Transcript Exotic BTZ black holes

On the exotic BTZ black holes
Baocheng Zhang
Based on papers PRL 110, 241302; PRD 88, 124017
Coauthor：P. K. Townsend
KITPC, 2014-6-25
Outlines
 (2+1) dimensional gravity
 BTZ black holes
 Exotic BTZ black holes
 Extension to BTZ BH in TMG
 Discussion and Conclusion
(2+1) dimensional gravity
Why do we want to study (2+1) dimensional gravity?
 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
 3D Einstein-Hilbert action can be written as
which is different from 4D action
And the former is equivalent to an ISO(2,1) ChernSimons 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); Zweidreibein gravity (Bergshoeff, Haan, Hohm, Merbis, & Townsend,
2013); ……
(2+1) dimensional gravity
(2+1) dimensional gravity
 It is more interesting to consider the (2+1) d EinsteinHilbert 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
BekensteinHawking entropy
 First/second/third laws
 Inner mechanics (Detournay, 2012)
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)
S=SL  S R 
 r
2G
 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
Non-BH entropy!
The entropy is proportional to the area of inner horizon!
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!
State counting
 Such entropy was obtained before by the method of
conical singularity (Solodukhin, 2006) and Wald’s Noether
charge method extended to the case of parityviolation (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 leftmoving 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
Extotic G
truncation
Conformal G
subgroup
 Negative central charge and holomorphic factorization
 Area-Law; Higher-spin extension
S  A  A
SH  f  y  A
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.