On the effects of relaxing On the effects of relaxing the asymptotics of gravity in three dimensions in three dimensions Ricardo Troncoso Centro de Estudios Científicos (CECS) Valdivia, Chile

Asymptotically AdS spacetimes 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 conditions General 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 conditions 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)

No hair conjecture is violated Hairy black holes Solitons Relaxed asymptotic conditions General Relativity with scalar fields: Relaxing the asymptotic conditions enlarges the space of allowed solutions 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 conditions Topologically massive gravity M. 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 conditions 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 [hep-th] A. Maloney, W. Song, A. Strominger [hep-th]

BHT Massive Gravity 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, [hep-th]

BHT Massive Gravity Special case: Reminiscent of what occurs for the EGB theory for dimensions D>4 Unique maximally symmetric vacuum [A single fixed (A)dS radius l] Solutions of constant curvature :

Einstein-Gauss-Bonnet 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-Bonnet Spherically symmetric solution (Boulware-Deser): Generic case: Special case:

Einstein-Gauss-Bonnet 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 ?

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

Negative cosmological constant T 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 Case of :

Black hole a single event horizon located at provided b > 0 : the bound is saturated when the horizon coincides with the singularity

The singularity is surrounded by an event horizon provided b < 0 : The bound is saturated at the extremal case Black hole

Negative cosmological constant 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:

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

Conserved charges Charges are finite The central charge is twice the standard value of Brown-Henneaux Abbott-Deser Deser-Tekin charges

Conserved charges Charges are finite The central charge is twice the standard value of Brown-Henneaux Abbott-Deser Deser-Tekin charges

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

Thermodynamics 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

Entropy 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 solitons and 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 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 constant T The 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

Thermodynamics Extremal case: Wick-rotated to Also to The metric for the Euclidean black hole (instanton) reads Both temperatures coincide:

Gravitational 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 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 constant A Asymptotically locally flat black hole Case of :