Differential rotation of an electrically conducting fluid in general relativity

Abstract

Starting from the Ricci identity for the 4-velocity vectoru a ,a mathematical identity is derived, in terms of the kinematic quantities and the Riemann curvature tensor, for ω,a H a , the derivative along a magnetic field line of the magnitude of the vorticity of an electrically conducting fluid. Maxwell's field equations are not used in the derivation and the result is true for a fluid with finite, infinite, or even nonuniform, electric conductivity. Previous results [3–5] derived for an infinitely conducting fluid are obtained as special cases. By expressing the Riemann curvature tensor in terms of the Ricci and Weyl tensors, Einstein's field equations are introduced and the role played by the free gravitational field is examined. It is found that ω,aHa does not depend on the electric part of the Weyl tensor and that, for an infinitely conducting fluid satisfying certain “steady state” conditions, ω,aHa is independent of that part of the curvature determined locally through Einstein's field equations.