# The trace of the trace of the energy-momentum tensor-dependent Einstein's field equations

## TL;DR

This paper derives a new form of $f(R,T)$ gravity equations by focusing on the trace of the field equations, revealing the theory's unimodular nature and reducing arbitrariness in matter Lagrangian choice.

## Contribution

It introduces a novel approach to $f(R,T)$ gravity by using the trace of the field equations to determine the matter Lagrangian, establishing the theory as unimodular.

## Key findings

- Derived a form of matter Lagrangian eliminating arbitrariness.
- Showed $f(R,T)$ gravity is inherently unimodular.
- Proposed a new version of the $f(R,T)$ gravity theory.

## Abstract

The $f(R,T)$ gravity field equations depend generically on both the Ricci scalar $R$ and trace of the energy-momentum tensor $T$. Within the assumption of perfect fluids, the theory carries an arbitrariness regarding the choice of the matter lagrangian density $\mathcal{L}$, not uniquely defined. Such an arbitrariness can be evaded by working with the trace of the theory field equations. From such an equation, one can obtain a form for $\mathcal{L}$, which does not carry the arbitrariness. The obtained form for $\mathcal{L}$ shows that the $f(R,T)$ gravity is unimodular. A new version of the theory is, therefore, presented and forthcoming applications are expected.