General-relativistic free decay of magnetic fields in a spherically symmetric body

K.-H. Rädler, H. Fuchs, U. Geppert, M. Rheinhardt, T. Zannias
2001 Physical Review D, Particles and fields  
The decay of a magnetic field penetrating a compact spherical electrically conducting body and continuing in its nonconducting surroundings is systematically studied. The body, considered as a rough model of a compact spherical star, is assumed to be nonrotating and showing no internal motion, and so the metric of the spacetime is static and spherically symmetric. Starting from the absolute space formalism of curved-space electrodynamics the initial value problem for the magnetic field is
more » ... ated. The concept of poloidal and toroidal fields is used to reduce the equations describing this problem to equations for the defining scalars of the magnetic field. By expansion of them in a series of spherical harmonics equations are derived for functions of the radial and time coordinates. A solution of these equations for the outer space is given. For the case of time-independent conductivity of the body, the equations for the interior of the body are reduced to ordinary differential equations which pose eigenvalue problems of the Sturm-Liouville type. After these reductions the solution of the initial value problem for the magnetic field is given as a superposition of magnetic field modes decaying exponentially in time. The shape of the modes is determined by the eigenfunctions of the Sturm-Liouville problems mentioned, and the decay rates by the corresponding eigenvalues. Explicit results, mainly gained by solving the relevant equations numerically, are given for the simple extreme case of constant density of the body. Their most striking feature is that all growth rates decrease with the growing compactness of the body. Furthermore, some concentration of the magnetic field in the inner parts occurs for high compactness. The consequences of our findings for the magnetic-field evolution in neutron stars are discussed as well as the implications for dynamo models.
doi:10.1103/physrevd.64.083008 fatcat:duewomy3qnao3dogzoqcgg7dae