On the black hole limit of rotating discs and rings

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.Comment: 18 pages, 4 figures; Dedicated to Gernot Neugebauer on the occasion of his 70th birthda

Gyromagnetic factor of rotating disks of electrically charged dust in general relativity

“…also used in the study of the uncharged disk [21,22,27,28]. This parameter is related to the redshift Z c of a photon emitted at the centre of the disk and measured at infinity via γ = Z c /(1 + Z c ).…”

“…Based on the algorithm introduced in [27], the authors of [25,26] were able to calculate the solution in terms of a high order post-Newtonian expansion in the parameter γ. In particular, [26] provided strong evidence that, analogous to the uncharged case [28], the limit γ → 1 leads to the extreme Kerr-Newman black hole.…”

On the black hole limit of rotating discs of charged dust

“…Interestingly, it turns out that the power series expansion at γ = 0 could be used to make a statement about the leading order behaviour for the power series expansion at γ = 1. The uncharged case is discussed analytically in [20]. For this purpose we Figure 4: The quantity S Ω in the limit γ → 1 for increasing expansion orders N and for a Páde approximation P [4,16] in √ γ (dashed dotted line) (a), the convergence criterion for increasing n (b) and the coefficient functions |c n | (c).…”

Acceleration of particles by acceleration horizons

“…the constant of integration in (31) is set to zero; it is implied that q > 1. Direct calculations based on coordinate transformations (11), (12) show that for the critical particlẽ…”

“…For instance, the Bertotti-Robinson space-time and its rotational analogue appear in the processes of different limiting transitions in the context of gravitational thermal ensembles [6] and can be relevant in the context of the AdS/CFT correspondence [7], [8] or the Kerr/CFT one [9]. They are also encountered in many other physical contexts connected with nonlinear electrodynamics [10], conformal mechanics [11], limiting transitions from rapidly rotating discs to black holes [12], etc. Acceleration horizons approximately describe an infinite throat of the extremal Kerr, Reissner-Nordström or Kerr-Newman black holes.…”