Thermodynamics of BlaThermodynamics of Black Holes from Equipartick Holes from Equipartition of Energy and Holotion of Energy and Holo

graphygraphyYu TianYu Tian

College of Physical Sciences, Graduate UnCollege of Physical Sciences, Graduate University of Chinese Academy of Sciencesiversity of Chinese Academy of Sciences{Based on PRD 81 (2010) 104013 [1002.127{Based on PRD 81 (2010) 104013 [1002.127

5], joint work with X.-N. Wu}5], joint work with X.-N. Wu}

Gravity as entropic force (E. Verlinde, Gravity as entropic force (E. Verlinde, On the Origin of Gravity and the Laws of Newton, [1001.0785]) [1001.0785])

Key points: Key points: One dimension of the space is emergOne dimension of the space is emerg

ent.ent. Holography.Holography. ““Equipartition of energy” on the hEquipartition of energy” on the h

olographic screen.olographic screen. Gravity is an entropic force.Gravity is an entropic force.

Entropic forceEntropic force

Newton’s 2nd law from entropic forceNewton’s 2nd law from entropic force

For horizon!

Holography!

Change of entropy on the screenChange of entropy on the screen

Entropic forceEntropic force

Temperature and accelerationTemperature and acceleration

Newton’s 2nd lawNewton’s 2nd law

Equipotential surface!

“Unruh effect”

Newton’s gravity as entropic forceNewton’s gravity as entropic force

31 2

2 3

n c AN

n G

Special case – Newton’s law of gravitySpecial case – Newton’s law of gravity3c A

NG

Microscopic D.O.F.?

1

2 BE Nk T “Equipartition rule”

2 2 1, 4 ,

2B

aE Mc A R k T

c

2

MmF ma G

R

No entropy here!

General case – the Poisson equationGeneral case – the Poisson equation

a

The change of entropy density due to arbitraryThe change of entropy density due to arbitrary displacement of a particledisplacement of a particle

dA

Equipotential surface o

r not?

Remarks:Remarks: When (and, generically, only when) tWhen (and, generically, only when) t

he holographic screen is an equipotehe holographic screen is an equipotential surface, the expression of the ential surface, the expression of the entropic force is consistent with Newtntropic force is consistent with Newton’s law of gravity.on’s law of gravity.

The (variation of) entropy density is The (variation of) entropy density is proportional to the (variation of) graproportional to the (variation of) gravitational potential, which suggests a vitational potential, which suggests a coarse-graining picture analogous to coarse-graining picture analogous to AdS/CFT.AdS/CFT.

Einstein’s gravity as entropic forceEinstein’s gravity as entropic force

: Killing vector

2aa

mN S

a a aF T S me

Static Stationary?

Einstein equations?

31 2

2 3

n cdN dA

n G

Komar mass inside the screen!

Questions:Questions:1. How to test these arguments, especia1. How to test these arguments, especia

lly the consistency between the (posslly the consistency between the (possible) local and global entropy variatioible) local and global entropy variation postulations in the relativistic case?n postulations in the relativistic case?

2. What is the relation (if any) between 2. What is the relation (if any) between Verlinde’s proposal and the traditioVerlinde’s proposal and the traditional space-time (black-hole) thermodynal space-time (black-hole) thermodynamics, especially the entropy on the namics, especially the entropy on the screen and the entropy of horizon?screen and the entropy of horizon?

Non-relativistic caseNon-relativistic case

The Laplace horizon:The Laplace horizon:

2

2h

GMr

c

The Verlinde temperature on the horizon:The Verlinde temperature on the horizon:

3

22 2 8hB B B

GM cT T

k c k cr k GM

“Match” the Hawking temperature!

4

2 4h

GM c

r GM

Higher dimensional case:Higher dimensional case:

1/( 3)

2

16( 1)

( 2)

n

hn

GMr c

n

Still “match”!

22

( 3)8 3

( 2) 2hnn h

n GM n

n r r

( 3)

4hB h

nT

k r

Entropy formulaEntropy formula

dA

2 3

2 4B

h h h

c k cS A

G

“Match” the Bekenstein entropy!

New choice of the relativistic New choice of the relativistic gravitational potentialgravitational potential

Verlinde’s potential is singular at the horizonVerlinde’s potential is singular at the horizon

Our choice of the gravitational potential (Our choice of the gravitational potential (YT & X.-YT & X.-N. Wu, PRD 81 (2010) 104013 [1002.1275])N. Wu, PRD 81 (2010) 104013 [1002.1275])

Have the same asymptotic behavior!

Verlinde’s proposal with our new potentialVerlinde’s proposal with our new potential

2a

B ak T Nc

2 3

2 4B

h h h

c k cS A

G

General spherical screen for SchwarzschildGeneral spherical screen for Schwarzschild

2 Bt

GM k cS Mr

r

Recall the Bekenstein entropy boundRecall the Bekenstein entropy bound

2 BkS Erc

So our (relativistic) entropy just satSo our (relativistic) entropy just saturates the Bekenstein bound.urates the Bekenstein bound.

Two simple models to check the Two simple models to check the formula of variation of entropyformula of variation of entropy

Thin shell modelThin shell model

R

Rm

= t

Charged dust shell modelCharged dust shell model

R

Rm

q

Open questions:Open questions: How about the general case?How about the general case?

