Dark energy from
scalar fields and conformal invariance
in extra dimensions
Ingunn Kathrine Wehus
Thesis submitted for the degree of
Doctor Scientiarum
Department of Physics
University of Oslo
October 2006
c 2006, Ingunn Kathrine Wehus
Copyright Takk!
this thesis is written in my blood, sweat and tears, but also with a lot of
joy. it is the result of four years as a phd student in the theory group in the
physics department at university of oslo.
first of all i want to thank my supervisor, professor finn ravndal, for
introducing me to such an interesting field of physics. in addition to being
a great physicist prof ravndal has the gift of constantly encouriging people.
i also want to thank prof licia verde and the rest of the astronomy group at
upenn for making my stay in philadelphia the fall 2004 highly enjoyable. and
naturally i want to thank the physics department and the research council
of norway for paying me for doing what i most wanted.
lots of thanks goes to my friends and fellow students and colleagues here
in the physics building for all help and support during the years. especially
thanks to joakim for all the physics discussions, and to hans kristian for
really fast proof-reading in the final spurt.
mest vil ækk takke famelie å kjente fårr at dåkke he gitt megg et liv dei
siste fire åran. tusen takk te alle heime -bodde på lista åh på lagshuse :-)
blinderen, oktober 2006
ingunn kathrine wehus
Contents
I
Introduction
1
1 Prologue
3
2 Physics
2.1 Gravity minimally coupled to matter . . . . . . . . . .
2.1.1 General Relativity minimally coupled to matter
2.1.2 Modified gravity minimally coupled to matter .
2.2 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Scalar field . . . . . . . . . . . . . . . . . . . .
2.2.2 Electromagnetism . . . . . . . . . . . . . . . .
2.3 Gravity non-minimally coupled to matter . . . . . . .
2.4 Conformal transformations . . . . . . . . . . . . . . . .
2.5 Conformal invariance . . . . . . . . . . . . . . . . . . .
2.5.1 Scalar field . . . . . . . . . . . . . . . . . . . .
2.5.2 Electromagnetism . . . . . . . . . . . . . . . .
2.6 Weyl transformations . . . . . . . . . . . . . . . . . . .
2.6.1 Weyl transformations in General Relativity . .
2.7 Einstein frame and Jordan frame . . . . . . . . . . . .
2.8 Quantization . . . . . . . . . . . . . . . . . . . . . . .
3 Cosmology
3.1 Constructing a cosmological model . . . . . . .
3.2 The old standard model . . . . . . . . . . . . .
3.2.1 Standard cosmological assumptions . . .
3.2.2 Acceleration and flatness . . . . . . . . .
3.3 Modifications . . . . . . . . . . . . . . . . . . .
3.4 Modifying classical physics . . . . . . . . . . . .
3.4.1 Modified Matter – Dark energy . . . . .
3.4.2 ΛCDM – the new standard model of the
3.4.3 Modified gravity . . . . . . . . . . . . .
3.4.4 Extreme makeover . . . . . . . . . . . .
3.5 Modifying cosmological assumptions . . . . . .
3.6 Quantum effects . . . . . . . . . . . . . . . . .
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4 Epilogue
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List of papers
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Bibliography
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II
37
Papers
Part I
Introduction
Chapter 1
Prologue
When Ask and Embla a dark night sit under their apple tree staring at the
stars, several existential questions come to mind
• What makes the apple fall in my head?
• What am I?
• Am I alone in the universe?
• Is there more out there than what I can see?
• Is a constant always constant?
• How do we know that other people feel the same as us?
• Am I only a three-dimensional ant in a higher-dimensional world?
• Is everything random?
• What is left if you take away everything?
Being a physicist means you can work with all of them!
4
Prologue
Gravitation
What makes the apple fall in my head?
The gravitational force pulls all massive objects towards each other. Gravitational interactions is supposed to reach infinitely far and to travel with the
fastest speed available, the speed of light. This means that gravity has to be
propagated by massless particles, called gravitons. Two galaxies separated
by an entire universe feel the gravitational pull of each other, due to the
gravitational field present everywhere.
Gravitation is described by Einstein’s theory of General Relativity. This
theory elegantly unite space and time into one concept, spacetime. When we
live in three spatial dimensions and add the time dimension this spacetime
becomes four-dimensional. According to Einstein this spacetime is no longer
some kind of absolute background, in which we can put massive particles
and then calculate the interactions between them while the spacetime itself is
unchanged. On the contrary, it is the mass distribution that forms spacetime.
Einstein’s equations tell us that the geometry, or curvature, of spacetime is
proportional to its mass distribution. For a given mass distribution we must
find the corresponding spacetime geometry according to Einstein’s equations.
More massive objects makes spacetime more curved.
Nothing which invalidates General Relativity has ever been measured.
Matter
What am I?
The vague term matter is used to describe the stuff we are all made of, or
more generally, everything in the universe besides gravity.
Matter includes the atoms which build mankind as well as the light we
see on the sky. Matter which moves slowly compared to the speed of light,
like atoms in human beings or a cluster of galaxies on the sky, is labeled
cold. Matter which speed is comparable to the speed of light, like light itself
and electromagnetic radiation in general, is referred to as hot. Matter which
we can see glowing is named bright while dark matter is invisible for both
human eyes and other electromagnetic instruments.
Cosmology
Am I alone in the universe?
From a large scale point of view our home is named the universe. And for
us inhabiting it, the universe is our whole world. To ask what is outside
5
universe is comparable to asking what was before dawn of time or who has
created God. On the other hand, what we as physicists can ask, is how our
universe has developed, and how it all will end.
When Hubble in 1929 measured that the galaxies surrounding us are
moving away from us, the possibility of a dynamical universe was revealed.
Ever since then the questions about the universe’s past, present and future
have been asked. This kind of questions is it cosmologists try to answer.
The basis of it all are Einstein’s equations.
The big bang models are the heart of the standard cosmological model
today. The big bang is the event which started our era of time. At this
time event, maybe 13 billions years ago, some kind of explosion ignited the
expansion of spacetime. The initial expansion is believed to have been so
strong that even today, the universe is still expanding.
This ongoing stretching of space itself makes the galaxies positioned at
different points in space move away from each other, the same way as the
raisins in a bun get more separated when the dough rises. In the dough the
rising is initiated by the yeast, but what was acting as the universe’s yeast
physicists are still trying to figure out.
Dark energy
Is there more out there than what I can see?
Eight years ago astronomers measured that the galaxies far away from us are
not only leaving us, they are leaving us faster and faster. The stretching of
space has a positive acceleration! This is hard to explain if one assume that
the universe is only made up of matter particles which are pulled towards
each other due to gravitational forces. This gravitational pull should slow
down the initial expansion, making the universe’s expansion have negative
acceleration.
The common way to explain the acceleration of the universe without
altering Einstein’s equations is to postulate that the universe must be filled
with something exotic which, contrary to ordinary matter, is being pulled
away from itself due to gravitational forces. It must experience some kind of
negative gravity. This mystical and unseen something is referred to as dark
energy.
The simplest version of dark energy is Einstein’s cosmological constant.
This corresponds to some constant dark energy being present all over universe, and its negative gravitational pull is responsible for the universe’s
positive acceleration. It has got its name from its initial entrance to the
world of physics as being a constant added to Einstein’s equations by Einstein himself.
This constant was originally put by hand into the Einstein equations to
6
Prologue
allow for a static universe. If Einstein had not put the cosmological constant
into his equations he would have predicted the expansion of the universe
14 years before Hubble measured it. Because of this Einstein later referred
to the cosmological constant as the biggest blunder of his life. The same
cosmological constant is nowadays used to account for the acceleration of
the universe, in what is the new standard model in cosmology – the ΛCDMmodel, an example of a big bang model.
The second simplest version of dark energy, and the most popular alternative to a cosmological constant, is a scalar field called quintessence.
Scalar fields
Is a constant always constant?
A scalar field is the second simplest thing to an universal constant. It can be
described as a “constant” which varies with time and space. A scalar field is
a scalar function of spacetime. For each point in spacetime the scalar field
has some specific magnitude. A vector field on the other hand has in each
point of spacetime both a magnitude and a direction. The electromagnetic
field is an example of a vector field. The gravitational field is an example of
a tensor field, having an even more complex structure.
The Standard Model in particle physics includes a scalar field called the
Higgs field. Scalar fields in the early universe are often called inflation while
scalar fields in late time universe are referred to as quintessence. Scalar fields
can have different cosmological behaviour due to different types of self interactions, or potential. In late-time cosmology the scalar field is supposed to
imitate a cosmological constant, and thereby causing the universe’s positive
acceleration.
At small scales the distribution of mass in the universe varies enormously
from one point in the center of a star to another point in the middle of
nowhere. But viewing the mass distribution from larger and larger scales the
mass get smeared out. At cosmological scales the distribution of matter like
galaxies and clusters of galaxies in the universe looks the same in all points
and directions in space. In cosmology it is therefore common to assume that
that the universe looks the same in all points of space, so that a cosmological
scalar field is space-independent, and only varies with time. Quintessence
can therefore be described as a ’time-varying cosmological constant’.
Although commonly used in various fields of physics, scalar fields have
never been experimentally detected.
7
Conformal invariance
How do we know that other people feel the same as us?
Performing a conformal transformation of our spacetime means that we
change our standard measuring rods and clocks. This change in the measuring rods and clocks is not the same from point to point in spacetime. We
multiply the local measuring rods at each point in spacetime by a scaling
factor which generally varies from point to point. In other words, we multiply
them by a spacetime-dependent scalar field.
What makes up a physical system is often divided into matter and gravity.
As described above the challenges in modern cosmology are often faced by
keeping the gravity part standard, but allowing for a non-standard matter
part like dark energy.
General Relativity together with dark energy means we have standard
gravity together with non-standard matter. By now performing a conformal
transformation of our geometry, that is changing the measuring rods and
clocks as described above, we find that our system is transformed into another system where we have standard non-exotic matter but non-standard
gravity.
These two descriptions are often viewed as two versions of the same
theory. The frame where we have standard gravity is labeled Einstein frame,
while the other frame where the matter part is standard is called Jordan
frame.
If our theory is conformally invariant we know that both the Jordan
frame version and the Einstein frame version describe the same physics. If
our theory is not conformally invariant, the question of which frame is the
physical one is debated.
From its definition, the transformations we do when moving between
Einstein frame and Jordan frame involve scalar fields. When moving between
frames like this, many extra-dimensional theories naturally give us scalar
fields in our four-dimensional spacetime.
Extra dimensions
Am I only a three-dimensional ant in a higher-dimensional world?
The possibility that we live in a world with more than four spacetime dimensions has always been a fascinating thought.
In physics it all started with Nordström in 1914. Nordström had then
already constructed a gravitational theory of his own, where the gravitational interactions were mediated by a scalar field rather than gravitons.
By assuming that space has one extra dimension he got a unified theory of
8
Prologue
electromagnetism and his scalar gravitational theory.
Theodor Kaluza was in 1919 the first to construct a higher-dimensional
theory based on the correct gravity model, Einstein’s General Relativity.
Like Nordström he assumed that spacetime has four spatial dimensions instead of three. In this higher-dimensional spacetime he introduced Einstein’s
gravity and nothing else. Then he noticed that, seen from our ordinary fourdimensional spacetime, this corresponded to both gravity and electromagnetism and a scalar field. He had found a unified theory of both gravity and
electromagnetism, which also included a scalar field.
In order for this to work, the extra, fifth, dimension has to be small and
compact. What does this mean? How can a dimension be small? Imagine
a garden hose. When watched from a large distance the surface of the hose
looks one-dimensional. But as we zoom in, we see that for each point in this
large dimension, we can walk a short loop into an extra dimension. The fact
that we return to our starting point without changing direction means that
the dimension is compact. In the same way, we can imagine that for each
point in our four-dimensional spacetime, there is a small loop extending into
a fifth dimension.
Many theories within modern physics, like string theory and brane worlds,
assume that we live in a higher-dimensional spacetime, although in most aspects we only see the usual four of them.
Quantum field theory
Is everything random?
Until now we have not mentioned quantization. Quantum behaviour is one
of the fundamental aspects of physics, and quantum theory is about as old
as general relativity. Quantum mechanics tells us that fundamental particles
do not follow the same deterministic mechanical laws as large billiard balls.
A quantum particle can have infinitely many options for how to move, and
we can only forecast the probability for a single particle to follow a given
path.
Zooming in on a fundamental matter particle like an atom or an electron,
this is not a miniature billiard ball but a small oscillator. The same way as a
guitar string only can produce given tones, the oscillating elementary particle
can only oscillate at given discrete frequencies. These oscillations give the
particle its energy. We say the energy is quantized. The energy of an atom
can only take distinct discrete values. The energy of only one level can not be
measured, only the difference between two levels. Using quantum mechanics
we can calculate the value of these energy levels and the gap between any
two levels matches experimental values.
We call the energy of the lowest possible state for an atom to be in, the
9
zero-point energy. If we want to we can adjust the theoretical zero-point
energy to be zero by subtracting this same zero-point energy from every
energy level. Since only energy differences between levels can be measured
in particle physics, the value of the zero-point energy is irrelevant.
Similar to the quantization of the energy of an atom, the energy associated with a field, like the electromagnetic or gravitational or scalar field,
is also quantized. The field quanta are the smallest possible bits, similar to
atoms in a human body. The quantum of the electromagnetic field is the
photon which carries light and other types of electromagnetic radiation. The
quantum of the gravitational field is the graviton, which carries gravitation.
Until now only the matter part of physics has gotten a consistent quantum
description. How to quantize gravity is still an open question. We still assume that the gravitational field may be quantized, meaning gravitons will
behave as quantum particles.
Casimir energy
What is left if you take away everything?
Like fundamental particles, the quanta of a quantum field also behave
as harmonic oscillators, making small fluctuations. For a quantum field all
energy states have infinite energy, but the gap between any two states is still
finite. The lowest possible energy state of a quantum field is infinite. We
call this the vacuum energy, and say it is due to quantum fluctuations of the
field. Since it is associated with a field, which from its definition fills all of
spacetime, the vacuum energy is present all over universe, also the otherwise
empty parts.
Again the vacuum energy, or zero-point energy, can not be measured from
its interactions with matter. We only measure energy differences. However,
if we insert two parallel plates in an otherwise empty vacuum, the quantum
fluctuations exhibit a force on the plates. Two metal plates in vacuum are
pulled together by the vacuum fluctuations of the electromagnetic field. The
force is inversely proportional to a power of the distance between the plates.
This force was first calculated theoretically by Casimir in 1948, and it has
later been experimentally verified. The finite force corresponds to a finite
addition to the infinite vacuum energy between the plates. We call this
additional vacuum energy due to the plates for the Casimir energy. Moving
the plates away from each other makes the Casimir energy become zero.
If our universe contains extra compact dimensions, these will act as confining plates for the quantum fields propagating in these extra dimensions,
and a Casimir energy will be generated. This Casimir energy from extra dimensions might be an alternative to other kinds of dark energy in cosmology.
10
Prologue
Chapter 2
Physics
In a D-dimensional spacetime with a metric g µν we describe our system by
an action integral
Z
√
S = dD x −gL
(2.1)
where L is the Lagrangian for the given system, which generally is a function
of both gµν and various other fields. Varying the action and demanding that
the variation shall be zero gives us the equations of motion. Varying the
action with respect to the metric means that when we let
the action will change by
gµν → gµν + δgµν
(2.2)
S → S + δS
(2.3)
where the variation δS is given by
δS =
∂S
∂S
∂S
δgµν +
δgµν,σ +
δgµν,σρ + · · ·
∂gµν
∂gµν,σ
∂gµν,σρ
(2.4)
and similarly for Lagrangians containing higher derivatives of g µν . We need
the variation of the determinant of the metric, which is given by
√
1√
1√
δ −g =
−gg µν δgµν = −
−ggµν δg µν
(2.5)
2
2
2.1
Gravity minimally coupled to matter
It is common to divide the Lagrangian into a gravity part and a matter part,
Z
√ (2.6)
S = dD x −g Lg + Lm
The gravity Lagrangian Lg is only a function of gµν and its derivatives, the
matter Lagrangian Lm is a function of gµν and matter fields only, and not
a function of derivatives of the metric. This means that matter and gravity
are minimally coupled.
12
Physics
2.1.1
General Relativity minimally coupled to matter
The gravity Lagrangian for Einstein’s theory of General Relativity is given
by
1
(2.7)
Lg = R
2
where R is the Ricci scalar, the trace of the Ricci tensor, R = g µν Rµν . The
Ricci scalar is a function of the metric g µν , its first derivatives and its second
derivatives. However, we have the identity
g µν
δRµν
=0
δg αβ
(2.8)
We let the gravity Lagrangian in (2.6) be Einstein gravity (2.7) and we vary
this action with respect to the metric. Taking advantage of (2.8) and (2.5)
we find
!
Z
√
∂L
1
1
m
δg µν (2.9)
dD x −g
Rµν − gµν R − gµν Lm − 2 µν
δS =
2
2
∂g
We see that demanding δS = 0 independent of the variation δg µν gives the
equations of motion
Eµν = Tµν
(2.10)
where the gravity tensor derived from the Einstein gravity Lagrangian (2.7),
the Einstein tensor, is defined by
1
Eµν = Rµν − gµν R
2
(2.11)
and the energy-momentum tensor derived from the matter Lagrangian L m
is given by
∂Lm
(2.12)
Tµν = −2 µν + gµν Lm
∂g
The matter Lagrangian is a function of other fields besides g µν . Variation of
the action (2.6) with respect to these other fields gives us additional equations
of motion for the matter fields.
2.1.2
Modified gravity minimally coupled to matter
Next we abandon General Relativity and explore more generalized theories
of gravity, often referred to as modified gravity. Presuming gravity is still
minimally coupled to matter we again have (2.6)
Z
√ S = dD x −g Lg + Lm
(2.13)
However, this time Lg is not given by (2.7), it is a general scalar function
of the metric. It may depend on both the Ricci scalar R, the squared Ricci
2.2 Matter
13
tensor Rµν Rµν and the squared Riemann tensor Rµνσρ Rµνσρ . In so-called
f (R)-theories, or nonlinear theories of gravity, L g is a general function of the
Ricci scalar. Varying the action with respect to the metric we end up with
a set of equations of motion we can write like
Gµν = Tµν
(2.14)
where now Gµν is the gravity tensor derived from the modified gravity Lagrangian. It is defined from
1
δS =
2
2.2
Z
√ dD x −g Gµν − Tµν δg µν
(2.15)
Matter
We will discuss two types of fundamental matter fields, scalar and electromagnetic.
2.2.1
Scalar field
The minimal Lagrangian for a scalar field is a function of the metric and
first order derivatives of the scalar field.
1
Lm = Lφ = − g µν φ,µ φ,ν
2
(2.16)
From (2.12) we find the corresponding energy-momentum tensor
1
Tµνφ = φ,µ φ,ν − gµν φ,α φ,α
2
(2.17)
which implies that for the scalar field minimally coupled to Einstein gravity
we have the Einstein equations
Eµν = Tµνφ
(2.18)
The equations of motion for φ are found by varying the action
S=
Z
√
1
R − g µν φ,µ φ,ν
dD x −g
2
(2.19)
with respect to φ,
22 φ = 0
This is the Klein-Gordon equation for a free scalar field.
(2.20)
14
Physics
2.2.2
Electromagnetism
The Lagrangian for a free electromagnetic field reads
1
1
LA = − F 2 = − g µν g αβ Fµα Fνβ
4
4
(2.21)
Here the electromagnetic field is given by the gauge field A µ
Fµν = Aν,µ − Aµ,ν
(2.22)
Variation with respect to the metric gives us
1
A
Tµν
= Fµα Fν α − gµν Fαβ F αβ
4
(2.23)
It is the gauge field Aµ , and not Fµν , which is the fundamental field in the
Lagrangian. Variation with respect to A µ gives the equations of motion
Fµν;ν = 0
(2.24)
These are the source-free Maxwell equations.
2.3
Gravity non-minimally coupled to matter
Looking at the Brans-Dicke action [1]
Z
ω µν
D √
S = d x −g φR − g φ,µ φ,ν + 2Lm
φ
(2.25)
we notice that it can not be written on the form (2.6) with a gravity Lagrangian only depending on the metric and its derivatives, and a matter Lagrangian not containing derivatives of the metric. In this theory gravity is
non-minimally coupled to a scalar field. The parameter ω is a constant.
Varying this action we find
ω
φEµν = Tµνφ + φ;µν − gµν 22 φ + Tµνm
(2.26)
φ
where Tµνφ corresponds to the minimal scalar field Lagrangian (2.16) and T µνm
is the energy-momentum tensor for the additional matter term L m . Here we
have used that for a term in the action like
Z
√
(2.27)
SφR = dD x −gφR
variation with respect to the metric gives us [2].
δSφR =
Z
!
√
2
d x −g φEµν − φ;µν − gµν 2 φ δg µν
D
(2.28)
2.4 Conformal transformations
15
Contracting equation (2.26) we find
(D − 2)R − (D − 2)
ω µν
2(D − 1) 2
2
g φ,µ φ,ν −
2 φ + Tm = 0
φ2
φ
φ
(2.29)
Here T m = Tµνm g µν is the trace of the energy-momentum tensor for the
additional matter fields. Variation of the action (2.25) with respect to the
scalar field φ gives the equation of motion for φ
R−
ω µν
ω
g φ,µ φ,ν + 2 22 φ = 0
φ2
φ
(2.30)
Combining (2.29) and (2.30) we are left with
D − 1 − (D − 2)ω 22 φ − T = 0
(2.31)
Another example of a non-minimal coupling between matter and gravity
is the conformally coupled scalar field which will be discussed in section 2.5.1.
2.4
Conformal transformations
A conformal transformation of the metric is a rescaling on the form
gµν → Ω2 gµν
(2.32)
where Ω = Ω(xµ ) is a scalar function of the spacetime coordinates x µ . This
means that the metric is multiplied by a scaling factor which generally varies
from point to point in spacetime. We have changed our measuring rods and
clocks, and thereby our metric, but our coordinates are the same. We do not
perform any coordinate transformations.
Two metrics connected by a conformal transformation are said to be
conformal.
2.5
Conformal invariance
The transformation (2.32) written on the form (2.2) corresponds to
or
δgµν = Ω2 − 1 gµν
(2.33)
δg µν = Ω−2 − 1 g µν
(2.34)
Performing a transformation (2.32) of the metric, the matter part of the
action
Z
√
(2.35)
Sm = dD x −gLm
16
Physics
change by
1
=−
2
δSm
√
dD x −g Ω−2 − 1 Tµν g µν
Z
(2.36)
Any matter Lagrangian corresponding to a traceless energy-momentum tensor
leaves the action invariant under conformal transformations. From the Einstein equations (2.14) a conformal invariant theory also demands G = 0,
where G is the trace of the gravity tensor.
A conformally invariant theory allows us to change our measuring rods
as described in section 2.4 without affecting the equations of physics.
2.5.1
Scalar field
By taking the trace of (2.17)
T
φ
=
D
1−
2
(2.37)
φµ φµ
we see that only in D = 2 spacetime dimensions is the action of a minimally
coupled scalar field conformally invariant.
However, a theory in which General Relativity is coupled to a scalar field
can be made conformally invariant by introducing an extra term proportional
to Rφ2 [3] in the Lagrangian
Z
√
1
R − ξRφ2 − g µν φ,µ φ,ν
(2.38)
S = dD x −g
2
The constant ξ is given in D spacetime dimensions as ξ =
variation of the extra term in the action
Z
√
S∆ = −ξ dD x −gφ2 R
D−2
4(D−1) .
The
(2.39)
with respect to the metric gives
δS∆ =
Z
√
dD x −g
!
− ξφ2 Eµν − ∆Tµνφ δg µν
where ∆Tµνφ is the Huggins term [4]
∆Tµνφ = ξ gµν 22 (φ2 ) − (φ2 );µν
(2.40)
(2.41)
This gives the Einstein equations
(1 − ξφ2 )Eµν = Tµνφ + ∆Tµνφ
(2.42)
where Tµνφ is the ordinary scalar field energy-momentum tensor (2.23) from
D−2
the combined energy-momentum
the minimal coupling case. For ξ = 4(D−1)
2.6 Weyl transformations
17
tensor Tµνφ + ∆Tµνφ is traceless, corresponding to a conformally invariant
theory. Then also Eµν and Rµν are traceless. Varying the action (2.38)with
respect to φ gives the equation of motion
22 φ − ξRφ = 0
(2.43)
Since R = 0 in a conformal theory the equation of motion for φ is the same
as for the minimally coupled scalar field (2.20).
2.5.2
Electromagnetism
Taking the trace of the electromagnetic energy-momentum tensor (2.23) we
find
D
TA = 1−
Fµν F µν
(2.44)
4
Our usual four spacetime dimensions is the only number of dimensions for
which Maxwell’s theory of electromagnetism is conformally invariant. Because electromagnetism is a gauge theory, and we need to conserve the gauge
invariance of the action, we can not add a term to make the theory conformally invariant in every dimension, as we could for the scalar field.
2.6
Weyl transformations
A Weyl transformation is a conformal transformation (2.32) of the metric.
We write
gµν → g̃µν = Ω2 gµν
(2.45)
and this gives us for the determinant of the new metric g̃ µν as a function of
the determinant of the old metric gµν
p
√
−g̃ = ΩD −g
(2.46)
Calculating the Ricci tensor we find
R̃µν = Rµν − (D − 2)Ω−1 Ω;µν − Ω−1 gµν Ω;αβ g αβ
+ 2(D − 2)Ω−2 Ω,µ Ω,ν − (D − 3)Ω−2 gµν Ω,α Ω,α
(2.47)
Contracting this we get the equation for the transformation of the Ricci
scalar
R̃ = Ω−2 R − 2(D − 1)Ω−3 g µν Ω;µν − (D − 1)(D − 4)Ω−4 g µν Ω,µ Ω,ν (2.48)
This equation for the Ricci scalar may be rewritten as
R̃ = Ω−2 R − 4
D − 1 − D+2 2 D−2
Ω 2 2 Ω 2
D−2
(2.49)
18
Physics
Here we must remember that the D’Alambertian operator is with respect to
the old metric gµν . We can also write down the expression for the transformation of the Einstein tensor
i
h
Ẽµν = Eµν + (D − 2)Ω−1 gµν Ω;αβ g αβ − Ω;µν
(2.50)
1
+ 2(D − 2)Ω−2 Ω,µ Ω,ν + (D − 2)(D − 5)Ω−2 gµν Ω,α Ω,α
2
We observe that in D = 2 dimensions the Einstein tensor is invariant under
a Weyl transformation.
2.6.1
Weyl transformations in General Relativity
For simplicity we look at Einstein gravity minimally coupled to matter. Prior
to the Weyl transformation (2.45) we have the Einstein equations (2.10)
Eµν = Tµν
(2.51)
Eµν is the Einstein tensor for the old metric g µν and Tµν is derived from the
original matter Lagrangian Lm using (2.12).
Conformal theory
In a conformal theory, we know that the Einstein equations take the same
form before and after the conformal transformation
Ẽµν = T̃µν
(2.52)
The energy-momentum tensor, the Einstein tensor and the Ricci tensor are
all traceless. Using R = 0 = R̃ in equation (2.49) we get
22 Ω
D−2
2
=0
(2.53)
while the transformation of the Einstein tensor simplifies to
1
Ẽµν = Eµν − (D − 2)Ω−1 Ω;µν + 2(D − 2)Ω−2 Ω,µ Ω,ν − (D − 2)Ω−2 gµν Ω,α Ω,α
2
(2.54)
Using this together with (2.51) and (2.52) we can write down T˜µν as function
of Tµν . In a conformal theory we can always deduce how matter transforms
under a conformal transformation since we know that the equations of motion
are always the same.
