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

Understand

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.