Entropic force for generic Entropic force for generic configurations (even in the configurations (even in the spherical case)?spherical case)?

Thermodynamic relations for Thermodynamic relations for our (relativistic) entropy?our (relativistic) entropy?

Alternative schemeAlternative scheme

Note that in Verlinde’s original analysis Note that in Verlinde’s original analysis and our discussions above, the “positiand our discussions above, the “position” of the screen is fixed during the qon” of the screen is fixed during the quasistatic processes, but in the ordinaruasistatic processes, but in the ordinary black-hole thermodynamics, it is the y black-hole thermodynamics, it is the Killing-horizon condition Killing-horizon condition 112 of th2 of the “screen” that is fixed during the que “screen” that is fixed during the quasistatic processes.asistatic processes.

So, a natural generalization of the ordinarSo, a natural generalization of the ordinary black-hole thermodynamics is to consy black-hole thermodynamics is to consider the quasistatic processes that keep ider the quasistatic processes that keep fixed the gravitational potential fixed the gravitational potential (or (or ) ) of the screen (Y.-X. Chen & J.-L. Li, of the screen (Y.-X. Chen & J.-L. Li, First law of thermodynamics on holographic screens in entropic force frame, [1006.[1006.1442]).1442]).

Take the ansatzTake the ansatz

of spherically symmetric metrics.of spherically symmetric metrics.Generalizing Smarr’s arguments, one obtGeneralizing Smarr’s arguments, one obt

ains the generalized first law of thermodains the generalized first law of thermodynamicsynamics

associated to general spherical screen, wassociated to general spherical screen, with ith TT the Verlinde temperature and the Verlinde temperature and SS eq equal to ual to AA4 everywhere (not only on the h4 everywhere (not only on the horizon) in the framework of Einstein graorizon) in the framework of Einstein gravity.vity.

22 2 2 2( )

( )

drds f r dt r d

f r

i ii

dM TdS dQ

Remarks:Remarks: Here Here MM is the ADM mass of the sp is the ADM mass of the sp

ace-time, not the Komar mass insace-time, not the Komar mass inside the screen. Equipartition rule.ide the screen. Equipartition rule.

Here the entropy Here the entropy SS violates the Be violates the Bekestein bound (outside the horizokestein bound (outside the horizon), but saturates the holographic n), but saturates the holographic bound.bound.

It is not clear how to realize VerliIt is not clear how to realize Verlinde’s assumption of entropy varnde’s assumption of entropy variation, and so gravity as an entropiation, and so gravity as an entropic force, within this scheme.ic force, within this scheme.

GeneralizationsGeneralizations Dynamical case with spherical symmetryDynamical case with spherical symmetry

R.-G. Cai, L.-M. Cao & N. Ohta, R.-G. Cai, L.-M. Cao & N. Ohta, Notes on Entropy Force in General Spherically Symmetric Spacetimes, Phys. Rev. D 81 (2010) 084012.

Define the surface gravity and temperature

for a general spherical screen (r const.) in a general spherically symmetric spacetime

1,

2 2a

aD D r T

By Einstein equations, there exists a generalized equipartition rule

where ere EE is the Misner-Sharp (or Hawki is the Misner-Sharp (or Hawking-Israel) energy inside the screen, ng-Israel) energy inside the screen, ww t the work densityhe work density

and and VV the “volume” the “volume”

1 1

2 3B

nE Nk T wV

n

1

2aaw T

12

1nnV r

n

Remarks:Remarks: The generalized equipartition rule holds The generalized equipartition rule holds

instantaneously for a general spherical sinstantaneously for a general spherical screen.creen.

The Misner-Sharp (or Hawking-Israel) eThe Misner-Sharp (or Hawking-Israel) energy is different from the Komar energnergy is different from the Komar energy even in general static case.y even in general static case.

The temperature The temperature TT here is different fro here is different from the Verlinde temperature even in genm the Verlinde temperature even in general static case.eral static case.

Upon redefinition of the temperature Upon redefinition of the temperature TT as the “effective” one, the generalized as the “effective” one, the generalized equipartition rule equipartition rule superficially superficially becombecomes the “standard” one.es the “standard” one.

Stationary case Stationary case ((YT & X.-N. Wu, PRD YT & X.-N. Wu, PRD 81 (2010) 104013 [1002.1275])81 (2010) 104013 [1002.1275])

The choice of Killing vectorThe choice of Killing vector

Concluding remarks & open questionsConcluding remarks & open questions1. From the holographic point of view, the gravit1. From the holographic point of view, the gravit

y can be regarded as an entropic force, and the y can be regarded as an entropic force, and the Einstein equations can be (at least partially) givEinstein equations can be (at least partially) given by the equipartition of energy on the hologren by the equipartition of energy on the holographic screen.aphic screen.

2. Various definitions of temperature and quasilo2. Various definitions of temperature and quasilocal mass/energy arise in this framework (or its cal mass/energy arise in this framework (or its generalizations).generalizations).

3. How to reconcile the two (or more) types of en3. How to reconcile the two (or more) types of entropy arising in this framework?tropy arising in this framework?

4. Generalization to theories other than Einstei4. Generalization to theories other than Einstein’s?n’s?

5. …5. …

Thank you!Thank you!