The Holographic Principal and its Interplay with Cosmology
Transcript of The Holographic Principal and its Interplay with Cosmology
The Holographic Principal and its Interplay with CosmologyT. Nicholas KypreosFinal Presentation: General Relativity09 December, 2008
T.N. Kypreos
What is the temperature of a Black Hole?
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ds2 = −(1− 2M
r
)dt2 + 1
1− 2Mr
dr2 + r2dΩ
ds2 = − u2
4M2dt2 + 4du2 + dX2
⊥
τ =t
4Mρ = 2u ds2 = −ρ2dτ2 + dρ2 + dX2
⊥
for simplicity, use the Schwarzschild metric
take a scenario with a stationary observer just outside of the horizon such that
rewrite the metric
this configuration is known as Rindler Coordinates
r = 2M +u2
2M
[1],[3],[6],[7]
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Unruh effectAn observer moving with uniform acceleration through the Minkowski vacuum ovserves a thermal spectrum of particles
β(u) = 2πρ = 4πu = 4π√
2M(r − 2M)
β(r′ →∞) = 4π√
2Mr
β(r′ →∞) = 8πM
β(r′ →∞) =1
kT= 8πM
this is subsequently red-shifted
take the limit as
β(r′) = 4π√
2M(r − 2M)√
1− 2Mr′
1− 2Mr
recalling thermodynamics
finally, the temperature
at the horizonʼs edge
T =!c3
8πGMkbhuman units
R = 2M
T =1
8πkbM
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What is the Entropy?
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dS =dQ
T= 8πMdQ
dS = 8πMdM = 8πd(M2) S = πR2 =A
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exploiting the equivalence principal, take theenergy added as a change in mass dM
R = 2M
• the entropy of the black hole is proportional to the area of the horizon
letting kb = 1
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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
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LH ∼ 1036 A ∼ L2H ∼ 1072 S ∼ T 3L3
H ∼ 1099
Example with GUT scale inflation:
T ∼ 10−3
Not consistent with Holographic Principle
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The Holographic Principle
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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.
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Expanding Flat Universe• common solution to any intro to general relativity text --
including our own
• provides simple test for the holographic principal
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Start with the metric for a flat, adiabatically expanding universe:
ds2 = −dt2 + a2(t)dxidxi
ρ
ρ+ 3H(1 + γ) = 0 where H =
a
ap = γρ
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Radiation Dominated (FRW) Universe
• integral starts a 0 -- looks like we cheated
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Solutions to this are known... some highlights:
a(t) = t3
3(γ+1) let a(1) = 1 ρ = 42(1+γ)2
1a3(1+γ)
To get the size of the universe, integrate a null geodesic to the boundary (which can be chosen as radial)
χH(t) = a(t)∫ t0
dt′
a(t′) = a(t)rH(t) such that rH(t) = dta(t)
So what’s the answer? These solutions will vary depending on the value of gamma, so let’s take .γ > − 1
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χH = a(t)rH(t) =3(γ + 1)3γ + 1
t
We get around the singularity at 0 by starting the integral at t=1. This corresponds to Planck time.
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Donʼt Forget about the entropy
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χH(1) ∼ 1At the start time, we haveThis is at least getting us somwhere. We can use this to limit the entropy density since the volume of space will also be ~1
S
A= σ ∼ 1
we know the area of the horizonand the entropy of the horizon
∼ χ2H
∼ σr2H
Recall this is an adiabatic expansion...S
A(t) ∼ σ
r3H
χ2H
= σtγ−1γ+1
Holographic Principle is upheld
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What does this mean?• found a case where the holographic principal holds...
• solution holds for
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−1 < γ < 1
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Closed Universe• closed universes tend to collapse...
• worth verifying with the Holographic Principle...
• consider the case where the universe is dominated by dark matter:
• area vanishes at
• is regular...
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ds2 = −dt2 + a2(t)(dχ2 + sin2 χdΩ)
p! ρ a = amax sin2(χH
2
)
S
A= σ
2χH − sin 2χH
2a2(χH) sin2 χH
χH = π
χH = π
Holographic Principal is violated
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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
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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
• can no longer access the quantum states inside
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S′ = SM + SBH
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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.
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Flat Universe Revisited• Lets take the situation where we have the universe
extending into Anti de Sitter Space with the flat universe from earlier.
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ρ =ρ0
a3(γ+1)− λ
a = ±√
ρ0
a3(γ+1)− λa2rewriting the Friedman equation yields
this has turning points when ρ0 = λa3(γ+1)
LH(a = 0) =B
(γ
2(γ+1) , 1/2)
3(γ + 1)√
λ
solving for the turning points yields
Λ < 0
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Revenge of the Flat Universe
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LH ∼ λ−(3γ+1)/6(γ+1) " 1leaving...
• holographic principle becomes violated
• violation for holographic principal happens well before planck limit
• most of the success of the holographic principle comes from Anti de Sitter space / Conformal Field Theory (AdS/CFT) Correspondence
•
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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
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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
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Backup Slides
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Phase space of solutions
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Laws of Black Hole Thermodynamics
Stationary black holes have constant surface gravity.
1.
2. The horizon area is increasing with time
3. Black holes with vanishing surface gravity cannot exist.
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dM =κ
8πdA + ΩdJ + ΦdQ
http://en.wikipedia.org/wiki/Black_hole_thermodynamics