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On the effects of relaxing the asymptotics of gravityPowerPoint Presentation

On the effects of relaxing the asymptotics of gravity

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the asymptotics of gravity

in three dimensions

Ricardo Troncoso

Centro de Estudios Científicos (CECS) Valdivia, Chile

Criteria: M. Henneaux and C. Teitelboim, CMP (1985)

- They are invariant under the AdS group
- The fall-off to AdS is sufficiently slow
- so as to contain solutions of physical interest
- At the same time, the fall-off is sufficiently fast
- so as to yield finite charges

Brown-Henneaux asymptotic conditions

General Relativity in D = 3 (localized matter fields)

J. D. Brown and M. Henneaux, CMP (1986)

- Asymptotic symmetries are enlarged
from AdS to the conformal group in 2D

- Canonical charges (generators) depend only on the metric and its derivatives
- Their P.B. gives two copies of the Virasoro algebra with central charge

General Relativity with scalar fields

M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, PRD (2002)

M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, PRD (2004)

M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, AP (2007)

- Scalar fields with slow fall-off: with
- Relaxed asymptotic conditions for the metric (slower fall-off)
- Same asymptotic symmetries (2D conformal group)
- Canonical charges (generators) acquire a contribution from the matter field
- Their P.B. gives two copies of the Virasoro algebra with the same central charge

General Relativity with scalar fields:

Relaxing the asymptotic conditions

enlarges the space of allowed solutions

- No hair conjecture is violated
- Hairy black holes
- Solitons

Hair effect:

Topologically massive gravity

M. Henneaux, C. Martínez, R. Troncoso PRD (2009)

- AdS waves are included
- Admits relaxed asymptotic conditions for
- Same asymptotic symmetries (2D conformal group)
- For the range the relaxed terms
do not contribute to the surface intergrals (Hair)

- Their P.B. gives two copies of the Virasoro algebra
with central charges

Topologically massive gravity at the chiral point

D. Grumiller and N. Johansson, IJMP (2008)

M. Henneaux, C. Martínez, R. Troncoso PRD (2009)

E. Sezgin, Y. Tanii 0903.3779 [hep-th]

A. Maloney, W. Song, A. Strominger 0903.4573 [hep-th]

- Admits relaxed asymptotic conditions with logarithmic behavior
(so called “Log gravity”)

- Same asymptotic symmetries (2D conformal group)
- The relaxed term does contribute to the surface intergrals
(at the chiral point “hair becomes charge”,

and the theory with this b.c. is not chiral )

- Their P.B. gives two copies of the Virasoro algebra
with central charges

Bergshoeff-Hohm-Townsend (BHT) action:

E. A. Bergshoeff, O. Hohm, P. K. Townsend, 0901.1766 [hep-th]

Field equations

(fourth order)

Linearized theory:

Massive graviton with two helicities (Fierz-Pauli)

Solutions of constant curvature :

Special case:

Unique maximally symmetric vacuum

[A single fixed (A)dS radius l]

Reminiscent of what occurs for the EGB theory

for dimensions D>4

D > 4 dimensions

- Second order field equations
- Generically admits two maximally symmetric solutions

Special case:

Unique maximally symmetric vacuum

[A single fixed (A)dS radius l]

Special case:

- Slower asymptotic behavior
- Relaxed asymptotic conditions
- The same asymptotic symmetries and finite charges
- J. Crisóstomo, R. Troncoso, J. Zanelli, PRD (2000)
- Enlarged space of solutions:
- new unusual classes of solutions in vacuum:
- static wormholes and gravitational solitons
- G. Dotti, J. Oliva, R. Troncoso, PRD (2007)
- D. H. Correa, J. Oliva, R. Troncoso JHEP (2008)

Does BHT massive gravity theory

possess a similar behavior ?

BHT massive gravity at the special point

- The field eqs. admit the following Euclidean solution
- D. Tempo, J. Oliva, R. Troncoso, CECS-PHY-09/03

- The metric is conformally flat
- Once the instanton is suitably Wick-rotated, the Lorentzian metric describes:
- Asymptotically locally flat and (A)dS black holes
- Gravitational solitons and wormholes in vacuum
- The rotating solution is found boosting this one

Negative cosmological constant

Case of :

- The solution describes asymptotically AdS black holes

- c : mass parameter (w.r.t. AdS)
- b : “gravitational hair”
- it does not correspond to any global charge
- generated by the asymptotic symmetries

b > 0 :

a single event horizon located at provided

the bound is saturated when the horizon coincides with the singularity

b < 0 :

The singularity is surrounded by an event horizon provided

The bound is saturated at the extremal case

Negative cosmological constant

Hair effect:

- For a fixed mass (c) BTZ:
- adding b>0 shrinks the black hole
- adding b<0 increases the black hole
- the ground state changes
- (c is bounded by a negative value)
- for negative c a Cauchy horizon appears

- Same asymptotic symmetries as for Brown-Henneaux (Conformal group in 2D)

Abbott-Deser Deser-Tekin charges

- Charges are finite
- The central charge is twice the standard value of
- Brown-Henneaux

Abbott-Deser Deser-Tekin charges

- Charges are finite
- The central charge is twice the standard value of
- Brown-Henneaux

The integration constant b is not related to any global charge associated with the asymptotic symmetries:

- Thus, b can be regarded as “pure gravitational hair”.

The metric for the Euclidean black hole reads

The solution is regular provided

- Extremal case: Wick-rotated to
- Also to wormhole covering space (see below)

Wald’s formula:

For the black hole:

- Extremal black hole has vanishing entropy
- (as expected semiclassically)
- First law is fulfilled:
- Cross check for both Deser-Tekin and Wald formulae
- No additional charge is required for b (since it is hair)

and wormholes

From the Euclidean black hole, Wick rotating the angle:

(Like the AdS soliton from the toroidal black hole on AdS)

Note that for the metric reduces to

The wormhole is constructed making

Wormhole metric:

- Neck radius is a modulus parameter
- No energy conditions are be violated

From the Euclidean black hole, Wick rotating the angle

and rescaling time, in the generic case, the metric reads:

This spacetime is regular everywhere provided

The soliton fulfills the relaxed asymptotic conditions described above

The mass is given by:

- Note that the soliton is devoid of gravitational hair

Positive cosmological constant

Case of :

- The solution describes black hole on dS spacetime

- Black hole provided b > 0 (exists due to the hair)
- event and cosmological horizons: ,
- mass parameter bounded from above:
- saturated in the extremal case

Both temperatures coincide:

The metric for the Euclidean black hole (instanton) reads

- Extremal case: Wick-rotated to
- Also to

From the Euclidean black hole, Wick rotating the angle:

Note that for the metric reduces to

Otherwise:

This spacetime is regular everywhere provided

Euclidean action for the three-sphere (Euclidean dS):

Vanishes for the rest of the solutions

Vanishing cosmological constant

Case of :

- Asymptotically locally flat black hole

- For b >0 and c > 0: event horizon at

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