Symmetries of the Energy-Momentum Tensor: Some Basic Facts

TL;DR

This paper examines different definitions of matter collineations in static spherically symmetric spacetimes and compares their implications for the energy-momentum tensor's symmetries.

Contribution

It analyzes the differences between three methods of defining matter collineations and compares their results in specific spacetime geometries.

Findings

Different definitions lead to non-equivalent matter collineation conditions.

The paper clarifies the implications of each definition for the symmetry properties.

Results depend on the choice of the Lie derivative form used in the definition.

Abstract

It has been pointed by Hall et al. [1] that matter collinations can be defined by using three different methods. But there arises the question of whether one studies matter collineations by using the ${\cal L}_\xi T_{ab}=0$, or ${\cal L}_\xi T^{ab}=0$ or ${\cal L}_\xi T_a^b=0$. These alternative conditions are, of course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition.