On the black hole limit of rotating discs and rings

TL;DR

This paper investigates how rotating fluid bodies, like discs and rings, approach the black hole limit in Einstein's theory, revealing a universal behavior of their multipole moments as they near extremal Kerr black holes.

Contribution

It provides a detailed analysis of the black hole limit for rotating discs and rings using series expansions and numerical methods, highlighting universal multipole moment behavior.

Findings

Multipole moments approach extremal Kerr values near the black hole limit.

Universal behavior of multipole moments observed across different solutions.

Series expansion and numerical results support the black hole transition scenario.

Abstract

Solutions to Einstein's field equations describing rotating fluid bodies in equilibrium permit parametric (i.e. quasi-stationary) transitions to the extreme Kerr solution (outside the horizon). This has been shown analytically for discs of dust and numerically for ring solutions with various equations of state. From the exterior point of view, this transition can be interpreted as a (quasi) black hole limit. All gravitational multipole moments assume precisely the values of an extremal Kerr black hole in the limit. In the present paper, the way in which the black hole limit is approached is investigated in more detail by means of a parametric Taylor series expansion of the exact solution describing a rigidly rotating disc of dust. Combined with numerical calculations for ring solutions our results indicate an interesting universal behaviour of the multipole moments near the black hole limit.