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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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slide1

On the effects of relaxing

the asymptotics of gravity

in three dimensions

Ricardo Troncoso

Centro de Estudios Científicos (CECS) Valdivia, Chile

slide2

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
slide3

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
slide4

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
slide5

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:

slide6

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

slide7

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

slide8

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)

slide9

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

slide10

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]

slide11

Einstein-Gauss-Bonnet

Spherically symmetric solution (Boulware-Deser):

Generic case:

Special case:

slide12

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)
slide13

Does BHT massive gravity theory

possess a similar behavior ?

slide14

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
slide15

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
slide16

Black hole

b > 0 :

a single event horizon located at provided

the bound is saturated when the horizon coincides with the singularity

slide17

Black hole

b < 0 :

The singularity is surrounded by an event horizon provided

The bound is saturated at the extremal case

slide18

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
slide19

Relaxed asymptotic conditions

  • Same asymptotic symmetries as for Brown-Henneaux (Conformal group in 2D)
slide20

Conserved charges

Abbott-Deser Deser-Tekin charges

  • Charges are finite
  • The central charge is twice the standard value of
  • Brown-Henneaux
slide21

Conserved charges

Abbott-Deser Deser-Tekin charges

  • Charges are finite
  • The central charge is twice the standard value of
  • Brown-Henneaux
slide22

Conserved charges

Black hole mass:

  • The divergence cancels at the special point
  • The mass is For GR:
slide23

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”.
slide24

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)
slide25

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)
slide26

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
slide27

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
slide28

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
slide29

Thermodynamics

Both temperatures coincide:

The metric for the Euclidean black hole (instanton) reads

  • Extremal case: Wick-rotated to
  • Also to
slide30

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

slide31

Euclidean action

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

Vanishes for the rest of the solutions

slide32

Vanishing cosmological constant

Case of :

  • Asymptotically locally flat black hole
  • For b >0 and c > 0: event horizon at
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