General properties of $\mathbf {f(R)}$ gravity vacuum solutions

TL;DR

This paper explores the fundamental characteristics of vacuum solutions in $f(R)$ gravity, establishing conditions under which these solutions resemble well-known cosmological models like FRW metrics.

Contribution

It proves that the gradient of the curvature scalar is an eigenvector of the Ricci tensor and classifies vacuum solutions based on the nature of this gradient.

Findings

Gradient of R is an eigenvector of Ricci tensor

Time-like gradient implies a Generalized FRW metric

In four dimensions, solutions reduce to FRW metrics

Abstract

General properties of vacuum solutions of $f(R)$ gravity are obtained by the condition that the divergence of the Weyl tensor is zero and $f''\neq 0$. Specifically, a theorem states that the gradient of the curvature scalar, $\nabla R$, is an eigenvector of the Ricci tensor and, if it is time-like, the space-time is a Generalized Friedman-Robertson-Walker metric; in dimension four, it is Friedman-Robertson-Walker.