Phenomenological Description of a Conservative Continuous Medium

Abstract

The energy–momentum–stress density evidently plays the role of source for the gravitational field and is in many ways analogous to the electric charge in electromagnetic theory. However, because matter in all its forms is coupled to the gravitational field, this source can be much more complicated than electric charge. Many times we may wish to find the gravitational field produced by a certain material system without knowing or being able to write an action functional for the system. We then need a general description of the system and its dynamical behavior that will enable us to keep track of its energy and momentum content and obtain an energy–momentum–stress density for it without necessarily knowing its structure in all fundamental respects. An example of such a description, which covers a wide range of practical cases, is the phenomenological treatment of a conservative continuous medium. By “conservative” we mean that there are no irreversible dissipative processes at work. Once having found the energy–momentum–stress density for a conservative medium, we shall find it not difficult to introduce dissipative mechanisms either again phenomenologically or, if that is insufficient, by the use of distribution functions and all the paraphernalia of the Boltzmann and other types of transport equations.