Non-conformal theory
For a non-conformal theory we can still change our measuring rods according
to (2.45) and calculate how the gravity part of the action change according
2.7 Einstein frame and Jordan frame
to (2.49). But to transform the matter part of the action may be non-trivial.
To find the energy-momentum tensor after the transformation as a function
of the original one, we must know the matter Lagrangian as a function of
the metric. Energy-momentum tensors corresponding to different types of
matter will generally transform different from each other.
2.7
Einstein frame and Jordan frame
A theory which can be described by an action on the form of (2.6), where
Lg = 21 R, is called a Einstein frame theory. In Einstein frame we have
standard Einstein gravity minimally coupled to some kind of matter. In
Jordan frame, on the other hand, we have a standard matter Lagrangian
while the gravity part is non-standard. There may also be non-minimal
couplings between matter and gravity.
By performing a Weyl transformation of the metric, we can move between
a Jordan frame theory and the corresponding Einstein frame theory. But
only for a conformally invariant theory are the Einstein frame theory and
the Jordan frame theory the same theory.
For a non-conformal theory we can use Weyl transformations to transform between Einstein frame and Jordan frame. The equations of motion
will then look different in the two frames, meaning that the laws of physics change when we introduce new measuring rods. However, the original
equations of motion for the original metric are mathematically the same as
the new equations for the new metric. We can always choose to work with
the simplest equations, but we must remember that since the two metrics
describe different physical systems only one of the metrics can be the physical one. If we choose the Jordan frame metric to be the physical one, the
Einstein frame metric is reduced to a mathematical quantity, a function of
the physical metric. To change our mind, and say that the Einstein frame
metric shall be the physical one, is the same as introducing a whole new
theory.
Only for a conformally invariant theory are both metrics physical. In
a conformally invariant theory we can not distinguish between conformal
metrics. Since all physics is invariant under the change of measuring rods
we can not tell which set of measuring rods we are using.
The Brans-Dicke theory discussed in section 2.3 is an example of a theory
written in Jordan Frame,
Z
ω µν
µν
D √
(2.55)
S = d x −g φR − g φ,µ φ,ν + 2Lm (g )
φ
Here the notation Lm (g µν ) is used to remind us that the matter Lagrangian
is a function of the metric g µν in addition to matter fields. The Weyl transformation
2
gµν → φ− D−2 gµν
(2.56)
19
20
Physics
gives us the corresponding Einstein frame theory
Z
2
1 µν
1
D−1
D √
µν
S̃ = d x −g R −
+ω
g φ,µ φ,ν + 2 2 Lm φ D−2 g
D−2
φ2
φ
(2.57)
Lm is now a function of the new metric but the functional form is the same.
2.8
Quantization
The classical field equations described until now are the starting point for
quantization. Keeping these field equations but promoting the classical fields
to operator fields we arrive at the quantum field theory. Both the traditional
canonical quantization and Feynman’s path integrals are best described in
the white “bible” of Peskin and Schroeder [5].
Chapter 3
Cosmology
3.1
Constructing a cosmological model
The standard recipe for making a cosmological model can be summarized as
follows:
1. Start with your favorite action and derive the fundamental equations
of motion. Alternatively, take a guess on some equations of motion
which may or may not be derived from an action.
2. Use your assumptions based on observations and/or intuition to simplify these equations of motion.
3.2
The old standard model
We start with the standard master equation in four spacetime dimensions
Z
√
S = d4 x −g Lg + Lm
(3.1)
where our system is the universe as a whole. For the gravity part, which
is described by General Relativity, L g = 12 R, this action corresponds to the
standard Einstein equations (2.10)
Eµν = Tµν
(3.2)
where Tµν is given from (2.12).
3.2.1
Standard cosmological assumptions
On cosmological scales the distribution of galaxies and clusters of galaxies
looks smooth all over the universe. We assume that the universe looks the
same in all directions, that space is isotropic. We also assume that it looks
22
Cosmology
the same in all points in space, in other words, the universe has no center.
This means that in addition to being isotropic, the universe is homogeneous.
The most general metric for a homogeneous and isotropic spacetime has
two free parameters. We use the Robertson-Walker ansatz for the geometry
of spacetime
dr 2
2
3
2
2
2
+ r dΩ3
ds = −dt + a (t)
(3.3)
1 − kr 2
Here the two free parameters are the scale factor a and the curvature constant
k for the spatial geometry. In a spherical geometry k = 1, for euclidean flat
space k = 0, while k = −1 corresponds to negatively curved, hyperbolic
space. The scale factor a is space-independent, it only varies with time.
On cosmological scales the motion of the individual galaxies may be
ignored compared to the expansion of spacetime. We make the assumption
that the matter content in universe today may be described as a pressureless
perfect fluid. A perfect fluid has parameters ρ and p. ρ is the energy density
and p is pressure. The energy-momentum tensor is written like
pf
Tµν
= (ρ + p)uµ uν + p gµν
(3.4)
where uµ is the 4-velocity of the fluid. The fluid is not moving with respect
to space, it is the space in itself that is stretching, so the 4-velocity has
components uµ = [1, 0, 0, 0]. Given the Robertson-Walker metric (3.3) the
energy-momentum tensor has components
pf
Tµν
a2
p , a2 r 2 p , a2 r 2 sin2 θ p
= diag ρ ,
1 − kr 2
(3.5)
pf
Demanding Tµν
to be divergence-free gives us
ȧ
ρ̇ + 3 (ρ + p) = 0
a
(3.6)
The perfect fluid energy-momentum tensor together with (3.3) gives two
independent components of the Einstein equations (3.2)
ȧ2
k
+3 2 =ρ
2
a
a
k
ä ȧ2
−2 − 2 − 2 = p
a a
a
3
(3.7)
(3.8)
These equations were found by Friedmann in 1922 and are referred to as the
Friedmann equations. Taking the derivative of (3.7) and using (3.8) gives
us (3.6). For p being zero or proportional to ρ these equations can easily be
solved.
3.3 Modifications
3.2.2
23
Acceleration and flatness
Adding the two Friedmann equations (3.7–3.8) we find
ä
1
ρ + 3p
=−
2
a
6M
(3.9)
We see that this equation is independent of curvature. In order to have positive acceleration we must have a negative pressure p < −ρ/3. Pressureless
cold matter always gives negative acceleration for the expansion of universe.
If we trust the measurements of a present accelerating universe [6, 7, 8], we
need to modify the above cosmological model.
Another view of the failure of the cosmological model presented above is
seen by considering the observed flatness of space [9]. When putting k = 0
in equation (3.7) we find
ρm
=1
(3.10)
Ωm ≡
3H 2
where we have introduced the relative density Ω m measuring the matter
density’s fraction of the critical density 3H 2 needed to have a spatial flat
universe according to (3.7). However, summing the observed mass of all
the luminous matter in the sky, such as stars, only gives us Ω m ≈ 0.04.
Adding the best estimate of the amount of cold dark matter in the universe
we can increase Ωm to around 0.3 but we are still far from fulfilling (3.10).
Something is wrong.
3.3
Modifications
If we want to modify this model we have several theoretical possibilities:
1. Change classical physics
• matter (Lm )
• gravity (Lg )
• extreme makeover (S)
2. Adjust cosmological assumptions
• homogeneity
• isotropy
• perfect fluid description
3. Include quantum effects
24
Cosmology
3.4
Modifying classical physics
This means we are modifying the equations of motion or, equivalently, the
action. We can either keep the action on the form (3.1) and only modify
Lm and/or Lg , or we can imagine the action being modified so thoroughly
that it can not be written in the form of (3.1). Our fundamental action may
for instance be higher-dimensional or we can have non-minimal couplings
between matter and gravity.
3.4.1
Modified Matter – Dark energy
The most common way of modifying the matter Lagrangian to obtain a
cosmology which allows for positive acceleration, is to introduce a new matter
component called dark energy in addition to the cold matter,
Lm → LDE + Lm
(3.11)
Here Lm represents the pressureless perfect fluid matter. Our new master
equation is therefore
Z
√
1
(3.12)
S = d4 x −g R + LDE + Lm
2
The simplest implementation of dark energy is the cosmological constant
described in section 3.4.2.
The second most popular thing to do is to introduce a scalar field into
our universe
1
LDE = Lφ = − g µν φ,µ φ,ν − V (φ)
(3.13)
2
This is called the quintessence field. Putting this into (3.12) we have the
Einstein equations
1
pf
Eµν = φ,µ φ,ν − gµν φ,α φ,α − gµν V (φ) + Tµν
2
(3.14)
Assuming the scalar-field to be homogeneous in space the scalar field energymomentum tensor may be written in the form of a perfect fluid, with
1
ρ = φ̇2 + V (φ)
2
1
p = φ̇2 − V (φ)
2
(3.15)
(3.16)
We notice that when the potential energy V (φ) dominates over the kinetic
energy 21 φ̇2 , a scalar field dominated universe gives positive acceleration.
Other modifications of the matter Lagrangian include a Chaplygin gas
[10][11] having p ∼ 1ρ .
3.4 Modifying classical physics
3.4.2
25
ΛCDM – the new standard model of the universe
The simplest modification of (3.1) is to reintroduce Einstein’s cosmological
constant Λ to Einsteins equations, giving us
S=
Z
√
1
d4 x −g R − Λ + Lm
2
(3.17)
This results in the following equations of motion
m
Eµν + Λgµν = Tµν
(3.18)
Using the same cosmological assumptions as a above, namely isotropy,
homogeneity and Lm describing a perfect fluid, our modified Friedmann
equations read
k
ȧ2
+3 2 =ρ+Λ
a2
a
ä ȧ2
k
−2 − 2 − 2 = p − Λ
a a
a
3
(3.19)
(3.20)
We see that introducing a cosmological constant corresponds to adding a new
perfect fluid with constant density ρ Λ = Λ and negative pressure pΛ = −ρΛ .
We can view this as some constant vacuum energy density being present
all over the universe, even when no other matter is present. This vacuum
energy is only observable through its gravitational effect. We see from (3.9)
that a universe dominated by this vacuum energy will have positive acceleration. Since CMB observations suggest a spatially flat cosmology, k is put
to zero in (3.19-3.20). The Friedmann equations can then again be easily
solved [12]. This new standard model of the universe, called the ΛCDM
model, can describe all cosmological observations we have today. However,
since we have no physical explanation of this vacuum energy or cosmological
constant, several other descriptions of dark energy have been suggested.
3.4.3
Modified gravity
An alternative to introducing dark matter in order to explain the observed
acceleration of the universe is to modify Einstein’s theory of gravity, as
discussed in section 2.1.2. For instance, Carroll et al. [13] suggest adding
a term going like 1/R to the Einstein gravity Lagrangian, while Nojiri and
Odintsov [14] study ln R -terms.
The non-specific generalized f (R)-theories have been studied by, e.g., Capozziello et al. [15]. These kind of models have been extensively investigated
during the last three years, see for instance [16] and references therein.
26
Cosmology
3.4.4
Extreme makeover
Several cosmological models including non-minimal couplings between matter and gravity have been suggested. Faraoni [17] has showed that a conformally coupled scalar field with a particular potential give p < −ρ, known
as phantom energy [18]. Phantom energy models will have extremal positive
acceleration according to (3.9)
Steinhardt and Turok [19, 20] have proposed a cyclic universe model
inspired by higher-dimensional brane-worlds. The effective four-dimensional
theory is a theory in which matter is non-minimally coupled to gravity.
3.5
Modifying cosmological assumptions
Several people have started to investigate whether abandoning some of the
standard cosmological assumptions described in section 3.2.1 can explain all
observations of today without introducing dark energy or modified gravity.
This started around 2000 with Tomita [21, 22] and Cèlèrier [23], who explored
whether an inhomogeneous matter dominated universe may show accelerated
expansion when considered in a homogeneous framework.
During the last year, the interest for inhomogeneous universe model has
boosted. There have been several ideas regarding what kind of inhomogeneities are needed, from superhorizon perturbations, see, e.g., Kolb et al. [24],
to the tiny perturbations of, e.g., Buchert [25] and Räsanen [26].
Following Tomita, Alnes et al. [27] have investigated models where we
live off-center in a spherically symmetric underdense region of space, see for
example [28] and references therein. For a short review of inhomogeneous
universe models see [29].
Based on observed anisotropies in the CMB spectrum, Jaffe et al. [30]
have proposed that we live in a Bianchi type VII h universe. This cosmology is
homogeneous, but abandon isotropy. However, the model needs dark energy
to explain acceleration, and is also ruled out by the small-scale observations
which shows that the universe is flat.
3.6
Quantum effects
Vacuum energy has been suggested as a physical explanation of the cosmological constant. Interpreting the constant energy density ρ Λ as a quantum
field vacuum energy eliminates the need of putting Λ by hand into (3.1) or
(3.2). The classical action is kept (3.2), but when the underlying fundamental
physical theory is quantized we get a constant vacuum energy density. The
only problem is that this vacuum energy density is infinitely large. Trying
to regulate the vacuum energy by introducing a Planck-scale cutoff gives a
value for ρΛ about 120 orders of magnitude larger then the observed value.
3.6 Quantum effects
Assuming extra compact dimensions, the vacuum energy from the quantum
fields propagating in these dimensions gets an extra, finite term – the Casimir
energy. In a D-dimensional spacetime with one compact dimension of size
L, the Casimir energy density is proportional to L −D and similar for more
compact dimensions. Since the size of the extra dimensions generally varies
with time, this corresponds to a time-varying cosmological “constant”, or a
scalar field. This may give us a cosmology in accordance with current observations of an expanding universe. An example of this is the six-dimensional
model by Albrecht et al. [31, 32] from 2002.
27
28
Cosmology
Chapter 4
Epilogue
Cosmology today is in trouble
The main advantage of the standard cosmological model today – the ΛCDM
model – is that it is the simplest model fitting all data. As it has been
formulated, the ΛCDM model has only two problems: the Λ and the CDM.
Keeping the ΛCDM model means that we must for now accept that about
95% of the content of universe is both unseen and unknown.
Cosmology today is interesting
The equivalence of the above statement is to say that cosmology has never
been more interesting than today. There is a multitude of theoretical possibilities to investigate and no answer book. And with new and better data
pouring in, we will be able to better distinguish between models.
Cosmology today needs fundamental physics
First when cosmology is completely founded on fundamental physics, cosmology can truly be understood. It is interesting that the largest physical
system, the universe as a whole, seems to be among those we know least
about.
30
Epilogue
List of papers
Paper I Dynamics of the scalar field in 5-dimensional Kaluza-Klein theory.
I. K. Wehus and F. Ravndal.
Int.J.Mod.Phys. A19, 4671 (2004).
Paper II Geometrical constraints on dark energy.
A. K. D. Evans, I. K. Wehus, Ø. Grøn, and Ø. Elgarøy.
Astron.Astrophys. 430, 399 (2005).
Paper III Black-body radiation in extra dimensions.
H. Alnes, F. Ravndal, and I. K. Wehus.
[quant-ph/0506131].
Paper IV Electromagnetic Casimir energy with extra dimensions.
H. Alnes, K. Olaussen, F. Ravndal, and I. K. Wehus.
Phys.Rev. D (In press) (2006), [quant-ph/0607081].
Paper V Resolution of an apparent inconsistency in the higher-dimensional
electromagnetic Casimir effect.
H. Alnes, K. Olaussen, F. Ravndal, and I. K. Wehus.
[hep-ph/0610081].
Paper VI Gravity coupled to a scalar field in extra dimensions.
I. K. Wehus and F. Ravndal.
(2006), [gr-qc/0610048].
Bibliography
[1] C. Brans and R. H. Dicke. Mach’s principle and a relativistic theory of
gravitation. Physical Rewiew, 124(3):925–935, nov 1961.
[2] K. Tywonik and F. Ravndal, Scalar Field Fluctuations between Parallel
Plates quant-ph/0408163.
[3] C. G. Callan, S. Coleman, and R. Jackiw, A New Improved Energy Momentum Tensor Ann. Phys. N.Y.59, 42 (1970).
[4] E. Huggins, Ph.D. thesis, Caltech, 1962 (unpublished).
[5] M.E. Peskin and D.V.Schroeder. An Introduction to Quantum Field
Theory. Perseus Books, 1995.
[6] A. G. Riess et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron.J. 116, 1009
(1998).
[7] S. Perlmutter et al. Measurements of Omega and Lambda from 42 HighRedshift Supernovae. Astrophys. J.517, 565 (1999).
[8] A. G. Riess et al. Type Ia Supernova Discoveries at z>1 From the Hubble
Space Telescope: Evidence for Past Deceleration and Constraints on
Dark Energy Evolution. Astrophys.J. 607, 665 (2004).
[9] D. N. Spergel et al. First Year Wilkinson Microwave Anisotropy Probe
(WMAP) Observations: Determination of Cosmological Parameters Astrophys.J.Suppl. 607, 175 (2003).
[10] A. Kamenshchick, U. Moschella, and V. Pasquier. An alternative to
quintessence. Phys. Lett.B511, 265 (2001).
[11] N. Bilic, G.G. Tupper, and R. Viollier. Unification of Dark Matter and
Dark Energy: the Inhomogeneous Chaplygin Gas. 2002, Phys. Lett. B,
535, 17
[12] Ø. Grøn. A new standard model of the universe. Eur. J. Phys.23, 135
(2002).
34
BIBLIOGRAPHY
[13] S.M. Carrol, V. Duvvuri, M. Trodden, and M.S. Turner. Is Cosmic
Speed-Up Due to New Gravitational Physics? Phys.Rev. D70, 043528
(2004).
[14] S. Nojiri and S.D. Odintsov. Modified gravity with ln R terms and cosmic acceleration. Gen.Rel.Grav. 36, 1765 (2004).
[15] S. Capozziello, V. F. Cardone, S. Carloni, and A. Troisi. Curvature
quintessence matched with observational data. Int.J.Mod.Phys. D12 ,
1969 (2003).
[16] S. Capozziello, V.F. Cardone, and A. Troisi. Reconciling dark energy
models with f(R) theories. Phys.Rev. D71, 043503 (2005).
[17] V. Faraoni. Big Smash of the universe. Phys.Rev. D68, 063508 (2003).
[18] R.R. Caldwell. A Phantom Menace?
Cosmological consequences
of a dark energy component with super-negative equation of state.
Phys.Lett.B545,23 (2002).
[19] P.J. Steinhardt and N. Turok. A Cyclic Model of the Universe.
Science295-5572, 1436 (2002).
[20] P.J. Steinhardt and N. Turok. Cosmic Evolution in a Cyclic Universe.
Phys.Rev. D65, 126003 (2002).
[21] K. Tomita. Bulk flows and CMB dipole anisotropy in cosmological void
models Astrophys. J. 529, 26 (2000).
[22] K. Tomita. Anisotropy of the Hubble Constant in a Cosmological Model
with a Local Void on Scales of 200 Mpc. Prog. Theor. Phys (2001).
[23] M.-N. Cèlèrier. Do we really see a cosmological constant in the supernovae data ? Astron.Astrophys 353,63 (2000).
[24] E. W. Kolb, S. Matarrese, A. Notari, and A. Riotto. hep-th/0503177
(2005).
[25] T. Buchert. On globally static and stationary cosmologies with
or without a cosmological constant and the Dark Energy problem.
Class.Quant.Grav.23, 817 (2006).
[26] S. Räsanen. Accelerated expansion from structure formation astroph/0607626 (2006)
[27] H. Alnes, M. Amarzguioui and Ø. Grøn. An inhomogeneous alternative
to dark energy? Phys.Rev. D73, 083519 (2006).
BIBLIOGRAPHY
[28] H. Alnes and M. Amarzguioui. The supernova Hubble diagram for offcenter observers in a spherically symmetric inhomogeneous universe.
astro-ph/0610331 (2006)
[29] M.-N. Cèlèrier. Accelerated-like expansion: inhomogeneities versus dark
energy astro-ph/0609352 (2006)
[30] T. R. Jaffe et al. Evidence of vorticity and shear at large angular scales
in the WMAP data: a violation of cosmological isotropy? Astrophys.J.
629, L1 (2005)
[31] A. Albrecht, C. P. Burgess, F. Ravndal, and C. Skordis. Exponentially
large extra dimensions. Phys.Rev. D65, 123506 (2002).
[32] A. Albrecht, C. P. Burgess, F. Ravndal, and C. Skordis. Natural quintessence and large extra dimensions. Phys.Rev. D65, 123507 (2002).
35
36
BIBLIOGRAPHY
Part II
Papers
Paper I
October 21, 2004 11:6 WSPC/139-IJMPA
02060
International Journal of Modern Physics A
Vol. 19, No. 27 (2004) 4671–4685
c World Scientific Publishing Company
DYNAMICS OF THE SCALAR FIELD IN
FIVE-DIMENSIONAL KALUZA KLEIN THEORY
INGUNN KATHRINE WEHUS and FINN RAVNDAL
Institute of Physics, University of Oslo, N-0316 Oslo, Norway
Received 24 February 2003
Using the language of differential forms, the Kaluza–Klein theory in 4 + 1 dimensions is
derived. This theory unifies electromagnetic and gravitational interactions in four dimensions when the extra space dimension is compactified. Without any ad hoc assumptions
about the five-dimensional metric, the theory also contains a scalar field coupled to the
other fields. By a conformal transformation the theory is transformed from the Jordan
frame to the Einstein frame where the physical content is more manifest. Including a
cosmological constant in the five-dimensional formulation, it is seen to result in an exponential potential for the scalar field in four dimensions. A similar potential is also found
from the Casimir energy in the compact dimension. The resulting scalar field dynamics
mimics realistic models recently proposed for cosmological quintessence.
Keywords: Extra dimensions; gravity; conformal transformations; unified theories; scalar
fields.
1. Introduction
A unified formulation of Einstein’s theory of gravitation and theory of electromagnetism in four-dimensional space–time was first proposed by Kaluza1 by assuming a
pure gravitational theory in a five-dimensional space–time. The metric components
were to be independent of the fifth coordinate.2 This so-called “cylinder condition”
was a few years later explained by Klein by invoking arguments from the newly
established quantum mechanics when the extra dimension was compactified on a
circle S 1 with a microscopic radius.3 In the following years it was studied by a large
number of authors and also extended to higher dimensions in order to incorporate
non-Abelian gauge theories.4 In the last couple of years it has seen a rebirth as the
mathematical implementation of the exciting possibility that our Universe can have
extra dimensions which in principle can be of macroscopic size.5 – 9
The field content of the original Kaluza–Klein theory is given by the fivedimensional metric ḡM N where the indices M , N = 0, 1, 2, 3, 4. Denoting the indices
of our four-dimensional space–time by Greek indices, its metric is given by the
components ḡµν while the electromagnetic vector potential is given by the components ḡµ4 = ḡ4µ . The remaining spatial component ḡ44 was for no good reason set
4671
October 21, 2004 11:6 WSPC/139-IJMPA
4672
02060
I. K. Wehus & F. Ravndal
equal to a constant by Kaluza1 and kept that way by the subsequent authors. This
assumption was apparently first abandoned by Jordan10 and then later by Thiry11
who showed that ḡ44 corresponds to a scalar field in our space–time. At that time
there was no need or place for such a field, but today we realize that it is a generic
feature of almost any extension of Einstein’s theory of gravity. From the experimental point of view it can be related to the physics behind the dark energy12 in
the Universe in the form of a cosmological constant13 or variable quintessence.14
The main motivation behind the present contribution is to investigate some of
the physical properties and manifestations of such a scalar field. In order to make
the presentation more accessible, we present the reduction of the five-dimensional
theory down to our four-dimensional space–time in more detail than is generally
available in the literature. We will take advantage of the simplifications which follow
from using differential forms as was first done by Thiry.11
After this reduction, we find that the scalar field couples directly to the gravitational field. To get this part on the standard form, we perform a conformal transformation so that the gravitational part is described by the usual Einstein–Hilbert
action. In this frame the scalar field couples only to the Maxwell field. It is shown
that this form of the full action can be obtained by performing a conformal transformation on the five-dimensional metric. The classical equations of motion for the
different fields are then derived and discussed.
If there is a cosmological constant in the five-dimensional space–time, we show
that it gives rise to an exponential potential for the scalar field. Such potentials
are phenomenologically very interesting in cosmology where they can drive the
observed acceleration of the Universe.14 A potential of the same form also follows
from the Casimir energy in the compactified, extra dimension. Including both of
these effects, we have an effective potential which is the sum of two exponential
terms. Adjusting the parameters in this potential, one can then obtain a realistic
cosmology of an accelerating Universe.15,16
2. Reduction to Four Dimensions
In the five-dimensional space–timea we have the line element ds̄2 = ḡM N dxM dxN
where the coordinates are xM = (xµ , x4 ≡ y). It is convenient to define ḡ44 = h
and ḡ4µ = hAµ where the fields h(x) and Aµ (x) at this stage depend on all five
coordinates. Our four-dimensional space–time is orthogonal to the basis vector ~e 4
in the extra direction. It is therefore spanned by the four basis vectors
~e µ⊥ = ~eµ − ~eµk = ~eµ −
and is endowed with the metric
gµν = ~eµ⊥ · ~eν⊥ = ḡµν −
a The
ḡµ4
~eµ · ~e4
~e4 = ~eµ −
~e 4
~e4 · ~e4
ḡ44
(1)
ḡ4µ ḡ4ν
= ḡµν − hAµ Aν .
ḡ44
(2)
flat metric in this space–time has the diagonal components ηM N = (−1, 1, 1, 1, 1).
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In this way we then have the following splitting of the five-dimensional metric
"
#
gµν + hAµ Aν hAµ
ḡM N =
(3)
hAν
h
and correspondingly for the contravariant components
#
" µν
−Aµ
g
MN
ḡ
=
−Aν h−1 + Aµ Aµ
(4)
which satisfy ḡM N ḡ N P = δM P . The line element in this space–time thus takes the
form
ds̄2 = ḡµν dxµ dxν + 2ḡµ4 dxµ dx4 + ḡ44 dx4 dx4
= gµν dxµ dxν + h(dy + Aµ dxµ )2 .
(5)
The appearance of the gauge potential 1-form in the last term is characteristic
for Kaluza–Klein theories also with additional extra dimensions.4 At this stage we
impose the cylinder condition ḡM N,4 = 0 which means that the fields gµν (x), Aµ (x)
and h(x) are independent of the fifth coordinate y.
2.1. Vierbeins and connection forms
In the four-dimensional space–time we introduce the vierbeins V µ̂ν and their
inverse V µν̂ satisfying V µ̂λ V λν̂ = δ µ̂ν̂ and V µν̂ V ν̂λ = δ µλ . The metric is thus
gµν = V α̂µ V β̂ν ηα̂β̂ where ηα̂β̂ is the four-dimensional Minkowski metric in this
space–time. Using these vierbeins we find the orthonormal basis 1-forms as
ω µ̂ = V µ̂ν dxν .
(6)
In the fifth direction we see from the line element (5) that the basis 1-form is
√
ω 4̂ = h(dy + Aµ dxµ ) .
(7)
Now using standard methods,17 we can calculate the connection 1-forms and the
curvature 2-forms needed to find the Riemann and Ricci tensors which enter the
Einstein–Hilbert action. As an example, we find
dω 4̂ =
1 −1/2
h
h,µ dxµ ∧ (dy + Aµ dxµ ) + h1/2 Aµ,ρ dxρ ∧ dxµ
2
4̂
4̂
since d2 y = 0. The result can be written in the form dω 4̂ = −Ω̄ µ̂ ∧ ω µ̂ where Ω̄ µ̂
is the corresponding five-dimensional connection 1-form. By using the antisymmetry
of dxρ ∧ dxµ to introduce the field strength Fµν = Aν,µ − Aµ,ν , we find
1
1√
4̂
h,µ̂ ω 4̂ ,
(8)
hFµ̂ν̂ ω ν̂ +
Ω̄ µ̂ =
2
2h
where h,µ̂ = h,ν V νµ̂ . Using the antisymmetry in the two orthonormal indices, we
µ̂
then also have the value for Ω̄
4̂
which we will need in the following.
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The exterior derivative of the remaining basis forms is similarly given in five
dimensions as
µ̂
µ̂
dω µ̂ = −Ω̄
ν̂
∧ ων̂ − Ω̄
∧ ω 4̂
4̂
while taken in the four-dimensional space–time it would just be dω µ̂ = −Ωµ̂ ν̂ ∧ ω ν̂
where Ωµ̂ ν̂ now is the the connection 1-form in four dimensions. We thus have the
relationship
1√
µ̂
hFν̂ µ̂ ω4̂
(9)
Ω̄ ν̂ = Ωµ̂ ν̂ +
2
between the connections in these two space–times.
2.2. Curvature forms and tensors
The curvature 2-forms are defined by the structure equation
1 µ̂
R
ω ρ̂ ∧ ωσ̂
(10)
2 ν̂ ρ̂σ̂
in the four-dimensional space–time and correspondingly in five dimensions. Their
components Rµ̂ ν̂ ρ̂σ̂ form the Riemann curvature tensor in the orthonormal frame
we are working in. In the same frame the corresponding expansion of the connection
forms is
Rµ̂ ν̂ = dΩµ̂ ν̂ + Ωµ̂ λ̂ ∧ Ωλ̂ ν̂ ≡
Ωµ̂ ν̂ = Ωµ̂ ν̂σ dxσ ,
(11)
where Ωµ̂ ν̂σ are the connection coefficients. These will enter the calculation together
with partial derivatives of the field strengths F µ̂ν̂ to give the covariant derivative
F µ̂ν̂;σ = F µ̂ν̂,σ + Ωµ̂ ρ̂σ F ρ̂ν̂ − Ωρ̂ ν̂σ F µ̂ρ̂
(12)
and similarly for the other components.
We calculate first the curvature form R̄µ̂ 4̂ from
µ̂
R̄µ̂ 4̂ = dΩ̄
µ̂
4̂
+ Ω̄
ν̂
ν̂
∧ Ω̄
4̂
.
Along the direction ω ρ̂ ∧ ω σ̂ we then find the curvature tensor components
1√
1
R̄µ̂ 4̂ρ̂σ̂ =
hFσ̂ ρ̂ ;µ̂ + √ 2h;µ̂ Fσ̂ ρ̂ + h;ρ̂ Fσ̂ µ̂ − h;σ̂ Fρ̂ µ̂
2
4 h
(13)
(14)
and similarly
R̄µ̂ 4̂ρ̂4̂ =
1 µ̂ν̂
1 ;µ̂
1
F Fρ̂ν̂ −
h ;ρ̂ + 2 h;µ̂ h;ρ̂
4
2h
4h
(15)
in the direction of ω ρ̂ ∧ ω 4̂ . These are all the components of the Riemann tensor
involving the fifth index, since the Riemann tensor is antisymmetric in the first two
and in the last two indices. We have here introduced the notation Fσ̂ρ̂;µ̂ = Fσ̂ ρ̂;ν V νµ̂
for the covariant derivative of the field tensor. We have also simplified the result
using the Bianchi identity Fσ̂ ρ̂;µ̂ + Fρ̂µ̂;σ̂ + Fµ̂σ̂;ρ̂ = 0.
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The remaining components of the curvature tensor follow now from the 2-form
µ̂
R̄µ̂ ν̂ = dΩ̄
µ̂
ν̂
+ Ω̄
λ̂
∧ Ω̄
λ̂
µ̂
ν̂
+ Ω̄
4̂
∧ Ω̄
4̂
ν̂
(16)
which can be evaluated along the same lines. It gives
1
R̄µ̂ ν̂ ρ̂σ̂ = Rµ̂ ν̂ ρ̂σ̂ + h 2Fν̂ µ̂ Fρ̂σ̂ − Fρ̂ µ̂ Fσ̂ ν̂ − Fσ̂ µ̂ Fν̂ ρ̂
(17)
4
which has the correct antisymmetry in the first and last two indices. We also notice
that the symmetry R̄µ̂ν̂ ρ̂σ̂ = R̄ρ̂σ̂ µ̂ν̂ is satisfied, as well as R̄µ̂ [ν̂ ρ̂σ̂] = 0.
The Ricci curvature tensor is defined in four dimensions to be Rν̂ σ̂ = Rµ̂ ν̂ µ̂σ̂
and similarly in five dimensions. It is symmetric in its two indices. We find its
components to be
1
1 ;µ̂
1 ;µ̂
R̄4̂4̂ = hF µ̂ν̂ Fµ̂ν̂ −
h
+
h h;µ̂ ,
(18)
4
2h ;µ̂ 4h2
1√
3
R̄ν̂ 4̂ =
(19)
hFν̂ µ̂ ;µ̂ + √ Fν̂ µ̂ h;µ̂ ,
2
4 h
1
1
1
R̄ν̂ σ̂ = Rν̂ σ̂ − hF µ̂ν̂ Fµ̂σ̂ −
h;ν̂ σ̂ + 2 h;ν̂ h;σ̂ .
(20)
2
2h
4h
For the scalar curvature R̄ = R̄µ̂ µ̂ expressed in a coordinate basis we then have
1
1
1
(21)
R̄ = R − hF µν Fµν − ∇2 h + 2 (∇µ h)2
4
h
2h
in agreement with Thiry.11 Here we have introduced the ∇-operator for the covariant derivative and ∇2 = ∇µ ∇µ is the four-dimensional d’Alembertian operator.
Needless to say, it is the appearance of the Maxwell Lagrangian in this higherdimensional curvature first derived by Kaluza and Klein, that we still do not understand the full significance of.
2.3. Einstein Hilbert action and equations of motion
These geometrical considerations become physical when we postulate that gravitation in the five-dimensional space is governed by the corresponding Einstein–Hilbert
action
Z
√
1 3
S = M̄
(22)
d5 x −ḡR̄ ,
2
where M̄ is the Planck mass in this space and R̄ is the Ricci curvature scalar (21). In
principle there can also be an additional term here corresponding to a cosmological
constant. Its implications will be considered in Subsec. 4.1. From the 4 + 1 split
of the metric in (3) we see that its determinant is simply ḡ = hg where g is the
determinant of the four-dimensional metric gµν . Using this in the action (22) and
then integrating out the fifth coordinate, we find the action
Z
√ √
1
1
1
1
(23)
S = M 2 d4 x −g h R − hF µν Fµν − ∇2 h + 2 (∇µ h)2 ,
2
4
h
2h
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when the extra dimension is a microscopic circle of radius a so that M 2 = 2πaM̄ 3
becomes the ordinary, four-dimensional Planck constant. The two last terms form
a total derivative,
√ 1 2
√
1
h ∇ h − 2 (∇µ h)2 = 2∇2 h .
(24)
h
2h
Assuming that h disappears far away these terms can be neglected from the action.
We are therefore left with the final result
Z
√
1 2
1 µν
4 √
S= M
d x −g h R − hF Fµν
(25)
2
4
which is the Kaluza–Klein action.
In the general case where the scalar field h(x) varies with position, the effective
gravitational constant given by the coefficient of the Ricci scalar in (25), is no longer
a constant, but varies with time and position in the four-dimensional space–time.
Electromagnetic interactions described by the Maxwell part, will similarly have
a variable coupling strength. For this reason the theory seems to be in disagreement with present-day observations although there have been recent indications
that the fine-structure constant may vary over cosmological time scales. 18 Kaluza–
Klein theory thus belongs to a wider class of fundamental theories characterized
by the extension of Einstein’s tensor theory of gravity to include also the effect of
scalar interactions. Such scalar-tensor theories of gravitation were constructed by
Jordan19 and later shown by Brans and Dicke20 to be compatible with gravitational
experiments and cosmological tests.
The classical equations of motion for the three fields can be derived from the
action (25). But it is simpler to use the five-dimensional action (22) which gives
rise to the equation of motion R̄M N = 0.
We have already the components of the Ricci tensor in orthonormal basis (18)–
(20). We now transform these to coordinate basis using the√ vierbeins V µ̂ν , along
√
with the rest of the fünfbein components V µ̂4 = 0, V 4̂µ = hAµ and V 4̂4 = h,
which we read out from (6) and (7). We then find
1
1
1
(∇µ h)2 ,
(26)
R̄44 = h2 F µν Fµν − ∇2 h +
4
2
4h
3
1
R̄µ4 = h∇ν Fµν + Fµν ∇ν h
2
4
1
1
1
+ Aµ h2 F µν Fµν − ∇2 h +
(27)
(∇µ h)2 ,
4
2
4h
1
1
1
hF σµ Fσν −
∇µ ∇ν h + 2 (∇µ h)(∇ν h)
2
2h
4h
1 2
1
1 2 µν
2
+ Aµ Aν h F Fµν − ∇ h +
(∇µ h)
4
2
4h
3
3
1
1
+ Aν h∇σ Fµσ + Fµσ ∇σ h + Aµ h∇σ Fνσ + Fνσ ∇σ h . (28)
2
4
2
4
R̄µν = Rµν −
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By equating these components of the Ricci scalar to zero we find the corresponding
four-dimensional equations of motion. With the help of the identity (24), we find
from (26) for the scalar field
√
1
∇2 h = h3/2 F µν Fµν
4
(29)
while (27) combined with (29) gives the equation of motion for the Maxwell field,
∇µ Fµν = −
3
Fµν ∇µ h .
2h
(30)
Finally, for the gravitational field we find from (28) when using (29) and (30)
Rµν =
√
1
1 σ
hF µ Fσν + √ ∇µ ∇ν h .
2
h
(31)
These equations can also be found in Wesson.21 Introducing the Einstein curvature
tensor Eµν = Rµν − 21 Rgµν , we see that the last equation can be written as
√
√
1
1
1
Eµν = h F σµ Fσν − gµν F ρσ Fρσ + √ ∇µ ∇ν h − gµν ∇2 h
(32)
2
4
h
when we express the Ricci scalar R in terms of the Maxwell and scalar fields using
(21) with R̄ = 0. In the first term we recognize the energy–momentum tensor of
the electromagnetic field, while the last term must be the corresponding entity for
the scalar field in this representation.
From the equation of motion (29) for the scalar field we see that it can take a
constant value, which can be chosen to be h = 1, provided the accompanying gauge
field satisfies the special condition F µν Fµν = 0. The magnitudes of the electric
and magnetic fields must therefore be the same everywhere. This rather strong and
unnatural condition was imposed for many years in investigations of the Kaluza–
Klein theory4 since there did not seem to be a real physical need for a scalar field on
the same footing as the electromagnetic and gravitational fields. Today, however,
the situation is different. In fact, scalar fields are at the core of the Higgs mechanism
in particle physics, cosmological inflation in the early universe and dark energy in
the late universe. In the following we will therefore keep the scalar field nonconstant
and study some of its physical implications.
3. Conformal Transformations to the Einstein Frame
A basic assumption in Einstein’s general theory of relativity is that all observers
are equipped with standard measuring rods and clocks. The properties of these
rods and clocks are coded into the components of the metric tensor gµν and has
the consequence that the gravitational action is just given by the volume integral
of the Ricci curvature scalar. This is obviously the case for the underlying, fivedimensional theory described by (22). But in the resulting,√four-dimensional theory
(25) we see that this term is multiplied by the scalar field h which is generally not
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constant. This corresponds to using nonstandard measuring rods and clocks. We can
now adjust these by changing the metric at every point by a Weyl transformation
gµν → Ω2 gµν
(33)
and choosing the scale factor Ω(x) appropriately. In a D-dimensional space–time
this results in the corresponding change
Rµν → Rµν − Ω−1 ∇2 Ωgµν − (D − 2)Ω−1 ∇µ ∇ν Ω
− (D − 3)Ω−2 (∇ρ Ω)2 gµν + 2(D − 2)Ω−2 ∇µ Ω∇ν Ω
(34)
of the Ricci tensor. This gives a change in the Ricci scalar
R → Ω−2 R − 2(D − 1)Ω−3 ∇2 Ω − (D − 1)(D − 4)Ω−4 (∇µ Ω)2
(35)
as shown in Ref. 22.
3.1. Weyl transformations
√
√
When we are in D = 4 dimensions −g → Ω4 −g while the Ricci scalar changes
according to (35). The first term in (25) thus changes as
Z
Z
√
√
√ √
d4 x −g hR → d4 x −gΩ4 h Ω−2 R + · · · .
(36)
For the coefficient of R to take the canonical value we must therefore choose
Ω = h−1/4 . Including the Maxwell term in (25) and the second term of the Weyl
transformation (35) we thus find the transformed Kaluza–Klein action
Z
15 1
1 23 µν
31 2
1 2
2
4 √
∇ h−
(∇µ h) . (37)
d x −g R − h F Fµν +
S= M
2
4
2h
8 h2
It can be simplified by combining the last two terms by a partial integration which
results in
Z
1 2
1 3 µν
3 1
2
4 √
2
(∇µ h) .
S= M
(38)
d x −g R − h F Fµν −
2
4
8 h2
After this Weyl transformation we are now in the Einstein frame. By construction
the gravitational part in the first term of the action has now the canonical form. The
last term describes a massless scalar field which is coupled to the electromagnetic
field in the second term. This is the physical content of the Einstein frame theory.
For similar transformations between the Jordan frame and the Einstein frame for
dilatonic brane-worlds see Ref. 23.
We could have achieved the same result by performing the Weyl transformation
(33) directly on the five-dimensional metric appearing in (22). Since we now have
√
√
−ḡ → Ω5 −ḡ, we find that the transformation needed is Ω = h−1/6 . Again
using (35) where now also the last term contributes and the gradients act in five
dimensions, we find
Z
Z
√
√
1
4 1 2 0
0 2
∇
h
−
d5 x −ḡR̄ → d4 x −g R̄ +
(∇
h
)
.
(39)
µ
3 h0
h0 2
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For later convenience we have here denoted the scalar field h0 instead of h. By
using (21) for the five-dimensional scalar curvature, we find the transformed action
integral
Z
√
1
1
1 1 2 0 1 1
0 2
(∇
h
)
.
(40)
S = M 2 d4 x −g R − h0 F µν Fµν +
∇
h
−
µ
2
4
3 h0
2 h0 2
Again we can use a partial integration to combine the two last terms, as in (38). The
final result for the Kaluza–Klein action after a five-dimensional Weyl transformation
is then
Z
√
1 1
1
1
0 2
.
(41)
S = M 2 d4 x −g R − h0 F µν Fµν −
(∇
h
)
µ
2
4
6 h0 2
We see that there is full agreement between the four-dimensional Weyl transformation gµν → h−1/2 gµν and the five-dimensional counterpart ḡM N → h−1/3 ḡM N . If
we put h0 = h3/2 in (41) this equation is transformed into (38).
This is also easily understood directly from the five-dimensional metrical structure (3). After the four-dimensional Weyl transformation the five-dimensional metric becomes


#
" −1
3
3
gµν + h 2 Aµ Aν h 2 Aµ
h 2 gµν + hAµ Aν hAµ
1
.
= h− 2 
3
3
2
2
hAν
h
h Aν
h
Introducing here h0 = h3/2 we then have the Weyl transformation of the fivedimensional metric used above.
3.2. Canonical fields
Although the gravitational part of the action now has the standard form, the kinetic
energies of scalar and electromagnetic fields do not have their canonical forms.
However, this is simple to achieve. In the action (41) we introduce
h0 = e
√
6φ/M
,
(42)
where now φ(x) is a scalar field with canonical normalization. Similarly, we redefine
the electromagnetic field by
√
Aµ → 2Aµ /M
(43)
and the Kaluza–Klein action takes its final form
Z
1 2
1 √6φ/M µν
1
2
4 √
S = d x −g M R − e
F Fµν − (∇µ φ) .
2
4
2
(44)
When the scalar field φ has values much less than the Planck mass M , both the
electromagnetic field and the scalar field are seen to be free.
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3.3. Equations of motion in Einstein frame
1
Now that we have performed the Weyl transformation ḡM N → h− 3 ḡM N the
equations of motion are no longer the same as Eqs. (29), (30) and (32) in the
Jordan frame. We can find the new equations of motion from the transformed fourdimensional action (41). Varying this action with respect to h gives
1
3 2
h Fµν F µν + (∇µ h)2 .
4
h
Similarly the equation for Aµ is found to be
∇2 h =
1
∇ν Fµν = − Fµν ∇ν h
h
while varying with respect to gµν gives us
1 scalar
1
el.mag.
hTµν
+ 2 Tµν
.
Eµν =
2
3h
(45)
(46)
(47)
el.mag.
Here Tµν
= F αµ Fαν − 41 F ρσ Fρσ gµν is the ordinary electromagnetic energy–
scalar
momentum tensor, while Tµν
= ∇µ h∇ν h− 21 (∇σ h)2 gµν is the energy–momentum
tensor for a free scalar field. By performing the canonical transformations (42) and
(43) we then obtain the equations of motion in canonical normalization:
r
3 √6φ/M
2
Fµν F µν ,
(48)
e
∇ φ=
8M 2
√
6
ν
∇ Fµν = −
Fµν ∇ν φ ,
(49)
M
i
1 h √
el.mag.
scalar
+ Tµν
.
(50)
Eµν = 2 e 6φ/M Tµν
M
We can also get Eqs. (45)–(47) by varying the five-dimensional action after the
0
0
Weyl transformation. This gives R̄M
N = 0, where R̄M N is the transformed Ricci
2
− 31
and we get the following
tensor which we find from (34). In our case Ω = h
expression for the transformed Ricci tensor:
5 ¯
1 ¯
¯N h
(∇P h)2 ḡM N −
∇M h∇
4h2
12h2
1 ¯2
1 ¯ ¯
∇M ∇N h +
∇ hgM N .
(51)
+
2h
6h
Here we have barred the covariant derivative to remind the reader that it is the
covariant derivative with respect to the five-dimensional metric. The relationship
¯ 2 h = ∇2 h +
between the d’Alembertian operator in four and five dimensions is ∇
1
2
2h (∇µ h) . By computing a couple of Christoffel symbols we find
0
R̄M
N = R̄M N −
¯4 ∇
¯ 4 h = 1 (∇µ h)2 ,
∇
2
¯4 ∇
¯ν h = 1 (∇µ h)2 Aν − 1 h∇µ hF µ ,
∇
ν
2
2
(52)
(53)
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¯µ ∇
¯ν h = ∇µ ∇ν h + 1 (∇σh)2 Aµ Aν
∇
2
1
− h∇ρ h(Aµ Fνρ + Aν Fµρ ) .
2
4681
(54)
0
When we now equate the various components of R̄M
N to zero and use the expressions (26)–(28) for the untransformed Ricci tensor, we end up with Eqs. (45)–(47).
To get the last equation we must also use the fact that the transformed Ricci scalar
R̄0 = 0.
4. Potential Energy for the Scalar Field
So far the scalar field is massless and will therefore modify the gravitational interactions over cosmological distances. This is surely unwanted and is easily avoided
by slightly enlarging the theory. By including a cosmological constant in five dimensions we will see that the scalar field develops a potential and thus also a nonzero
mass. Alternatively, we will see that the Casimir vacuum energy induced by the
presence of the compact, fifth dimension also generates a similar potential.
4.1. Cosmological constant in five dimensions
With a cosmological constant Λ̄ in the original, five-dimensional theory, the fundamental action (22) is replaced by
Z
√
1
(55)
S = M̄ 3 d5 x −ḡ(R̄ − 2Λ̄) .
2
Going through the same compactification as before, followed by the Weyl transformation in five dimensions, it immediately follows that
Z
1
√
1
1 1
1
0 2
0− 3
.
(56)
(∇
h
)
−
2
Λ̄h
S = M 2 d4 x −g R − h0 F µν Fµν −
µ
2
4
6 h0 2
Introducing here the canonically normalized fields, it takes the more informative
form:
Z
√
1 √6φ/M µν
1 2
1
2
2
4 √
− 23 φ/M
. (57)
S = d x −g M R − e
F Fµν − (∇µ φ) − M Λ̄e
2
4
2
The five-dimensional cosmological constant is thus seen to correspond to an exponential potential in four dimensions. Its absolute sign is directly set by the sign
of Λ̄.
Such an exponential potential for a scalar field addition to Einstein’s tensor
theory was first considered by Wetterich.24 It has been much studied since then in
connection with models for quintessence.14 Its cosmological evolution is completely
characterized by the coefficient of φ in the exponent.
October 21, 2004 11:6 WSPC/139-IJMPA
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02060
I. K. Wehus & F. Ravndal
4.2. Casimir energy from the compact dimension
The cosmological constant is a contribution to the vacuum energy from physics on
short scales, i.e. scales shorter than the size
Z 2πa √
√
(58)
L=
dy h = 2πae 2/3φ/M
0
of the compact dimension. But including quantum effects, the Casimir energy will
contribute to the vacuum energy at the scale L due to the confinement of the
massless field quanta in a space with a compact, fifth dimension. The corresponding
momentum is therefore quantized with the values p = (2π/L)n where n = 0, ±1,
±2, . . . , ±∞. In (58) we have chosen to calculate L after the four-dimensional Weyl
transformation, which means that
√ the correspondence between φ(x) and h(x) is
√
given by h = (e 6φ/M )2/3 = e2 2/3φ/M . The calculation of the Casimir energy is
done in lowest order perturbation theory where the field quanta represent small
oscillation around the ground state ḡM N = ηM N which follows from the classical
equations of motion.
Considering first the contribution from one such massless mode, it gives rise to
the Casimir energy
Z
∞
V
d3 k X p 2
E0 =
k + (2πn/L)2 ,
(59)
2
(2π)3 n=−∞
where V is a finite 3-volume. Using dimensional regularization, we do the momentum integral in d dimensions where we have the general formula
Z
dd k
1
Γ(N − d/2) 2 d/2−N
= (4π)−d/2
(m )
.
(60)
(2π)3 (k 2 + m2 )N
Γ(N )
In our case N = −1/2 and we find
d+1
∞
d+1 V X
2πn
−d/2 Γ − 2 E0 =
.
(4π)
2 n=−∞
L
Γ − 12
The sum over the compact quantum number n is still divergent. It is made finite
with zeta-function regularization which gives
∞
X
n=−∞
nd+1 = 2ζ(−d − 1) .
Taking now d → 3, we have
(61)
V 2
π Γ(−2)ζ(−4) .
(62)
L4
Although Γ(−2) is infinite and ζ(−4) is zero, their product is finite as follows from
the reflection formula
z −z/2
1 − z −(1−z)/2
Γ
π
ζ(z) = Γ
π
ζ(1 − z)
(63)
2
2
E0 = −
October 21, 2004 11:6 WSPC/139-IJMPA
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Dynamics of the Scalar Field in Five-Dimensional Kaluza–Klein Theory
4683
for zeta-functions. We put z = −4 in this formula and use it to rewrite E0 . The
corresponding vacuum energy density E0 = E0 /V L in five dimensions is then
E0 = −
3ζ(5)
.
4π 2 L5
(64)
This is in agreement with the result of Applequist and Chodos25 obtained by a more
indirect approach. It also follows from the calculations of Ambjørn and Wolfram26
who studied Casimir energies in different geometries. More recently the corresponding Casimir energy has been calculated by Albrecht, Burgess, Ravndal and Skordis 27
in a similar Kaluza–Klein theory in six dimensions of which two are compact in
order to derive a corresponding scalar potential. Elizalde et al. have calculated the
corresponding Casimir energy for an anti-de Sitter background.28
In five dimensions the graviton has five physical degrees of freedom. The total
Casimir energy is thus the above calculated result (64) times five. With the length L
expressed by the scalar field as in (58), we put this energy into the five-dimensional
action integral and obtain the potential
√
1 4
15ζ(5) −6√ 2 φ/M
3
(65)
e
V (φ) = 5E0 (2πa) h h− 4 = −
(2π)6 a4
in the Einstein frame. This potential term enters the Kaluza–Klein action instead
of the last term in (57). It has the same exponential form as the contribution from
the cosmological constant, but the exponent is six times as large.
In general the potential will get contributions both from the small-scale cosmological constant and from the Casimir energy. It will thus have the form
V (φ) = Ae−αφ + Be−βφ ,
(66)
where the exponents are fixed but the coefficients are unknown. We only know that
at least one of the prefactors A or B must be negative in this particular theory.
Including also higher order quantum corrections to the Casimir energy, the
simple exponential result will be modified. In the six-dimensional theory of Albrecht,
Burgess, Ravndal and Skordis27 the radiative corrections add up to a polynomial
in the field φ. This modification can be important when such potentials are used in
models for cosmological quintessence.29
5. Conclusions
The original, five-dimensional theory of Kaluza and Klein is the simplest example of
the more elaborate theories used today to describe physics with extra dimensions.
In addition to unifying the electromagnetic and gravitational fields, it also contains
a scalar field which codes the size of the single extra dimension here. Similar theories
with additional compact dimensions will contain a corresponding scalar field which
is usually called the radion.6 This represents an extension of Einstein’s tensor theory
of gravity and can have important, cosmological consequences. In this connection
the radion field appears as quintessence which can give rise to acceleration of the
October 21, 2004 11:6 WSPC/139-IJMPA
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I. K. Wehus & F. Ravndal
Universe at late times. This depends on the field dynamics which is governed by its
effective potential which appears in the Einstein frame.
Here it is pointed out that this effective potential will in the lowest order approximation used here be a sum of two exponential terms. One is resulting from the
small-scale cosmological constant in the higher-dimensional space–time while the
other is induced as a Casimir energy due to one or more compact dimensions. It
has been shown by others that potentials of such a form allow for a consistent
description of the evolution of the Universe since radiation domination until today
when the dark energy dominates and gives acceleration.15,16 It would be even more
satisfactory if also the inflationary mechanism could be explained by similar physics
from extra dimensions.
References
1. T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Phys. Mat. Klasse, 966 (1921).
2. G. Nordström, Phys. Zeit. 15, 504 (1914), had earlier used the same idea to formulate
an in four dimensions unified theory of electromagnetism and his scalar theory of
gravitation, based on Maxwell’s theory in a five-dimensional space–time.
3. O. Klein, Zeit. F. Phys. 37, 895 (1926); Nature 118, 516 (1926).
4. T. Appelquist, A. Chodos and P. G. O. Freund, Modern Kaluza–Klein Theories
(Addison-Wesley, Menlo Park, California, 1987).
5. I. Antoniadis, Phys. Lett. B246, 377 (1990).
6. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B429, 263 (1998); Phys.
Rev. D59, 086004 (1999).
7. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B436, 257
(1998).
8. T. Han, J. D. Lykken and R. Zhang, Phys. Rev. D59, 105006 (1999).
9. G. F. Giudice, R. Rattazzi and J. D. Wells, Nucl. Phys. B544, 3 (1999).
10. P. Jordan, Ann. D. Physik 1, 219 (1947).
11. Y. Thiry, Comptes Rendus 226, 216 (1948).
12. P. J. E. Peebles and B. Ratra, astro-ph/0207347.
13. S. Weinberg, astro-ph/0005265.
14. For a recent summary, see M. Doran and C. Wetterich, astro-ph/0205267.
15. T. Barreiro, E. J. Copeland and N. J. Nunes, Phys. Rev. D61, 127301 (1999).
16. A. S. Majumdar, Phys. Rev. D64, 083503 (2001).
17. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (W. H. Freeman and
Company, New York, 1973).
