Linearized Field Equations and Extra Force in $f(R,{\bf T}^{(n)})$ Extended Gravity

TL;DR

This paper explores an extended gravity theory involving higher-order invariants of the energy-momentum tensor, analyzing gravitational wave modes and demonstrating the presence of an extra force causing non-geodesic motion of test particles.

Contribution

It introduces a generalized $f(R,{f T}^{(n)})$ gravity model, studies its gravitational wave polarization modes, and derives the explicit form of the extra force affecting test particle trajectories.

Findings

Additional polarization modes appear with dynamical scalar fields.

Test particles experience a non-geodesic motion due to an extra force.

The theory reduces to $f(R,T)$ gravity when $n=1$.

Abstract

We consider an extended theory of gravity with Lagrangian $\mathcal{L} = f(R,{\bf T}^{(n)})$, with ${\bf T}^{(n)}$ being a $2n$-th order invariant made of contractions of the energy-momentum tensor. When $n=1$ this theory reduces to $f(R,T)$ gravity, where $T$ accounts for the trace of the energy-momentum tensor. We study the gravitational wave polarization modes, from which it results that when the matter Lagrangian contains dynamical scalar fields minimally coupled to the geometry, further polarization modes arise with respect to General Relativity. Finally we show that the motion for test particles is non-geodesic and we explicitly obtain the extra-force.