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On the definition of energy for a continuum, its conservation laws, and the energy-momentum tensor. (English) Zbl 1357.83002

Summary: 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 general space-time. 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 on the fields.

MSC:

83A05 Special relativity
83C40 Gravitational energy and conservation laws; groups of motions
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
81T20 Quantum field theory on curved space or space-time backgrounds
53Z05 Applications of differential geometry to physics
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
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References:

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