Geometric equations of state in Friedmann-Lema\^{i}tre universes admitting matter and Ricci Collineations

TL;DR

This paper explores geometric equations of state in Friedmann-Lemaître universes with matter and Ricci collineations, deriving physically viable models that depend on geometric conditions rather than observer-dependent assumptions.

Contribution

It introduces a geometric approach to defining equations of state in cosmology using Ricci collineations, bypassing observer-dependent constraints.

Findings

Derived linear and non-linear equations of state.

Found solutions satisfying energy conditions.

Identified physically viable cosmological models.

Abstract

As a rule in General Relativity the spacetime metric fixes the Einstein tensor and through the Field Equations (FE) the energy-momentum tensor. However one cannot write the FE explicitly until a class of observers has been considered. Every class of observers defines a decomposition of the energy-momentum tensor in terms of the dynamical variables energy density ($\mu$), the isotropic pressure ($p$), the heat flux $q^a$ and the traceless anisotropic pressure tensor $\pi_{ab}$. The solution of the FE requires additional assumptions among the dynamical variables known with the generic name equations of state. These imply that the properties of the matter for a given class of observers depends not only on the energy-momentum tensor but on extra a priori assumptions which are relevant to that particular class of observers. This makes difficult the comparison of the Physics observed by different classes of observers for the {\it same} spacetime metric. One way to overcome this unsatisfactory situation is to define the extra condition required among the dynamical variables by a geometric condition, which will be based on the metric and not to the observers. Among the possible and multiple conditions one could use the consideration of collineations. We examine this possibility for the Friedmann-Lema\^{i}tre-Robertson-Walker models admitting matter and Ricci collineations and determine the equations of state for the comoving observers. We find linear and non-linear equations of state, which lead to solutions satisfying the energy conditions, therefore describing physically viable cosmological models.