On the effects of relaxingOn the effects of relaxing

the the asymptoticsasymptotics of gravity of gravity

in three dimensionsin three dimensions

Ricardo TroncosoRicardo Troncoso

Centro de Estudios Científicos (CECS) Centro de Estudios Científicos (CECS) Valdivia, Valdivia, ChileChile

Asymptotically AdS Asymptotically AdS spacetimesspacetimes

• 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

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

• 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

Brown-Henneaux asymptotic Brown-Henneaux asymptotic conditionsconditionsGeneral Relativity in D = 3 (localized matter fields)J. D. Brown and M. Henneaux, CMP (1986)

• 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

Relaxed asymptotic Relaxed asymptotic conditionsconditionsGeneral Relativity with scalar fieldsM. 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)

• No hair conjecture is violated

• Hairy black holes

• Solitons

Relaxed asymptotic Relaxed asymptotic conditionsconditions

General Relativity with scalar fields:

Relaxing the asymptotic conditionsenlarges the space of allowed solutions

Hair effect:Hair effect:

• 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

Relaxed asymptotic Relaxed asymptotic conditionsconditionsTopologically massive gravityM. Henneaux, C. Martínez, R. Troncoso PRD (2009)

• 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

Relaxed asymptotic Relaxed asymptotic conditionsconditionsTopologically massive gravity at the chiral pointD. 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]

BHT Massive BHT Massive GravityGravity

Field equations(fourth order)

Linearized theory: Massive graviton with two helicities (Fierz-Pauli)

Bergshoeff-Hohm-Townsend (BHT) action:E. A. Bergshoeff, O. Hohm, P. K. Townsend, 0901.1766 [hep-th]

BHT Massive BHT Massive GravityGravity

Special case:

Reminiscent of what occurs for the EGB theoryfor dimensions D>4

Unique maximally symmetric vacuum [A single fixed (A)dS radius l]

Solutions of constant curvature :

Einstein-Gauss-Einstein-Gauss-BonnetBonnet

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

D > 4 dimensions

Special case:

Unique maximally symmetric vacuum [A single fixed (A)dS radius l]

Einstein-Gauss-Einstein-Gauss-BonnetBonnetSpherically symmetric solution (Boulware-Deser):

Generic case:

Special case:

Einstein-Gauss-Einstein-Gauss-BonnetBonnet

• 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)

Special case:

Does BHT massive gravity theory Does BHT massive gravity theory

possess a similar behavior ?possess a similar behavior ?

• 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

BHT massive gravity at the special BHT massive gravity at the special pointpoint

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

Negative cosmological Negative cosmological constantconstant

• The solution describes asymptotically AdS black The solution describes asymptotically AdS black holesholes

•c : mass parameter (w.r.t. AdS)

•b : “gravitational hair”it does not correspond to any global chargegenerated by the asymptotic symmetries

Case of :

Black holeBlack hole

a single event horizon located at provided

b > b > 0 :0 :

the bound is saturated when the horizon coincides with the singularity

The singularity is surrounded by an event horizon provided

b < b < 0 :0 :

The bound is saturated at the extremal case

Black holeBlack hole

Negative cosmological Negative cosmological constantconstant

• 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

Hair effect:Hair effect:

Relaxed asymptotic Relaxed asymptotic conditionsconditions

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

Conserved chargesConserved charges

• Charges are finite

• The central charge is twice the standard value of Brown-Henneaux

Abbott-Deser Deser-Tekin charges

Conserved chargesConserved charges

• Charges are finite

• The central charge is twice the standard value of Brown-Henneaux

Abbott-Deser Deser-Tekin charges

Conserved chargesConserved charges

• The divergence cancels at the special point

• The mass is For GR:

Black hole mass:

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

Conserved Conserved chargescharges

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

ThermodynamicsThermodynamics

• Extremal case: Wick-rotated to

• Also to wormhole covering space (see below)

The metric for the Euclidean black hole reads

The solution is regular provided

EntropyEntropy

• 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)

Wald’s formula:

For the black hole:

Gravitational solitonsGravitational solitons

and wormholesand wormholes

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

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:

Gravitational solitonGravitational soliton

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 mass is given by:

• Note that the soliton is devoid of gravitational hair

The soliton fulfills the relaxed asymptotic conditions described above

Positive cosmological Positive cosmological constantconstant

• The solution describes black hole on dS spacetimeThe solution describes black hole on dS spacetime

Case of :

• Black hole provided b > 0 (exists due to the hair)

• event and cosmological horizons: , • mass parameter bounded from above:

• saturated in the extremal case

ThermodynamicsThermodynamics

• Extremal case: Wick-rotated to

• Also to

The metric for the Euclidean black hole (instanton) reads

Both temperatures coincide:

Gravitational solitonGravitational soliton

From the Euclidean black hole, Wick rotating the angle:

Note that for the metric reduces to

Otherwise:

This spacetime is regular everywhere provided

Euclidean actionEuclidean action

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

Vanishes for the rest of the solutions

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

Vanishing cosmological Vanishing cosmological constantconstant

• Asymptotically locally flat black holeAsymptotically locally flat black hole

Case of :