On the significance of the stress-energy tensor in Finsler spacetimes

TL;DR

This paper explores the definition and properties of the stress-energy tensor in Lorentz-Finsler spacetimes, emphasizing its anisotropic nature and revisiting conservation laws through geometric divergence concepts.

Contribution

It introduces a geometric framework for the stress-energy tensor in Finsler spacetimes, highlighting its anisotropic features and proposing a divergence based on the Chern connection.

Findings

Stress-energy tensor in Finsler spacetimes is anisotropic.

A geometric divergence consistent with Finsler geometry is formulated.

The Chern connection naturally arises in divergence computations.

Abstract

We revisit the physical arguments which lead to the definition of the stress-energy tensor $T$ in the Lorentz-Finsler setting $(M,L)$ starting at classical Relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes $T$ an anisotropic 2-tensor (i. e., a tensor for each $L$-timelike direction), in contrast with the energy-momentum vectors defined on $M$. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is revised from a geometric viewpoint and, then, the conservation laws of $T$ for each observer field are revisited. We introduce a natural {\em anisotropic Lie bracket derivation}, which leads to a divergence obtained from the volume element and the non-linear connection associated with $L$ alone. The computation of this divergence selects the Chern anisotropic connection, thus giving a geometric interpretation to previous choices in the literature.