Vorticity of a Perfect Fluid Undergoing a Gravitational Collapse

Abstract

The vorticity propagation equation for a perfect fluid in general relativity is derived in a form which is the same as that of Maxwell's equation for the magnetic field four-vector in relativistic magnetohydrodynamics. Starting from this result, an expression for the change of vorticity during a gravitational collapse is obtained in terms of the spatial geometry, using a procedure similar to that introduced by Cocke (1966) in relativistic magnetohydrodynamics. It is assumed that the equation of state of the fluid is p = 1Xp" where IX is a constant and p, is the total proper energy density. If t < IX :s;; 1, it is found that the vorticity tends to zero during an isotropic collapse, in agreement with a result obtained previously by Ellis (1973) using a different procedure. Nonisotropic collapses are also considered. The dynamical importance of vorticity in a gravitational collapse is examined by considering the behaviour of w2 /p,.