On the effects of relaxing the asymptotics of gravity

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

2. 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

3. 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

4. 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

5. 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:

6. 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

7. 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

8. 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)

9. 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

10. 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]

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

12. 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)

13. Does BHT massive gravity theory possess a similar behavior ?

14. 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

15. 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

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

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

18. 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

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

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

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

22. Conserved charges Black hole mass: • The divergence cancels at the special point • The mass is For GR:

23. 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”.

24. 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)

25. 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)

26. 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

27. 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

28. 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

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

30. 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

31. Euclidean action Euclidean action for the three-sphere (Euclidean dS): Vanishes for the rest of the solutions

32. Vanishing cosmological constant Case of : • Asymptotically locally flat black hole • For b >0 and c > 0: event horizon at