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  • 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?


    ds2 = (1 2Mr

    )dt2 + 1

    1 2Mrdr2 + r2d

    ds2 = u2

    4M2dt2 + 4du2 + dX2


    4M = 2u ds2 = 2d2 + d2 + 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



  • T.N. Kypreos 3

    Unruh effectAn observer moving with uniform acceleration through the Minkowski vacuum ovserves a thermal spectrum of particles

    (u) = 2 = 4u = 4

    2M(r 2M)

    (r ) = 4


    (r ) = 8M

    (r ) = 1kT

    = 8M

    this is subsequently red-shifted

    take the limit as

    (r) = 4

    2M(r 2M)

    1 2Mr

    1 2Mr

    recalling thermodynamics

    finally, the temperature

    at the horizons edge

    T =!c3

    8GMkbhuman units

    R = 2M

    T =1


  • T.N. Kypreos

    What is the Entropy?


    dS =dQ

    T= 8MdQ

    dS = 8MdM = 8d(M2) S = R2 =A


    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


    LH 1036 A L2H 1072 S T 3L3H 1099Example with GUT scale inflation:

    T 103

    Not consistent with Holographic Principle

  • T.N. Kypreos

    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

    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)dxidxi

    + 3H(1 + ) = 0 where H =


    ap =

  • T.N. Kypreos

    Radiation Dominated (FRW) Universe

    integral starts a 0 -- looks like we cheated


    Solutions to this are known... some highlights:

    a(t) = t3

    3(+1) let a(1) = 1 = 42(1+)21


    To get the size of the universe, integrate a null geodesic to the boundary (which can be chosen as radial)

    H(t) = a(t) t0


    a(t) = a(t)rH(t) such that rH(t) =dt


    So whats the answer? These solutions will vary depending on the value of gamma, so lets take . > 13H = a(t)rH(t) =

    3( + 1)3 + 1


    We get around the singularity at 0 by starting the integral at t=1. This corresponds to Planck time.

  • T.N. Kypreos

    Dont Forget about the entropy


    H(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


    A= 1

    we know the area of the horizonand the entropy of the horizon

    2H r2H

    Recall this is an adiabatic expansion...S

    A(t) r


    2H= t


    Holographic Principle is upheld

  • T.N. Kypreos

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


    1 < < 1

  • T.N. Kypreos

    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...


    ds2 = dt2 + a2(t)(d2 + sin2 d)

    p! a = amax sin2(H





    2H sin 2H2a2(H) sin2 H

    H =

    H =

    Holographic Principal is violated

  • T.N. Kypreos

    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

    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 inside13

    S = SM + SBH

  • T.N. Kypreos

    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

    Flat Universe Revisited Lets take the situation where we have the universe

    extending into Anti de Sitter Space with the flat universe from earlier.




    a =


    a2rewriting the Friedman equation yields

    this has turning points when 0 = a3(+1)

    LH(a = 0) =B


    2(+1) , 1/2)

    3( + 1)

    solving for the turning points yields

    < 0

  • T.N. Kypreos

    Revenge of the Flat Universe


    LH (3+1)/6(+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

  • T.N. Kypreos

    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


    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



    6. G t Hooft, Dimensional Reduction in Quantum Gravity, gr-qc/9310006

    7. Carroll, Spacetime and Geometry, 2004 Pearson


    9. Ross, Black Hole Thermodynamics, hep-th/0502195


  • T.N. Kypreos

    Backup Slides


  • T.N. Kypreos

    Phase space of solutions


  • T.N. Kypreos

    Laws of Black Hole Thermodynamics

    Stationary black holes have constant surface gravity.


    2. The horizon area is increasing with time

    3. Black holes with vanishing surface gravity cannot exist.


    dM =

    8dA + dJ + dQ