Time-Dependent Scalar Fields in Modified Gravities in a Stationary Spacetime

TL;DR

This paper investigates the existence of time-dependent scalar fields around stationary spacetimes in various modified gravity theories, extending previous no-hair theorems to include non-minimal couplings and different formulations of $f(R)$ gravity.

Contribution

It generalizes no-hair theorems to modified gravities with non-minimal scalar couplings, showing existence conditions for time-dependent scalar hair.

Findings

Time-dependent scalar hair does not exist in metric $f(R)$ gravity.

Time-dependent scalar hair may exist in Palatini $f(R)$ gravity.

Non-minimally coupled scalar fields can support time-dependent hair.

Abstract

Most no-hair theorems involve the assumption that the scalar field is independent of time. Recently in [Phys. Rev. D90 (2014) 041501(R)] the existence of time-dependent scalar hair outside a stationary black hole in general relativity was ruled out. We generalize this work to modified gravities and non-minimally coupled scalar field with an additional assumption that the spacetime is axisymmetric. It is shown that in higher-order gravity such as metric $f(R)$ gravity the time-dependent scalar hair doesn't exist. While in Palatini $f(R)$ gravity and non-minimally coupled case the time-dependent scalar hair may exist.