18. J. K. Webb et al., Phys. Rev. Lett. 87, 091301 (2001).
19. P. Jordan, Nature 164, 637 (1949); Schwerkraft und Weltall (Friedr. Vieweg & Sohn,
Braunschweig, 1952).
20. C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).
21. P. S. Wesson, Space, Time, Matter : Modern Kaluza–Klein Theory (World Scientific,
Singapore, 1999).
22. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space–Time (Cambridge University Press, Cambridge, 1973).
23. S. Nojiri, O. Obregon, S. D. Odintsov and V. I. Tkach, Phys. Rev. D64, 043505
(2001).
24. C. Wetterich, Nucl. Phys. B302, 668 (1988).
25. T. Appelquist and A. Chodos, Phys. Rev. D28, 772 (1983).
October 21, 2004 11:6 WSPC/139-IJMPA
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Dynamics of the Scalar Field in Five-Dimensional Kaluza–Klein Theory
4685
26. J. Ambjørn and S. Wolfram, Ann. Phys. (N.Y.) 147, 33 (1983).
27. A. Albrecht, C. P. Burgess, F. Ravndal and C. Skordis, Phys. Rev. D65, 123506,
123507 (2002).
28. E. Elizalde, S. Nojiri, S. D. Odintsov and S. Ogushi, hep-th/0209242.
29. A. Albrecht and C. Skordis, Phys. Rev. Lett. 84, 2076 (2000).
Paper II
Astronomy
&
Astrophysics
A&A 430, 399–410 (2005)
DOI: 10.1051/0004-6361:20041590
c ESO 2005
Geometrical constraints on dark energy
A. K. D. Evans1 , I. K. Wehus2 , Ø. Grøn3,2 , and Ø. Elgarøy1
1
2
3
Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern, 0315 Oslo, Norway
e-mail: oelgaroy@astro.uio.no
Department of Physics, University of Oslo, PO Box 1048, Blindern, 0316 Oslo, Norway
Oslo College, Faculty of Engineering, Cort Adelers gt. 30, 0254 Oslo, Norway
Received 3 July 2004 / Accepted 23 September 2004
Abstract. We explore the recently introduced statefinder parameters. After reviewing their basic properties, we calculate the
statefinder parameters for a variety of cosmological models, and investigate their usefulness as a means of theoretical classification of dark energy models. We then go on to consider their use in obtaining constraints on dark energy from present and
future supernovae type Ia data sets. We find that it is non-trivial to extract the statefinders from the data in a model-independent
way, and one of our results indicates that parametrizing the dark energy density as a polynomial of second order in the redshift
is inadequate. Hence, while a useful theoretical and visual tool, applying the statefinders to observations is not straightforward.
Key words. cosmology: theory – cosmology: cosmological parameters
1. Introduction
It is generally accepted that we live in an accelerating universe. Early indications of this fact came from the magnituderedshift relationship of galaxies (Solheim 1966), but the reality of cosmic acceleration was not taken seriously until the
magnitude-redshift relationship was measured recently using
high-redshift supernovae type Ia (SNIa) as standard candles
(Riess et al. 1998; Perlmutter et al. 1999). The observations can
be explained by invoking a contribution to the energy density
with negative pressure, the simplest possibility being Lorentz
Invariant Vacuum Energy (LIVE), represented by a cosmological constant. Independent evidence for a non-standard contribution to the energy budget of the universe comes from e.g. the
combination of the power spectrum of the cosmic microwave
background (CMB) temperature anisotropies and large-scale
structure: the position of the first peak in the CMB power spectrum is consistent with the universe having zero spatial curvature, which means that the energy density is equal to the critical
density. However, several probes of the large-scale matter distribution show that the contribution of standard sources of energy density, whether luminous or dark, is only a fraction of the
critical density. Thus, an extra, unknown component is needed
to explain the observations (Efstathiou et al. 2002; Tegmark
et al. 2004).
Several models describing an accelerated universe have
been suggested. Typically, they are tested against the SNIa data
on a model-by-model basis using the relationship between luminosity distance and redshift, dL (z), defined by the model.
Another popular approach is to parametrize classes of dark energy models by their prediction for the so-called equation of
state w(z) ≡ px /ρx , where px and ρx are the pressure and the energy density, respectively, of the dark energy component in the
model. One can then Taylor expand w(z) around z = 0. The current data allow only relatively weak constraints on the zerothorder term w0 to be derived. A problem with this approach is
that some attempts at explaining the accelerating Universe do
not involve a dark component at all, but rather propose modifications of the Friedmann equations (Deffayet 2001; Deffayet
et al. 2002; Dvali et al. 2000; Freese & Lewis 2002; Gondolo
& Freese 2003; Sahni & Shtanov 2003). Furthermore, it is possible for two different dark energy models to give the same
equation of state, as discussed by Padmanabhan (2002) and
Padmanabhan & Choudhury (2003).
Recently, an alternative way of classifying dark energy
models using geometrical quantities was proposed (Sahni et al.
2003, Alam et al. 2003). These so-called statefinder parameters are constructed from the Hubble parameter H(z) and its
derivatives, and in order to extract these quantities in a modelindependent way from the data, one has to parametrize H in
an appropriate way. This approach was investigated at length
in Alam et al. (2003) using simulated data from a SNAP1 -type
experiment. In this paper, we present a further investigation of
this formalism. We generalize the formalism to universe models with spatial curvature in Sect. 2, and give expressions for
the statefinder parameters in several specific dark energy models. In the same section, we also take a detailed look at how
the statefinder parameters behave for quintessence models, and
show that some of the statements about these models in Alam
et al. (2003) have to be modified. In Sect. 3 we discuss what can
1
see http://snap.lbl.gov
400
A. K. D. Evans et al.: Geometrical constraints on dark energy
be learned from current SNIa data, considering both direct χ2
fitting of model parameters to data, and statefinder parameters.
In Sect. 4 we look at simulated data from an idealized SNIa survey, showing that reconstruction of the statefinder parameters
from data is likely to be non-trivial. Finally, Sect. 5 contains
our conclusions.
(1)
äa
Ḣ
= − 2 − 1,
(2)
2
ȧ
H
where dots denote differentiation with respect to time t. The
proposed SNAP satellite will provide accurate determinations
of the luminosity distance and redshift of more than 2000 supernovae of type Ia. These data will permit a very precise determination of a(z). It will then be important to include also
the third derivative of the scale factor in our characterization of
different universe models.
Sahni and coworkers (Sahni et al. 2003; Alam et al.
2003) recently proposed a new pair of parameters (r, s) called
statefinders as a means of distinguishing between different
types of dark energy. The statefinders were introduced to characterize flat universe models with cold matter (dust) and dark
energy. They were defined as
q = −
...
a
Ḧ
Ḣ
=
+3 2 +1
aH 3 H 3
H
r−1
s = ·
3 q − 12
r =
H
x − 1.
H
Calculating r, making use of a = −a2 , we obtain
2
H
H
H 2
r(x) = 1 − 2 x +
+
x.
H
H
H2
3k
8πGa2
3
(H 2 + kx2 − Ωm0 H02 x3 ),
(9)
8πG
where and Ωm0 and Ωx0 are the present densities of matter and
dark energy, respectively, in units of the present critical density ρc0 = 3H02 /8πG. In the following, we will use the notation
Ωi ≡ 8πGρi (t)/3H 2 (t), Ωi0 ≡ Ωi (t = t0 ), where t0 is the present
age of the Universe, and also Ω = i Ωi . From Friedmann’s
acceleration equation
4πG ä
=−
(ρi + 3pi ),
(10)
a
3 i
where pi is the contribution to the pressure from component i,
it follows that
Ω
3 1 2
H2
k
q−
=
(H ) x − x2 − H 2 ·
(11)
px =
4πG
2
8πG 3
3
Hence, if dark energy is described by an equation of state px =
w(x)ρx , we have
w(x) =
1
k 2
2 2
3 (H ) x − H − 3 x
·
H 2 + kx2 − H02 Ωm0 x3
(12)
In the following subsections, we calculate statefinder parameters for universe models with different types of dark energy.
2.1. Models with an equation of state p = w (z)ρ
(4)
First we consider dark energy obeying an equation of state
of the form px = wρx , where w may be time-dependent.
Quintessence models (Wetterich 1988; Peebles & Ratra 1988),
where the dark energy is provided by a scalar field evolving in
time, fall in this category. The formalism in Sahni et al. (2003)
and Alam et al. (2003) will be generalized to permit universe
models with spatial curvature. Then Eq. (4) is generalized to
(5)
s=
(6)
The statefinder s(x), for flat universe models, is then found by
inserting the expressions (5) and (6) into Eq. (4). The generalization to non-flat models will be given in the next subsection.
The Friedmann equation takes the form2
8πG
k
(ρm + ρx ) − 2 ,
(7)
3
a
where ρm is the density of cold matter and ρx is the density
of the dark energy, and k = −1, 0, 1 is the curvature parameter
H2 =
2
This gives for the density of dark energy:
(3)
Introducing the cosmic redshift 1 + z = 1/a ≡ x, we have
Ḣ = −H H/a, where H = dH/dx, the deceleration parameter
is given by
q(x) =
(8)
=
The Friedmann-Robertson-Walker models of the universe have
earlier been characterized by the Hubble parameter and the
deceleration parameter, which depend on the first and second
derivatives of the scale factor, respectively:
ȧ
a
ρm = ρm0 a−3 .
ρx = ρc − ρm −
2. Statefinder parameters: Definitions
and properties
H =
with k = 0 corresponding to a spatially flat universe. The dust
component is pressureless, so the equation of energy conservation implies
Throughout this paper we use units where the speed of light c = 1.
r−Ω
,
3(q − Ω/2)
(13)
where Ω = Ωm + Ωx = 1 − Ωk , and Ωk = −k/(a2 H 2 ).
The deceleration parameter can be expressed as
1
1
[Ωm + (1 + 3w)Ωx ] = (Ω + 3wΩx ).
(14)
2
2
After differentiation of Eq. (2) and some simple algebra one
finds
q̇
(15)
r = 2q2 + q − ,
H
and further manipulations lead to
9
3 ẇ
r = Ωm + 1 + w(1 + w) Ωx −
Ωx ·
(16)
2
2H
q=
A. K. D. Evans et al.: Geometrical constraints on dark energy
2
Inserting Eq. (16) into Eq. (13) gives
s =1+w−
1 ẇ
·
3 wH
401
(17)
1.5
For a flat universe Ωm + Ωx = 1 and the expression for r simplifies to
1
r
3 ẇ
9
Ωx .
r = 1 + w(1 + w)Ωx −
2
2H
(18)
0.5
Note that for the case of LIVE, w = −1 = constant, and one
finds r = Ω, s = 0 for all redshifts. For a model with curvature and matter only one gets r = 2q = Ωm , s = 2/3. The
same result is obtained for a flat model with matter and dark
energy with a constant equation of state w = −1/3, which is
the equation of state of a frustrated network of non-Abelian
cosmic strings (Eichler 1996; Bucher & Spergel 1999). Thus,
the statefinder parameters cannot distinguish between these two
models. However, neither of these two model universes are
favoured by the current data (for one thing, they are both decelerating), so this is probably an example of academic interest
only.
For a constant w, and Ωm0 + Ωx0 = 1, the q–r plane for
different values of Ωx and w is shown in Fig. 1. Quintessence
with w = constant is called quiessence. The relation between q
and r for flat universe models with matter+quiessence is found
by eliminating Ωx between Eq. (14), with Ω = 1, and Eq. (16).
This gives
1
r = 3(1 + w)q − (1 + 3w),
2
(19)
which is the equation of the dotted straight lines in Fig. 1. When
Ωx = 1, all models lie on the solid curve given by
1
3
w+
2
2
9
r = w(1 + w) + 1,
2
or
(21)
r = 2q + q,
(22)
q =
2
(20)
in accordance with Eq. (15) since q̇ = 0 for these models. This
curve is the lower bound for all models with a constant w. For
−1 ≤ w ≤ 0, all matter+quiessence models will at any time
fall in the sector between this curve and the r = 1-line which
corresponds to ΛCDM. The results shown in Alam et al. (2003)
seem to indicate that all matter+quintessence models will fall
within this same sector as the matter+quiessence models do.
However, as we will show below, this is not strictly correct.
2.2. Scalar field models
If the source of the dark energy is a scalar field φ, as in the
quintessence models (Wetterich 1988; Peebles & Ratra 1988),
the equation of state factor w is
w=
φ̇2 − 2V(φ)
·
φ̇2 + 2V(φ)
(23)
0
–1 –0.8 –0.6 –0.4 –0.2
0
0.2
0.4
0.6
0.8
1
q
Fig. 1. The q − r-plane for flat matter+quiessence models. The horizontal curve has w = −1 (ΛCDM). Then w increases by 1/10 counterclockwise until we reach w = 1 in the upper right. When Ωx0 = 0
all models start at the point q = 0.5, r = 1 (Einstein-de Sitter model).
As Ωx0 increases every model moves towards the solid curve which
marks Ωx0 = 1. The crosses mark the present epoch.
Then,
V̇
φ̇2
+ 8πG 3 ,
H2
H
and furthermore,
r = Ω + 12πG
(24)
Ω 3
px
4πG 1 2
q − = wΩx = 4πG 2 = 2
φ̇ − V ·
2
2
2
H
H
(25)
Hence the statefinder s is
2 φ̇2 + 23 HV̇
s=
·
φ̇2 − 2V
For models with matter+quintessence+curvature,
Friedmann and energy conservation equations give
1 1
2
2
ρm − V(φ) + ρk
Ḣ = −3H +
3
2M 2 2
1 2
φ̇ = 3H 2 M 2 − ρm − V(φ) − ρk
2
ρ̇m = −3Hρm
ρ̇k = −2Hρk ,
(26)
the
(27)
(28)
(29)
(30)
and
q =
1
Ωm + 2Ωkin − Ωpot
2
(31)
MV
r = Ωm + 10Ωkin + Ωpot + 3 6Ωkin
·
ρc
(32)
As customary when discussing quintessence, we have introduced the Planck mass M 2 = 1/8πG. Furthermore, we have
defined Ωkin = φ̇2 /2ρc , and Ωpot = V(φ)/ρc . For an exponential potential, V(φ) = A exp(−λφ/M), looking at values at the
present epoch, and eliminating Ωpot0 , using Ωm0 +Ωkin0 +Ωpot0 +
Ωk0 = 1, one obtains
3
q0 = Ωm0 − (1 − Ωk0 ) + 3Ωkin0
(33)
2
r0 = (1 − Ωk0 ) + 9Ωkin0
−3λ 6Ωkin0 (1 − Ωk0 − Ωm0 − Ωkin0 )·
(34)
402
A. K. D. Evans et al.: Geometrical constraints on dark energy
12
1
10
0.8
8
6
0.6
4
r
r_0
0.4
2
0.2
0
–2
0
–4
–0.2
–6
–1
–0.5
0
0.5
1
1.5
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
q
q_0
2
Fig. 3. Time-evolution of q and r for models with matter and
quintessence with an exponential potential. The crosses mark the
present epoch. The diamond represents the present ΛCDM model.
The curve on top has λ = 0.2 and then λ increases by 0.2 for
each curve going counter-clockwise until we reach λ = 2 to
the right. The corresponding values for Ωkin today are Ωkin0 =
0.002, 0.01, 0.02, 0.04, 0.06, 0.09, 0.12, 0.165, 0.22, 0.29. The dotted
curve shows the area all matter+quiessence models must lie within
at all times. We see that all models will eventually move towards this
curve.
1.5
1
r_0
0.5
0
–0.5
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
q_0
Fig. 2. Present values of q and r for matter+quintessence with an exponential potential. Top panel: from top to bottom the different curves
have λ = −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. They all start at the point
(q0 (Ωkin = .73) = 1.595, r0 (Ωkin = .73) = 7.57) (matter+Zeldovich
gas (px = ρx )). As Ωkin decreases when we move to the left, they join
at the point (q0 (Ωkin = 0) = −0.595, r0 (Ωkin = 0) = 1) (ΛCDM,
marked with a diamond). The dotted curve shows the area all matter+quiessence models must lie within at all times. Bottom panel:
zoom-in of the figure above. Here the curve having λ2 = 2 is also
plotted (thick line).
By choosing for instance Ωm0 = 0.27 and Ωk0 = 0 we can
plot the values of q0 and r0 for varying Ωkin0 ; see Fig. 2. As
we can see from Eqs. (33)–(34), when Ωkin0 = 0, q0 and
r0 are independent of λ, and have the same values as in the
ΛCDM model. This is obvious, since taking away the kinetic
term will reduce quintessence to LIVE. However, when Ωkin0
is slightly greater then 0 we can make r0 as large or as small
as we like, by choosing |λ| sufficiently large. There is no reason all quintessence models should lie inside the constant-wcurve. However, in order to get an accelerating universe today
we must have λ2 < 2. But also for λ2 < 2 the present values of q0 and r0 can lie outside the constant-w-curve. In fact,
when we move on to the time-evolving statefinders, plotting
q and r as functions of time for given initial conditions, we
obtain plots like Fig. 3. Here we have chosen as initial conditions Ωm0 = 0.27 and Ωk0 = 0 as above, and h = 0.71. The
last initial condition, for the quintessence field, we have chosen to be φ0 = M/100 combined with the overall constant A
in the potential chosen to give Ωkin0 as stated in the caption
of Fig. 3. This corresponds to the universe being matter dominated at earlier times. When Ωpot0 Ωkin0 we have high acceleration today. Choosing Ωkin0 = 0 will again give us ΛCDM.
The three rightmost curves in the figure have λ2 > 2 and no
eternal acceleration, although the λ = 1.6 universe accelerates
today. It seems that in order to get a universe close to what we
observe, r and q for models with matter+quintessence with an
exponential potential will essentially lie within the same area
as matter+quiessence models. In Fig. 4 we have plotted the trajectories in the s0 –r0 -plane and the s0 –q0 -plane for the same
models as in Fig. 2, to be compared with Figs. 5c and 5d in
Alam et al. (2003).
Choosing instead a power-law potential V(φ) = Aφ−α gives
V = − αφ V and
q =
1
Ωm + 2Ωkin − Ωpot
2
M
r = Ωm + 10Ωkin + Ωpot − 3α
6Ωkin Ωpot .
φ
(35)
(36)
We see that for φ0 = M we get the same curves in the
q0 −r0 -plane when varying α as we got when varying λ in the
exponential potential, see Fig. 2. We also see that varying φ0 for
a given value of α is essentially the same as varying α. Figure 5
shows the q0 −r0 -plane for the case α = 2. Figure 6 shows an
example of time-evolving statefinders (φ0 = M, Ωkin0 = 0.05,
Ωm0 = 0.27 Ωk0 = 0, h = 0.71). If one compares this plot with
Fig. 1b in Alam et al. (2003), the two do not quite agree. Alam
et al. (2003) do not give detailed information about the initial
conditions for the quintessence field. Our initial conditions correspond to a universe which was matter-dominated up to now,
when quintessence is taking over.
A. K. D. Evans et al.: Geometrical constraints on dark energy
403
4
5
3
0
2
1
s_0
–5
r_0
0
–10
–1
–15
–2
–20
–3
–25
–4
–4
–2
0
2
4
6
8
–1
10
–0.5
0
0.5
1
1.5
q_0
r_0
2
Fig. 5. Present values of q and r for matter+quintessence with a powerlaw potential with α = 2. From top to bottom the different curves have
φ0 = 8M, 4M, 2M, M, M2 , M4 , M8 .The diamond represents the ΛCDM
model. The dotted curve shows the area all matter+quiessence models
must lie within at all times.
1
s_0
0
2
–1
0
r
–2
–0.6
–0.4
–0.2
0
0.2
0.4
–2
q_0
Fig. 4. Present values of the statefinder parameters and the deceleration parameter for models with matter and quintessence with an
exponential potential. The diamond represents the ΛCDM model.
Top panel: from left to right the different curves have λ =
−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5. Bottom panel: from top to bottom
the different curves have λ = −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5.
2.3. Dark energy fluid models
We will now find expressions for r and s which are valid even if
the dark energy does not have an equation of state of the form
px = wρx . This is the case e.g. in the Chaplygin gas models
(Kamenshchik, Moschella & Pasquier 2001; Bilic et al. 2002).
The expression for the deceleration parameter can be written as
1
px
q=
Ω,
(37)
1+3
2
ρx
and using this in Eq. (15) we find
3 ṗx
r = 1−
Ω
2 Hρx
1 ṗx
·
s = −
3H px
–4
–6
–1
–0.5
0
0.5
1
q
Fig. 6. Time-evolution of q and r for models with matter and
quintessence with a power-law potential. The crosses mark the present
epoch, the diamond represents the present ΛCDM model. All models
start out from the horizontal ΛCDM line and will eventually end up
as a de Sitter universe (q = −1, r = 1). The curve going deepest down
has α = 5 and moving upwards we have α = 4, 3, 2, 1. The dotted
curve shows the area all matter+quiessence models must lie within at
all times. Obviously the same is not the case for matter+quintessence
models.
The Generalized Chaplygin Gas (GCG) has an equation of state
(Bento et al. 2002)
(38)
(39)
For a universe with cold matter and dark energy one finds
9 ρx + px ∂px
r = 1+
Ω
(40)
2 ρm + ρx ∂ρx
ρx ∂px
s = 1+
·
(41)
px ∂ρx
p=−
A
,
ρα
(42)
and integration of the energy conservation equation gives
1
ρ = A + Ba−3(1+α) 1+α ,
(43)
where B is a constant of integration. This can be rewritten as
1
ρ = ρ0 As + (1 − As )x3(1+α) 1+α ,
(44)
404
A. K. D. Evans et al.: Geometrical constraints on dark energy
where ρ0 = (A+B)1/(1+α) , and As = A/(A+B). For a flat universe
with matter and a GCG, the Hubble parameter is given by
1
H 2 (x)
= Ωm0 x3 + (1 − Ωm0 ) As + (1 − As )x3(1+α) 1+α .
H0
3
β −1
(46)
where Sk (x) = sin x for k = 1, Sk (x) = x for k = 0, Sk (x) =
sinh x for k = −1, and
z
dz
·
(56)
I = H0
0 H(z)
(47)
where β = 3(1 + α), h(x) = H(x)/H0 , and
v = As + (1 − As )xβ
(48)
f (x) = Ωm0 x2 + (1 − Ωm0 )(1 − As )v
3
β −1
xβ−1 .
(49)
In the r−s plane, the GCG models will lie on curves given by
(see Gorini et al. 2003)
r =1−
9 s(s + α)
·
2
α
(50)
We note that a recent comparison of GCG models with SNIa
data found evidence for α > 1 (Bertolami et al. 2004).
2.4. Cardassian models
As an alternative to adding a negative-pressure component to
the energy-momentum tensor of the Universe, one can take the
view that the present phase of accelerated expansion is caused
by gravity being modified, e.g. by the presence of large extra dimensions. For a general discussion of extra-dimensional
models and statefinder parameters, see Alam & Sahni (2002).
As an example, we will consider the Modified Polytropic
Cardassian ansatz (MPC) (Freese & Lewis 2002; Gondolo &
Freese 2003), where the Hubble parameter is given by
H(x) = H0
Ωm0 x3 1 + u
1/ψ
,
(51)
with
−ψ
u = u(x) =
Ωm0 − 1
,
x3(1−n)ψ
(52)
and where n and ψ are new parameters (ψ is usually called q
in the literature, but we use a different notation here to avoid
confusion with the deceleration parameter). For this model, the
deceleration parameter is given by


3  1 + nu 
q(x) = 
(53)
−1
2 1+u 
and the statefinder r by

u(1 − n) − (1 + nu)
9 1 + nu 
r(x) = 1 −
1 +
4 1+u
1+u

2
(1 − n) u 
−2q
·
(1 + u)(1 + nu) 
1+z
Sk ( |Ωk0 |I),
√
H0 |Ωk0 |
dL =
β
3 Ωm0 x + (1 − Ωm0 )(1 − As ) v
x
−1
2
Ωm0 x3 + (1 − Ωm0 ) v3/β
x
3 x2 r(x) = 1 − 3 2 f (x) +
f (x),
2 h2 (x)
h (x)
q(x) =
The luminosity distance is given by
(45)
This leads to the following expressions for q(x) and r(x):
3
2.5. The luminosity distance to third order in z
(54)
(55)
The statefinder parameters appear when one expands the luminosity distance to third order in the redshift z. This expansion
has been carried out by Chiba & Nakamura (1998) and Visser
(2003). The result is

z 
1
1
dL ≈
1 + (1 − q0 )z − (1 + r0 − q0
H0
2
6


−3q20 − Ωk0 )z2 .
(57)
One can also find an expression for the present value of the
time derivative of the equation of state parameter w in terms of
the statefinder r0 . A Taylor expansion to first order in z gives
2
Ωm0 − r0
9
w(z) ≈ w0 − 1 + w0 (1 + w0 ) +
z.
(58)
3
2
Ωx0
3. Lessons drawn from current SNIa data
In this section we will consider the SNIa data presently available, in particular whether one can use them to learn about
the statefinder parameters. We will use the recent collection of
SNIa data in Riess et al. (2004), their “gold” sample consisting
of 157 supernovae at redshifts between ∼0.01 and ∼1.7.
3.1. Model-independent constraints
The approximation to dL in Eq. (57) is independent of the
cosmological model, the only assumption made is that the
Universe is described by the Friedmann-Robertson-Walker
metric. We see that, in addition to H0 , this third-order expansion of dL depends on q0 and the combination r0 − Ωk0 .
Fitting these parameters to the data, we find the constraints
shown in Fig. 7. The results are consistent with those of similar analyses in Caldwell & Kamionkowski (2004) and Gong
(2004). In Fig. 8 we show the marginalized distributions for
q0 and r0 − Ωk0 . We note that the supernova data firmly support an accelerating universe, q0 < 0 at more than 99% confidence. However, about the statefinder parameter r0 , little can
be learned without an external constraint on the curvature.
Imposing a flat universe, e.g. by inflationary prejudice or by
invoking the CMB peak positions, there is still a wide range of
allowed values for r0 . This is an indication of the limited ability of the current SNIa data to place constraints on models of
dark energy. There is only limited information on anything beyond the present value of the second derivative of the Hubble
parameter.
A. K. D. Evans et al.: Geometrical constraints on dark energy
5
405
2
1.5
4
1
3
w1
r0-Ωk0
0.5
2
0
-0.5
1
-1
0
-1.5
-1
-2
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
-2
-1.8
-1.6
q0
-1.2
-1
-0.8
-0.6
w0
Fig. 7. Likelihood contours (68, 95 and 99%) resulting from a fit of
the expansion of the luminosity distance to third order in z.
Fig. 9. Likelihood contours (68, 95 and 99%) for the coefficients w0
and w1 in the linear approximation to the equation of state w(z) of
dark energy, resulting from a fit of the expansion of the luminosity
distance to third order in z.
subsection we will consider the following models:
pdf
2
1
0
-1.4
−3
−1
1
3
5
q0 , r0−Ωk0
Fig. 8. Marginalized probability distributions for q0 (full line) and r0 −
Ωk0 (dotted line).
