Battery and Dynamo theory in the Kerr metric

Abstract

The generation and evolution of magnetic fields in the plasma accreting into a rotating black hole is studied in the 3+1 split of the Kerr metric. Attention is focused on effects of the gravitomagnetic potential. The gravitomagnetic force appears as battery term in the generalized Ohm's law. The gravitomagnetic battery is likely to saturate at higher field strength than the classical Biermann battery. The coupling of the gravitomagnetic potential with electric fields appears as gravitomagnetic current in Maxwell's equations. In the magnetohydrodynamic induction equation, this current re-appears as source term for the poloidal magnetic field, which can produce closed magnetic structures around an accreting black hole. In principle, even self-excited axisymmetric dynamo action is possible, which means that Cowling's anti-dynamo theorem does not hold in the Kerr metric. Finally, simulations of the alpha-Omega dynamo in accretion flows into the hole are presented. I assume a simple expression of alpha in this relativistic context. The modes of the dynamo are oscillating for dynamo numbers which are typical for accretion disks. In a zero angular momentum flow into a Kerr black hole there is still shear, i.e. the angular velocity, Omega, of the plasma equals the angular velocity of space, omega, and it has been speculated (Meier 1998) that even then a dynamo could operate. This is shown to be unlikely due to the rapid accretion.