Black hole hair in generalized scalar-tensor gravity: An explicit example

Abstract

In a recent Letter we have shown that in shift-symmetric Horndeski theory the scalar field is forced to obtain a nontrivial configuration in black hole spacetimes, unless a linear coupling with the Gauss-Bonnet invariant is tuned away. As a result, black holes generically have hair in this theory. In this companion paper, we first review our argument and discuss it in more detail. We then present actual black hole solutions in the simplest case of a theory with the linear scalar-GaussBonnet coupling. We generate exact solutions numerically for a wide range of values of the coupling and also construct analytic solutions perturbatively in the small coupling limit. Comparison of the two types of solutions indicates that non-linear effects that are not captured by the perturbative solution lead to a finite area, as opposed to a central, singularity. Remarkably, black holes have a minimum size, controlled by the length scale associated with the scalar-Gauss-Bonnet coupling. We also compute some phenomenological observables for the numerical solution for a wide range of values of the scalar-Gauss-Bonnet coupling. Deviations from the Schwarzschild geometry are generically very small.

Constraining some Horndeski gravity theories

“…In this paper, we shall work out perihelion precession and bending of light by a central massive object (which could be a black hole or the Sun) in the two static and spherically symmetric solutions, presented in [1,2]. Quite interestingly, the later solution is only valid for a black hole spacetime (which, being hairy, differs from the Schwarzschild solution) and as a consequence putting forward an intriguing qualitative departure from the General Relativity, which we shall discuss in detail later.…”

Lovelock Galileons and black holes

“…Early work adopting perturbative analysis with dilatonic couplings 3 were performed in [19][20][21][22][23], while numerical investigations were performed in [24]. Recent investigations on the subject were carried out in [25,26]. A different way to bifurcate no hair theorems is to involve scalar tensor interactions involving translational invariant Galileons such as the John term of Fab 4 which reads, G µ ν ∇ µ φ∇ ν φ where G µν is the 4 dimensional Einstein tensor.…”

Gravitation-Wave Emission in Shift-Symmetric Horndeski Theories

“…In the special case of SSHTs, black holes are the same as in GR if χ = 0 in Eq. (17) [52], but possess scalar hairs if χ = 0 [53][54][55]. We thus expect dipolar emission from binaries involving black holes in SSHTs with χ = 0 [49,56].…”