General Relativity solutions in modified gravity

TL;DR

This paper establishes three general conditions that modified gravity theories must satisfy to admit solutions identical to those of General Relativity, aiding in the classification and constraint of such theories using gravitational wave observations.

Contribution

It provides a comprehensive and general set of conditions applicable to various scalar-tensor theories, clarifying when they can reproduce GR solutions in different spacetime geometries.

Findings

Identifies three key conditions for modified gravity theories to admit GR solutions.

Classifies theories into three categories based on their solutions: no-hair, hairy, or both.

Offers a framework to constrain modified gravity theories with observational data.

Abstract

Recent gravitational wave observations of binary black hole mergers and a binary neutron star merger by LIGO and Virgo Collaborations associated with its optical counterpart constrain deviation from General Relativity (GR) both on strong-field regime and cosmological scales with high accuracy, and further strong constraints are expected by near-future observations. Thus, it is important to identify theories of modified gravity that intrinsically possess the same solutions as in GR among a huge number of theories. We clarify the three conditions for theories of modified gravity to allow GR solutions, i.e., solutions with the metric satisfying the Einstein equations in GR and the constant profile of the scalar fields. Our analysis is quite general, as it applies a wide class of single-/multi-field scalar-tensor theories of modified gravity in the presence of matter component, and any spacetime geometry including cosmological background as well as spacetime around black hole and neutron star, for the latter of which these conditions provide a necessary condition for no-hair theorem. The three conditions will be useful for further constraints on modified gravity theories as they classify general theories of modified gravity into three classes, each of which possesses i) unique GR solutions (i.e., no-hair cases), ii) only hairy solutions (except the cases that GR solutions are realized by cancellation between singular coupling functions in the Euler-Lagrange equations), and iii) both GR and hairy solutions, for the last of which one of the two solutions may be selected dynamically.