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

Black Holes 973221100–1–1–2–200–5–3–300 –200 –10001002003000200–2–6–4–2024650Fig. 2. Isometric embedding of the space-geometry of a (5 + 1)–dimensionalSchwarzschild black hole into six-dimensional Euclidean space, near the throat of theEinstein-Rosen bridge r =(2m) 1/3 ,with2m = 2. The variable along the vertical axisasymptotes to ≈±3.06 as r tends to infinity. The right picture is a zoom to the centre ofthe throat. The corresponding embedding in (3 + 1)–dimensions is known as the Flammparaboloid .solution for all initial values), see Refs. 22 and 51 for some results concerningthis question.In the extended Schwarzschild space-time the set {r =2m} is a nullhypersurface E , the Schwarzschild event horizon. The stationary Killingvector X = ∂ t extends to a Killing vector ˆX in the extended spacetimewhich becomes tangent to and null on E , except at the ”bifurcation sphere”right in the middle of Figure 1, where ˆX vanishes. The global properties ofthe Kruskal–Szekeres extension of the exterior Schwarzschild d spacetime,make this space-time a natural model for a non-rotating black hole.There is a rotating generalisation of the Schwarzschild metric, also discussedin the chapter by R. Price in this volume, namely the two parameterfamily of exterior Kerr metrics, which in Boyer-Lindquist coordinates taked The exterior Schwarzschild space-time (1) admits an infinite number of non-isometricvacuum extensions, even in the class of maximal, analytic, simply connected ones. TheKruskal-Szekeres extension is singled out by the properties that it is maximal, vacuum,analytic, simply connected, with all maximally extended geodesics γ either complete, orwith the curvature scalar R αβγδ R αβγδ diverging along γ in finite affine time.