Equations of state of continuous matter in general relativity

Abstract

The invariant forms that equations of state of continuous matter may take in general relativity, when the rheological behaviour of matter at any event may depend on previous rheological states through which that matter has passed, are discussed. A complete set of variables, needed in the general case to specify the relevant information about the matter at event x i , is first obtained as a set of space-tensors and then as a set of four-tensors orthogonal to the four-velocity at x i . These variables represent proper measures of deformation history, mechanical-stress history, temperature history, proper-time lag and physical constants of the material. An unambiguous definition is given of a physical constant (tensor) of the material (equation (122)). Elasticity, viscosity, and all possible combinations of these properties are within the scope of the theory. A detailed discussion is included of the processes of differentiation and integration of tensor quantities with respect to proper-time, following a particle along its world-line, such as will occur in equations of state in the general case. A convected integral with respect to proper-time is expressed (in equation (111)) in terms of displacement functions X 'm , which relate events x i and x 'm on the world-line of the same particle (such that x 'm is earlier than x i by an interval of proper-time t—t' ) through equations x 'm = X 'm (x i , t — t' ) . A convected derivative with respect to proper-time is expressed (in equation (82)) in terms of a Lie derivative defined with respect to the velocity vector field. Successive convected differentiation of a finite-strain tensor, defined in relation to an arbitrary reference configuration of a material element, gives rise to a sequence of rate-of-strain tensors.