Under the assumption of a spatially flat universe, Ωk0 = 0,
with Ωm0 = 0.3, one can use Eq. (58) to obtain constraints
on w0 and w1 in the expansion w(z) = w0 + w1 z of the equation of state of dark energy. The resulting likelihood contours
are shown in Fig. 9. As can be seen in this figure, there is no
evidence for time evolution in the equation of state, the observations are consistent with w1 = 0. The present supernova data
show a slight preference for a dark energy component of the
‘phantom’ type with w0 < −1 (Caldwell 2002). Note, however,
that the relatively tight contours obtained here are caused by
the strong prior Ωm0 = 0.3. It should also be noted that the
third-order expansion of dL is not a good approximation to the
exact expression for high z and in some regions of the parameter space.
1. The expansion of dL to second order in z, with h and q0 as
parameters.
2. The third-order expansion of dL , with h, q0 , and r0 − Ωk0 as
parameters.
3. Flat ΛCDM models, with Ωm0 and h as parameters to be
varied in the fit.
4. ΛCDM with curvature, so that Ωm0 , ΩΛ0 (the contribution
of the cosmological constant to the energy density in units
of the critical density, evaluated at the present epoch), and
h are varied in the fits.
5. Flat quiessence models, that is, models with a constant
equation of state w for the dark energy component. The parameters to be varied in the fit are Ωm0 , w, and h.
6. The Modified Polytropic Cardassian (MPC) ansatz, with
Ωm0 , q, n, and h as parameters to be varied.
7. The Generalized Chaplygin Gas (GCG), with Ωm0 , As , α,
and h as parameters to be varied.
8. The ansatz of Alam et al. (2003),
(59)
H = H0 Ωm0 x3 + A0 + A1 x + A2 x2 ,
where we restrict ourselves to flat models, so that A0 =
1 − Ωm0 − A1 − A2 . The parameters to be varied are Ωm0 , A1 ,
A2 , and h.
Note that these models have different numbers of free parameters. To get an idea of which of these models is actually preferred by the data, we therefore employ the Bayesian
Information Criterion (BIC) (Schwarz 1978; Liddle 2004).
This is an approximation to the Bayes factor (Jeffreys 1961),
which gives the posterior probability of one model relative to
another assuming that there is no objective reason to prefer one
of the models prior to fitting the data. It is given by
3.2. Direct test of models against data
B = χ2min + Npar ln Ndata ,
The standard way of testing dark energy models against data
is by maximum likelihood fitting of their parameters. In this
where χ2min is the minimum value of the χ2 for the given model
against the data, Npar is the number of free parameters, and Ndata
(60)
406
A. K. D. Evans et al.: Geometrical constraints on dark energy
Table 1. Results of fitting the models considered in this subsection to
the SNIa data.
Model
χ2min
2. order expansion
177.1
2
187.2
3. order expansion
162.3
3
177.5
Flat ΛCDM
163.8
2
173.9
# parameters
B
ΛCDM with curvature
161.2
3
176.4
Flat + constant EoS
160.0
3
175.2
MPC
160.3
4
180.5
GCG
161.4
4
181.6
Alam et al.
160.5
4
180.7
(A0 = A2 = 0), and w = −1/3 (A0 = A1 = 0), and the luminosity distance-redshift relationship is given by
dx
1 + z 1+z
·
(63)
dL =
3
H0 1
Ωm0 x + A0 + A1 x + A2 x2
Having fitted the parameters A0 , A1 , and A2 to e.g. supernova
data using Eq. (63), one can then find q and r by substituting
Eq. (62) into Eqs. (5) and (6):
1
A2 x2 + 2A1 x + 3A0
q(x) =
(64)
1−
2
Ωm0 x3 + A2 x2 + A1 x + A0
Ωm0 x3 + A0
,
(65)
r(x) =
Ωm0 x3 + A0 + A1 x + A2 x2
and furthermore the statefinder s is found to be
is the number of data points used in the fit. As a result of the
approximations made in deriving it, B is given in terms of the
minimum χ2 , even though it is related to the integrated likelihood. The preferred model is the one which minimizes B. In
Table 1 we have collected the results for the best-fitting models. When comparing models using the BIC, the rule of thumb
is that a difference of 2 in the BIC is positive evidence against
the model with the larger value, whereas if the difference is 6
or more, the evidence against the model with the larger BIC
is considered strong. The second-order expansion of dL is then
clearly disfavoured, thus the current supernova data give information, although limited, on r0 − Ωk0 . We see that there is
no indication in the data that curvature should be added to the
ΛCDM model. Also, the last three models in Table 1 seem to
be disfavoured. One can conclude that there is no evidence in
the current data that anything beyond flat ΛCDM is required.
This does not, of course, rule out any of the models, but tells
us that the current data are not good enough to reveal physics
beyond spatially flat ΛCDM. A similar conclusion was reached
by Liddle (2004) using a more extensive collection of cosmological data sets and considering adding parameters to the flat
ΛCDM model with scale-invariant adiabatic fluctuations.
3.3. Statefinder parameters from current data
If the luminosity distance dL is found as a function of redshift from observations of standard candles, one can obtain the
Hubble parameter formally from
H(x) =
d dL
dx x
−1
·
(61)
However, since observations always contain noise, this relation
cannot be applied straightforwardly to the data. Alam et al.
(2003) suggested parametrizing the dark energy density as a
second-order polynomial in x, ρx = ρc0 (A0 + A1 x + A2 x2 ), leading to a Hubble parameter of the form
H(x) = H0 Ωm0 x3 + A0 + A1 x + A2 x2 ,
(62)
and fitting A0 , A1 , and A2 to data. This parametrization reproduces exactly the cases w = −1 (A1 = A2 = 0), w = −2/3
s(x) =
2
A 1 x + A 2 x2
,
3 3A0 + 2A1 x + A2 x2
(66)
and the equation of state is given by
w(x) = −1 +
1 A1 x + 2A2 x2
·
3 A 0 + A 1 x + A 2 x2
(67)
The simulations of Alam et al. (2003) indicated that the
statefinder parameters can be reconstructed well from simulated data based on a range of dark energy models, so we will
for now proceed on the assumption that the parametrization in
Eq. (62) is adequate for the purposes of extracting q, r and s
from SNIa data. We comment this issue in Sect. 4.
In Fig. 10 we show the deceleration parameter q and the
statefinder r extracted from the current SNIa data. The error
bars in the figure are 1σ limits. We have also plotted the model
predictions for the same quantities (based on best-fitting parameters with errors) for ΛCDM, quiessence, and the MPC.
The figure shows that all models are consistent at the 1σ level
with q and r extracted using Eq. (62). Thus, with the present
quality of SNIa data, the statefinder parameters are, not surprisingly, no better at distinguishing between the models than
a direct comparison with the SNIa data. We next look at simulated data to get an idea of how the situation will improve with
future data sets.
4. Future data sets
We will now make an investigation of what an idealized SNIa
survey can teach us about statefinder parameters and dark energy, following the procedure in Saini et al. (2004).
A SNAP-like satellite is expected to observe ∼2000 SN
up to z ∼ 1.7. Dividing the interval 0 < x ≤ 1.7 into
50 bins, we therefore expect ∼40 observations of SN in each
bin. Empirically, SNIa are very good standard candles with a
small dispersion in apparent magnitude σmag = 0.15, and there
is no indication of redshift evolution. The apparent magnitude
is related to the luminosity distance through
m(z) = M + 5 log DL (z),
(68)
where M = M0 + 5 log[H0−1 Mpc−1 ] + 25. The quantity
M0 is the absolute magnitude of type Ia supernovae, and
A. K. D. Evans et al.: Geometrical constraints on dark energy
2
407
15
1
r (from data)
q (from data)
10
0
−1
5
0
−2
−3
0
1
−5
2
0
z
1
2
z
15
2
1
r (ΛCDM)
q (ΛCDM)
10
0
−1
5
0
−2
−3
0
1
−5
2
0
1
2
z
z
15
2
Fig. 11. Binned, simulated data set for a Cardassian model with ψ = 1,
n = −1 (upper curve), a flat ΛCDM universe with Ωm0 = 0.3 (middle
curve), and for a Generalized Chaplygin Gas with A s = 0.4, α = 0.7
(lower curve). The 1σ error bars are also shown.
1
r (quiessence)
q (quiessence)
10
0
−1
5
to the simulated dL , and hence our results give the ensemble
average of the parameters we fit to the simulated data sets.
0
−2
−3
0
1
−5
2
0
1
2
z
z
4.1. A ΛCDM universe
15
2
DL (z) = H0 dL (z) is the Hubble constant free luminosity distance. The combination of absolute magnitude and the Hubble
constant, M, can be calibrated by low-redshift supernovae
(Hamuy et al. 1993; Perlmutter et al. 1999). The dispersion in
the magnitude, σmag , is related to the uncertainty in the distance, σ, by
We first simulate data based on a flat ΛCDM model with
Ωm0 = 0.3, h = 0.7, giving the data points shown in the middle curve in Fig. 11. To this data set we first fit the quiessence
model, the MPC, the GCG, and the parametrization of H from
Eq. (62). Since all models reduce to ΛCDM for an appropriate
choice of parameters, distinguishing between them based on
the χ2 per degree of freedom alone is hard. Based on the bestfitting values and error bars on the parameters A0 , A1 , and A2
in Eq. (62) we can reconstruct the statefinder parameters from
Eqs. (64)–(66). In Fig. 12 we show the deceleration parameter and statefinder parameters reconstructed from the simulated
data. The statefinders can be reconstructed quite well in this
case, e.g. we see clearly that r is equal to one, as it should for
flat ΛCDM. In Fig. 13 we show the statefinders for a selection of models, obtained by fitting their respective parameters
to the data, and using the expressions for q and r for the respective models derived in earlier sections, e.g. Eqs. (46) and (47)
for the Chaplygin gas. Since all models reduce to ΛCDM for
the best-fitting parameters, their q and r values are also consistent with ΛCDM. Thus, if the dark energy really is LIVE, a
SNAP-type experiment should be able to demonstrate this.
ln 10
σ
=
σmag ,
dL (z)
5
4.2. A Chaplygin gas universe
1
0
r (MPC)
q (MPC)
10
−1
5
0
−2
−3
0
1
z
2
−5
0
1
2
z
Fig. 10. The deceleration parameter q and the statefinder r extracted
from current SNIa data using the Alam parametrization of H (top row),
for ΛCDM (second row), quiessence (third row), and the Modified
Polytropic Cardassian ansatz (bottom row)
(69)
and for σmag = 0.15, the relative error in the luminosity distance is ∼7%. If we assume that the dL we calculate for each
z value is the mean of all√dL s in that particular bin, the errors reduce from 7% to 0.07/ 40 ≈ 0.01 = 1%. We do not add noise
We have also carried out the same reconstruction exercise using simulated data based on the GCG with As = 0.4, α = 0.7,
see Fig. 11. Figure 14 shows q and r reconstructed using
the parametrization of H. The same quantities for the models
408
A. K. D. Evans et al.: Geometrical constraints on dark energy
Fig. 12. The statefinder parameters and the deceleration parameter for
the best-fitting reconstruction of the simulated data based on ΛCDM,
using the parametrization of Alam et al. The 1σ error bars are also
shown.
Fig. 14. The statefinder parameters and the deceleration parameter for
the best-fitting reconstruction of the simulated data based on the GCG,
using the parametrization of Alam et al. The 1σ error bars are also
shown.
Fig. 13. The statefinder parameters for a selection of models, evaluated
at the best-fitting values of their respective parameters to the simulated
ΛCDM dataset, with 1σ errors included.
Fig. 15. The statefinder parameters for a selection of models, evaluated
at the best-fitting values of their respective parameters to the simulated
Chaplygin gas data set, with 1σ errors included.
considered, based on their best-fitting parameters to the simulated data, are shown in Fig. 15. For the Cardassian model, the
best-fitting value for the parameter n, nbf , depends on the extent
of the region over which we allow n to vary. Extending this region to larger negative values for n moves nbf in the same direction. However, the minimum χ2 value does not change significantly. This is understandable, since H(x) for the MPC model
is insensitive to n for large, negative values of n. The quantities
r(x) and q(x) also depend only weakly on the allowed range for
n, whereas their error bars are sensitive to this parameter. We
chose to impose a prior n > −1, producing the results shown in
Fig. 15. The best-fitting values for ψ and n were, respectively,
0.06 and −0.94.
Figure 16 shows the deceleration parameter extracted from
the Alam et al. parametrization (full line), with 1σ error bars.
Also plotted is the best fit q(z) from the quiessence (squares),
Cardassian (triangles) and Chaplygin (asterisk) models. We
note that the q(z) from the Alam et al. parametrization has a
somewhat deviating behaviour from the input model, especially
A. K. D. Evans et al.: Geometrical constraints on dark energy
Fig. 16. Comparison of q(z) extracted using the parametrized H(z)
with q(z) for the various best-fitting models. The input model is a
GCG model with As = 0.4, α = 0.7. Error bars are only shown on
the values extracted using the Alam et al. parametrization, but in the
other cases they are roughly of the same size as the symbols. See text
for more details.
409
Fig. 17. Comparison of r(z) extracted using the parametrized H(z)
with r(z) for the various best-fitting models. The input model is a
GCG model with As = 0.4, α = 0.7. Error bars are only shown on
the values extracted using the Alam et al. parametrization, but in the
other cases they are roughly of the same size as the symbols. See text
for details.
at larger z. Also, no model can be excluded on the basis of their
predictions for q(z)
Figure 17 shows the same situation for the statefinder
parameter r(z). Note again that for large z, the recovered
statefinder from the Alam et al. parametrization does not correspond well with the input model. As with the case for q(z), the
quiessence and Cardassian models follow each other closely.
These, however, do not agree with the input model for low values of z (similar to the case for q(z) they diverge for low z).
Comparing the statefinder r for the quiessence and Cardassian
models with that of the input GCG model, indicates that, not
surprisingly, neither of them is a good fit to the data.
4.3. A Cardassian universe
We repeated the analysis described in Sects. 4.1 and 4.2, this
time based on an underlying Cardassian model. The values of
the input parameters were chosen to be ψ = 1, n = −1. The luminosity distance for this model is shown in Fig. 11. Figures 18
and 19 show, respectively, the deceleration parameter q(z) and
the statefinder r(z) for the input Cardassian model (triangles)
compared to the reconstructed parameters (full line) using the
Alam et al. parametrization for H(z). For clarity, only the error bars for the reconstructed parameters are shown. As before,
the error bars for the input model are roughly the size of the
symbols, except in the case of z = 0−0.7 for r(z) where they
are somewhat larger (up to two symbol sizes in each direction). We see that the deceleration parameter is reconstructed
quite well. However, the behaviour of the reconstructed r(z)
does not seem to agree well with the input model, although the
input model is more or less within the 1 σ errors bars of the
Fig. 18. Comparison of q(z) extracted using the parametrized H(z)
with q(z) for the input Cardassian model.
reconstructed statefinder. For the Cardassian universe, the discrepancy between input and reconstructed parameter is most
conspicuous for low z (z < 0.7). This further corroborates the
conclusion in Sect. 4.2 that a better parametrization for H(z)
is needed. The best fit quiessence and Chaplygin gas models
are not shown in these figures. We only remark in passing that
with the quiessence model we managed to reproduce the input
model quite well, while the Chaplygin gas model was a very
poor fit to these simulated data.
410
A. K. D. Evans et al.: Geometrical constraints on dark energy
state w(x) from SNIa data using Eq. (67). They found that this
parametrization forces the behaviour of w(x) onto a specific set
of tracks, and may thus give spurious evidence for redshift evolution of the equation of state. Since there are intrinsic correlations between the statefinders, finding an unbiased reconstruction procedure, and demonstrating that it really is so, is likely
to be very hard.
Acknowledgements. We acknowledge support from the Research
Council of Norway (NFR) through funding of the project 159637/V30
“Shedding Light on Dark Energy”. The authors wish to thank Håvard
Alnes for interesting discussions and the anonymous referee for valuable comments and suggestions.
References
Fig. 19. Comparison of r(z) extracted using the parametrized H(z)
with r(z) for the input Cardassian model.
The exercises in this subsection and the previous one indicate that there are potential problems with extracting the
statefinders from data in a reliable, model-independent way.
The fact that r extracted from the simulated data using the Alam
et al. parametrization deviaties from r(z) for the input model in
the two cases, indicates that one needs a better parametrization
in order to use statefinder parameters as empirical discriminators between dark energy models. In fact, a potential problem
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all derivatives of dynamical variables of order higher than the
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quantity, e.g. H, and it is hard to do this without introducing
bias in the correlation between r and q.
5. Conclusions
We have investigated the statefinder parameters as a means of
comparing dark energy models. As a theoretical tool, they are
useful for visualizing the behaviour of different dark energy
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Paper III
Black-body radiation with extra dimensions
Håvard Alnes1 , Finn Ravndal2 and Ingunn Kathrine Wehus3
Institute of Physics, University of Oslo, N-0316 Oslo, Norway.
Abstract
The general form of the Stefan-Boltzmann law for the energy density of black-body
radiation is generalized to a spacetime with extra dimensions using standard kinetic and
thermodynamic arguments. From statistical mechanics one obtains an exact formula. In a
field-theoretic derivation, the Maxwell field must be quantized. The notion of electric and
magnetic fields is different in spacetimes with more than four dimensions. While the energymomentum tensor for the Maxwell field is traceless in four dimensions, it is not so when there
are extra dimensions. But it is shown that its thermal average is traceless and in agreement
with the thermodynamic results.
1
Introduction
Since the introduction of string theory to describe all the fundamental interactions, the
possibility that we live in a spacetime with more than four dimensions, has steadily generated more interest both in elementary particle physics and nowadays also in cosmology.
We know that any such extra dimensions must be microscopic if they exist at all and many
efforts are already under way to investigate such a possibility[2].
Physical phenomena in spacetimes with extra spatial dimensions will in general be different
from what we know in our spacetime of four dimensions. Although the fundamental laws
in such spacetimes can easily be generalized, their manifestations will not be the same.
Even if it turns out that there are no extra dimensions, it is still instructive to investigate
what the physics then would be like.
Here we will take a closer look at the Maxwell field and then in particular its properties at
finite temperature in the form of black-body radiation. It is characterized by a pressure p
and an energy density ρ. In a spacetime with three spatial dimensions, these are related
by p = ρ/3. This can be derived by purely kinematic arguments which are used in the next
section to find the corresponding relation in a D-dimensional Minkowski spacetime. When
the radiation is in thermal equilibrium at temperature T , we then get by thermodynamic
arguments the generalization of the Stefan-Boltzmann law on the form ρ ∝ T D .
In the following section these results are derived more accurately using statistical mechanics. One obtains these results by just assuming that the radiation is composed of massless
havard.alnes@fys.uio.no
finn.ravndal@fys.uio.no
3
i.k.wehus@fys.uio.no
1
2
1
particles described by quantum mechanics. Except for an overall factor giving the spin
multiplicity, these results should then be the same for scalar particles with no spin, photons which have spin-1 and gravitons which are the massless spin-2 quanta of gravitation.
While our results for the pressure and density corresponds to a traceless thermal average
of the energy-momentum tensor, the general trace of the corresponding energy-momentum
tensors of these fields is not zero. This apparent paradox is discussed and resolved in the
last section where we concentrate on the Maxwell field and the corresponding photons at
finite temperature. The quantization of the field in a spacetime with dimensions D > 4
is a bit more complicated than in the usual case since the magnetic field can no longer
be represented by a vector. Also the number of independent directions of polarization
or helicity states will now be more than two. Landsberg and De Vos has calculated the
Stefan-Bolzmann constant in d-dimensional spaces, but including only two helicity states
instead of d − 1, as later noticed by Menon and Agrawal [1].
2 Thermodynamics
In an ordinary spacetime with D = 3 + 1 dimensions the pressure of black-body radiation
with energy density ρ is given as p = ρ/3. This is most easily derived from a simple
kinetic consideration of massless particles impinging on a plane wall and thereby being
reflected[3]. If the angle between the momentum of the incoming particle and the normal
to the plane is θ, then the kinetic pressure is
p = ρ h cos2 θ i
(1)
Rπ
where the average is taken over the full spherical angle 2π 0 dθ sin θ = 4π. In d = 3 spatial
dimensions one then finds h cos 2 θ i = 1/3 which gives the above result for the pressure.
With extra, non-compactified dimensions the pressure will again be given by (1). The
angular average is then a bit more cumbersome to evaluate and is worked out in the
Appendix. Not so surprising, we then find that the pressure is in general p = ρ/d where
d is the number of spatial dimensions.
Assuming that the black-body radiation is described by ordinary thermodynamics, we can
now also derive the temperature dependence of the energy density[4]. If the radiation fills
a volume V and is in equilibrium with temperature T , the total energy U = ρV will obey
the energy equation
∂U
∂V
=T
T
∂p
∂T
V
With p = ρ/d this gives
ρ=
T dρ
ρ
−
d dT
d
which simplifies to
dρ
dT
= (d + 1)
ρ
T
2
−p
(2)
One thus obtains for the energy density
ρ = CT D
(3)
where C is an integration constant and D = d + 1 is the dimension of the extended
spacetime.
3 Statistical mechanics
We will now consider the radiation as made up of massless particles moving in a ddimensional space obeying Bose-Einstein statistics and in thermal equilibrium at temperature T . It will be convenient to use units so that the speed of light c = 1 and Planck’s
constant h̄ = 1. The energy of one such particle with momentum k is then ω k = |k| = k
and the internal energy density will be given by
ρ=
Z
dd k
ωk
(2π)d eβωk − 1
(4)
with β = 1/(kB T ) where kB is the Boltzmann constant. The differential volume element
in momentum space is dd k = Ωd−1 k d−1 dk where the full solid angle Ωd−1 is given in the
Appendix. Making then use of the integral
Z
∞
dx
0
xn
= Γ(n + 1)ζ(n + 1)
ex − 1
where ζ(z) is Riemann’s zeta-function, we have
ρ=
Ωd−1
Γ(D)ζ(D)(kB T )D
(2π)d
(5)
when expressed in terms of the spacetime dimension D = d + 1. The temperature dependence is seen to be in agreement with the thermodynamic result (3). The pressure of the
gas is similarly obtained from the free energy as in the ordinary case for D = 4. Again
one finds p = ρ/d consistent with the kinetic argument in the first section.
Using the duplication formula for the Γ-function
22z− 2
1
Γ(2z) = √ Γ(z)Γ z +
,
2
2π
1
in the result (5) for the density, the formula for the corresponding pressure takes the
somewhat simpler form
p=
Γ(D/2)
ζ(D)(kB T )D
π D/2
(6)
As a check, we get in the ordinary case of four spacetime dimensions p = (π 2 /90)(kB T )4
since ζ(4) = π 4 /90. This is essentially the Stefan-Boltzmann law in these particular units.
3
If the massless particles making up the radiation gas has non-zero spin, these results must
be multiplied by the corresponding spin multiplicity factor.
4 Scalar quantum field theory
The massless particles in the previous section will be the quanta of a corresponding quantized field theory. Let us first consider the simplest case when these are spinless particles
described by a scalar field φ = φ(x, t). The corresponding Lagrangian is
L=
1 µν
1
η ∂µ φ∂ν φ ≡ (∂λ φ)2
2
2
(7)
where the Greek indices take the values (0, 1, 2, . . . , d). The metric in the D-dimensional
Minkowski spacetime is ηµν = diag(1, −1, −1, . . . , −1). From here follows the equation of
motion ∂ 2 φ = 0 which is just the massless Klein-Gordon equation.
We are interested in the energy density and pressure of the field. Both follow from the
corresponding canonical energy-momentum tensor[5]
1
Tµν = ∂µ φ∂ν φ − ηµν (∂λ φ)2
2
(8)
In a quantized theory it is the expectation value h T µν i which gives the corresponding
measured values. The energy density is therefore
ρ = h T00 i =
1 2
1
h φ̇ i + h (∇φ)2 i
2
2
(9)
Spherical symmetry implies that the expectation values of all the spatial components are
simply given as h Tmn i = pδmn where p is the pressure. Thus we find that the pressure is
given by the trace
p=
1
1
1
h Tnn i = h (∇φ)2 i + h φ̇2 − (∇φ)2 i
d
d
2
(10)
when we use the Einstein convention summing over all equal indices. These expectation
values are usually divergent, but the thermal parts will be finite for such a free field.
In order to quantize the field, we consider it to be confined to a finite volume V with
periodic boundary conditions. It can then be expanded in modes which are plane waves
with wave vectors k. The field operator takes the form
φ(x, t) =
X
k
s
i
1 h
ak ei(k·x−ωk t) + a†k e−i(k·x−ωk t)
2ωk V
(11)
where ak and a†k are annihilation and creation operators with the canonical commutator
[ak , a†k0 ] = δkk0
4
As in ordinary quantum mechanics, the number of particles in the mode with quantum
number k is given by the operator a†k ak . At finite temperature its expectation value
h a†k ak i equals the Bose-Einstein distribution function
nk =
1
eβωk − 1
(12)
This also equals h ak a†k i when we disregard the vacuum or zero-point contributions which
can be neglected at finite temperature. Similarly, the expectation values of the operator
products ak ak and the Hermitian adjoint a†k a†k are zero.
We can now find the energy density from (9). For the mode with wavenumber k the time
derivative φ̇ will pick up a factor ωk while the gradient ∇φ will pick up a corresponding
factor k. Thus we get
ρ=
X
k
1
(ω 2 + k2 )nk
2ωk V k
Now letting the volume go to infinity so that
X
k
→V
Z
dd k
(2π)d
and using k2 = ωk2 for massless particles, we reproduce the expression (4) from statistical
mechanics. We thus recover the same result for the thermal energy density. Similarly for
the pressure, the two last terms in (10) will cancel and the equation simplifies to p = ρ/d
as before.
The expectation value of the full energy-momentum tensor at finite temperature is now
diagonal with the components h Tµν i = diag(ρ, p, p, . . . , p). Since the pressure p = ρ/d,
we have then recovered the well-known fact that the expectation value of the trace is zero
for massless particles, h T µµ i = 0. What is a bit surprising is that the trace of the energymomentum tensor itself in (8) is generally not zero. In fact, we have T µµ = (1−D/2)(∂λ φ)2
which is only zero in D = 2 spacetime dimensions. In that case the massless scalar
field is said to have conformal invariance[7]. This is a fundamental symmetry in modern
string theories. But we know from thermodynamics and statistical mechanics that the
expectation value of the trace of Tµν is zero in all dimensions. From the above calculation
we see that here in quantum field theory, this comes about since we are dealing with
massless particles for which h φ̇2 i = h (∇φ)2 i. This is equivalent to h (∂λ φ)2 i = 0 which
follows from the equation of motion for the field after a partial integration.
Concerning conformal invariance, the massless scalar field is special. Many years ago
Callan, Coleman and Jackiw[6] showed that it can be endowed with conformal invariance
in any dimension, not only for D = 2. The corresponding energy-momentum tensor will
then be the canonical one in (8) plus a new term
∆Tµν = −
1 D − 2
∂µ ∂ν − ηµν ∂ 2 φ2
4D−1
5
(13)
Taking now the the trace of this improved energy-momentum tensor, it is found to be
zero when one makes use of the classical field equation ∂ 2 φ = 0. Needless to say, the
expectation value of the conformal piece (13) is zero at finite temperature. However,
this is not the case at zero temperature for the scalar Casimir energy due to quantum
fluctuations between two parallel plates[8]. The new term makes the energy density finite
between the plates and only when it is included, is the vacuum expectation value of the
trace of the energy-momentum tensor zero. For the scalar field, this can be achieved in
any dimension.
5 Maxwell field and photons
Black-body radiation is historically considered to be a gas of photons at finite temperature.
