  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.
