The Holographic Principal and its Interplay with Cosmology. T. Nicholas Kypreos Final Presentation: General Relativity 09 December, 2008

Transcription

1 The Holographic Principal and its Interplay with Cosmology T. Nicholas Kypreos Final Presentation: General Relativity 09 December, 2008

2 What is the temperature of a Black Hole? for simplicity, use the Schwarzschild metric ds 2 = ( 1 2M r ) dt M r dr 2 + r 2 dω take a scenario with a stationary observer just outside of the horizon such that rewrite the metric τ = r =2M + u2 2M ds 2 = u2 4M 2 dt2 +4du 2 + dx 2 t 4M ρ =2u ds 2 = ρ 2 dτ 2 + dρ 2 + dx 2 this configuration is known as Rindler Coordinates 2 [1],[3],[6],[7]

3 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 at the horizonʼs edge recalling thermodynamics finally, the temperature 3 1 2M r 1 2M r β(r )=4π 2Mr β(r )=8πM R =2M β(r )= 1 kt =8πM 1 c 3 T = 8πk b M T = 8πGMk b human units

4 What is the Entropy? ds = dq T =8πMdQ letting k b =1 exploiting the equivalence principal, take theenergy added as a change in mass dm ds =8πMdM =8πd(M 2 ) R =2M S = πr 2 = A 4 the entropy of the black hole is proportional to the area of the horizon 4

5 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: T 10 3 L H A L 2 H S T 3 L 3 H Not consistent with Holographic Principle 5

6 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. 6

7 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: ds 2 = dt 2 + a 2 (t)dx i dx i ρ ρ +3H(1 + γ) =0 where H = ȧ a p = γρ 7

8 Radiation Dominated (FRW) Universe Solutions to this are known... some highlights: a(t) =t (γ+1) let a(1) = 1 ρ = 2(1+γ) 2 a 3(1+γ) To get the size of the universe, integrate a null geodesic to the boundary (which can be chosen as radial) χ H (t) =a(t) t 0 dt a(t ) = a(t)r H(t) such that r H (t) = dt a(t) So what s the answer? These solutions will vary depending on the value of gamma, so let s take γ > (γ + 1) χ H = a(t)r H (t) = 3γ +1 t 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. 8

9 Donʼt Forget about the entropy At the start time, we have χ H (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 we know the area of the horizon χ 2 H and the entropy of the horizon σrh 2 S A = σ 1 Recall this is an adiabatic expansion... S A (t) σ r3 H χ 2 H = σt γ 1 γ+1 Holographic Principle is upheld 9

10 What does this mean? found a case where the holographic principal holds... solution holds for 1 < γ < 1 10

11 Closed Universe closed universes tend to collapse... worth verifying with the Holographic Principle... consider the case where the universe is ) dominated by dark matter: ds 2 = dt 2 + a 2 (t)(dχ 2 + sin 2 χdω) S A = σ 2χ H sin 2χ H 2a 2 (χ H ) sin 2 χ H p ρ a = a max sin 2 ( χ H 2 area vanishes at χ H = π χ H = π is regular... Holographic Principal is violated 11

12 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 12

13 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 S = S M + S BH 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 13

14 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 compared to current observations that give S

15 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. ρ = ρ 0 a λ 3(γ+1) ρ0 rewriting the Friedman equation yields ȧ = ± λa2 a3(γ+1) this has turning points when solving for the turning points yields L H (ȧ = 0) = ρ 0 = λa 3(γ+1) B ( γ 2(γ+1), 1/2 ) 3(γ + 1) λ 15

16 Revenge of the Flat Universe leaving... L H λ (3γ+1)/6(γ+1) 1 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 16

17 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 17

18 References 1. W. Fischler, L. Susskind, Hologragphy and Cosmology, hep-th/ Edward Witten, Anti De Sitter Space And Holography, hep-th/ Nemanja Kaloper, Andrei Linde, Cosmology Vs Holography, hep-th/ G t Hooft, Dimensional Reduction in Quantum Gravity, gr-qc/ Carroll, Spacetime and Geometry, 2004 Pearson Ross, Black Hole Thermodynamics, hep-th/

19 Backup Slides 19

20 Phase space of solutions 20

21 Laws of Black Hole Thermodynamics Stationary black holes have constant surface gravity. 1. dm = κ da + ΩdJ + ΦdQ 8π 2. The horizon area is increasing with time 3. Black holes with vanishing surface gravity cannot exist. 21