Exploring Thermodynamic Geometry of Black Holes

1. Thermodynamic Geometry and BTZ black holes Bhupendra Nath Tiwari IIT Kanpur in collaboration with T. Sarkar & G. Sengupta. This talk is mainly based on: On the thermodynamic geometry of BTZ black holes; J. High Energy Phys. JHEP 11 (2006) 015. Indian String Meeting, 2007.

2. Plan of the talk: • Thermodynamic Geometry. • BTZ black hole: thermodynamic Geometry (a) rotating BTZ black hole, (b) rotating BTZ black hole with “fluctuations”. • BTZ-CS black hole: thermodynamic Geometry (a) rotating BTZ-CS black hole, (b) rotating BTZ-CS black hole with “fluctuations”. • Conclusion.

3. Motivations • Black hole phase transitions Vs moduli space geometry. • Black hole thermodynamics and attractor mechanism. • Critical exponents of black hole phase transitions and higher derivative corrections. • Black hole entropy: Wald Vs Cardy. • Etc..

4. Thermodynamic Geometry • It’s a Riemannian geometric model based on the consideration of the theory of fluctuations along with the laws of equilibrium thermodynamics. • For any thermodynamic system, there exists equilibrium thermodynamic states which can be represented by points in the state space. • The distance between arbitrary two equilibrium states is inversely proportional to the fluctuations connecting the two states. In particular, “less probable” means “far apart”. • The metric in this state space is:

5. Consider and then the Taylor expansion up to second order: where is Ruppenier metric. • The probability distribution in Gaussian approximation takes the form: • With normalization: ,we have ,where

6. It is the thermodynamic scalar curvature that is proportional to the correlation volume which signifies the interaction(s) of the underlying statistical system. i.e., , where d is spatial dimension of the statistical system and the fixes the physical scale. • We illustrate the thermodynamic geometry with two thermodynamic variables: the mass M and the angular momentum J for BTZ black holes.

7. Rotating BTZ black hole • We choose: • The metric is defined by where and are lapse and shift functions and M, J, are mass angular momentum and the cosmological constant. • The horizons are located at: where

8. The mass M and angular momentum J are: and • The entropy is given by: where the outer horizon of BTZ is given by

9. The Ruppenier metric of the state space of BTZ black hole is defined by • The Christoffel connections are defined by • The only non-zero is , where

10. and • The Ricci scalar is: • The BTZ entropy gives R= 0, i.e. it is a non interacting system.

11. BTZ and Thermal fluctuations • The thermodynamic of BTZ black holes with small fluctuations in a canonical ensemble is stable if • Then the entropy in the micro-canonical ensemble is • Where is entropy in the canonical ensemble & C is the specific heat. • Set henceforth .

12. The Hawking temperature is • Writing in terms of entropy • The stability of canonical ensemble is just • In other words, the Hessian of internal energy w.r.t. extensive variable remains positive

13. This condition governs the situation away from the extremality or . • The BTZ micro-canonical entropy up to leading order is • Note that far from the extremality ,even at zero angular momentum, there is a finite value of thermo- dynamic scalar curvature unlike the non-rotating BTZ. • At low temp the quantum effects dominates and the above expansion does not hold anymore.

14. The R(S) Vs J graph is shown below: • For any value of M, we see:

15. Thermodynamic Geometry of BTZ-CS • Kraus, Solodukhin,… considers, the BTZ solution to gravitational action which includes Einstein- Hilbert & Chern- Simons terms. • In this case, the entropy can be written to be where k is the Chern- Simons coupling constant. • A stability bound on k is:

16. The presence of non- zero k modifies the M and J of the usual BTZ: where are mass and angular momentum of usual BTZ. • The above equation comes by the modified stress tensor of the theory using the Fefferman- Graham expansion of the BTZ metric.

17. The entropy of BTZ- CS may be written as ,where • The Ruppenier Geometry in this case turns out to be flat. So the BTZ-CS in non interacting statistical system.

18. BTZ-CS and Thermal fluctuations • The thermodynamic geometry of BTZ-CS can be described by the outer & inner horizons: • Expressing in terms of CS- corrected mass and angular momentum,

19. The mass is given by • The temperature is • The Specific heat is given by ,where

20. The underlying thermodynamical system is stable if C>0. • The corrected canonical entropy of BTZ-CS is where is the canonical BTZ-CS entropy. • In the large entropy limit , the entropy is • Interestingly the factor was first found by Carlip is reproduced and seems to be universal.

21. Notice that the large entropy limit is the stability bound, beyond which quantum effects dominate. • The Ricci scalar of the thermodynamic geometry with full corrections is positive definite. This we can see from the following figure R(S) Vs k for far from extremality.

23. We again notice that, • The Ricci scalar R(S) diverges at • This is a stability bound from Ruppenier geometry. • This bound was also obtained for given cosmological constant by Solodukhin.

24. Conclusion • The BTZ Ruppenier geometry remains flat, with and without CS terms, which shows that it is a non- interacting statistical system. • The small fluctuations produces interacting system as scalar curvature is non- zero. • The “ln- corrections” to canonical entropy of BTZ and BTZ-CS have same form. This illustrates the universality of the corrections such as of Carlip,… • The thermodynamic geometry of BTZ-CS plus arbitrary covariant higher derivatives is flat.

25. THANKS

26. Motivation: Basics • The laws of thermodynamics are not most fundamental but arise from the microscopic properties of the system. • The entropy plays a major role in the thermodynamics and statistical mechanics. • The state of equilibrium is defined as the state of maximum entropy. • The physical quantities describing a macroscopic body in equilibrium have certain deviations from the mean values. • We may consider a probability distribution of several thermodynamic quantities and their simultaneous small or thermal fluctuations from their mean values.

27. Plan Cum results: • We consider thermodynamic Geometry of BTZ black holes. • We show that thermodynamic scalar curvature of (a) rotating BTZ black hole is zero, (b) rotating BTZ black hole with “ln-fluctuations” is non-zero, (c) Universal correction of Carlip is reproduced. • The Similar results are true for BTZ-CS. • BTZ-CS with higher derivative corrections is thermodynamically flat. • We can analyze critical points of black hole phase transitions.

28. Actually the is anomaly in gravitational theory on 3D space-time is described by the action: ,where where K is second fundamental form of boundary. • The is the gravitational Chern-Simons term, given by Where the curvature is for torsion free Lorentz connection ,determined by ,where the o. n. basis is square root of the metric

29. Consider D= d+1 Euclidean dimensions, and in the Gaussian normal coordinates by foliating the space-time with d dimensional hyper surfaces labeled by , we have the metric: , where the is induced metric on the boundary. • Define extrinsic curvature of a fixed surface: • Then D dimensional Ricci scalar decomposes as , where • Now we have the bdry term in the action at fix surface, whose variation contains ,so we need to add Gibbon- Hawking term in the action. This gives us the modified stress tensor:

30. In the case of AdS the cosmological constant the solution to Einstein equations admits the expansion: • The boundary is placed at with metric conformal to • The convergence of action and stress tensor requires additional counter terms that are intrinsic to the boundary, which in the case of just the boundary cosmological constant: • So the modified stress tensor is where is inverse of