How the magnetic field behaves during the motion of a highly conducting fluid under its own gravity--A new theoretical, relativistic approach

Abstract

Within the context of general relativity we study in a fully covariant way the so-called Euler-Maxwell system of equations. In particular, on decomposing the aforementioned system into its 1 temporal and 1 + 2 spatial components at the ideal magnetohydrodynamic limit, we bring it in a simplified form that favors physical insight to the problem of a self-gravitating, magnetized fluid. Of central interest is the decomposition of Faraday's equation which leads to a new general solution governing the evolution of the magnetic field during the motion of the highly conducting fluid. According to the latter relation, the magnetic field generally grows or decays in proportion to the inverse cube law of the scale factor--associated with the continuous contraction or expansion of the fluid's volume respectively. The magnetic field's law of variation, which has remarkable implications for the motion of the whole fluid, is subsequently applied to homogeneous (anisotropic-magnetized) cosmological models--especially to the Bianchi I case--as well as to the study of homogeneous and anisotropic gravitational collapse in a magnetized environment. Concerning the cosmological application, we derive the evolution equations of Bianchi I spacetime permeated by large-scale magnetic fields (these equations reduce to their FRW counterparts at the small/large--scale limit). Also, the compatibility of the new evolution formula for the magnetic field with the standard cosmic nucleosynthesis constraint is examined. As for the application in astrophysics, our results predict that homogeneous gravitational implosion is impeded when the electric Weyl tensor (associated with tidal forces) along the magnetic forcelines overwhelms the magnetic energy density. Lastly, our model denotes that the satisfaction of the aforementioned criterion is ultimately driven into a problem of initial conditions.