The Gravitational Field of a Disk

Abstract

The general solution of the static field equations of general relativity is given for a disk of "counterrotating" dust particles. The only nonvanishing components of the energy-momentum tensor are Jo 0 and T x x , which are assumed to have 5-function singularities on the disk. Two representative families of solutions are considered, and it is shown that, for these solutions, physical considerations severely limit the strength of the gravitational potentials. The first family has surface density proportional to some power of 1 -p 2 . The requirement that the velocity of the dust particles should not exceed c places a bound on the gravitational red-shift of z-1.5803 for these models. The second family is that of the uniformly rotating disks defined by v 2 =p 2 co 2 e~i+. Bardeen has pointed out that these disks can have arbitrarily large red-shifts without violating the velocity condition. However, it is shown that their red-shift cannot exceed 1.9015 before their binding energy becomes negative. This work suggests that the largest gravitational red-shift to which counterrotating dust disks can give rise is of order of magnitude 1.

Morgan-Morgan-NUT disk space via the Ehlers transformation

“…in which U and k are functions of ρ and z. Morgan and Morgan [2,11] found the following solution for a finite disk (of mass M and radius a) without radial pressure in terms of the oblate ellipsoidal coordinates (ξ, η);…”

“…Exact thin disk solutions of Einstein field equations are discussed extensively in different contexts in the literature. The above static solution, first derived by Morgan and Morgan [2], is considered to represent the spacetime of a finite disk of counterrotating particles. Properties of static counterrotating disks have been studied in [17,18].…”

Relativistic static thin disks: The counterrotating model

“…These were first studied by Bonnor and Sackfield [1], obtaining pressureless static disks, and by Morgan and Morgan, obtaining static disks with and without radial pressure [2,3]. In connection with gravitational collapse, disks were first studied by Chamorro, Gregory and Stewart [4].…”

Note on the effect of a massive accretion disk in the measurements of black hole spins

“…A realistic thin disk composed of ordinary matter was obtained by Lemos and Letelier in [10] by making an inversion of a thin disk of the Morgan-Morgan family [11]. The disk has an inner edge at b and the metric function λ D solution of Eq.…”