Variational aspects of relativistic field theories, with application to perfect fluids

Abstract

By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully covariant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor T μν . By perturbing the Noether operator's definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwell's equations for F μν and Euler's equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) (c)]. General relativity is a theory which does have an unconstrained variational principle but the definition of Noether operator is more ambiguous than for other fields. We define a pseudotensorial operator which includes the Einstein and Landau-Lifschitz complexes as special cases and satisfies a certain criterion on the asymptotic behavior. Then our extremal theorem leads to a proof of the uniqueness of Minkowski space: It is the only asymptotically flat, stationary, vacuum solution to Einstein's equations having R 4 global topology and a maximal spacelike hypersurface. We next consider perfect fluid dynamics. The failure of the extremal-energy theorem elucidates why constraints have always been used in variational principles that lead to Euler's equations. We discuss their meaning and give what we consider to be the “minimally constrained” principle. A discussion of one constraint, “preservation of particle identity,” from the point of view of path-integral quantum mechanics leads to the conclusion that it is inapplicable to degenerate Bose fluids, and this gives immediately the well-known irrotational flow of such fluids. Finally, we develop a restricted extremal theorem for the case of perfect fluids with self-gravitation, which has the same form as before except that certain perturbations are forbidden in (c). We show that it is a generalization of the Bardeen-Hartle-Sharp variational principle for relativistic stellar structure. It may be useful in constructing nonaxisymmetric stellar models (generalized Dedekind ellipsoids). We also give the Newtonian versions of the main results here, and we show to what extent the extremal theorems extend to fields that may not even have a variational principle.