On the exotic BTZ black holes Baocheng Zhang Based on papers PRL 110, ; PRD 88, Coauthor ： P. K. Townsend KITPC,

Outlines  (2+1) dimensional gravity  BTZ black holes  Exotic BTZ black holes  Extension to BTZ BH in TMG  Discussion and Conclusion

 Understand the classical gravity further Singularity; cosmic censorship; closed timelike curves; ……  Gain an insight into quantum gravity Black hole solutions; gravitons (modified theory); quantization; AdS/CFT correspondence; …… (2+1) dimensional gravity Why do we want to study (2+1) dimensional gravity?

(2+1) dimensional gravity  3D Einstein-Hilbert action can be written as which is different from 4D action And the former is equivalent to an ISO(2,1) Chern- Simons action, but there is not this equivalence for the latter. (Witten, 1988)  There are two essential features for vacuum gravity: No local d.o.f. or propagating d.o.f. (Leutwyler, 1966) No black-hole solutions (Ida, 2000) So it is usually considered as dynamics of flat space. (Deser, Jackiw, & t’Hooft, 1984)

(2+1) dimensional gravity  As discussed, (2+1) d GR doesn’t include the propagating d.o.f., but one can find some modified models to change the situation within which the physical spin-2 modes are massive.  3D massive gravity models includes: Topological massive gravity (Deser, Jackiw, & Templeton, 1982) ; New massive gravity (Bergshoeff, Hohm, & Townsend, 2009) ; General massive gravity (Bergshoeff, Hohm, & Townsend, 2009) ; Zwei- dreibein gravity (Bergshoeff, Haan, Hohm, Merbis, & Townsend, 2013) ; ……

(2+1) dimensional gravity

 It is more interesting to consider the (2+1) d Einstein- Hilbert action with a negative cosmological constant,  This model is the difference of two special linear group Chern-Simons terms, (Witten 1988)  Chern-Simons field equations is equivalent to vacuum Einstein field equations.

Outlines  (2+1) dimensional gravity  BTZ black holes  Exotic BTZ black holes  Extension to BTZ BH in  Discussion and Conclusion

BTZ black holes  There are no asymptotically flat black holes of 3D GR but there are “BTZ” black holes, which are asymptotic to an AdS vacuum. (Banados, Teitelboim, & Zanelli, 1992)  The BTZ metric is locally isomorphic to the AdS vacuum, so any theory of 3D gravity admitting an AdS vacuum will also admit BTZ black holes.  Metric  Horizon  Mass and angular momentum (3D GR)

BTZ black holes  The most important feature is that it has thermodynamic properties analogous to (3+1) d black holes.  Temperature  Entropy  First/second/third laws  Inner mechanics (Detournay, 2012) Bekenstein- Hawking entropy

State counting  More important is to find the microscopic d.o.f. responsible for the entropy which is beyond the thermodynamics given by classical gravity theory.  Chern-Simons description provides an effective way to approach the purpose.  Asymptotic symmetries and AdS/CFT (Brown & Henneaux)  Cardy formula (Cardy, 1986)  Effective central charge (see review by Carlip, 2005)

State counting  For 3D GR, the central charges of dual CFT2 are  Using the Cardy formula and the relations we get the entropies  The statistical mechanics demands the thermodynamic entropy of BTZ black holes (Strominger, 1998; Birmingham, et al, 1998)  What states are we counting? (see review by Carlip, 2005)

Outlines  (2+1) dimensional gravity  BTZ black holes  Exotic BTZ black holes  Extension to BTZ BH in  Discussion and Conclusion

Exotic BTZ black holes  BTZ metric solves any field equations that admit AdS as a solution. For example, 3D conformal gravity. Mass M and angular momentum J of BTZ black holes given by i.e. the reverse of 3D GR! The black hole is exotic.  Other 3D gravity models were earlier found to have the property. (Carlip & Gegenberg, 1991; Carlip et al, 1995; Banados, 1998)  Entropy of exotic BTZ black hole can be computed (e.g. by Wald formula) and is The entropy is proportional to the area of inner horizon! Non-BH entropy!

How should we understand the exotic black holes? Why its mass and angular momentum interchange in the BTZ metric?

Exotic 3D EG  3D EG with AdS3 vacuum is a Chern-Simons theory for the AdS3 group, that is, (Achucarro & Townsend, 1986)  The normal 3D EG is the difference of the two special linear group Chern-Simons terms. (Witten, 1988)  The sum gives a parity-odd “exotic” action with the same field equations (Witten, 1988). The Lagrangian 3-forms is, where is torsion 2-form.

