Thermodynamic Geometry and BTZ black holes Bhupendra Nath

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 thermodynamic geometry of BTZ black holes; J. High Energy Phys. JHEP 11 (2006) 015. Indian String Meeting, 2007.

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.

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. .

Thermodynamic Geometry • It’s a Riemannian geometric model based on the consideration of 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: 4

• 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

• It is 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 thermodynamic geometry with two thermodynamic variables: the mass M and the angular momentum J for BTZ black holes.

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 7

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

• 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

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

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 .

• 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

• 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 thermodynamic scalar curvature unlike the non-rotating BTZ. • At low temp the quantum effects dominates and the above expansion does not hold anymore.

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

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:

• 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 theory using the Fefferman- Graham expansion of the BTZ metric.

• 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. 17

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, 18

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

• 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.

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

• 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. 23

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.

THANKS

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 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.

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 nonzero, (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. 27

• Actually the is anomaly in gravitational theory on 3 D spacetime is described by the action: , 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

• 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:

• In the case of Ad. S the cosmological constant the solution to Einstein equations admits the expansion: • The boundary is placed at to with metric conformal • 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 of is inverse