Gravitational convergence, shear deformation and rotation of magnetic forcelines

We consider the "kinematics" of spacelike congruences and apply them to a family of self-gravitating magnetic forcelines. Our aim is to investigate the convergence and the possible focusing of these lines, as well as their rotation and shear deformation. In so going, we introduce a covariant 1+2 splitting of the 3-dimensional space, parallel and orthogonal to the direction of the field lines. The convergence, or the divergence, of the latter is monitored by the associated Raychaudhuri equation, which is obtained by propagating the spatial divergence of a unitary spacelike vector along its own direction. Applied to a magnetic vector, the resulting expression shows that, although the convergence of the magnetic forcelines is affected by the gravitational pull of all the other sources, it is unaffected by the field's own gravity, irrespective of how strong the latter is. This rather counterintuitive result is entirely due to the magnetic tension, namely to the negative pressure the field exerts parallel to its lines of force. In particular, the magnetic tension always cancels out the field's energy-density input to the Raychaudhuri equation, leaving the latter free of any direct magnetic-energy contribution. Similarly, the rotation and the shear deformation of the aforementioned forcelines are also unaffected by the magnetic input to the total gravitational energy. In a sense, the magnetic lines do not seem to "feel" their own gravity no matter how strong the latter may be.
Comment: Typos corrected. Published version