Abstract
No-hair theorems for scalar-tensor theories imply that the trivial scalar field configuration is the unique configuration around stationary black hole spacetimes. The most basic assumption in these theorems is that a constant scalar configuration is actually admissible. In this paper, we classify shift-symmetric Horndeski theories according to whether or not they admit the trivial scalar configuration as a solution and under which conditions. Local Lorentz symmetry and the presence of a linear coupling between the scalar field and Gauss-Bonnet invariant plays feature prominently in this classification. We then use the classification to show that any theory without linear Gauss-Bonnet coupling that respects Local Lorentz symmetry admits all GR solutions. We also study the scalar hair configuration around black hole spacetimes in theories where the linear Gauss-Bonnet coupling is present. We show that the scalar hair of the configuration is secondary, fixed by the regularity of the horizon, and is determined by the black hole horizon properties.
Highlights
After more than 100 years since its introduction, general relativity (GR) is still the most successful theory describing gravitational interactions
We use the classification to show that any theory without linear Gauss-Bonnet coupling that respects local Lorentz symmetry admits all solutions of general relativity
We argued that Horndeski theories can be split in three classes: (i) theories that admit all of the spacetimes of GR with a constant scalar configuration; (ii) theories do not belong to the previous class but that admit flat space with constant scalar; (iii) theories in which the scalar has to be nontrivial in flat space or do not admit flat space at all, and they are Lorentz violating
Summary
After more than 100 years since its introduction, general relativity (GR) is still the most successful theory describing gravitational interactions. [8] that a linear coupling between a scalar and the GaussBonnet invariant respects shift symmetry and at the same time leads to a contribution to the scalar’s field equation that depends only on the Gauss-Bonnet invariant.1 The latter does not vanish in black hole spacetimes; it sources the scalar and makes a constant scalar configuration inadmissible. It was further shown there that this is the only coupling term with this property, which at the same time allows for a constant scalar configuration in flat space (the Gauss-Bonnet invariant vanished) The latter is a requirement if one wants the theory to respect local Lorentz symmetry (LLS). Sticking to our initial motivation, we use it to prove that a black hole in Horndeski theory cannot have an independent charge
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