Relativistic Disks. I. Background Models

Abstract

Relativistic kinetic theory i s used in conjunction with the theory of relativistic surface layers in order to study relativistic disks of matter. After a brief general discussion, attention i s restricted to the case of counter-rotating disks. The general surface stress-energy tensors of such disks a r e exhibited and a distribution function which generates these s t r e s senergy tensors i s deduced. This i s followed by a discussion of stability, and a criteria for the stability of particle orbits i s derived. Finally, the question of central red shift i s considered. It is shown that all counter-rotating disks without singularities a t the r i m will have a finite central red shift, but the question of the existence of a maximum central red shift remains open.

Vertical stability of circular orbits in relativistic razor-thin disks

“…The results of [11,12] point in the direction of the proof presented in this paper, whereas the results of [4,5,34,47] are mathematically inconsistent and the conclusions derived from them must be reformulated in light of the present approach. Since the stability criterion obtained and the adiabatic invariance analysis are both local, we can extend our formalism to black holes surrounded by razor-thin disks or rings [4,15,30,31,34], provided the metric outside the horizon has the form (5) and the orbit does not cross the horizon.…”

“…Such a statement, or the more general condition (14) or (27), formalizes, in the context of the theory of distributionvalued stress-energy tensors [23,24], the previous results [11,12] about the vertical stability of circular geodesics in the presence of a razor-thin disk. When restricted to some cases of interest, conditions (28) adopt a quite simplified form.…”

“…Exact solutions for the Einstein equations representing thin disks without radial pressure were first proposed by Morgan and Morgan [8,9]. The first applications of the theory of distributions to curved spacetimes representing razor-thin disks of self-gravitating matter were proposed in [10,11], together with an initial study of the stability of circular geodesics [11]. The stability of circular geodesics in smooth axially symmetric stress-energy distributions was first considered in [12], in which some remarks about their vertical stability in the limit of an infinitesimally thin disk are presented.…”

“…The presence of a δ-like singularity on the plane of the thin disk [14,15,19,[22][23][24] prevents the use of the geodesic deviation equation and the corresponding first-order perturbation approach, as first pointed out in [34]. The existing results about vertical stability of circular geodesics in relativistic thin disks consider only particular cases (dust disk, [11]) or are obtained as a limiting case of smooth stress-energy tensors [12,34]. Although some authors modeled the effects on the geodesic equations due to this δ-like singularity (e.g.…”

Conformastatic disk-haloes in Einstein-Maxwell gravity

“…A coordinate system that adapts naturally to a finite source and presents the required discontinuous behavior is given by the oblate spheroidal coordinates. Some examples of finite thin disks obtained from vacuum solutions expressed in these coordinates can be found in references [1,2,4,7], and from electrovacuum solutions in reference [38].…”