Rotating systems, universal features in dragging and anti-dragging effects, and bounds onto angular momentum

Abstract

We consider stationary, axially symmetric toroids rotating around spinless black holes, assuming the general-relativistic Keplerian rotation law, in the first post-Newtonian approximation. Numerical investigation shows that the angular momentum accumulates almost exclusively within toroids. It appears that various types of dragging (anti-dragging) effects are positively correlated with the ratio $M_\mathrm{D}/m$ ($M_\mathrm{D}$ is the mass of a toroid and $m$ is the mass of the black hole) - moreover, their maxima are proportional to $M_\mathrm{D}/m$. The horizontal sizes of investigated toroids range from c. 50 to c. 450 of Schwarzschild radii $R_\mathrm{S}$ of the central black hole; their mass $M_\mathrm{D} \in (10^{-4}m, 40m)$ and the radial size of the system is c. 500 $R_\mathrm{S}$. We found that the relative strength of various dragging (anti-dragging) effects does not change with the mass ratio, but it depends on the size of toroids. Several isoperimetric inequalities involving angular momentum are shown to hold true.