The Holographic Principal
and its Interplay with
Cosmology
T. Nicholas Kypreos
Final Presentation: General Relativity
09 December, 2008
What is the temperature of a Black Hole?
for simplicity, use the Schwarzschild metric
!
ds = − 1 −
2
2M
r
"
dt2 +
1
1− 2M
r
dr2 + r2 dΩ
take a scenario with a stationary observer just
outside of the horizon such that
rewrite the metric
t
τ=
4M
ρ = 2u
u2
r = 2M +
2M
2
u
2
2
2
ds2 = −
dt
+
4du
+
dX
⊥
4M 2
2
ds2 = −ρ2 dτ 2 + dρ2 + dX⊥
this configuration is known as Rindler Coordinates
T.N. Kypreos
2
[1],[3],[6],[7]
Unruh effect
An observer moving with uniform acceleration through the
Minkowski vacuum ovserves a thermal spectrum of particles
!
β(u) = 2πρ = 4πu = 4π 2M (r − 2M )
this is subsequently red-shifted
"
!
β(r ) = 4π 2M (r − 2M )
!
take the limit as
1− 2M
r!
1− 2M
r
√
β(r → ∞) = 4π 2M r
!
R = 2M
β(r! → ∞) = 8πM
at the horizonʼs edge
1
recalling thermodynamics β(r → ∞) = kT = 8πM
1
!c3
finally, the temperature T = 8πk M T =
8πGM kb
b
!
T.N. Kypreos
3
human units
What is the Entropy?
dQ
dS =
= 8πM dQ
T
letting kb = 1
exploiting the equivalence principal, take
theenergy added as a change in mass dM
dS = 8πM dM = 8πd(M ) R = 2M
2
A
S = πR =
4
• the entropy of the black hole is
2
proportional to the area of the horizon
T.N. Kypreos
4
Introduction
•
holographic principle is a contemporary question in
physics literature ( see references )
•
holographic principal provides complications when being
connected to cosmological models
•
will explore some of the successes, failures, and
“paradoxes” of the holographic principle
Example with GUT scale inflation:
2
72
S ∼ T 3 L3H ∼ 1099
T ∼ 10−3 LH ∼ 1036 A ∼ LH ∼ 10
Not consistent with Holographic Principle
T.N. Kypreos
5
The Holographic Principle
Seemingly Innocuous Statment: The description
of a spatial volume of a black hole is encoded on
the boundary of the volume.
Extreme Extension: The universe could be seen as
a 2-D image projected onto a cosmological horizon.
T.N. Kypreos
6
Expanding Flat Universe
• common solution to any intro to general relativity text -including our own
•
provides simple test for the holographic principal
Start with the metric for a flat, adiabatically expanding universe:
ds2 = −dt2 + a2 (t)dxi dxi
ρ̇
+ 3H(1 + γ) = 0
ρ
T.N. Kypreos
ȧ
where H =
a
7
p = γρ
Radiation Dominated (FRW) Universe
Solutions to this are known... some highlights:
a(t) = t
3
3(γ+1)
let a(1) = 1
ρ=
4
1
2(1+γ)2 a3(1+γ)
To get the size of the universe, integrate a null geodesic to the
boundary (which can be chosen as radial)
! t dt!
dt
χH (t) = a(t) 0 a(t! ) = a(t)rH (t) such that rH (t) = a(t)
So what’s the answer? These solutions will vary depending
on the value of gamma, so let’s take γ > − 13 .
3(γ + 1)
χH = a(t)rH (t) =
t
3γ + 1
integral starts a 0 -- looks like we cheated
We get around the singularity at 0 by starting the integral
at t=1. This corresponds to Planck time.
•
T.N. Kypreos
8
Donʼt Forget about the entropy
At the start time, we have χH (1) ∼ 1
This is at least getting us somwhere. We can use this
to limit the entropy density since the volume of space
will also be ~1
2
∼
χ
we
know
the
area
of
the
horizon
H
S
2
=σ∼1
∼ σrH
and
the
entropy
of
the
horizon
A
Recall this is an adiabatic expansion...
3
γ−1
S
rH
(t) ∼ σ 2 = σt γ+1
A
χH
Holographic Principle is upheld
T.N. Kypreos
9
What does this mean?