These massless particles are the quanta of the Maxwell field. In a spacetime with D = d+1
dimensions, the electromagnetic potential is a vector A µ (x) with D components. A gauge
transformation is defined as Aµ → Aµ + ∂µ χ where χ(x) is a scalar function. It leaves
the Faraday field tensor Fµν = ∂µ Aν − ∂ν Aµ invariant as in D = 4 dimensions. Thus the
Lagrangian for the field also takes the same form,
1
1 2
L = − Fαβ F αβ ≡ − Fαβ
4
4
(14)
and is obviously also gauge invariant. While the components F 0j form an electric vector
E with d components, the d(d − 1)/2 magnetic components F ij no longer form a vector.
The number of electric and magnetic components are equal only when D = 4.
From the above Lagrangian one gets the corresponding energy-momentum tensor
1
2
Tµν = Fµλ F λν + ηµν Fαβ
4
(15)
as in the ordinary, four-dimensional case[5]. The energy density is therefore
T00 = Ei2 +
1 2
1
1
Fij − 2Ei2 = Ei2 + Fij2
4
2
4
(16)
while the pressure in the radiation will follow from the spatial components
1
Tmn = −Em En + Fmk Fnk − δmn Fij2 − 2Ei2
4
(17)
In order to calculate the expectation values of these quantities, the field must be quantized.
This is most convenient to do in the Coulomb gauge ∇·A = 0. In vacuum, the component
A0 = 0 and we have D − 2 degrees of freedom, each corresponding to a independent
polarization vector eλ . Corresponding to the expansion of the scalar field operator in
(11), we now have for the electromagnetic field
A(x, t) =
X
kλ
s
i
1 h
eλ akλ ei(k·x−ωk t) + e∗λ a†kλ e−i(k·x−ωk t)
2ωk V
6
(18)
The polarization vectors for a mode with wavenumber k are orthonormalized so that
e∗λ · eλ0 = δλλ0 together with k · eλ = 0 and thus satisfy the completeness relation
X
λ
e∗λi eλj = δij −
ki kj
k2
(19)
Since the electric field E = −Ȧ, we then find that each polarization component has the
same expectation value as the scalar field in the previous section, i.e.
h Ei2 i = (D − 2)
X
k
ωk2
nk
2ωk V
(20)
where nk is the Bose-Einstein density (12). Similarly, for the magnetic components F ij =
∂i Aj − ∂j Ai we obtain
h Fij2 i = 2(D − 2)
X
k
k2
nk
2ωk V
(21)
when we make use of the above completeness relation for the polarization vectors. Since
the photons are massless, we again have ω k = |k| and therefore h Fij2 i = 2h Ei2 i. As
expected, we then find that the density of the photon gas is D − 2 times the density of the
scalar gas. We also see that the expectation value of the last term in (17) is zero. Defining
again the pressure as h Tmn i = pδmn , it follows then that p = ρ/d as it should be.
Even though the expectation value of the energy-momentum tensor for the electromagnetic
field is found to be zero at finite temperature, the tensor itself in (15) has a trace T µµ =
2 /4. It is therefore traceless only in the ordinary case of D = 4 spacetime
(D − 4)Fµν
dimensions when the field has conformal invariance. In contrast to the scalar case, the
Maxwell field will not have this symmetry in spacetimes with extra dimensions and there
is no corresponding, improved energy-momentum tensor with zero trace. As we have seen
above, this has no serious implications for the photon gas at finite temperature. But at zero
temperature when one calculates the electromagnetic Casimir energy, the tracelessness of
Tµν plays an important role[9]. A similar calculation of this vacuum energy with extra
dimensions would therefore be of interest.
6 Conclusion
Black-body radiation is defined to be a free gas of massless particles at finite temperature.
Purely kinetic arguments then relates the pressure and density by p = ρ/d where d is the
spatial dimension of the volume containing the gas. This is equivalent to saying that the
finite-temperature expectation value of the energy-momentum tensor of the gas is zero.
Describing these particles as quanta of a field theory, we have shown that this tracelessness
has very little to do with the trace of the corresponding energy-momentum tensor of the
field.
7
There are many different ways to derive the energy-momentum tensor. The correct result
follows from the most direct derivation starting from the Lagrangian L of the field in a
curved spacetime with metric gµν instead of the Minkowski metric ηµν . From a variational
principle one then finds that the energy-momentum tensor will have the general form[7]
Tµν = 2
∂L
− gµν L
∂g µν
(22)
From the scalar Lagrangian (7) this gives the energy-momentum tensor (8) and similarly
for the Maxwell field with the Lagrangian (14). What we have shown for these fields at
finite temperature can be summed up in the statement that the expectation value h L i
in the last term of Tµν is zero. This follows directly from the equations of motion and
is also true for massive fields. The average must be taken over an infinite volume or a
finite volume with periodic boundary conditions so that the surface terms from the partial
integrations vanish.
Here we have only considered the unphysical situation where the extra dimensions are
assumed to be infinite in extent like the ones we know. In a realistic situation they must
be compactified or curled up at a very small scale that is consistent with present-day
observations. The black-body radiation laws derived above will then be modified and end
up as small corrections to the four-dimensional results. Such an investigation will be more
demanding and is not taken up here.
We want to thank Professor R. Jackiw for several useful comments. This work has been
supported by grants no. 159637/V30 and 151574/V30 from the Research Council of Norway.
Appendix
In a d-dimensional Euclidean space one can specify a point on the unit sphere by d − 1
angles, (φ, θ1 θ2 , . . . , θd−2 ). Here φ is an azimuthal angle with the range 0 ≤ φ < 2π while
all the polar angles θn vary in the range 0 ≤ θn < π. The differential solid angle is then
given as
dΩd−1 = 2π
d−2
Y
sinn θn dθn
n=1
which gives for the full solid angle in d dimensions
Ωd−1 = 2π
d−2
YZ π
n=1 0
sinn θn dθn =
2π d/2
Γ(d/2)
when we make use of the integral
Z
π
dθ sinn θ =
0
√ Γ( n+1
2 )
π n+2
Γ( 2 )
8
For the pressure in black-body radiation we now need the average over this solid angle of
h cos2 θd−2 i = 1 − h sin2 θd−2 i. With these integral formulas it is now straightforward to
show that
h sin2 θd−2 i =
Γ( d2 )Γ( d+1
2 )
d+2
Γ( d−1
2 )Γ( 2 )
=
d−1
d
and thus h cos2 θd−2 i = 1/d as used in the first section of the main text.
References
[1] P. T. Landsberg and Alexis De Vos, J. Phys. A: Math. Gen. 22, 1073 (1989); V. J.
Menon and D. C. Agrawal, J. Phys. A: Math. Gen. 31, 1109 (1998);
[2] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Today, 55N2, 35 (2002); L.
Randall, Science, 296, 1422 (2002); J. Hewett and M. Spiropulu, Ann. Rev. Nucl.
Part. Sci., 52, 397 (2002).
[3] For a clear presentation of this argument, see for instance S. Gasiorowicz, The Structure of Matter: A Survey of Modern Physics, Addison-Wesley (1979).
[4] M.W. Zemansky, Heat and Thermodynamics, McGraw-Hill Book Company (1957).
[5] J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley (1967).
[6] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. N.Y. 59, 42 (1972).
[7] M. Forger and H. Römer, Ann. Phys. N.Y. 309, 306 (2004).
[8] B. deWitt, Phys. Rep. 19C, 295 (1975); K. Tywoniuk and F. Ravndal, quantph/0408163.
[9] L.S. Brown and G.J. Maclay, Phys. Rev. 184, 1272 (1969).
9
Paper IV
Electromagnetic Casimir energy with extra dimensions
H. Alnes, F. Ravndal and I.K. Wehus
Department of Physics, University of Oslo, N-0316 Oslo, Norway.
and
K. Olaussen
Department of Physics, NTNU, N-7491 Trondheim, Norway.
Abstract
We calculate the energy-momentum tensor due to electromagnetic vacuum fluctuations between two parallel hyperplanes in more than four dimensions, considering
both metallic and MIT boundary conditions. Using the axial gauge, the problem can
be mapped upon the corresponding problem with a massless, scalar field satisfying
respectively Dirichlet or Neumann boundary conditions. The pressure between the
plates is constant while the energy density is found to diverge at the boundaries when
there are extra dimensions. This can be related to the fact that Maxwell theory is
then no longer conformally invariant. A similar behavior is known for the scalar field
where a constant energy density consistent with the pressure can be obtained by improving the energy-momentum tensor with the Huggins term. This is not possible for
the Maxwell field. However, the change in the energy-momentum tensor with distance
between boundaries is finite in all cases.
1
Introduction
When a classical field is quantized, the modes that can be excited are solutions of the
classical wave equation and are labeled by different quantum numbers. These modes will
depend on the imposed boundary conditions and will therefore be influenced by the presence of confining boundaries. Each such mode has a zero-point energy which contributes
to the total vacuum energy of the field. As a result, these vacuum fluctuations give the
ground state of the system an energy which depends on the presence of nearby boundaries.
For two parallel and perfectly conducting plates placed in vacuum Casimir[1] showed that
for the electromagnetic field this energy corresponds to an attractive pressure
P =−
π2
240L4
(1)
between the plates, separated by a distance L. This macroscopic quantum effect was
for a long time in doubt, even after the first experimental verifications by Sparnaay[2].
Today this Casimir force is measured to high precision[3] and even effects of non-zero
temperatures are being investigated[4].
1
Together with this experimental progress, modern regularization methods to remove the
unphysical divergences endemic in these calculations now make them much simpler than
previously[5]. One can then investigate more detailed properties of the effect like how
the energy or stresses are distributed between the plates. One must then calculate the
vacuum expectation value of the full energy-momentum tensor as was done by Lütken and
Ravndal[6]. This confirmed a previous calculation by de Witt[7] of the fluctuations of the
electric and magnetic fields near a metallic boundary where they diverge but in such a way
that the energy density remains finite. Since Maxwell theory is conformally invariant in
D = 4 spacetime dimensions, one can directly relate this energy density to the attractive
pressure[8].
Recently the Casimir energy has been invoked to explain the dark energy which seems to
drive the present acceleration of the Universe[9]. It appears in particular in cosmological
models with extra dimensions[10]. In these models the confinement of the fluctuating
fields is provided by compactification of the extra dimensions. The energy appears as
a cosmological constant in our four-dimensional Universe and is given numerically by a
formula of the same form as (1) with L given by the size of the compactified dimensions.
The calculation of Casimir energies in higher-dimensional spacetimes was first done by
Ambjørn and Wolfram[11]. They calculated the global energies equivalent to the Casimir
force and thus obtained no knowledge of how the energy is distributed between the hyperplanes. Since the force due to electromagnetic fluctuations is expected to be proportional
to the force due to fluctuations of a massless scalar field, only the effects of this kinematically simpler field was investigated. Each mode has a momentum k T transverse to the
normal of the plates plus a component k z = nπ/L with n = 1, 2, . . . in the direction of the
normal. If d is the number of spatial dimensions, the vacuum energy between the plates
per (d − 1)-dimensional hyperarea will follow from the divergent integral
∞
E=
1X
2 n=1
Z
dd−1 kT
(2π)d−1
q
k2T + (nπ/L)2
(2)
We can now do the transverse integration by dimensional regularization. The remaining
sum over n is then done by analytical continuation of the Riemann zeta function ζ R (z) to
give
E=−
Γ(−d/2)ζR (−d) π d
L
2(4π)d/2
It can be simplified using the reflection formula
Γ(s/2)π −s/2 ζR (s) = Γ((1 − s)/2)π −(1−s)/2 ζR (1 − s)
(3)
for the zeta function. Then we can write E = E 0 L when we introduce the energy density
E0 = −
Γ(D/2)ζR (D)
(4π)D/2 LD
2
(4)
Defining now the pressure between the plates by P = −∂E/∂L, it is then simply
P = (D − 1)E0
(5)
where D = 1 + d is the spacetime dimension. Taking D = 4 and multiplying the result by
two for the two polarization degrees of the photon, we recover the original result (1). In
D-dimensional spacetime we must for the same reason multiply the result (5) by D − 2 to
obtain the electromagnetic Casimir force.
The spatial distribution of the vacuum energy can be obtained from the energy-momentum
tensor. For the scalar field it is
Tµν = ∂µ φ∂ν φ − ηµν L
(6)
where the massless Lagrangian is L = (1/2)(∂ λ φ)2 choosing the metric to be ηµν =
diag(1, −1, · · · , −1). Since its trace T µµ = (1 − D/2)(∂λ φ)2 is zero only in D = 2 dimensions, it is in general not conformally invariant in higher dimensions. Calculating now
the Casimir energy density h T00 i in for example D = 4 spacetime dimensions, one then
obtains a result which diverges at the plates after regularization[12]. When integrated, it
will thus not reproduce the total Casimir energy corresponding to the force (5).
For the scalar field this apparent problem can be solved. One can improve[13] the above
energy-momentum tensor in any spacetime dimension D > 2 by adding the Huggins
term[14]
∆Tµν = −
1D−2
(∂µ ∂ν − ηµν ∂ 2 )φ2
4D−1
(7)
The energy-momentum tensor is then traceless and conformal invariance has been restored.
Since this new term is a divergence, it will not contribute to the force, but change the
distribution of energy around the plates. Vacuum expectation values of all components
of the energy-momentum tensor are now constant between the plates and zero outside as
already noticed by de Witt[7], Milton[15] and others[12].
However, the situation for the electromagnetic field is somewhat different. It has the
energy-momentum tensor
1
Tµν = Fµα F αν + ηµν Fαβ F αβ
4
(8)
2
where Fµν = ∂µ Aν − ∂ν Aµ is the Faraday tensor. From the trace T µµ = (−1 + D/4)Fαβ
we see that it is conformally invariant only in D = 4 spacetime dimensions. In this case
the Casimir energy is also constant between the plates[6] as for the scalar field. But for
dimensions D > 4 it is no longer clear how the energy is distributed since there is no way
to construct a gauge-invariant analogue of the Huggins term in Maxwell theory.
In the following we will investigate this problem in more detail. In D = 4 dimensions one
can choose the transverse gauge and expand the classical field in electromagnetic multipoles. This is cumbersome in higher dimensions since the field then has more magnetic
3
than electric components. From the geometry of the problem it is more natural to choose
the axial gauge nµ Aµ = Az = 0 where the D-vector nµ = (0, 0, . . . , 0, 1) is normal to the
plates and is called the z-direction. The Faraday tensor then has a correlator which can
be directly obtained from scalar field theory in the same geometry. In the next chapter
we therefore derive a general expression for the scalar field correlators, both in the case
of Neumann and Dirichlet boundary conditions. These can then be used to calculate
the expectation value of the scalar energy-momentum tensor (6) and the relevance of the
Huggins term is discussed.
For the electromagnetic field considered in Chapter 3, we need to know the boundary
conditions. These can be of the metallic type used for the standard Casimir force in
D = 4 or the QCD version used in the MIT bag model for confinement of quarks. In
the axial gauge we find that these two possibilties correspond to Dirichlet and Neumann
conditions for the corresponding scalar field. The fluctuations of the different components
of the Faraday tensor are then calculated with particular attention to the energy density
and the pressure between the plates. While the pressure is found to be constant and in
agreement with the global Casimir force, the energy density diverges at the plates.
In the last chapter this problem, which no longer can be cured with a Huggins term, is discussed and compared with similar divergences in other systems. With physical boundaries
that only confines fluctuations with frequencies below a certain cut-off, all field fluctuations
should reach a finite value when the boundaries are approached.
2
Scalar fields
It will be very convenient to denote by a bar any vector or tensor component orthogonal
to the unit normal nµ of the plates, which is taken to be in the z-direction. Thus for a
full D-vector we write A = (Aµ ) = (Aµ̄ , Az ). Note that the (D − 1)-vector Ā = (Aµ̄ ) also
includes the time component A0 . The metric can thus be written as η µν = η̄ µν − nµ nν
where η̄ is the projection of η onto the barred subspace. The field operator for a massless
scalar field satisfying the Dirichlet boundary conditions φ(x̄, z = 0, L) = 0 in D = d + 1
spacetime dimensions will then be
r ∞ Z d−1 r
i
2 X d kT
1 h
¯
¯
−ik·x̄
†
ik·x̄
a
(k
)
e
+
a
(k
)
e
sin(nπz/L)
(9)
φ(x) =
n T
n T
L
(2π)d−1 2ωn
n=1
q
Here k̄ = (ωn , kT ) with the frequency ωn = k2T + m2n where mn = kz = nπ/L. Had we
instead chosen the Neumann boundary condition ∂ z φ(x̄, z = 0, L) = 0, we just have to
make the replacement sin(nπz/L) → cos(nπz/L) in the sum. Then there
p should also be
a n = 0 mode to be included in the sum with normalization constant 1/L. But with
the regularization we will use in the following, it will not contribute and is therefore not
further considered.
4
2.1
Feynman correlator
In the above field operator for the scalar Dirichlet modes we have used a normalization
which corresponds to [an (kT ), a†n0 (k0T )] = δnn0 (2π)d−1 δ(kT − k0T ) for the annihilation and
creation operators. From this we find the Feynman propagator
GD (x, x0 ) = hΩD |T φ(x)φ(x0 )| ΩD i
Z d
∞
d k̄ 2 X sin(nπz/L) sin(nπz 0 /L) −ik̄·(x̄−x̄0 )
e
= i
(2π)d L
k̄ 2 − m2n + iε
n=1
(10)
Here we integrate over all components of the d-dimensional Lorentz vector k̄. Assuming
¯ ≡ x̄ − x̄0 to be spacelike, we may choose a coordinate system where it has no components
in the time direction. We can then rotate k 0 to the imaginary axis and find
Z d X
∞
2
sin(nπz/L) sin(nπz 0 /L) ik̄·¯
d k̄
0
GD (x, x ) =
e
(11)
L (2π)d
k̄ 2 + m2n
n=1
(where now k̄ is a Euclidean vector. In order to evaluate the sum, we consider the function
∞
g(z, z 0 ) =
2 X sin(nπz/L) sin(nπz 0 /L)
L
k̄ 2 + m2n
(12)
n=1
which solves the differential equation
d2
2
− 2 + k̄ g(z, z 0 ) = δ(z − z 0 )
dz
(13)
on [0, L] with Dirichlet boundary conditions. As can be verified by insertion, the solution
is
g(z, z 0 ) =
sinh k̄z< sinh k̄(L − z> )
k̄ sinh k̄L
(14)
where z< = min(z, z 0 ) and z> = max(z, z 0 ). Expanding the hyperbolic functions, one then
finds
∞
X
1 −k|z−z
0|
0
0
0
¯
¯
¯
¯
¯
g(z, z 0 ) =
e
− e−k(z+z ) − e−k(2L−z−z ) + e−k(2L−|z−z |)
e−2j kL
2k̄
j=0
∞ Z
h
i
X
1
0
0
dkz
eikz (z−z −2jL) − eikz (z+z −2jL) .
(15)
=
2
2
2π k̄ + kz
j=−∞
The last equality is verified by evaluating the k z integral by contour integration. Inserting
now this partial result into (11) and using rotational invariance, we find
∞ Z
X
dd+1 k 1 i[kz (z−z 0 −2jL)+k̄·¯]
i[kz (z+z 0 −2jL)+k̄·¯
]
e
−
e
(16)
GD (x, x0 ) =
(2π)d+1 k 2
j=−∞
5
where now the (d + 1)-dimensional vector k = ( k̄, kz ). Each of the integrals are given by
the generalized Coulomb potential
Z n
d k 1 ik·(x−x0 )
Γ(n/2 − 1)
0
(17)
e
= n/2
Vn (x − x ) =
n
2
(2π) k
4π |x − x0 |n−2
in n = d+1 spatial dimensions. Had we instead considered Neumann boundary conditions,
the sine function in (9) would have been replaced by the corresponding cosine function.
The only change would then have been that the last term in (16) came in with opposite
sign. Introducing the D-vectors zj = (¯
, z − z 0 − 2jL) and z̃j = (¯
, z + z 0 − 2jL) of lengths
Rj = (zj2 )1/2 and R̃j = (z̃j2 )1/2 , we can now write the result for both correlators as
G(x, x0 )N/D =
∞ h
X
j=−∞
Vd+1 (Rj ) ± Vd+1 (R̃j )
i
(18)
where the upper sign is for Neumann and the lower for Dirichlet boundary conditions.
For a massive field we would have found a similar result, but with the Coulomb potential
replaced by the corresponding generalized Yukawa potential.
In fact, almost every student of introductory electrostatics could have written down this
result immediately by realizing that the problem is equivalent to calculating the potential
of a point charge between parallel plates in D = d+1 spatial dimensions, using the method
of images to enforce the boundary conditions.
2.2
Energy-momentum tensor
The term Vd+1 (R0 ) in (18) is equal to the free correlator G 0 (x, x0 ) = h0 |T φ(x)φ(x0 )| 0i,
where | 0i is the bulk vacuum. It diverges in the limit x 0 → x. But defining now the
physical vacuum expectation value of the energy-momentum tensor by the point-split
limit
h Tµν (x)i = lim
[hΩ |Tµν (x0 , x)| Ωi − h0 |Tµν (x0 , x)| 0i]
0
x →x
(19)
its contribution is removed. The finite expectation values will then follow from the regularized correlator GD (x, x0 ) − G0 (x, x0 ) which contains the effects of the plates. We will
continue to denote it by GD (x, x0 ) in the following and it is given by (18) when we in the
first part leave out the j = 0 term. Since we have assumed that x − x 0 is non-zero and
spacelike, the time-ordering symbol in the correlator can be ignored and it satisfies the
Klein-Gordon equation (∂¯2 − ∂z2 + m2 )G(x, x0 ) = 0 for both boundary conditions.
For the Dirichlet vacuum expectation value of the scalar energy momentum tensor (6), we
first need the part
1
h ∂xµ̄ φ(x)∂xν̄0 φ(x0 )iD = −∂xµ̄ ∂xν̄ GD (x, x0 ) = − η µ̄ν̄ ∂¯2 GD (x, x0 )
d
1 µ̄ν̄ 2
2
=
η (m − ∂z )GD (x, x0 )
d
6
(20)
using Lorentz invariance. Similarly, it follows that
h ∂z φ(x)∂z 0 φ(x0 )iD = −∂z2 GN (x, x0 )
(21)
since the two parts in the correlator (18) has opposite symmetry under the exchange
z → z 0 . For vacuum expectation value of the point-split Lagrangian
1
L(x, x0 ) = [ηµ̄ν̄ ∂xµ̄ φ(x)∂xν̄0 φ(x0 ) − ∂z φ(x)∂z 0 φ(x0 ) − m2 φ(x)φ(x0 )]
2
we thus find
i
1 h
h L(x, x0 )iD = ∂z2 GN (x, x0 ) − GD (x, x0 )
(22)
2
The point-split expressions for the canonical energy-momentum tensor (6) are thus found
to be
h1
i
m2
1
GD − ∂z2 GD + (GN − GD )
(23)
h Tµ̄ν̄ iD = ηµ̄ν̄
d
d
2
1
(24)
h Tzz iD = − ∂z2 (GN + GD )
2
Corresponding results for the Neumann expectation values are obtained by the exchange
D ↔ N . The physical limit x → x0 can now be taken where a resulting z-dependence
can only come from the last sum in the correlators (18). But for the pressure P = h T zz i
we see that this will cancel out in the sum G D + GN so that the pressure is constant
between the plates. This is physical reasonable and is also the case for the fluctuations of
a massive field. It follows directly from the conservation of the energy-momentum tensor.
The expectation values of the other other components of the energy-momentum tensor in
(23) will in general be dependent on the position z between the plates.
Let us now calculate the pressure in the massless limit. We will then need the double
derivative ∂z2 (GD + GN ) which follows directly from (18) in the limit x → x 0 as
∂z2 (GN
+ GD ) = 2 lim
0
z →z
=
∞
X
0
∂z2 Vd+1 (Rj )
j=−∞
2(D − 1)Γ(D/2)
ζR (D)
(4π)D/2 LD
∞
1
Γ((d − 1)/2) X0
= 2d(d − 1)
d+1
(d+1)/2
|2jL|
4π
j=−∞
(25)
where the 0 denotes that j = 0 is excluded from sum. Using this in (24) we reproduce
exactly the standard pressure (5) obtained from the total energy. It is seen to be the same
for both boundary conditions.
The energy density E = h T00 i between the plates follows from (23). When the mass m = 0,
we then need to calculate in addition the quantity
∂z2 (GN
− GD ) = 2 lim
0
z →z
=
∞
X
∂z2 Vd+1 (R̃j )
j=−∞
(D − 1)Γ(D/2)
fD (z/L)
(4π)D/2 LD
7
(26)
when we introduce the function
fD (z/L) =
∞
X
j=−∞
1
|j + z/L|D
(27)
Notice that the term j = 0 is now to be included. The sum can be expressed by the
Hurwitz zeta function
ζH (s, a) =
∞
X
1
(n + a)s
n=0
(28)
which allows us to write
fD (z/L) =
∞
X
j=0
∞
X
1
1
+
D
(j + z/L)
(j − z/L)D
j=1
= ζH (D, z/L) + ζH (D, 1 − z/L)
When D = 4 the same position-dependent term was derived on this form by Kimball[15].
But when the spacetime dimension D is an even number, we can express the result in
terms of the digamma function ψ(x) using the relation
k−1
d
(−1)k
ψ(x)
(29)
ζH (k, x) =
(k − 1)! dx
We then have
fD (z/L) =
1
(D − 1)!
d
dx
D−1 h
ψ(x) − ψ(1 − x)
i
(D = even)
where x = z/L. This simplifies even more since ψ(x) − ψ(1 − x) = −π cot(πx), which
allows us to write
πD
d D−1
fD (z/L) =
cot θ
(D = even)
(30)
−
Γ(D)
dθ
with θ = πz/L. For the ordinary Casimir effect in D = 4 spacetime dimensions, this
function also appeared in the calculation of the electromagnetic field using another regularization and choice of gauge[6]. This follows from writing f 4 (z/L) = (π 4 /3)F (θ) which
gives
3
2
−
F (θ) =
(31)
sin4 θ sin2 θ
This function will then characterize all position-dependent expectation values when D = 4.
Collecting the above results, we now have the scalar vacuum energy density in arbitrary
spacetime dimensions
i
Γ(D/2) h
ED/N = −
ζ
(D)
±
(D/2
−
1)f
(z/L)
(32)
R
D
(4π)D/2 LD
8
where the lower sign is for Neumann boundary conditions. While the first term corresponds
to a constant density, the last term gives a position-dependent contribution which in
general diverges at the position of the plates, i.e. where z = 0 and z = L. Only in the
special case D = 2 when the scalar field has conformal invariance, will it be absent. When
integrated over the volume between the plates, the first term alone is seen to give the
total Casimir energy(4). The last term gives a divergent contribution to the same energy
and should be absent. No such term was found in the calculation of the pressure. It
is consistent with just the first part of the energy density which alone gives the correct
Casimir force.
2.3
Huggins term
A free, massless scalar field can couple to gravity in a conformally invariant way. The resulting energy-momentum tensor will thus be traceless[7][13]. It differs from the canonical
expression (6) by the extra Huggins term (7). Using the equation of motion ∂ 2 φ = 0, one
then finds that the improved energy-momentum tensor indeed is traceless.
When we now want to evaluate the Huggins term for the vacuum between the two plates
using the above point-split regularization, we interpret
∂µ ∂ν φ2 = (∂µ ∂ν + ∂µ ∂ν0 + ∂µ0 ∂ν + ∂µ0 ∂ν0 )φ(x)φ(x0 )
where the primed derivatives are with respect to x 0 . This gives the ground state expectation value
D−2 2
1
∂ (GN − GD )
(33)
h ∆Tµν iD = η̄µν
2
D−1 z
which is seen to be proportional to h Li. The Huggins correction has no components in
the z-direction, leaving the pressure unaltered. The z-dependent terms cancel against the
same terms in the canonical part (23) so that the resulting energy density will be constant.
