Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

Abstract

An explicit global and unique isometric embedding into hyperbolic 3-space, H 3 , of an axisymmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H 3 of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the U (1) isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into H 3 , for arbitrary values of the angular momentum. For this example, considering a quotient of H 3 by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding which cannot be made global.

Strong-field tidal distortions of rotating black holes. II. Horizon dynamics from eccentric and inclined orbits

“…As mentioned previously, work in progress indicates that embedding the horizon in the globally hyperbolic space H 3 , following Ref. [32], is an elegant way to get around this restriction.…”

“…As such, we confine our embedding visualizations in this paper to the range 0 ≤ a/M < √ 3/2. Work in progress indicates that an elegant way to lift this restriction will to be embed the horizon's distorted geometry in the globally hyperbolic space H 3 [32].…”

“…We will use σ +,× in much of our discussion of results, especially in Sec. V. With the tetrad and gauge that we use, the perturbed shear is governed by the equation [25] 32) where the derivative operator D ≡ l µ ∂ µ . Let us expand σ as we expanded Ψ 0 :…”

Shape of rotating black holes

“…If L ≤ 5l 12 there is nothing to prove. Assume then that L > 5l 12 and assume without loss of generality that the middle point between l 1 and l 2 (that is…”

“…To get a better graphical understanding one could isometrically embed them into Euclidean space. This can be done for small values of J , obtaining then nice oblate spheroidal shapes [12], but there is a maximum value of J (less than M) after which isometric embeddings into Euclidean space are no more available [3] . A detailed discussion of these issues is presented in [12] including an analysis of isometric embeddings of horizons into the hyperbolic space.…”