Some remarks on the definition of classical energy and its conservation laws

Mayeul Arminjon (1)

1 3SR - Laboratoire sols, solides, structures - risques [Grenoble]

In classical non-relativistic theories, there is an exact local conservation equation for the energy, having the form of the continuity equation for mass conservation, and this equation occurs from the power equation. We illustrate this by the example of Newtonian gravity for self-gravitating elastic bodies. In classical special-relativistic theories, there is also an exact local conservation equation for the energy, though it comes from the definition of the energy-momentum tensor. We then study that definition in a general spacetime: Hilbert's variational definition is briefly reviewed, with emphasis on the boundary conditions. We recall the difference between the local equation verified by Hilbert's tensor T in a curved spacetime and the true local conservation equations discussed before. We ask if the addition of a total divergence may change T and find that the usual formula giving T is not generally valid when the matter Lagrangian depends on the derivatives of the metric. We end with a result proving uniqueness of the energy density and flux, when both depend polynomially on the fields.

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]

Mayeul Arminjon. Some remarks on the definition of classical energy and its conservation laws. 2015. ⟨hal-01213871v1⟩