On the exotic BTZ black holes Baocheng Zhang Based on papers PRL 110, ; PRD 88, Collaborated with Prof. P. K. Townsend 郑州,

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+NCC)

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)

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

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  Two different Lagrangians for 3D Einstein field equation 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.