[Paper Review] The rigidly rotating disk of dust and its black hole limit

Reinhard Meinel|ArXiv.org|Mar 26, 1997

Cosmology and Gravitation Theories3 citations

TL;DR

This paper presents the exact global solution for a rigidly rotating disk of dust in general relativity, derived using soliton theory techniques applied to the axisymmetric, stationary vacuum Einstein equations. It demonstrates that such a disk has a maximum mass when angular momentum is fixed, beyond which it transitions to a Kerr black hole, confirming long-standing conjectures by Bardeen and Wagoner.

ABSTRACT

The exact global solution of the Einstein equations [Neugebauer & Meinel, Phys. Rev. Lett. 75 (1995) 3046] describing a rigidly rotating, self-gravitating disk is discussed. The underlying matter model is a perfect fluid in the limit of vanishing pressure. The solution represents the general-relativistic analogue of the classical Maclaurin disk. It was derived by applying solution techniques from soliton theory to the axisymmetric, stationary vacuum Einstein equations. In contrast to the Newtonian solution, there exists an upper limit for the total mass of the disk - if the angular momentum is fixed. At this limit, a transition to a rotating black hole, i.e., to the Kerr solution occurs. Another limiting procedure leads to an interesting cosmological solution. These results prove conjectures formulated by Bardeen and Wagoner more than twenty-five years ago.

Motivation & Objective

• To derive an exact global solution for a self-gravitating, rigidly rotating disk of dust in general relativity.
• To investigate the physical limits of such a disk, particularly its maximum mass under fixed angular momentum.
• To explore the transition from the disk solution to a rotating black hole, i.e., the Kerr solution.
• To confirm theoretical conjectures by Bardeen and Wagoner regarding the existence of a black hole limit for rotating disks.

Proposed method

• The solution is constructed by applying inverse scattering methods from soliton theory to the axisymmetric, stationary vacuum Einstein equations.
• The matter model is a perfect fluid in the limit of vanishing pressure, representing a dust disk.
• The solution is derived as a nonlinear superposition of soliton solutions, ensuring exactness and global validity.
• The method allows for the inclusion of rotation and self-gravity in a fully relativistic framework.
• The analysis includes the asymptotic behavior and boundary conditions at the axis and at spatial infinity.
• The transition to the Kerr black hole is studied through a limiting procedure as the disk's mass approaches its upper bound.

Experimental results

Research questions

• RQ1 What is the maximum mass a rigidly rotating dust disk can achieve for a fixed angular momentum in general relativity?
• RQ2 Does the solution for a rotating dust disk naturally evolve into a Kerr black hole in the limit of maximal mass?
• RQ3 How does the relativistic disk solution differ from the Newtonian Maclaurin disk in terms of mass and rotational limits?
• RQ4 What is the role of soliton theory in constructing exact solutions for stationary, axisymmetric spacetimes with matter?
• RQ5 Can the conjectures by Bardeen and Wagoner regarding the black hole limit of rotating disks be rigorously confirmed?

Key findings

• The rigidly rotating disk of dust has a finite upper mass limit when angular momentum is held constant, unlike its Newtonian counterpart.
• At this mass limit, the solution smoothly transitions to the Kerr black hole solution, confirming the physical continuity of the spacetime geometry.
• The solution is exact and global, derived via inverse scattering methods applied to the Einstein equations.
• The transition to the Kerr solution occurs through a limiting procedure that preserves the physical and geometric consistency of the spacetime.
• The results provide rigorous confirmation of conjectures by Bardeen and Wagoner made over 25 years prior.
• An alternative limiting procedure yields a cosmological solution, indicating the solution's broader applicability beyond black hole physics.