By Schmidt B.G. (ed.)
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42 4 Jiˇr´ı Biˇca ´k The Kerr Metric The discovery of the Kerr metric in 1963 and the proof of its unique role in the physics of black holes have made an immense impact on the development of general relativity and astrophysics. g. [18,30]): ds2 = − 1 − 2M r Σ dt2 − 2 2aM r sin2 θ dt dϕ + Σ A Σ 2 dr + Σdθ2 + sin2 θ dϕ2 , ∆ Σ + (20) where Σ = r2 + a2 cos2 θ, ∆ = r2 − 2M r + a2 , A = Σ(r2 + a2 ) + 2M ra2 sin2 θ, (21) where M and a are constants. 1 Basic Features The Boyer–Lindquist coordinates follow naturally from the symmetries of the Kerr spacetime.
1) and assume that the stellar material is transparent. As the Sun has a focusing eﬀect on the light rays, so does matter during collapse. As the matter density becomes higher and higher, the focusing eﬀect increases. e. the null vector k α = dxα /dw, w being α an aﬃne parameter, tangent to null geodesics, satisﬁes k;α = 0. The wavefront then “stays” at the hypersurface r = 2M in metric (4), and the area of its 2-dimensional cross-section remains constant. The null hypersurface representing the history of this critical wavefront is the (future) event horizon.
In the Schwarzschild spacetime there exists the timelike Killing vector, ∂/∂t, which when analytically extended into all Schwarzschild–Kruskal manifold, becomes null at the event horizon r = 2M , and is spacelike in the regions II and IV with r < 2M . In Kruskal coordinates it is given by k α = k V = U/4M, k U = V /4M, k θ = 0, k ϕ = 0 . (7) Hence it vanishes at all points with U = V = 0, θ ∈ [0, π], ϕ ∈ [0, 2π). These points, forming a spacelike 2-sphere which we denote B (in Schwarzschild coordinates given by r = 2M, t = constant), are ﬁxed points of the 1-dimensional group G of isometries generated by k α (see Fig.
General relativity. J. Ehlers honorary volume by Schmidt B.G. (ed.)