General-relativistic free decay of magnetic fields in a spherically symmetric body - art. no. 083008

The decay of a magnetic field penetrating a compact spherical electrically conducting body and continuing in its nonconducting surroundings is systema tically studied. The body, considered as a rough model of a compact spheric al star, is assumed to be nonrotating and showing no internal motion, and s o the metric of the spacetime is static and spherically symmetric. Starting from the absolute space formalism of curved-space electrodynamics the init ial value problem for the magnetic field is formulated. The concept of polo idal and toroidal fields is used to reduce the equations describing this pr oblem to equations for the defining scalars of the magnetic field. By expan sion of them in a series of spherical harmonics equations are derived for f unctions 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 ord inary differential equations which pose eigenvalue problems of the Sturm-Li ouville type. After these reductions the solution of the initial value prob lem for the magnetic field is given as a superposition of magnetic field mo des decaying exponentially in time. The shape of the modes is determined by the eigenfunctions of the Sturm-Liouville problems mentioned, and the deca y rates by the corresponding eigenvalues. Explicit results, mainly gained b y solving the relevant equations numerically, are given for the simple extr eme case of constant density of the body. Their most striking feature is th at all growth rates decrease with the growing compactness of the body. Furt hermore, some concentration of the magnetic field in the inner parts occurs for high compactness. The consequences of our findings for the magnetic-fi eld evolution in neutron stars are discussed as well as the implications fo r dynamo models.