Spherically symmetric solutions in higher-derivative gravity

Abstract

Extensions of Einstein gravity with quadratic curvature terms in the action arise in most effective theories of quantised gravity, including string theory. This article explores the set of static, spherically symmetric and asymptotically flat solutions of this class of theories. An important element in the analysis is the careful treatment of a Lichnerowicz-type 'no-hair' theorem. From a Frobenius analysis of the asymptotic small-radius behaviour, the solution space is found to split into three asymptotic families, one of which contains the classic Schwarzschild solution. These three families are carefully analysed to determine the corresponding numbers of free parameters in each. One solution family is capable of arising from coupling to a distributional shell of matter near the origin; this family can then match on to an asymptotically flat solution at spatial infinity without encountering a horizon. Another family, with horizons, contains the Schwarzschild solution but includes also non-Schwarzschild black holes. The third family of solutions obtained from the Frobenius analysis is nonsingular and corresponds to 'vacuum' solutions. In addition to the three families identified from near-origin behaviour, there are solutions that may be identified as 'wormholes', which can match symmetrically on to another sheet of spacetime at finite radius.1

Abdus Salam and Quadratic Curvature Gravity: Classical Solutions

“…This gives the following results, 15 for the general α, β theory: (s, t) = (1, −1) with 5 free parameters (10a) (s, t) = (0, 0) with 3 free parameters (10b) (s, t) = (2, 2) with 6 free parameters (10c)…”

“…the counterparts of the classic Schwarzschild solution in Einstein theory. The recent results presented here have been obtained in collaboration with Hong Lü, Alun Perkins and Chris Pope [14][15][16] . We consider the gravitational action…”

“…The presence of a horizon together with asymptotic flatness yields a substantial simplification of the relevant equations however, owing to a 'no-hair' theorem 15,18 for the trace of the field equation (2).…”

Higher-derivative gravity with non-minimally coupled Maxwell field

“…In fact in four dimensions, gravity extended with quadratic curvature invariants was shown to be renormalizabe [12,13]. Recently a new static black hole over and above the usual Schwarzschild black hole was obtained in the four-dimensional theory [14,15]. When there is a cosmological constant, higher-derivative gravities in AdS backgrounds can have a critical point in the parameter space for which the ghost modes become log modes and may be truncated out by some strong boundary conditions.…”

Cosmology in Poincaré gauge gravity with a pseudoscalar torsion

“…In fact, starting from a well behaved Lagrangian 1 2 R + αR 2 + βR µν R µν in quadratic gravity [57][58][59][60][61][62][63] and string theory [64] and adding a quadratic term of torsion γT µ νρ T µ νρ a good toy model can be obtained [65], where we derive to give the gravitational field equation and the cosmological evolutional equations. When the macroscopic spacetime average of the spin vanishes, the cosmological equations are found to split into two families.…”