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phys documents (dragged) 22 - 16 Physics Formulary by ir...

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16 Physics Formulary by ir. J.C.A. Wevers r > 2 m : u = r 2 m - 1 exp r 4 m cosh t 4 m v = r 2 m - 1 exp r 4 m sinh t 4 m r < 2 m : u = 1 - r 2 m exp r 4 m sinh t 4 m v = 1 - r 2 m exp r 4 m cosh t 4 m r = 2 m : here, the Kruskal coordinates are singular, which is necessary to eliminate the coordinate singularity there. The line element in these coordinates is given by: ds 2 = - 32 m 3 r e - r/ 2 m ( dv 2 - du 2 ) + r 2 d Ω 2 The line r = 2 m corresponds to u = v = 0 , the limit x 0 → ∞ with u = v and x 0 → -∞ with u = - v . The Kruskal coordinates are only singular on the hyperbole v 2 - u 2 = 1 , this corresponds with r = 0 . On the line dv = ± du holds d θ = d ϕ = ds = 0 . For the metric outside a rotating, charged spherical mass the Newman metric applies: ds 2 = 1 - 2 mr - e 2 r 2 + a 2 cos 2 θ c 2 dt 2 - r 2 + a 2 cos 2 θ r 2 - 2 mr + a 2 - e 2 dr 2 - ( r 2 + a 2 cos 2 θ ) d θ 2 - r 2 + a 2 + (2 mr - e 2 ) a 2 sin 2 θ r 2 + a 2 cos 2 θ sin 2 θ d ϕ 2 + 2 a (2 mr - e 2 ) r 2 + a 2 cos 2 θ sin 2 θ ( d ϕ
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