We review the energy concept in the case of a continuum or a system of
fields. First, we analyze the emergence of a true local conservation equation
for the energy of a continuous medium, taking the example of an isentropic
continuum in Newtonian gravity. Next, we consider a continuum or a system of
fields in special relativity: we recall that the conservation of the
energy-momentum tensor contains two local conservation equations of the same
kind as before. We show that both of these equations depend on the reference
frame, and that, however, they can be given a rigorous meaning. Then we review
the definitions of the canonical and Hilbert energy-momentum tensors from a
Lagrangian through the principle of stationary action in a general spacetime.
Using relatively elementary mathematics, we prove precise results regarding the
definition of the Hilbert tensor field, its uniqueness, and its tensoriality.
We recall the meaning of its covariant conservation equation. We end with a
proof of uniqueness of the energy density and flux, when both depend
polynomially of the fields.
Keywords: energy conservation; conservation equation; special relativity;
general relativity; Hilbert tensor; variational principleComment: 35 pages in 12pt article format. Published version is Open Access at
Publisher (see DOI below). This version (V3) is a review article that exposes
in detail the results of V2 and also presents results not discussed there. V1
and V2 are successive versions of a conference talk. Sect. 3.5 of V2 contains
results which are not there in V