In fact, when m = 0 we have for both sets of boundary conditions that
h Tµν + ∆Tµν i = E0 (η̄µν + (D − 1)nµ nν )
(34)
when expressed in terms of the energy density (4). The pressure is simply D − 1 times
this constant energy density. This is a direct consequence of the energy-momentum tensor
now being traceless.
3
Maxwell fields
2 /4 is gauge invariant. In a general
The electromagnetic Lagrangian density L = −F µν
spacetime it can be written as
1
1 2
L = Ei2 − Bij
2
4
9
(35)
where Ei = −(Ȧi +∂i A0 ) are the components of the electric field vector while the magnetic
field is given by the antisymmetric tensor B ij = ∂i Aj − ∂j Ai .
We want to calculate the vacuum expectation value of the corresponding energy-momentum tensor (8) between the plates. As already explained in the introduction, it is then
most natural to work in the axial gauge n µ Aµ = 0. Since the plate normal vector nµ only
has a component along the z-axis, this requires the component A z = 0. The component
A0 is no longer a canonical variable, but depends on the others via the Maxwell equation
∂i F i0 = 0, which gives
A0 = −∆−1 ∂i Ȧi
(36)
where the operator ∆ = ∂i2 . There are thus D − 2 independent degrees of freedom in a
D-dimensional spacetime described by the spatial field components A i where i 6= z. We
can then express the full Lagrangian in terms of these fields. After partial integrating and
neglecting surface terms, we find it to be
Z
i
h 1
(37)
dd x Ȧi δij − ∂i ∆−1 ∂j Ȧj − Ai ∂i ∂j − δij ∆ Aj
L=
2
As usual, the first or electric part acts like a kinetic energy while the magnetic part acts
like a potential energy.
3.1
Boundary conditions and the correlator
In order to quantize this theory, we must impose boundary conditions for the electromagnetic field components on the confining plates. For the original Casimir effect in d = 3
dimensions one had metallic plates in mind where one could impose the standard constraints n × E = 0 and n · B = 0 on the elecric and magnetic field vectors. For the more
abstract case we have in mind here, we could just as well consider the MIT boundary
condition nµ Fµν = 0 proposed for the quark bag model ensuring color confinement[16]. In
terms of components, this is equivalent to the two conditions n × B = 0 and n · E = 0.
They are therefore just the electromagnetic duals of the metallic boundary conditions.
For our problem under consideration the MIT boundary condition can most directly be
imposed. With the normal vector nµ along the z-axis, it is equivalent to F µ̄z = 0 in our
previous index notation. Now in the axial gauge this is simply equivalent to the Neumann boundary condition ∂z Ai (x̄, z = 0, L) = 0, and defining ∆−1 in (36) with Neumann
boundary conditions.
With this index notation, the metallic boundary conditions can also easily be generalized
to higher dimensions by noticing that in d = 3 dimensions they correspond to F µ̄0 = 0.
In the axial gauge this is achieved in all dimensions by requiring A i (x̄, z = 0, L) = 0, i.e.
Dirichlet conditions, and defining ∆ −1 in (36) with Dirichlet boundary conditions. In this
way we can take directly over many of the previous results for the scalar field.
10
The field components Aµ̄ obey the classical wave equation ∂ 2 Aµ̄ −∂µ̄ (∂ ν̄ Aν̄ ) = 0. Solutions
will be of the form Aµ̄ (x) = aµ̄ (x̄)b(z) where the function b(z) ∝ cos(nπz/L) when we
impose MIT, i.e. Neumann, boundary conditions and b(z) ∝ sin(nπz/L) for metallic or
Dirichlet boundary conditions. With these boundary conditions the remaining functions
aµ̄ (x̄) satisfy (∂¯2 + m2n )aµ̄ − ∂µ̄ (∂ ν̄ aν̄ ) = 0 which are just the equations of motion for a
massive vector field with mass mn = nπ/L. With this observation, we then immediately
have the correlator
Dµ̄ν̄ (x, x0 ) = hΩD |T Aµ̄ (x)Aν̄ (x0 )| ΩD i
Z d
∞
kµ̄ kν̄
d k̄ 2 X sin(nπz/L) sin(nπz 0 /L)
0
¯
= i
η
−
e−ik·(x̄−x̄ ) (38)
µ̄ν̄
(2π)d L n=1
m2n
k̄ 2 − m2n + iε
A corresponding result is obtained with Neumann boundary conditions. As before, we
then drop the mode with n = 0.
3.2
Electromagnetic expectation values
We are now in the position of calculating the value of the energy-momentum tensor (8)
between the two plates. First we need the expectation value
2
ηᾱµ̄ ηβ̄ ν̄ − ηᾱν̄ ηβ̄ µ̄ ∂¯2 GD (x, x0 )
h Fµ̄ν̄ (x)Fᾱβ̄ (x0 )iD =
d
2 (39)
ηᾱµ̄ ηβ̄ ν̄ − ηᾱν̄ ηβ̄ µ̄ ∂z2 GD (x, x0 )
=
D−1
since the massless correlator satisfies the free wave equation ( ∂¯2 − ∂z2 )GD = 0. The
structure of this result follows directly from antisymmetry of the field tensor and Lorentz
invariance in the barred directions. As expected, it is simply given by the scalar correlator.
In the same way we also find
1 D−2
h Fµ̄z (x)Fν̄z (x0 )iD = ηµ̄ν̄ ∂z2 − ∂¯2 GD (x, x0 ) =
ηµ̄ν̄ ∂z2 GN (x, x0 )
d
D−1
(40)
2 /4 − F 2 /2 is thereThe point-split expectation value of the Lagrangian density L = −F µ̄ν̄
µ̄z
fore
h
i
1
h L(x, x0 )iD = (D − 2)∂z2 GN (x, x0 ) − GD (x, x0 )
(41)
2
in analogy with (22).
From these results we can now read off the fluctations of the vacuum fields when the limit
x → x0 is taken. For this purpose we combine (25) and (26) which give
Γ(D/2)
1
2
∂z GN/D = (D − 1)
ζR (D) ± fD (z/L)
(42)
2
(4π)D/2 LD
11
in this limit. With µ̄ = ν̄ = 0 in (40) we then find for the z-component of the electric field
1
Γ(D/2)
2
(43)
ζR (D) + fD (z/L)
h Ez iD = (D − 2)
2
(4π)D/2 LD
The other components follow from (39) which gives
2Γ(D/2)
1
2
h E i iD = −
ζR (D) − fD (z/L)
2
(4π)D/2 LD
(44)
where there is no summation over the index i 6= z on the left-hand side. For the fluctations
2 i = −h E 2 i and h B 2 i = −h E 2 i
in the magnetic components we similarly find h B iz
D
z D
ij D
i D
where again there is no summation over the indices i, j 6= z. These relations also hold for
the case of Neumann boundary conditions except for a change of signs in the last term of
(43) and (44).
These vacuum field fluctuations where first calculated by Lütken and Ravndal[6] for the
ordinary Casimir effect with D = 4 spacetime dimensions in the Coloumb gauge and a
different regularization. Using ζR (4) = π 4 /90 and the function (31) for f4 (z/L) in the
above general results, we find
π2
1
h Ez2 i =
(45)
F
(θ)
+
48L4
15
for Dirichlet boundary conditions, where θ = πz/L as before. The two other transverse
components are given by (44) as
1
π2
2
2
F (θ) −
(46)
h Ex i = h E y i =
48L4
15
Fluctuations of the transverse magnetic components are then h B x2 i = h By2 i = −h Ez2 i
while for the normal compoent we have h B z2 i = −h Ex2 i.
Since the pressure is given by the expectation value of T zz = Fz µ̄ F µ̄z + L, it now follows
as
1
P = − (D − 2)∂z2 (GN + GD ) = (D − 2)(D − 1)E0
2
(47)
when expressed in terms of the energy density (4). It is again constant between the plates
and a factor D − 2 times the scalar pressure (24). This is exactly as expected since the
Maxwell field has D − 2 scalar degrees of freedom.
For the other components of the energy-momentum tensor we similarly find
i
1 D − 2 2h
h Tµ̄ν̄ iD/N = −
∂z GN + GD ± (D − 4)(GN − GD ) ηµ̄ν̄
2D−1
i
Γ(D/2) h
= −(D − 2)
ζ
(D)
±
(D/2
−
2)f
(z/L)
ηµ̄ν̄
R
D
(4π)D/2 LD
12
(48)
where the lower sign is for Neumann boundary conditions. It is only for D = 4 that the
last, position-dependent term will be absent. And it is also then that Maxwell theory is
conformally invariant.
The above results for the electromagnetic field are very similar to what we found for the
canonical, massless scalar field in the previous section. In that case the theory could
be made conformally invariant with an improved energy-momentum tensor which gives a
constant energy density. But for the Maxwell field there is no way to construct a local and
gauge-invariant analogue of a similar Huggins term to cancel out the position-dependent
part of (48). Thus the total Casimir energy obtained by integrating the energy-density
is divergent and therefore looks different from what follows from the regularized sum of
the zero-point energies of all modes. However, the difference turns out to be an infinite
constant independent of the distance between plates.
3.3
Discussion and conclusion
The massless and free, canonical scalar field theory is not conformally invariant in other
dimensions than D = 2. And it is only then that the Casimir energy density is constant and
gives a finite integrated energy. The same satisfactory situation is also possible in higher
dimensions when the theory is extended by making it conformally invariant, corresponding
to adding the Huggins term to the energy-momentum tensor.
This has been well-known for a long time, but not very well understood from a physical
point of view. One of the most recent and detailed discussions of this phenomenon has been
undertaken by Fulling who has attempted to understand the divergences in the canonical
theory at a deeper level[17]. One can isolate the problem to the lack of commutativity
between regularization of the integrated energy and the integration of the regulated energy
density. This is perhaps not so surprising from a mathematical point of view, but hard
to accept physically since the energy density is a physical quantity and should be tied up
with the total energy of system. In other systems like the Casimir energy for a sphere, the
energy density again diverges at the surface[18], but this is understood from its non-trivial
geometry as first discussed by Deutsch and Candelas[19]. Since then the problem has been
addressed by Fulling[17] and Milton with collaborators[20]. For a plane boundary there
should be no such geometric complications.
The electromagnetic Casimir effect for D = 4 is very similar to the scalar effect for D = 2.
But for dimensions D > 4 there is no Huggins term for the electromagnetic case to cure the
problem. From the point of view of the Casimir force alone, this is not a problem because
the pressure is given by the expectation value h T zz i which is constant in all dimensions
and equal to the force. But at first sight this force has little to do with the integral of the
energy density h T00 i which will always diverge at the plates when D > 4.
This becomes especially clear when we just consider the electromagnetic fluctuations
around one plate. The induced energy density can then be obtained from equation (48)
13
by taking the limit L → ∞. The pressure will then be zero on both sides of the plate
while the other components become
h Tµ̄ν̄ iD/N = ∓(D − 2)(D/2 − 2)
Γ(D/2)
(4π)D/2 |z|D
ηµ̄ν̄
(49)
since fD = (L/z)D in this limit, as follows from the definition (27). It is non-zero on both
sides of the plate and diverges when we approach it. The situation is analogous to the
diverging energy density surrounding a classical pointlike electron. Thus, the behaviour
(49) is related to the intrinsic structure of a single plate, and the correponding integrated
(infinite) energy is part of the energy required to make that plate. It does not contribute to
the Casimir force. Thus, to find a connection between Casimir force and energy density it
is sufficient to investigate the changes in energy density as two plates are brought together
from infinite distance. We thus define T µ̄ν̄ as the expression (48) subtracted contributions
like (49) from plates at z = (0, L), taking into account both sides of each plate. We find
that

for z < 0,
 ∓(D/2 − 2)(L/(L − z))D
Γ(D/2)
˜
η
×
Tµ̄ν̄ (z) = −(D − 2)
ζ
(D)
±
(D/2
−
2)
f
(z/L)
for
0 < z < L, (50)
µ̄ν̄
D
 R
(4π)D/2 LD
D
∓(D/2 − 2)(L/z)
for z > L,
where f˜D (z/L) = ζH (D, 1 + z/L) + ζH (D, 2 − z/L). This quantity Tµ̄ν̄ is finite everywhere,
and its integrated energy agrees perfectly with the Casimir force since the integrals over
the z-dependent terms in (50) cancel each other. The consistency between the various
approches to the Casimir effect, i.e. total energy from mode sum (2), the pressure term
(47) and the change in energy density (50), gives support to the belief in a Casimir force
which is essentially independent of the details of the plates.
Of course, the interesting problem of the intrinsic and finite structure of a single plate remains. But this is similar to the problem of resolving the divergences caused by pointlike
objects in quantum field theory. One obvious approach would be to modify the boundary
conditions. If they were made softer so that they didn’t affect fluctuations with wavelengths below a certain cut-off λc , one would expect that the resulting energy density
would be modified and finite for distances |z| < λ c away from the plate. This has actually
been investigated by Graham et al. where a more physical boundary is described by an
additional field[21].
In the more unphysical case where the Neumann or Dirichlet boundary conditions are
replaced by periodic boundary conditions when D > 4, there would be no problems of
these kinds. The energy density is then constant, giving a total energy consistent with
the force between the plates. This is equivalent to the problem of photons in thermal
equilibrium. Even if the trace of the energy-momentum tensor is non-zero when D > 4,
the pressure P in this blackbody radiation is given by the energy density ρ by the standard
expression P = ρ/d where d = D − 1 is the number of spatial dimensions[22].
But imposing such periodic boundary conditions, would be equivalent to just avoiding the
problem. In conclusion, we must admit that the total vacuum fluctuations near confining
14
boundaries is still not completely understood, but is very likely to depend on the microscopic details of those boundaries. This is especially the case for the electromagnetic field
in spacetimes with more than four dimensions. Fortunately the Casimir force seems to be
rather insensitive to those details.
Acknowledgement: We want to thank a referee for an insightful comment which helped
to clarify the conclusion presented above. This work has been supported by the grants
NFR 159637/V30 and NFR 151574/V30 from the Research Council of Norway.
References
[1] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
[2] M.J. Sparnaay, Physica 24, 751 (1958).
[3] S.K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); U. Mohideen and A. Roy ibid. 81,
4549 (1998); H.B. Chan et al. Science 291, 1941 (2001); G. Bressi et al. Phys. Rev.
Lett. 88, 041804 (2002).
[4] V.V. Nesterenko, G. Lambiase and G. Scarpetta, Riv. Nuovo Cim. 27N6, 1 (2004);
I. Brevik, J.B. Aarseth, J.S. Høye and K.A. Milton, quant-ph/0410231.
[5] For more general reviews, see for instance M. Bordag, U. Mohideen and V.M.
Mostepanenko, Phys. Rep. 353, 1 (2001); K.A. Milton, J. Phys. A37, R209 (2004).
[6] C.A. Lütken and F. Ravndal, Phys. Rev. A31, 2082 (1985).
[7] B. DeWitt, Phys. Rep. 19C, 295 (1975).
[8] L.S. Brown and G.J. Maclay, Phys. Rev. 184, 1272 (1969).
[9] A.G. Riess et al., Astron. J. 116 1009 (1998); S. Perlmutter et al., Astrophys. J. 517,
565 (1999).
[10] E. Ponton and E. Poppitz, JHEP 0106, 019 (2001); A. Albrecht, C.P. Burgess, F.
Ravndal and C. Skordis, Phys. Rev. D65, 123507 (2002); K.A. Milton, Grav. Cosmol.
9, 66 (2003).
[11] J. Ambjørn and S. Wolfram, Ann. Phys. N.Y. 147, 1 (1983).
[12] K. Tywoniuk and F. Ravndal, quant-ph/0408163.
[13] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. N.Y. 59, 42 (1972).
[14] E. Huggins, Ph.D. thesis, Caltech, 1962 (unpublished).
[15] K.A. Milton, Phys. Rev. D68, 065020 (2003).
15
[16] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9,
3471 (1974).
[17] S.A. Fulling, J. Phys. A36, 6857 (2003)
[18] K. Olaussen and F. Ravndal, Nucl. Phys. B192, 237 (1981).
[19] D. Deutsch and P. Candelas, Phys. Rev. D20, 3063 (1979).
[20] K.A. Milton, I. Cavero-Pelaèz and J. Wagner, J. Phys. A39, 6543 (2006).
[21] N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, O. Schröder and H. Weigel, Nucl.
Phys. B677, 379 (2004).
[22] H. Alnes, I.K. Wehus and F. Ravndal, quant-ph/0506131.
16
Paper V
Resolution of an apparent inconsistency in the
higher-dimensional electromagnetic Casimir effect
H. Alnes, F. Ravndal and I.K. Wehus
Department of Physics, University of Oslo, N-0316 Oslo, Norway.
and
K. Olaussen
Department of Physics, NTNU, N-7491 Trondheim, Norway.
Abstract
The vacuum expectation value of the electromagnetic energy-momentum tensor
between two parallel plates in spacetime dimensions D > 4 is calculated in the axial
gauge. While the pressure between the plates agrees with the global Casimir force,
the energy density is divergent at the plates and not compatible with the total energy
which follows from the force. However, subtracting the divergent self-energies of the
plates, the resulting energy is finite and consistent with the force. In analogy with the
corresponding scalar case for spacetime dimensions D > 2, the divergent self-energy of
a single plate can be related to the lack of conformal invariance of the electromagnetic
Lagrangian for dimensions D > 4.
Two parallel, metallic plates separated by the distance L in vacuum, will interact due to
the modifications of the quantum fluctuations of the electromagnetic field caused by the
boundary conditions at the plates. The resulting force was first calculated by Casimir[1]
who found it to be given by the attractive pressure P = −π 2 /240L4 . Using the conformal symmetry of the electromagnetic field in D = 4 spacetime dimensions, Brown
and Maclay[2] later obtained the vacuum expectation values of all the components of the
electromagnetic energy-momentum tensor
Tµν = Fµα F αν − ηµν L
(1)
2 is the standard Lagrangian. While these expectation values were
where L = −(1/4)Fαβ
constant between the plates, the corresponding fluctuations of the separate electric and
magnetic fields were found by Lütken and Ravndal to be in general non-constant and
actually divergent as one approaches one of the plates[3]. These divergences are caused
by imposing ideal boundary conditions valid for arbitrarily small wavelengths of the field.
A physical boundary would only affect fluctuations down to a finite wavelength which is
expected to result in an increasing, but finite value of the fluctuations near the plates.
The quantitative effects of such more realistic boundary conditions have been investigated
during the last few years but a complete and satisfactory description is still lacking[4].
Casimir forces in spacetimes with dimensions D > 4 were first systematically calculated by
Ambjørn and Wolfram[5]. For the electromagnetic field between two parallel hyperplanes
1
with separation L, the attractive pressure was found to be
P = −(D − 1)(D − 2)
Γ(D/2)ζR (D)
(4π)D/2 LD
(2)
where the factor D − 2 is the number of physical degrees of freedom in the field resulting
from gauge invariance. If the energy density between the plates is constant, it would just
be this pressure divided by the factor D − 1. This is the case when D = 4 and it is of
interest to see if it holds also in the more general case D > 4. For this purpose we calculate
in the following the separate fluctuations of the electric and magnetic components of the
field which then allows us to find all the vacuum expectation values of the components of
the energy-momentum tensor (1).
Today these quantum effects could be of relevance for stacks of parallel branes where the
electromagnetic field is replaced by one or more of the abelian Ramond-Ramond fields.
Any divergent energy density would then have serious implications for the stability of such
configurations due to the resulting large gravitational interactions.
The electromagnetic field tensor Fµν = ∂µ Aν − ∂ν Aµ in D = d + 1 spacetime dimensions
has d electric components Ei = F0i and d(d − 1)/2 magnetic components B ij = Fij . For
the geometry under consideration, the simplest and most natural choice of gauge is the
axial gauge nµ Aµ = 0 where the unit D-vector nµ is normal to the plates. Taking this
along the z-axis, we thus have Az = 0. The component A0 is no longer a free variable, but
depends on the others via the Maxwell equation ∂ i F i0 = 0. It gives A0 = −∆−1 ∂i Ȧi where
the operator ∆ = ∂i2 . There are thus D − 2 independent degrees of freedom described by
the spatial field components Ai where i 6= z. The full Lagrangian then follows as
Z
h i
1
dd x Ȧi δij − ∂i ∆−1 ∂j Ȧj − Ai ∂i ∂j − δij ∆ Aj
(3)
L=
2
after a few partial integrations and neglecting surface terms.
In order to quantize the system, we must solve the classical wave equation following from
the Lagrangian. For this purpose we impose the boundary condition n µ Fµν = 0 at the
plates. This is the same as for the MIT quark bag where it had a physical justification[6].
Here it is just taken for convenience. In the axial gauge it gives ∂ z Ai = 0 at the plates which
is the Neumann boundary condition for each physical field component A i (x) = Ai (t; xT , z).
We then have the general mode expansion
r ∞ Z
nπz 2 X dd−1 kT
ikT ·xT
Ai (t; xT , z) =
A
(t,
k
)e
cos
(4)
in
T
L
(2π)d−1
L
n=1
p
which satisfies the wave equation and the boundary conditions. The factor
2/L is a
normalization factor. In the mode sum we have dropped a term with n = 0 since it will
not contribute to any physical results after regularization.
Quantization can now be done in the standard way. We introduce orthonormal polarization
vectors eλ normal to the wavevector kT and a longitudinal polarization vector e L along this
2
direction. The coordinate components A in of the field are then replaced by the polarization
components (Aλn , ALn ). After quantization at t = 0 the transverse components can then
be written on the standard form as
r
i
1 h
Aλn (kT ) =
aλn (kT ) + a†λn (−kT )
(5)
2ωn
where ωn2 = k2T + kz2 with kz = πn/L. The creation and annihilation operators now have
the standard commutator
[aλn (kT ), aλ0 n0 (k0T )] = δλλ0 δnn0 (2π)d−1 δ(kT − k0T )
However, the longitudinal component
r
h
i
1
ωn
aLn (kT ) + a†Ln (−kT )
ALn (kT ) =
2ωn kz
(6)
(7)
contains an extra factor when the corresponding creation and annihilation operators have
the same canonical commutator (6). The full field operator (4) is then expressed in terms
of these new operators corresponding to definite polarization states.
The field fluctuations between the two plates can now easily be calculated. As a simple
example, consider Ez = −∂z ∆−1 ∂j Ȧj . If we isolate the mode with quantum numbers
(n, kT ), we find the operator
r r
nπz 2
1 (ikj )(−iωn )
ωn
ikT ·xT
−1
e
cos
a
e
+ H.c.
a
e
+
∆ ∂j Ȧj =
Ln Lj
λn λj
L 2ωn
ωn2
kz
L
(8)
acting on the vacuum state. Here we have used that ∆ gives k 2T + kz2 = ωn2 in momentum
space. We see that the transverse modes will not contribute here since they satisfy the
orthogonality condition eλ · kT = 0. However, for the longitudinal mode we have instead
eL · kT = kT and it will give a non-zero contribution. The derivative ∂ z gives a factor kz
and cos(nπz/L) → sin(nπz/L). For this mode alone we thus get the fluctuation
h Ez2 i|n,kT =
2 1 2
nπz
kT sin2
L 2ωn
L
(9)
Including all the modes, we thus have for the full fluctuation of this electric field component
∞
h Ez2 i =
1X
L
n=1
Z
dd−1 kT
kz2
nπz
ω
−
sin2
n
(2π)d−1
ωn
L
(10)
when we write kT2 = ωn2 − kz2 . For the other components we similarly find
∞
h Ei2 i
=
1X
L
n=1
Z
dd−1 kT
kz2
nπz
cos2
ωn (d − 2) +
ωn
L
(2π)d−1
3
(11)
where there is an implied sum over the transverse index i. The magnetic field fluctuations
can be obtained the same way and become
∞ Z
1 X dd−1 kT
nπz
kz2
2
h Biz i =
(d − 2) sin2
(12)
ωn +
L
(2π)d−1
ωn
L
n=1
∞ Z
1 X dd−1 kT
kz2
nπz
2
h Bi<j i =
ωn (d − 2) −
(d − 2) cos2
(13)
d−1
L
ω
L
(2π)
n
n=1
when we again sum over the indices i and j.
Using now a combination of dimensional and zeta-function regularization as previously
used when D = 4[7], we can write the result on the form
∞ Z
1
Γ(D/2)
1 X dd−1 kT
2nπz
ωn 1 ± cos
(14)
ζR (D) ± fD (z/L)
=−
2L
(2π)d−1
L
2
(4π)D/2 LD
n=1
and
∞
1 X
2L n=1
Z
dd−1 kT kz2
(2π)d−1 ωn
2nπz
1 ± cos
L
1
Γ(D/2)
ζR (D) ± fD (z/L)
= −(D − 1)
2
(4π)D/2 LD
(15)
Here ζR (D) is the Riemann zeta-function while f D (z/L) depends on the distance z from
the plates. When the spacetime dimension D is even, it can be written on the compact
form
d D−1
πD
cot θ
(D = even)
(16)
−
fD (z/L) =
Γ(D)
dθ
where θ = πz/L. But when D is odd, no such closed expression is easily derived. However,
using a different regularization based on the corresponding point-split Green’s functions,
one finds in general[8]
fD (z/L) =
∞
X
j=−∞
1
= ζH (D, z/L) + ζH (D, 1 − z/L)
|j + z/L|D
(17)
where ζH (D, z/L) is the Hurwitz zeta-function. When D is even, this can be shown to
agree with (16).
The regularized fluctuations of the electric field normal to the plates thus become
(D − 2)Γ(D/2)
1
2
h Ez i =
ζR (D) − fD (z/L)
2
(4π)D/2 LD
(18)
while for the transverse components we find
h Ei2 i
(D − 2)Γ(D/2)
1
= −2
ζR (D) + fD (z/L)
2
(4π)D/2 LD
4
(19)
For the magnetic fluctuations we similarly have
2
h Biz
i
1
(D − 2)Γ(D/2)
ζR (D) − fD (z/L)
= −(D − 2)
2
(4π)D/2 LD
(20)
(D − 2)Γ(D/2)
1
= (D − 3)
ζR (D) + fD (z/L)
2
(4π)D/2 LD
(21)
and
2
h Bi<j
i
Notice again that in these expressions we have summed over the transverse indices i and
j, each taking D − 2 different values. All these correlators are seen to diverge near the
plates z → 0 or z → L where the function f D (z/L) diverges. This is the same phenomenon
which has previously been seen in D = 4 dimensions[3].
The pressure betwen the plates due to these fluctuations is defined by P = h T zz i. From
2 − E 2 + L where now
(1) we have Tzz = Biz
z
(D − 2)Γ(D/2)
1
fD (z/L)
h Li = − (D − 1)
2
(4π)D/2 LD
(22)
2 i from above, the z-dependence from the
Together with the values for h Ez2 i and h Biz
function fD (z/L) cancels out in the pressure and gives the expected value (2).
So far there are no inconsistencies in the obtained results. But when we now calculate the
energy density E = h T00 i between the plates, with T00 = Ei2 + Ez2 − L, we obtain
E =−
(D − 2)Γ(D/2)
[ζR (D) − (D/2 − 2)fD (z/L)]
(4π)D/2 LD
(23)
The z-dependence in the last term is non-zero when D > 4 and makes the energy density
diverge like z −D with distance z from the plates. As a result, the total energy of the system
is infinite, a result which seems to be impossible to reconcile with the finite Casimir force
(2). In fact, (2) corresponds to having a constant energy density equal to the first term
in (23). This apparent inconsistency has been verified in a different approach based on
Green’s function methods[8].
