On the stability of axisymmetric systems to axisymmetric perturbations in general relativity. I. The equations governing nonstationary, and perturbed systems

Abstract

Axisymmetric systems in general relativity are considered. The field and the fluid equations that are appropriate to general nonstationary (but axisymmetric) systems are first derived. They are then specialized to yield the equations which govern stationary equilibrium. The equations which determine the evolu tion of small departures from equilibrium are also obtained. Related matters that are considered include the Landau-Lifshitz complex, the conserved quantities, and the constancy of the baryon number and the angular momentum (per baryon) of a fluid element as it moves. The theory is developed with a view toward establishing criteria for the stability of rotating systems to axisymmetric perturbations.