Effective Field Theory of Perturbations on Arbitrary Black Hole Backgrounds with Spacelike Scalar Profile

TL;DR

This paper develops a model-independent effective field theory framework for analyzing perturbations around black holes with spacelike scalar profiles in scalar-tensor theories, enabling tests of gravity beyond general relativity in strong-field regimes.

Contribution

It introduces a novel EFT approach for scalar-tensor theories with spacelike scalar fields on arbitrary black hole backgrounds, incorporating residual symmetries and consistency relations.

Findings

Derived the EFT action invariant under $(2+1)$d diffeomorphisms.

Applied the framework to a black hole with time-varying mass.

Provided a tool for model-independent tests of gravity in strong fields.

Abstract

We develop the effective field theory (EFT) of perturbations in the context of scalar-tensor theories with a spacelike scalar profile on arbitrary black hole backgrounds. Our construction of the EFT is based on the fact that in the unitary gauge, where the scalar field is chosen as one of the spatial coordinates, the background scalar field spontaneously breaks the diffeomorphism invariance along the direction of its gradient. The residual symmetry on a timelike hypersurface of constant scalar field is referred to as the $(2+1)$d diffeomorphism invariance. We then derive a set of consistency relations, imposed on the EFT parameters, by requiring that the EFT action in the unitary gauge be invariant under the $(2+1)$d diffeomorphisms. For concreteness, we apply the EFT to study the background dynamics of a class of non-static and spherically symmetric solutions, focusing in particular on a black hole solution with a time-varying mass. We emphasize that our EFT framework is broadly applicable to any black hole background as long as the scalar field remains spacelike throughout the spacetime region of interest. This formulation provides a model-independent approach for testing scalar-tensor theories as gravity beyond general relativity in the strong-gravity regime.