Hilbert’s energy–momentum tensor extended

Abstract

In classical field models, Hilbert’s energy–momentum tensor for a matter field is defined by a variational derivative of a Lagrangian with respect to the metric tensor. Constitutive relationships between fundamental field variables are a subject that is dealt with by a wide range of models in solid-state physics. In this case, a higher-order constitutive tensor replaces the second-order metric tensor. A similar premetric description with a linear constitutive relation has recently been provided for classical field models of gravity and electrodynamics. In this paper, we analyze the extension of the Hilbert definition of the energy–momentum tensor to models with general linear constitutive law. In order to treat integrals on a differential manifold covariantly, we need the differential forms description. It follows that the Lagrangian, electromagnetic current, and energy–momentum current must all be represented as twisted 4-forms, 3-forms, and vector-valued 3-forms, respectively. The constitutive relation then appears as a linear map on forms. We derive a commutative variation identity for an arbitrary linear map, enabling direct variation processes without having to deal with the individual components. We present evidence for the close relationship between the explicit form of the energy–momentum current of Hilbert’s type and the commutative-variation identity. A variety of field models with a general linear constitutive law are subject to this construction.