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

T.N. Kypreos

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

T.N. Kypreos

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

T.N. Kypreos

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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T.N. Kypreos

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