Black holes in vector-tensor theories

Abstract

We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordström (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.

Speed of gravitational waves and black hole hair

“…To proceed we assume that h(r) = f (r). Making use of the field equations calculated in [29] (translating from a vectortensor theory to a shift-symmetric scalar-tensor theory such that A µ → ∂ µ φ , i.e. A 0 = X 0 = 0, A 1 = dφ /dr), we find the following hairy solution:…”

“…It can be easily shown that spherically symmetric BHs can indeed have hair. For instance, [29] shows the solution when G 3 = 0, G 4 = M 2 P /2, in which case:…”

“…In contrast, since the Lagrangian L 6 couples the vector field non-minimally to curvature through L µναβ this Lagrangian corresponds to an intrinsic vector mode, and as such does not contribute to the background equations of motion for a homogenous and isotropic cosmological background (where A µ = A 0 (t), 0 ) [102]. In this case, with nonminimal coupling to curvature, one solution is that of a stealth Schwarzschild black hole with a non-trivial profile for the background vector field [29]:…”

“…no new charge), but the black hole is dressed with a non-trivial additional field profile. For example, some stealth black hole solutions in scalar-tensor and vector-tensor theories can be found in [28,29]. In this situation, non-dynamical tests will not be able to distinguish these models from GR and, therefore, for the purpose of this paper, these examples will not be considered as hairy solutions.…”

Eikonal quasinormal modes of black holes beyond general relativity

“…Focusing, for obvious reasons, on the coupled case, we adopt a largely theory-agnostic approach and postulate a pair of wave equations for the tensorial and the extra 'scalar' field with parametrised potentials. Although far from representing the most general situation [23], our parametrised model provides a useful benchmark for describing perturbed non-GR black holes; in addition it has the merit of including as a special case at least a pair of modified theories of gravity, namely, dynamical Chern-Simons gravity [16] and the sixth-order Proca theory [23,25].…”