A Variational Principle for Rotating Stars in General Relativity

A variational principle for the structure of differentially rotating stars in general relativity is developed both as an action principle from which the Einstein equations and the equation of hydrostatic equilibrium can be derived and as an energy-extremization principle. We show that the equilibrium configuration of the star and its gravitational field extremizes the gravitational mass of momentarily stationary, axisymmetric configurations which satisfy the initial-value equations of general relativity and in which each ring of matter has a fixed number of baryons, a fixed angular momentum, and a fixed entropy per baryon. A simple positive-definite expression for the gravitational mass of such a momentarily stationary configuration is obtained directly from the initial-value equations. Ways in which the different forms of the variational principle can be used in numerical calculations of the structure of rotating stars in a general relativity are discussed, as well as some analytic results relating to stability against convection.