Dirty rotating black holes: Regularity conditions on stationary horizons

Abstract

We consider generic, or ''dirty'' (surrounded by matter), stationary rotating black holes with axial symmetry. The restrictions are found on the asymptotic form of metric in the vicinity of nonextremal, extremal and ultraextremal horizons, imposed by the conditions of regularity of increasing strength: boundedness on the horizon of the Ricci scalar, of scalar quadratic curvature invariants, and of the components of the curvature tensor in the tetrad attached to a falling observer. We show, in particular, that boundedness of the Ricci scalar implies the ''rigidity'' of the horizon's rotation in all cases, while the finiteness of quadratic invariants leads to the constancy of the surface gravity. We discuss the role of quasiglobal coordinate r that is emphasized by the conditions of regularity. Further restrictions on the metric are formulated in terms of subsequent coefficients of expansion of metric functions by r. The boundedness of the tetrad components of curvature tensor for an observer crossing the horizon is shown to lead in the horizon limit to diagonalization of the Einstein tensor in the frame of a zero angular momentum observer on a circular orbit (ZAMO frame) for horizons of all degrees of extremality.

Bañados-Silk-West effect with nongeodesic particles: Extremal horizons

“…Then, using that E, L, and u z are bounded (15), while u r and X are O(N), and ω H = const (which follows from regularity [13]), it is easy to see that all the terms in (74) and (75) are automatically finite, so…”

“…IV B. 5 of [13], for which ∂ r ω H = ∂ 2 r ω H = 0, and thus s can be equal to 2 or 3. Correspondingly, the BSW-2 effect can be realized near such horizons.…”

Ultra-high energy collisions of nonequatorial geodesic particles near dirty black holes

“…Near the horizon, ∆ ∼ N 2 , so r is the analog of the quasiglobal coordinate used in the spherically symmetric case [11]. It is worth noting that the form of the metric somewhat differs from that in the Gauss normal coordinates used in [12], [13], [14]. It is more convenient for our purposes and, in particular, facilitates the comparison to the Kerr metric.…”

Unbounded energies of debris from head-on particle collisions near black holes

“…Say, for the Kerr metric, expansion goes in powers of r − r + , where r is the Boyer-Lindquist coordinate and r + is the horizon radius. For extremal black holes, N ∼ r − r + , so expansion in terms of r − r + is equivalent to the expansion in terms of N. For the nonextremal black holes, N 2 ∼ r − r + , so expansion would start from the terms N 2 (see [18] for details). In particular, for the extremal Kerr metric,…”

“…where ω H is the horizon value of ω and B i is some model-dependent coefficient [18]. Say, for the Kerr metric, expansion goes in powers of r − r + , where r is the Boyer-Lindquist coordinate and r + is the horizon radius.…”