Kinematics and Dynamics of $f(R)$ Theories of Gravity

TL;DR

This paper extends the equations of relativistic fluid dynamics to general $f(R)$ theories of gravity, demonstrating that key theorems from general relativity also apply in this broader context.

Contribution

It generalizes fluid dynamic equations and key theorems from general relativity to $f(R)$ gravity, providing a new framework for analyzing relativistic fluids.

Findings

Friedmann–Lemaître–Robertson–Walker conditions derived for $f(R)$ theories

Ehlers–Geren–Sachs theorem extended to $f(R)$ gravity

Perfect fluid spacetimes with specific conditions imply FLRW geometry

Abstract

We generalise the equations governing relativistic fluid dynamics given by Ehlers and Ellis for general relativity, and by Maartens and Taylor for quadratic theories, to generalised $f(R)$ theories of gravity. In view of the usefulness of this alternative framework to general relativity, its generalisation can be of potential importance for deriving analogous results to those obtained in general relativity. We generalise, as an example, the results of Maartens and Taylor to show that within the framework of general $f(R)$ theories, a perfect fluid spacetime with vanishing vorticity, shear and acceleration is Friedmann--Lema\^{\i}tre--Robertson--Walker only if the fluid has in addition a barotropic equation of state. It then follows that the Ehlers--Geren--Sachs theorem and its ``almost'' extension also hold for $f(R)$ theories of gravity.