It is tempting to explain this problem by the imposed boundary conditions. We have used
the MIT boundary condition which is equivalent to letting the electromagnetic vector potential satisfy Neumann boundary conditions in the axial gauge. Had we instead imposed
metallic boundary conditions, equivalent to Dirichlet boundary conditions for the vector
potential in the axial gauge, the only change in the above results would be the replacement of the mode functions cos(nπz/L) with sin(nπz/L) in (4) so that f D → −fD in the
above results. Needless to say, the problem would remain. Only for periodic boundary
conditions, as for finite temperature, would the disturbing term be absent[9]. But this is
not necessarily satisfying from a physical point of view. A more mathematical discussion
of such divergences near confining boundaries has been initiated by Fulling but here only
scalar fields are considered[10].
5
A physical explanation of the above conumdrum becomes apparent when we take the limit
L → ∞ and thus consider the quantum fluctuations around a single plate. From (23) we
then find the energy density
E1 = (D − 2)(D/2 − 2)
Γ(D/2)
(4π)D/2 |z|D
(24)
which is non-zero on both sides of the plate and diverges when we approach it. This
situation is analogous to the diverging energy density surrounding a pointlike electron.
It is intrinsic to a single plate and should not contribute to the interaction between the
plates induced by the same vacuum fluctuations. To see the connection with the Casimir
force, we should subtract the self-energy (24) for both plates from the full energy density
(23), taking into account both sides of each plate. We thus obtain the interaction energy
density

for z < 0,
 (D/2 − 2)(L/(L − z))D
Γ(D/2)
˜
˜D (z/L) for 0 < z < L,
×
(25)
E = −(D − 2)
ζ
(D)
−
(D/2
−
2)
f
R

(4π)D/2 LD
(D/2 − 2)(L/z)D
for z > L,
where now
f˜D (z/L) = ζH (D, 1 + z/L) + ζH (D, 2 − z/L)
(26)
It is seen to be finite everywhere, even at the plates. When integrating over the full
volume, the z-dependent terms cancels out as follows from
Z 1
Z ∞
Z 0
dx
dx
˜(x) +
−
dx
f
=
D
D
(1
−
x)
x
0
1
−∞
"
#
∞ X
1
2
1
1+
−
=0
(27)
D−1
(n + 2)D−1 (n + 1)D−1
n=0
Only the z-independent term in (25) contributes and agrees perfectly with the total energy
corresponding to the Casimir force.
A similar and somewhat simpler system is the Casimir energy induced by a massless scalar
field in the same geometry. One will then find a very similar result for the energy density
as obtained here[8]. It diverges near the plates for all spacetime dimensions D > 2. Again
this can be attributed to a divergent self-energy of each plate. However, when D = 2
there are no such divergences and zero self-energy. But this is also the dimension in which
the scalar theory has conformal invariance. In higher dimensions D > 2 it is possible to
make the scalar theory retain this invariance by adding a conformal term. The resulting,
improved energy-momentum tensor[12] then contains an additional piece discovered by
Huggins[13] and makes it traceless. Including the Huggins term, the divergent part of the
energy density corresponding to the last term in (23) drops out as s first noticed by de
Witt when D = 4[11].
6
For the electromagnetic field we have used the canonical energy-momentum tensor (1)
which has the trace T µµ = (4 − D)L. It is zero for D = 4 which reflects the well-known
fact that Maxwell theory is then conformally invariant. There are then no diverences
in the Casimir energy. Thus it is natural to relate the apparent inconsistency in the
electromagnetic Casimir energy when D > 4 to the lack of conformal invariance. It does
not seem to be possible to construct an improved energy-momentum tensor in this case
because gauge invariance forbids the existence of any corresponding local Huggins term.
From this point of view the divergent, electromagnetic self-energy can therefore not be
removed. For this to be done, one needs a more realistic description of the boundary plates
along the lines considered by others[4].
This work has been supported by the grants NFR 159637/V30 and NFR 151574/V30 from
the Research Council of Norway.
References
[1] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
[2] L.S. Brown and G.J. Maclay, Phys. Rev. 184, 1272 (1969).
[3] C.A. Lütken and F. Ravndal, Phys. Rev. A31, 2082 (1985).
[4] K.A. Milton, J. Phys. A37, R209 (2004); N. Graham, R.L. Jaffe, V. Khemani, M.
Quandt, O. Schröder and H. Weigel, Nucl. Phys. B677, 379 (2004); K.A. Milton, I.
Cavero-Pelaèz and J. Wagner, J. Phys. A39, 6543 (2006).
[5] J. Ambjörn and S. Wolfram, Ann. Phys. N.Y. 147, 1 (1983).
[6] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9,
3471 (1974).
[7] K. Tywonik and F. Ravndal, quant-ph/0408163.
[8] H. Alnes, K. Olaussen, F. Ravndal and I.K. Wehus, quant-ph/0607081.
[9] H. Alnes, F. Ravndal and I.K. Wehus, quant-ph/0506131.
[10] S.A. Fulling, J. Phys. A36, 6857 (2003)
[11] B. de Witt, Phys. Rep. 19C, 295 (1975).
[12] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. N.Y. 59, 42 (1972).
[13] E. Huggins, Ph.D. thesis, Caltech, 1962 (unpublished).
7
Paper VI
Gravity coupled to a scalar field in extra dimensions
Ingunn Kathrine Wehus and Finn Ravndal
Department of Physics, University of Oslo, P.O.Box 1048 Blinderen, N-0316 Oslo, Norway.
E-mail: i.k.wehus@fys.uio.no
Abstract. In d + 1 dimensions we solve the equations of motion for the case of gravity
minimally or conformally coupled to a scalar field. For the minimally coupled system the
equations can either be solved directly or by transforming vacuum solutions, as shown before
in 3 + 1 dimensions by Buchdahl. In d + 1 dimensions the solutions have been previously
found directly by Xanthopoulos and Zannias. Here we first rederive these earlier results, and
then extend Buchdahl’s method of transforming vacuum solutions to d + 1 dimensions. We
also review the conformal coupling case, in which d + 1 dimensional solutions can be found by
extending Bekenstein’s method of conformal transformation of the minimal coupling solution.
Combining the extended versions of Buchdahl transformations and Bekenstein transformations
we can in arbitrary dimensions always generate solutions of both the minimal and the conformal
equations from known vacuum solutions.
1. Introduction
Theories in which gravity couples to a scalar field are common in extra-dimensional problems.
A few examples are Kaluza-Klein, Jordan and Brans-Dicke theories, as well as string theory in
general. Consequently, the corresponding Einstein equations have been studied extensively for
the last sixty years, with particular emphasis on searching for black hole solutions.
Minimally coupled scalar fields in 3 + 1 dimensions were first studied by Fisher [1] in 1948,
and later by Bergmann and Leipnik [2] and by Janis, Newman and Winicour [3]. These were
all solving the equations directly. However, in 1959 Buchdahl showed that it is always possible
to generate a solution for the minimal coupling case by means of a particular transformation [4]
of a vacuum solution metric. The problem was later revisited by Janis, Robinson and Winicour
[5], who also included electromagnetism in their solutions.
The extension from 3 + 1 dimensions to d + 1 dimensions was first done by Xanthopoulos and
Zannias [6], using a direct solution technique. In the present paper we generalize Buchdahl’s
transformation method to arbitrary dimensions to solve the same problem.
Solutions of the conformally coupled equations in 3 + 1 dimensions were first found by
Bocharova, Bronnikov and Melnikov in 1970 [7], using a direct approach. However, these results
were published in a Russian journal, and did therefore not get the attention of physicists in the
western world. As a result, the solutions were re-discovered independently by Bekenstein [8]
in 1974, using a conformal transformation method. These solutions included a black hole-like
solution [9], known as the BBMB (Bocharova–Bronnikov–Melnikov–Bekenstein) black hole. The
first direct solution in the West was found by Frøyland [10] in 1982, who demonstrated that the
metric coincides with the extremal Reissner-Nordström solution. In a later work, Xanthopoulos
and Zannias [11] further showed that the BBMB solution is unique.
Again, the extension to d + 1 dimension was also done by Xanthopoulos, this time together
with Dialynas [12]. They found solutions for the conformally coupled case by extending
Bekenstein’s method of transforming minimal solutions. They also demonstrated that the BBMB
black hole only exists in 3 + 1 dimensions.
These issues have been investigated in collaboration with Tangen [13]. One goal of this
investigation was to extend Buchdahl’s and Bekenstein’s methods of generating solutions by
conformal transformations to spacetimes with extra dimensions. However, during the course of
the project, we found that most results had already been obtained by others. We will here give a
pedagogical introduction to the field based on the more formal approach of Tangen. In particular,
we will show how the Buchdahl transformation can be extended to arbitrary dimensions.
2. Gravity coupled to a scalar field
In D = d + 1 dimensions the theory of gravity coupled to a free scalar field is described by the
action
Z
√
1
R − ξRφ2 − g µν φ,µ φ,ν
S = dD x −g
(1)
2
We work in natural units where c = 1 = M D , MD being the D-dimensional reduced Planck
mass. For instance, in four spacetime dimensions we have M 4−2 = 8πG3 = 1.
Putting the parameter ξ to zero gives us the common case where the scalar field is minimally
coupled to gravity, while keeping ξ non-zero allows for more general couplings. In this paper we
will also be interested in the conformal coupling case defined by ξ = d−1
4d . Varying the action
with respect to φ gives the equation of motion for the scalar field
2 φ − ξRφ = 0
(2)
while varying with respect to the metric g µν gives the Einstein equations
(1 − ξφ2 )Eµν = Tµνφ + ∆Tµνφ
(3)
Here Tµνφ is the ordinary scalar field energy-momentum tensor for the minimal coupling (ξ = 0)
case
1
Tµνφ = φ,µ φ,ν − gµν φ,α φ,α
(4)
2
and ∆Tµνφ is the Huggins term [14] coming from the extra term ξRφ 2 in the Lagrangian [15]
∆Tµνφ = ξ gµν 2 (φ2 ) − (φ2 );µν
(5)
To summarize, the total energy-momentum tensor is given by
1
Tµν = φ,µ φ,ν − gµν φ;α φ;α + ξ gµν 2 (φ2 ) − (φ2 );µν + ξφ2 Eµν
2
(6)
Only for ξ = d−1
4d will Tµν be traceless and the theory will be conformally invariant. By
contracting equation (3) we see that in this case also the Ricci scalar vanishes, R = 0. Taking
the trace of equation (4), we notice that only in 1 + 1 dimensions is a minimally coupled scalar
field conformally invariant.
2.1. Statical, spherical symmetric solutions
We are searching for black hole-like solutions and are only interested in static and spherical
symmetrical solutions. The most general static and spherical symmetric metric in d + 1
dimensions can be written
ds2 = −e2α(r) dt2 + e2β(r) dr 2 + e2γ(r) r 2 dΩd2
(7)
where α, β and γ are unknown functions of the radial coordinate r, and dΩ 2d is the solid angle
element in d − 1 dimensions. Putting γ = 1 gives us Schwarzschild coordinates in which the
equations of motion often take the simplest form. In our case the solutions can often not
be explicitly written down in Schwarzschild coordinates and it is better to work in isotropic
coordinates, defined by β(r) = γ(r). In Schwarzschild coordinates the Einstein tensor has the
independent components
e2β−2α Ett =
Err =
(d − 1) 0 (d − 2)(d − 1) 2β
(e − 1)
β +
r
2r 2
(d − 1) 0 (d − 2)(d − 1) 2β
(e − 1)
α −
r
2r 2
(8)
(9)
The scalar field configuration must also be static and spherically symmetric, so φ ,µ can only have
one non-zero component, φ,r ≡ φ0 . Using this when calculating the energy-momentum tensor
we find for the minimal case
1
e2β−2α Tttφ = Trrφ = φ02
(10)
2
and for the Huggins term
φ
2β−2α
00
0 d−1
02
0
e
∆Ttt = −2ξ φφ + φφ
(11)
−β +φ
r
d−1
∆Trrφ = 2ξφφ0
(12)
+ α0
r
We also find the following expression for the D’Alambertian operator
h
d−1 i
2 = e−2β ∂r2 + (α0 − β 0 +
)∂r
r
(13)
Rtt = e−2α 2 α
(14)
Using this we get the following simple expression for the first component of the Ricci tensor
while the Ricci scalar may be written
R = −22 α +
i0
d − 1 h d−2 −2β
1
−
e
r
r d−1
(15)
3. Minimal coupling
3.1. Fundamental equations
Putting ξ = 0 gives us the minimal coupling case. Equation (2) then reduces to 2 φ = 0, and
using equation (13) we obtain the equation of motion for φ,
φ00 = −(α0 − β 0 +
d−1 0
)φ
r
(16)
For a non-constant scalar field this can easily be integrated to give
φ0 = Ce−α+β r −(d−1)
(17)
where C is a constant of integration. We further notice from (14) that R tt = 0 gives us 2 α = 0
so for both α and φ non-zero we have
α0 = Kφ0
(18)
for some constant K. When using (8-9) and (10) the Einstein equations can be simplified to
e2β − 1 =
φ0 2 =
r
α0 − β 0
d−2
d−1 0
α + β0
r
(19)
(20)
Using equation (18) to substitute for φ 0 , and then eliminating β and β 0 from equations (17), (19)
and (20) we are left with a first-order differential equation for α. But this can not be explicitly
solved to find α, indicating that the general solution of the minimally coupled equations can not
be explicitly written i Schwarzschild coordinates. We can however look at two special cases.
First we notice that for constant φ equation (20) imply α 0 + β 0 = 0. Then equation (19) can
be rewritten as
h
i0
r d−2 1 − e−2β
=0
(21)
which may be integrated explicitly, and we end up with the trivial Schwarzschild vacuum solution
for the metric [16] in d + 1 dimensions [17]
Bs
Bs −1 2
2
2
ds = − 1 − d−2 dt + 1 − d−2
dr + r 2 dΩd2
(22)
r
r
were the integration constant Bs is canonically normalized to give Newtonian gravity in the
Gd M
large r limit, Bs = 2(d−2)
with Gd being the d-dimensional Newtonian gravitational constant,
and M is the mass of the black hole.
Second, for the special case α = 0, equation (19) can be rewritten as
i0
h
=0
(23)
r 2(d−2) 1 − e−2β
giving
e−2β = 1 −
C0
(24)
r 2(d−2)
Inserting this into (20) we see that for φ 02 to be positive, the integration constant C 0 has to be
negative. We put C 0 = −A2 and integrate φ0 to get
!
r
r
d−1
A2
A
φ=±
(25)
1 + 2(d−2) − d−2 + C 00
ln
d−2
r
r
Since we want φ to go to zero for large r when the metric approaches flat Minkowski space we
choose the integration constant C 00 = 0. Our final solution thus reads
−1
A2
2
2
ds = −dt + 1 + 2(d−2)
dr 2 + r 2 dΩd2
(26)
r
!
r
r
A2
A
d−1
1 + 2(d−2) − d−2
ln
(27)
φ = ±
d−2
r
r
Xanthopoulos and Zannias [6] found a general two-parameter solution in arbitrary dimensions
by solving the equations of motion in isotropic coordinates. This solution may be written as
# 2 "
#2a
#− 2a
"
2(d−2) d−2
d−2
d−2 − r d−2
d−2 − r d−2
r
r
r
2
0
0
0
dr 2 + r 2 dΩd2 (28)
dt
+
1
−
ds2 = −
d−2
d−2
2(d−2)
r
r d−2 + r0
r d−2 + r0
r
d−1
r d−2 − r0d−2
φ =
(29)
(1 − a2 ) ln
d−2
r d−2 + r0d−2
"
where r0 and a are arbitrary constants. The parameter a can run between 0 and 1, and a = 1
corresponds to the Schwarzschild metric (22) plus a constant scalar field solution. a = 0 gives
the upper sign version of (26-27). As showed by Xanthopoulos and Zannias this latter solution
is a naked singularity, and no value of a yields a black hole solution [6].
3.2. Buchdahl transformations
We will now show that for a given solution of the d+1 dimensional Einstein equations in vacuum,
one can always generate a solution of the same equations minimally coupled to a scalar field. In
3+1 dimensions this was first shown by Buchdahl [4] and later by Janis, Robinson and Winicour
[5]. For the general d + 1 dimensional case see Tangen [13].
For a metric on the form
i
i
ds2 = −e2V (x ) dt2 + e−2V (x ) ĥij xi xj
(30)
ĥij being a d-dimensional spatial metric and V is a function of spatial coordinates only, the
Ricci tensor reads
i
h 2
ˆ V − (d − 3)V,i V ,i = e2V 2 V
(31)
R00 = e4V i
h 2
ˆ V − (d − 3)V,i V ,i
(32)
Rij = R̂ij + (d − 3)Vˆ;ij + (d − 5)V,i V,j + ĥij The hat denotes quantities derived using the metric ĥij which also is used to raise indexes. If
the metric (30) is a solution of the Einstein equations in vacuum we must have R µν = 0, which
implies
ˆ 2 V − (d − 3)V,i V ,i = 0
R̂ij + (d − 3)Vˆ;ij + (d − 5)V,i V,j = 0
(33)
(34)
We now introduce a new metric
i
i
ds̄2 = −e2U (x ) dt2 + e−2U (x ) h̃ij xi xj
(35)
where again U is a function of spatial coordinates only, and the spatial metric h̃ij is conformal
to the spatial vacuum metric ĥij . We want our metric (35) to be a solution of the minimally
coupled equations. For this metric the Einstein equations for gravity minimally coupled to a
static scalar field, R̄µν = φ,µ φ,ν , reads
˜ 2 U − (d − 3)U,i U ,i = 0
R̃ij + (d − 3)U˜;ij + (d − 5)U,i U,j = φ,i φ,j
(36)
(37)
while we have the equation of motion for the scalar field φ
h 2
i
˜ φ − (d − 3)φ,i U ,i
2 φ̄ = 0 = e2U (38)
Since the Ricci tensor in d dimensions transforms as
2
˜ Ω
R̂µν = R̃µν − (d − 2)Ω−1 Ω˜;µν − Ω−1 h̃µν + 2(d − 2)Ω−2 Ω,µ Ω,ν − (d − 3)Ω−2 h̃µν Ω,α Ω,α
(39)
under a Weyl transformation ĥij = Ω2 h̃ij of the metric, we find that when setting
U
= aV
ĥij
2
(40)
2 d−3
d−2
1−a
U
a
= Ω h̃ij = e
h̃ij
r
d − 1 1 − a2
U
φ = ±
d − 2 a2
(41)
(42)
the equations (33-34) are transformed into (36-37). The equation of motion for φ (38) is also
fulfilled. We see from (42) that a necessary constraint is a 2 ≤ 1. In conclusion, given a vacuum
solution of the Einstein equations on the form (30), we can always find a solution of the Einstein
equations for a minimally coupled scalar field given by
ds2 = −e2V a dt2 + e−2V b h̃ij xi xj
r
d−1
φ = ±
(1 − a2 )V
d−2
where a is an arbitrary constant and b = a+d−3
d−2 .
We adopt the higher-dimensional Schwarzschild solution (22) written in the form
ds2 = −e2V dt2 + e−2V dr 2 + e2V r 2 dΩd2
where
eV =
r
1−
B
r d−2
(43)
(44)
(45)
(46)
as our vacuum solution. Using the above transformation, we end up with the following twoparameter set of solutions
a+d−3 ds2 = −e2V a dt2 + e−2V d−2 dr 2 + e2V r 2 dΩd2
r
1 d−1
B
2
φ = ±
(1 − a ) ln 1 − d−2
2 d−2
r
(47)
(48)
Renaming the constant B = 4r0d−2 and making the coordinate transformation to isotropic
coordinates
2
d−2
r → r r d−2 − r0d−2
(49)
we see when choosing the upper sign in (48) that these are exactly the same solutions as those
found by Xanthopoulos and Zannias [6].
4. Conformal coupling
4.1. Fundamental equations
We now consider the case ξ = d−1
4d for which the system is conformally invariant. Since T µν ,
Eµν and Rµν are traceless in a conformal theory, equation (2) still reduces to 2 φ = 0 and the
Einstein equations in Schwarzschild coordinates can be written
φ0 2
φφ0 α0
+
=
1−
d(d − 1)
d
φφ0
φ0 2
d−1
0
+
α +
=
1−
(d − 1)
d
r
i
d − 1 2 h 2 0 d − 2 2β
φ
β + 2 (e − 1)
4d
r
r
i
h
d − 1 2 2 0 d − 2 2β
φ
α − 2 (e − 1)
4d
r
r
(50)
(51)
Further, since 2 φ = 0, equations (16) and (17) are still valid.
When trying to solve the above equations, it is tempting to choose α 0 + β 0 = 0. Then we get
from adding (50) and (51)
d+1 0 2
φφ00 =
(φ )
(52)
d−1
with solution
A
φ=
(53)
d−1
(r − B) 2
where A and B are constants. Combining with the φ-equation (17) which for the case α 0 +β 0 = 0
reads
φ0 = Ce−2α r −(d−1)
(54)
we can solve for e2α and find
d−3
B d+1
2
r− 2
e2α = e−2β = 1 −
r
(55)
Here we have chosen the integration constant C in (54) such that the metric approaches
Minkowski for large r. But only for d = 3 dimensions is this a solution of the Einstein equations
2
(50) and (51). In 3 + 1 dimensions we find A 2 = Bξ = 6B 2 and the metric given by (55) reduces
to the d = 3 version of the extremal Reissner-Nordström metric[18, 19][17]
B 2 2
B −2 2
ds = − 1 − d−2 dt + 1 − d−2
dr + r 2 dΩd2
r
r
2
(56)
The corresponding scalar field solution is
φ=
√ B
6
r−B
(57)
To get canonical normalization we must put B = B s /2. For d 6= 3 the d+1-dimensional extremal
Reissner-Nordström metric is not a solution of the combined gravity and scalar field equations.
In this case there are no solutions having α 0 + β 0 = 0.
4.2. Bekenstein transformations
Introducing a new metric g̃µν in the form of the following conformal transformation of the old
metric gµν
p
4
(58)
g̃µν = cosh d−1 ( ξφ) gµν
together with a redefinition of the scalar field
p
1
ψ = √ tanh( ξφ)
ξ
(59)
brings us from the minimal coupling case of the Einstein plus scalar field equations in the old
metric gµν
Eµν (g) = Tµνφ (g)
2 φ = 0
(60)
(61)
to the conformal coupling case of the same equations for the new metric g̃ µν
(1 − ξψ 2 )Eµν (g̃) = Tµνψ (g̃) + ∆Tµνψ (g̃)
2
˜ ψ = 0 = R
(62)
(63)
Here we use the fact that the minimal scalar field energy-momentum tensor (4) does not change
under a conformal transformation like (58)
Tµνφ (g̃) = Tµνφ (g)
(64)
and that the Einstein tensor under a conformal transformation g̃ µν = Ω2 gµν in D dimensions
change like
h
i
Ẽµν = Eµν + (D − 2)Ω−1 gµν 2 Ω − Ω;µν
1
+ 2(D − 2)Ω−2 Ω,µ Ω,ν + (D − 2)(D − 5)Ω−2 gµν Ω,α Ω,α
2
The result is the same if we instead of transformations (58-59) use
p
4
g̃µν = sinh d−1 ( ξφ) gµν
p
1
ψ = √ coth( ξφ)
ξ
(65)
(66)
(67)
The latter correspond to 1 − ξψ 2 being negative while the first transformations (58-59) is used
for positive 1−ξψ 2 . The transformations (58-59) and (66-67) were first found in 3+1 dimensions
by Bekenstein [8]. Maeda [20] showed very generally that Lagrangians with arbitrary couplings
between φ and R in arbitrary dimensions can always be transformed to a minimally coupled
Einstein Frame theory by means of a conformal transformation. The specific extension of
equations (58-59) and (66-67) to arbitrary dimensions was done by Xanthopoulos and Dialynas
[12].
We now take as our minimal solution the solution (46-48) found in section 3.2 which we write
like
ds2 = −e2V a dt2 + e−2V b dr 2 + e2V (1−b) r 2 dΩd2
r
d−1
φ = ±
(1 − a2 )V
d−2
(68)
(69)
Performing the transformations (58-59) we arrive at a two-parameter solution of the conformal
equations
2V c
4
i
e
+ e−2V c d−1 h 2V a 2
2
ds =
−e dt + e−2V b dr 2 + e2V (1−b) r 2 dΩ2
(70)
2
p
e4V c − 1
ξψ = tanh(±2V c) = ± 4V c
(71)
e
+1
where the constant c is given by
d−1
c=
4
s
1 − a2
d(d − 2)
(72)
The conformal solution (70-71) has the same two parameters a and B as the minimal solution
(46-48). V is still given by (46) and we still have b = a+d−3
d−2 .
1
Choosing now the particular solution given by c = 14 , corresponding to a = d−1
and b = d−2
d−1 ,
the metric (70) simplifies to
2
ds =
eV + 1
2
4
d−1
−dt2 + e−2V dr 2 + r 2 dΩ2
(73)
In order to write this particular solution in Schwarzschild coordinates, we introduce a new radial
coordinate R given by
2
q

 d−1
B
1 + 1 − rd−2

R=
r
(74)
2
Then the metric (73) can be written as
2
ds = −
R
r(R)
2
2
dt +
2
d−1
R
r(R)
d−1
2
d−3
+
d−1
!−2
dR2 + R2 dΩ2
(75)
where r(R) is given implicitly from (74). When choosing the lower sign in (71) the scalar field
φ now takes the form
d−1
p
r(R) 2
−1
(76)
ξψ =
R
Only in d = 3 dimensions can (74) be solved explicitly to give
1
1
B
=
1−
r
R
4R
(77)
so that we can write (75) and (76) in Schwarzchild coordinates
B 2 2
B −2
ds = − 1 −
dt + 1 −
dR2 + R2 dΩ2
4R
4R
√
B/4
ψ =
6
R − B/4
2
(78)
(79)
which is the same BBMB solution with the extremal Reissner-Nordström metric as we found
in last section (56-57). As demonstrated by Xanthopoulos and Zannias [6] and Xanthopoulos
and Dialynas [12], this is the only known black hole solution for gravity coupled to scalar fields
except for the trivial solutions where φ is constant and the metric is a vacuum black hole. The
BBMB solution has been extensively studied by for instance Zannias [21] who has shown it not
to have a continuous new parameter, and therefore not to contradict the no scalar hair-theorem
[22][23][24]. For an overview see [25].
5. Conclusions
Given a vacuum solution of the Einstein equations, solutions of the equations for gravity coupled
either minimally or conformally to a massless scalar field can be generated in arbitrary spacetime
dimensions. To obtain a minimal solution we perform a generalized Buchdahl transformation
on our vacuum metric. To obtain a conformal solution we perform a generalized Bekenstein
transformation on this minimal solution.
In the search for static and spherical symmetric black hole-like solutions we choose the
Schwarzschild black hole as our seeding metric. This gives us both Xanthopoulos and Zannias’
minimal solutions and Xanthopoulos and Dialynas’ conformal solutions. It is known that only
in 3 + 1 dimensions do these solutions include a black hole, namely the BBMB black hole where
the metric is the extremal Reissner-Nordström metric. This makes us wonder what is special
with the four-dimensional spacetime we normally call home.
Acknowledgments
We wish to thank Kjell Tangen and Hans Kristian Eriksen for useful discussions. This work has
been supported by grant NFR 151574/V30 from the Research Council of Norway.
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