Magnetic fields of current loops around black holes

Uncertainties of astrophysical models are often tied to magnetic fields. One of the scenarios where magnetic fields are supposed to be crucial is the disc accretion onto a compact object. This problem is nowadays mostly being solved by powerful MHD codes, but there is still some room for analytical work in highly symmetric situations, such as the stationary and axisymmetric one.

I suggest to study, analytically as far as possible, the magnetic fields of current loops around stationary black holes. My master student Zuzana Vlasáková recently checked and compared, in the Schwarzschild and Kerr backgrounds, various formulas which appeared in the literature as the solution of that problem (see e.g. [1]). The formulas might now be employed to obtain a magnetic field of an "accretion disc" modelled as a continuous distribution of circular equatorial current loops. Another possible extension would be to consider a "heavy" (non-test) current loop.

It would in any case be good to compare the above EM problem with its gravitational counterpart, represented by the space-time of a black hole encircled by a rotating ring or disc. Recently we have been studying such a configuration perturbatively and my PhD student Pavel Čížek derived, in a closed form, the linear perturbation of Schwarzschild due to a rotating light finite thin disc [2]. Crucial in this result was writing the Green functions of the "circular" problem in closed forms (using elliptic integrals); it may also serve as a hint for the EM problem. Papers by B. Linet (mainly [3]) should also be helpful.

This is a theoretical project, likely without numerical programming, but the knowledge of Maple or Mathematica is very welcome.

[1] Moss I. G., Phys. Rev. D 83 (2011) 124046

[2] Čížek P., Semerák O., ApJ Suppl. 232 (2017) 14

[3] Linet B., J. Phys. A: Math. Gen. 12 (1979) 839