1. {Based on PRD 81 (2010) 104013 [1002.1275], joint work with X.-N. Wu}
Thermodynamics of Black Holes from Equipartition of Energy and Holography
Yu Tian
College of Physical Sciences, Graduate University of Chinese Academy of Sciences
{Based on PRD 81 (2010) [ ], joint work with X.-N. Wu}

2. Gravity as entropic force (E
Gravity as entropic force (E. Verlinde, On the Origin of Gravity and the Laws of Newton, [ ])
Key points:
One dimension of the space is emergent.
Holography.
“Equipartition of energy” on the holographic screen.
Gravity is an entropic force.

3. Entropic force

4. Newton’s 2nd law from entropic force
Holography!
For horizon!

5. Equipotential surface!
Change of entropy on the screen
Equipotential surface!
Entropic force
Temperature and acceleration
“Unruh effect”
Newton’s 2nd law

6. Newton’s gravity as entropic force
Special case – Newton’s law of gravity
Microscopic D.O.F.?
“Equipartition rule”

7. General case – the Poisson equation
No entropy here!

8. Equipotential surface or not?
The change of entropy density due to arbitrary
displacement of a particle
Equipotential surface or not? dA

9. Remarks:
When (and, generically, only when) the holographic screen is an equipotential surface, the expression of the entropic force is consistent with Newton’s law of gravity.
The (variation of) entropy density is proportional to the (variation of) gravitational potential, which suggests a coarse-graining picture analogous to AdS/CFT.

10. Einstein’s gravity as entropic force
: Killing vector

11. Komar mass inside the screen!
Einstein equations?
Static Stationary?

12. Questions:
1. How to test these arguments, especially the consistency between the (possible) local and global entropy variation postulations in the relativistic case?
2. What is the relation (if any) between Verlinde’s proposal and the traditional space-time (black-hole) thermodynamics, especially the entropy on the screen and the entropy of horizon?

13. “Match” the Hawking temperature!
Non-relativistic case
The Laplace horizon:
The Verlinde temperature on the horizon:
“Match” the Hawking temperature!

14. Higher dimensional case:
Still “match”!

15. “Match” the Bekenstein entropy!
Entropy formula dA
“Match” the Bekenstein entropy!

16. Have the same asymptotic behavior!
New choice of the relativistic gravitational potential
Verlinde’s potential is singular at the horizon
Our choice of the gravitational potential (YT & X.-N. Wu, PRD 81 (2010) [ ])
Have the same asymptotic behavior!

17. Verlinde’s proposal with our new potential

18. General spherical screen for Schwarzschild
Recall the Bekenstein entropy bound
So our (relativistic) entropy just saturates the Bekenstein bound.

19. Two simple models to check the formula of variation of entropy
Thin shell model R Rm

20.  = t

21. Charged dust shell modelRm q

24. Open questions:
How about the general case?
Entropic force for generic configurations (even in the spherical case)?
Thermodynamic relations for our (relativistic) entropy?

25. Alternative scheme
Note that in Verlinde’s original analysis and our discussions above, the “position” of the screen is fixed during the quasistatic processes, but in the ordinary black-hole thermodynamics, it is the Killing-horizon condition   12 of the “screen” that is fixed during the quasistatic processes.
So, a natural generalization of the ordinary black-hole thermodynamics is to consider the quasistatic processes that keep fixed the gravitational potential  (or ) of the screen (Y.-X. Chen & J.-L. Li, First law of thermodynamics on holographic screens in entropic force frame, [ ]).

26. Take the ansatz of spherically symmetric metrics.
Generalizing Smarr’s arguments, one obtains the generalized first law of thermodynamics associated to general spherical screen, with T the Verlinde temperature and S equal to A4 everywhere (not only on the horizon) in the framework of Einstein gravity.

27. Remarks:
Here M is the ADM mass of the space-time, not the Komar mass inside the screen. Equipartition rule.
Here the entropy S violates the Bekestein bound (outside the horizon), but saturates the holographic bound.
It is not clear how to realize Verlinde’s assumption of entropy variation, and so gravity as an entropic force, within this scheme.

28. Dynamical case with spherical symmetry
Generalizations
Dynamical case with spherical symmetry
R.-G. Cai, L.-M. Cao & N. Ohta, Notes on Entropy Force in General Spherically Symmetric Spacetimes, Phys. Rev. D 81 (2010)
Define the surface gravity and temperature for a general spherical screen (r  const.) in a general spherically symmetric spacetime

29. By Einstein equations, there exists a generalized equipartition rule where E is the Misner-Sharp (or Hawking-Israel) energy inside the screen, w the work density and V the “volume”

30. Remarks:
The generalized equipartition rule holds instantaneously for a general spherical screen.
The Misner-Sharp (or Hawking-Israel) energy is different from the Komar energy even in general static case.
The temperature T here is different from the Verlinde temperature even in general static case.
Upon redefinition of the temperature T as the “effective” one, the generalized equipartition rule superficially becomes the “standard” one.

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

32. The choice of Killing vector

33. Concluding remarks & open questions
1. From the holographic point of view, the gravity can be regarded as an entropic force, and the Einstein equations can be (at least partially) given by the equipartition of energy on the holographic screen.
2. Various definitions of temperature and quasilocal mass/energy arise in this framework (or its generalizations).
3. How to reconcile the two (or more) types of entropy arising in this framework?
4. Generalization to theories other than Einstein’s?
5. …

34. Thank you!