Identity for scalar-valued functions of tensors and its applications to energy-momentum tensors in classical field theories and gravity

TL;DR

This paper proves a general tensor identity for scalar functions of tensors, extending previous work, and applies it to analyze energy-momentum tensors in classical field theories and gravity, including generalized Einstein equations.

Contribution

It introduces a new tensor identity for scalar-valued functions of tensors, applicable regardless of internal symmetries, and demonstrates its use in gravitational energy-momentum analysis and generalized Einstein equations.

Findings

The identity generalizes Rosenfeld's earlier work.

It confirms the zero-energy universe conjecture.

Provides a unified framework for energy-momentum tensors in various gravity theories.

Abstract

We prove a theorem on scalar-valued functions of tensors, where ``scalar'' refers to absolute scalars as well as relative scalars of weight $w$. The present work thereby generalizes an identity referred to earlier by Rosenfeld in his publication ``On the energy-momentum tensor''. The theorem provides a $(1,1)$-tensor identity which can be regarded as the tensor analogue of the identity following from Euler's theorem on homogeneous functions. The remarkably simple identity is independent of any internal symmetries of the constituent tensors, providing a powerful tool for deriving relations between field-theoretical expressions and physical quantities. We apply the identity especially for analyzing the metric and canonical energy-momentum tensors of matter and gravity and the relation between them. Moreover, we present a generalized Einstein field equation for arbitrary version of vacuum space-time dynamics -- including torsion and non-metricity. The identity allows to formulate an equivalent representation of this equation. Thereby the conjecture of a zero-energy universe is confirmed.