (iω̃)
i
dx e
x D−3
D−3
0
And with i = eiπ/2 this yields
Z ∞
RSD
2RSD
2RSD
−1
πω
−iω
−iω̃x iω 2RSD
D−3 Γ
dx e
x D−3
= e D−3 ω̃
i
ω .
D−3
0
The second integral becomes with ν = δ + iω2RSD /(D − 3) + 1
Z ∞
2RSD
2RSD
−iω
−1
−iω̃x iω 2RSD
D−3
Γ i
dx e
x D−3 = (iω̃)
ω+1
D−3
0
2RSD
−i πω RSD −iω 2RSD 2RSD
D−3 iω
e D−3 ω̃
Γ i
ω
=
ω̃
D−3
D−3
2RSD
2RSD ω πω RD−3
2RSD
SD
−iω D−3
=
e
ω̃
Γ i
ω .
(D − 3)ω̃
D−3
Thus,
αωω̃
RSD
=
π(D − 3)
r
ω πω RSD −iω 2RSD
2RSD
D−3 Γ
i
e D−3 ω̃
ω
ω̃
D−3
(F.24)
(F.25)
(F.26)
For the Bogoliubov coefficient βωω̃ll0 mm0 , we have
Z
↔
∗
βωω̃ll0 mm0 = −(φ̃ω̃l0 m0 , uωlm ) = −i (φ̃ω̃lm ∂ U uωlm ))r D−2 dU dΩD−2
Z 0
↔
i
0 2RSD
dU e−iω̃U ∂ U (−U )iω D−3 δll0 δm,−m0
=− √
4π ω ω̃ −∞
= βωω̃ δll0 δm,−m0 ,
(F.27)
#
.
146
Appendix F. Scalar Products in the Schwarzschild Geometry
where
βωω̃
"
#
Z 0
Z
2RSD
2RSD
2RSD ω 0
−1
dU e−iω̃U (−U )iω D−3 .
= √
dU e−iω̃U (−U )iω D−3 −1 − ω̃
D
−
3
4π ω ω̃
−∞
−∞
(F.28)
In the same manner as for the previous coefficient we may transform this into
"
#
Z
Z ∞
2RSD
2RSD
2RSD ω ∞
−1
dx eiω̃x xiω D−3 −1 − ω̃
βωω̃ = √
dx eiω̃x xiω D−3 . (F.29)
D
−
3
4π ω ω̃
0
0
with x = −U . By letting µ = ε − iω̃ and ν = δ + iω2RSD /(D − 3) the first
integral may be calculated to be
Z ∞
2RSD
2RSD
−1
−iω
iω̃x iω 2RSD
D−3 Γ
dx e x D−3
ω
= (−iω̃)
i
D−3
0
2RSD
RSD
2RSD
−iω D−3
−πω D−3
ω̃
Γ i
ω .
(F.30)
=e
D−3
And the second integral to be
Z ∞
2RSD
2RSD
−iω D−3
−1
iω̃x iω 2RSD
D−3
dx e x
ω+1
= (−iω̃)
Γ i
D−3
0
2RSD ω −πω RSD −iω 2RSD
2RSD
D−3
D−3
=−
ω̃
Γ i
e
ω .
(D − 3)ω̃
D−3
(F.31)
Thus,
βωω̃
RSD
=−
π(D − 3)
r
ω −πω RSD −iω 2RSD
2RSD
D−3 ω̃
D−3 Γ
i
e
ω .
ω̃
D−3
(F.32)
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