100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

100 P. T. ChruścielFig. 3. A coordinate representation 81 of the outer ergosphere r =˚r + ,theeventhorizonr = r + , the Cauchy horizon r = r − , and the inner ergosphere r =˚r − with the singularring in Kerr space-time. Computer graphics by Kayll Lake. 66symmetry cos θ = ±1. Note that ∂˚r ± /∂θ ≠ 0 at those axes, so the ergospherehas a cusp there. The region bounded by the outermost horizonr = r + and the outermost ergosphere r =˚r + is called the ergoregion, withX spacelike in its interior. We refer the reader to Refs. 15 and 79 for anexhaustive analysis of the geometry of the Kerr space-time.Fig. 4. Isometric embedding in Euclidean three space of the ergosphere (the outer hull),and part of the event horizon, for a rapidly rotating Kerr solution. The hole arises due tothe fact that there is no global isometric embedding possible for the event horizon whena/m > √ 3/2. 81 Somewhat surprisingly, the embedding fails to represent accurately thefact that the cusps at the rotation axis are pointing inwards, and not outwards. Computergraphics by Kayll Lake. 66The hypersurfaces {r = r ± } provide examples of null acausal boundaries.Causality theory shows that such hypersurfaces are threaded by a