Consistency Condition of Spherically Symmetric Solutions in $f(R)$ Gravity

TL;DR

This paper investigates the conditions under which spherically symmetric solutions in $f(R)$ gravity are consistent with the modified field equations, highlighting the necessity of an extra metric condition for consistency.

Contribution

It clarifies the consistency conditions for spherically symmetric solutions in $f(R)$ gravity within the metric formalism, emphasizing the role of an additional metric condition.

Findings

Spherical solutions are generally consistent with $f(R)$ gravity equations.

An extra metric condition is required for full consistency.

The results clarify constraints on spherically symmetric solutions in modified gravity.

Abstract

In this work we study the spherical symmetric solutions of $f(R)$ gravity in the metric formalism. We show that for a generic $f(R)$ gravity, the spherical symmetric solution is consistent with the modified gravity equations except in the case of imposing an extra condition for the metric.