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On the effects of relaxing the asymptotics of gravity. in three dimensions. Ricardo Troncoso. Centro de Estudios Científicos (CECS) Valdivia, Chile. Asymptotically AdS spacetimes. Criteria: M. Henneaux and C. Teitelboim, CMP (1985). They are invariant under the AdS group

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On the effects of relaxing the asymptotics of gravity

On the effects of relaxing

the asymptotics of gravity

in three dimensions

Ricardo Troncoso

Centro de Estudios Científicos (CECS) Valdivia, Chile


On the effects of relaxing the asymptotics of gravity

Asymptotically AdS spacetimes

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


On the effects of relaxing the asymptotics of gravity

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


On the effects of relaxing the asymptotics of gravity

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)

  • 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


On the effects of relaxing the asymptotics of gravity

Relaxed asymptotic conditions

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:


On the effects of relaxing the asymptotics of gravity

Relaxed asymptotic conditions

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


On the effects of relaxing the asymptotics of gravity

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 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


On the effects of relaxing the asymptotics of gravity

BHT Massive Gravity

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)


On the effects of relaxing the asymptotics of gravity

BHT Massive Gravity

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


On the effects of relaxing the asymptotics of gravity

Einstein-Gauss-Bonnet

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]


On the effects of relaxing the asymptotics of gravity

Einstein-Gauss-Bonnet

Spherically symmetric solution (Boulware-Deser):

Generic case:

Special case:


On the effects of relaxing the asymptotics of gravity

Einstein-Gauss-Bonnet

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)


On the effects of relaxing the asymptotics of gravity

Does BHT massive gravity theory

possess a similar behavior ?


On the effects of relaxing the asymptotics of gravity

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


On the effects of relaxing the asymptotics of gravity

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


On the effects of relaxing the asymptotics of gravity

Black hole

b > 0 :

a single event horizon located at provided

the bound is saturated when the horizon coincides with the singularity


On the effects of relaxing the asymptotics of gravity

Black hole

b < 0 :

The singularity is surrounded by an event horizon provided

The bound is saturated at the extremal case


On the effects of relaxing the asymptotics of gravity

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


On the effects of relaxing the asymptotics of gravity

Relaxed asymptotic conditions

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


On the effects of relaxing the asymptotics of gravity

Conserved charges

Abbott-Deser Deser-Tekin charges

  • Charges are finite

  • The central charge is twice the standard value of

  • Brown-Henneaux


On the effects of relaxing the asymptotics of gravity

Conserved charges

Abbott-Deser Deser-Tekin charges

  • Charges are finite

  • The central charge is twice the standard value of

  • Brown-Henneaux


On the effects of relaxing the asymptotics of gravity

Conserved charges

Black hole mass:

  • The divergence cancels at the special point

  • The mass is For GR:


On the effects of relaxing the asymptotics of gravity

Conserved charges

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”.


On the effects of relaxing the asymptotics of gravity

Thermodynamics

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)


On the effects of relaxing the asymptotics of gravity

Entropy

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)


On the effects of relaxing the asymptotics of gravity

Gravitational solitons

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


On the effects of relaxing the asymptotics of gravity

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 soliton fulfills the relaxed asymptotic conditions described above

The mass is given by:

  • Note that the soliton is devoid of gravitational hair


On the effects of relaxing the asymptotics of gravity

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


On the effects of relaxing the asymptotics of gravity

Thermodynamics

Both temperatures coincide:

The metric for the Euclidean black hole (instanton) reads

  • Extremal case: Wick-rotated to

  • Also to


On the effects of relaxing the asymptotics of gravity

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


On the effects of relaxing the asymptotics of gravity

Euclidean action

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

Vanishes for the rest of the solutions


On the effects of relaxing the asymptotics of gravity

Vanishing cosmological constant

Case of :

  • Asymptotically locally flat black hole

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