Document

Report
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
• 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
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
Relaxed asymptotic conditions
General Relativity with scalar fields:
Relaxing the asymptotic conditions
enlarges the space of allowed solutions
Hair effect:
• No hair conjecture is violated
• Hairy black holes
• Solitons
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
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
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)
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
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]
Einstein-Gauss-Bonnet
Spherically symmetric solution (Boulware-Deser):
Generic case:
Special case:
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)
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
Black hole
b>0:
a single event horizon located at
provided
the bound is saturated when the horizon coincides
with the singularity
Black hole
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
Relaxed asymptotic conditions
• Same asymptotic symmetries as for Brown-Henneaux
(Conformal group in 2D)
Conserved charges
Abbott-Deser
Deser-Tekin charges
• Charges are finite
• The central charge is twice the standard value of
Brown-Henneaux
Conserved charges
Abbott-Deser
Deser-Tekin charges
• Charges are finite
• The central charge is twice the standard value of
Brown-Henneaux
Conserved charges
Black hole mass:
• The divergence cancels at the special point
• The mass is
For GR:
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”.
Thermodynamics
The metric for the Euclidean black hole reads
The solution is regular provided
• Extremal case:
• Also to
Wick-rotated to
wormhole covering space (see below)
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)
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
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
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
Thermodynamics
Both temperatures coincide:
The metric for the Euclidean black hole (instanton) reads
• Extremal case:
• Also to
Wick-rotated to
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
Vanishing cosmological constant
Case of :
• Asymptotically locally flat black hole
• For b >0 and c > 0: event horizon at

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