A Relativistic Extension of Cowling's Theorem.

Abstract

Cowling's theorem within the framework of non relativistic dynamo theory states that an axisymmetric magnetic field can not supported by dynamo action due to an axisymmetric conducting fluid flow. In this work we shall examine whether effects of spacetime curvature may modify the conclusion of this theorem. We study an axially symmetric magnetic field B generated by an axisymmetric conducting fluid on a non singular static, spherically symmetric and asymptotically flat with one end spacetime. For such background geometry at first we derive a set of equations describing the evolution of the poloidal and toroidal components of the magnetic field generated by an axisymmetric conducting fluid flow. The background curvature influence the evolution of the poloidal and toroidal component via the Ricci curvature and non local contributions manifesting themselves as the gradient in the magnitude of the timelike and axial Killing vector fields. Despite the presence of those contributions, we shall show that an axisymmetric magnetic field can not maintained by dynamo action as long as:1) The axially symmetric fluid flow is divergence free.2) The background geometry correspond to a static spherical star of constant density with compactness ratio ε = 2GM/(c2R), in the range ε ∊ [0, 8/9].For an arbitrary static spherical stellar model the conclusion remains intact provided specific integrals involving the eq of state are negative definite.