• found a case where the holographic principal holds...
• solution holds for −1 < γ < 1
T.N. Kypreos
10
Closed Universe
• closed universes tend to collapse...
• worth verifying with the Holographic Principle...
ds2 = −dt2 + a2 (t)(dχ2 + sin2 χdΩ)
S
2χH − sin 2χH
=σ 2
A
2a (χH ) sin2 χH
•
consider the case where the universe
dominated by
! χis "
H
2
p
!
ρ
a
=
a
sin
dark matter:
max
2
area vanishes at χH = π
•
•
χH = π is regular...
Holographic Principal is violated
T.N. Kypreos
11
What Does All This Math Tell Us?
• topologically closed universes could not exist
• collapsing universes cannot happen in a manner that
preserves the second law of thermodynamics
•
•
entropy is always increasing
if the universe is collapsing, the surface area must
eventually vanish
T.N. Kypreos
12
So what about Thermodynamics?
• in the advent that the holographic principal cannot be
consistent with the second law of thermodynamics, there
are two choices
•
•
holographic principal is wrong?
thermodynamics is wrong?
•
propose “Generalized Second Law of Thermodynamics”
•
effectively adding a correction term to work around writing
off the entropy inside a black hole as unmeasurable
S = SM + SBH
!
•
can no longer access the quantum states inside
T.N. Kypreos
13
Cosmological Inflation
• holographic principal and inflation are compatible
depending on the rate of inflation
•
could be that holographic principal bounds inflation or vice
versa
•
currently holographic bounds put S ≤ 10120 compared to
current observations that give S ≤ 1090.
T.N. Kypreos
14
Flat Universe Revisited Λ < 0
• Lets take the situation where we have the universe
extending into Anti de Sitter Space with the flat universe
from earlier.
ρ0
ρ = 3(γ+1) − λ
a
!
ρ0
2
−
λa
rewriting the Friedman equation yields ȧ = ±
a3(γ+1)
ρ0 = λa3(γ+1)
!
"
γ
solving for the turning points yields
B 2(γ+1)
, 1/2
√
LH (ȧ = 0) =
3(γ + 1) λ
this has turning points when
T.N. Kypreos
15
Revenge of the Flat Universe
leaving... LH ∼ λ−(3γ+1)/6(γ+1) " 1
•
•
holographic principle becomes violated
•
most of the success of the holographic principle comes
from Anti de Sitter space / Conformal Field Theory
(AdS/CFT) Correspondence
violation for holographic principal happens well before
planck limit
•
T.N. Kypreos
16
Conclusions
•
holographic principle discovered in uncovering the
temperature of a black hole
•
has found a home in string theory and quantum gravity
research ( AdS/CFT correspondence)
•
seems to have more of the characteristics of an anomaly
than cause for quantum gravity
•
does not work cosmologically for topologically closed
universe / flat universe going to Anti de Sitter Space
T.N. Kypreos
17
References
1.
W. Fischler, L. Susskind, Hologragphy and Cosmology, hep-th/9806039
2.
Edward Witten, Anti De Sitter Space And Holography, hep-th/9802150
3.
Nemanja Kaloper, Andrei Linde, Cosmology Vs Holography, hep-th/9904120
4.
http://apod.nasa.gov/apod/ap011029.html
5.
http://www.nasa.gov/topics/universe/features/wmap_five.html
6.
G ’t Hooft, Dimensional Reduction in Quantum Gravity, gr-qc/9310006
7.
Carroll, Spacetime and Geometry, 2004 Pearson
8.
http://www.nasa.gov/images/content/171881main_NASA_20Nov_516.jpg
9.
Ross, Black Hole Thermodynamics, hep-th/0502195
T.N. Kypreos
18
Backup Slides
T.N. Kypreos
19
Phase space of solutions
T.N. Kypreos
20
Laws of Black Hole
Thermodynamics
Stationary black holes have constant surface
gravity.
κ
1. dM =
dA + ΩdJ + ΦdQ
8π
2. The horizon area is increasing with time
3. Black holes with vanishing surface gravity
cannot exist.
http://en.wikipedia.org/wiki/Black_hole_thermodynamics
T.N. Kypreos
21