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

TL;DR

This paper studies rotating toroids around black holes in general relativity, revealing universal features of dragging effects, their dependence on toroid size and mass ratio, and establishing related inequalities.

Contribution

It introduces a detailed analysis of dragging effects in relativistic toroids, highlighting their proportionality to mass ratio and size dependence, with new inequalities involving angular momentum.

Findings

Dragging effects correlate with the mass ratio M_D/m.

Maxima of dragging effects are proportional to M_D/m.

Isoperimetric inequalities involving angular momentum are validated.

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.