Exotic EG has exotic BH  It was shown that 3D EG is equivalent to a Chern-Simons gauge theory with the 1-form potential,  For every there is a conserved charge, defined as holonomy of asymptotic U(1) connection [CQG 12, 895 (1995)]  For normal 3D EG we have  For exotic 3D EG we have So mass and angular momentum are reversed!

For such exotic entropy, whether it still has the thermodynamic significance?

Thermodynamics  The Hawking temperature and the angular momentum of BTZ black hole are which are geometrical and model-independent.  For generality, consider the mass and angular momentum  It was shown that the only form of entropy satisfies the first law of black hole thermodynamics  Note that the cases and correspond, respectively, to normal and exotic BTZ black holes.

Thermodynamics  The event horizon is a Killing horizon for the Killing vector  At horizon, we have which implies  For exotic BTZ black holes, it changes into  Through the calculation, we have the second law, which means the SL is valid for the exotic BTZ black hole!

 Such entropy was obtained before by the method of conical singularity (Solodukhin, 2006) and Wald’s Noether charge method extended to the case of parity- violation (Tachikawa, 2007).  For such exotic entropy, what is its microscopical interpretation through Cardy formula directly? State counting

 For exotic 3D BTZ black hole, we have  The weak cosmic censorship condition needed for the existence of the event horizon.  Thus we would have which implies that Cardy formula would give an imaginary entropy. The normal statistical mechanics is invalid!

A matter of convention  Within thermodynamic approximation, the left and right moving modes of CFT do not interact. This is exactly true if partition function of 3D Einstein gravity factorizes holomorphically: [Maloney & Witten, 2010]  Given factorization, we can change the conventions for left- moving modes, so that all energies are negative and all states have negative norm. This gives (which is known to be the case for conformal 3D gravity). Now we have  Appling the Cardy formula, we find

Exotic statistical mechanics  Thermodynamics of exotic black holes is normal, so we expect the formula is still valid.  So from the perspective of partition function, we have the exotic statistical mechanics in star contrast to the normal case  Now we have the entropy of exotic BTZ black hole This is the thermodynamic entropy which is the first time to obtain the statistical exotic black hole entropy!

Outlines  (2+1) dimensional gravity  BTZ black holes  Exotic BTZ black holes  Extension to BTZ BH in TMG  Discussion and Conclusion

Modified Cardy formulas  Whether the modified Cardy formula can be obtained with a fundamental method in CFT without recourse to the thermodynamic relation?  Using Carlip’s method to obtain

Cardy formula  Partition function on a torus  Using modular invariance and saddle approxi., the density of states can be gotten as  If the central charge for the right mover is still positive, the same process led to the Cardy formula by taking the logarithm of the exponential term of the density of states; that is

Modified Cardy formula  Now we begin to calculate the case with negative central charge, and the density of states is  After using modular invariance, we have  Define  Using saddle point aproxi., the extremal point is

Modified Cardy formula  Then the f function can be calculated as  The density of states  This leads to the modified Cardy formula,

Extension  Topological massive gravity:  Its BTZ black hole solution:  Compared with NEG and EEG, Identification of the parameters is dependent on concrete theories!

Extension  Central charges: [Hotta, et al, 2008]  According to the previous modified Cardy formulas to calculate the entropy in different ranges of coupling parameter, we get a consistent form,  It is noticed that in the range the entropy is negative, which is consistent with the range of negative mass.

Extension  In previous calculation, we can get a term that gives the logarithmic correction of entropy associated with the outer horizon,  For the 3D Einstein gravity, we obtain  For TMG, we obtain  This means that Chern-Simons term doesn’t influence the logarithmic correction of entropy

Outlines  (2+1) dimensional gravity  BTZ black holes  Exotic BTZ black holes  Extension to BTZ BH in TMG  Discussion and Conclusion

Discussion  Why BTZ black holes of conformal gravity are exotic  Negative central charge and holomorphic factorization  Area-Law; Higher-spin extension Extotic GConformal G truncation subgroup

Conclusions  The BTZ black holes of exotic 3D EG are exotic.  Thermodynamics of exotic black holes is normal.  Exotic black hole entropy needs exotic statistical mechanics.  Extension to TMG is feasible.