Exact black hole solutions in shift symmetric scalar-tensor theories

Abstract

We derive a variety of exact black hole solutions in a subclass of Horndeski's scalartensor theory possessing shift symmetry, φ → φ + c, and reflection symmetry, φ → −φ. The theory admits two arbitrary functions of X := −(∂φ) 2 /2, and our solutions are constructed without specifying the concrete form of the two functions, implying that black hole solutions in specific scalar-tensor theories found in the literature can be extended to a more general class of theories with shift symmetry. Our solutions include a black hole in the presence of an effective cosmological constant, the Nariai spacetime, the Lifshitz black hole, and other nontrivial solutions, all of which exhibit nonconstant scalar-field profile.

Slowly rotating black hole solutions in Horndeski gravity

“…For a static scalar field (q = 0), Eqs. (20)- (22) reproduce the results obtained in [43,44]; for reflection-symmetric theories, they reduce to the results of [21]. For slowly rotating solutions at linear order in the BH angular velocity, the only nonvanishing component of the equations of motion yields a second-order ordinary differential equation for the variable ω(r):…”

“…(46) is finite. For reflection-symmetric theories (G 3 = G 5 = 0), this latter requirement simplifies to the condition that (B/A) ′ should be finite [21].…”

Lovelock Galileons and black holes

“…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. There when one considers additionally a linear time dependence of the scalar field [27] (see also [28][29][30][31][32][33][34], and [35][36][37][38][39][40][41]) it was shown that the time dependence yielded analytic GR like black holes with additionally well defined scalars on the black hole horizon. For a recent review on black holes and scalar fields see [42] and [43].…”

Slowly rotating neutron stars in the nonminimal derivative coupling sector of Horndeski gravity

“…where κ = (16πG) −1 , α and η are two parameters controlling the strength of the minimal and nonminimal kinetic couplings [34][35][36][37][38][39][40][41]. One important feature of this model is that the shift symmetry φ → φ + φ 0 implies that the equation of motion for the scalar field can be written as the current conservation law ∇ µ J µ = 0.…”