Qualitative analysis of the Tolman metrics within the unimodular framework

TL;DR

This paper explores the behavior of Tolman metrics within trace-free (unimodular) gravity, revealing unique features like an extra integration constant affecting mass and equations of state, with qualitative analysis through graphical plots.

Contribution

It provides a qualitative analysis of Tolman metrics in unimodular gravity, highlighting differences from standard Einstein equations and the role of an additional integration constant.

Findings

Extra integration constant influences gravitational mass and equation of state.

Energy conservation law is modified but can be supplemented.

Graphical analysis illustrates qualitative behavior of physical quantities.

Abstract

We investigate the behaviour of the Tolman metrics within the formalism of the trace-free (or unimodular) gravity. While this approach is similar to the standard Einstein field equations, some subtlety arises. The effective number of independent field equations is reduced by one on account of the density and pressure appearing as an inseparable entity the inertial mass density. Further energy is not conserved within the trace-free theory but the conservation law may be supplemented to the field equations. This presentation of the field equations offers a different avenue to determine the density and pressure explicitly. It turns out that an extra integration constant is always in evidence. While this constant has little impact on the dynamics and energy conditions, it makes a significant impact on the gravitational mass and equation of state. Graphical plots are generated to analyse the behaviour of physical quantities qualitatively.