The General Theory of Relativity A Mathematical Exposition
The General Theory of Relativity: A Mathematical Exposition will serve readers as a modern mathematical introduction to the general theory of relativity. Throughout the book, examples, worked-out problems, and exercises (with hints and solutions) ar
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Black Holes
5.1 Spherically Symmetric Black Holes The phenomenon of a massive body imploding into a black hole is a fascinating topic to investigate. Nowadays, the possible detection of supermassive black holes in the centers of galaxies [96,187] makes these studies extremely relevant. A historical survey and review of current research in black hole physics may be found in [72]. Investigations on black holes historically started from the Schwarzschild metric of (3.9) expressed, with slightly different notation, as 2 2 2m 2 2m 1 .db ' /2 1 db t ; ds 2 D 1 .db r /2 C b r db C sin2 b b r b r n o bWD b D r; b ; b ' ;b t W 0 < 2m < b r; 0 < b < ; < b ' < ; 1 < b t < 1 : (5.1) Clearly, some of the metric tensor components in (5.1) are undefined for b r D 2m; the Schwarzschild radius. However, the corresponding orthonormal components of the curvature tensor in (3.12) are real-analytic for 0 < b r : Therefore, the space–time geometry is completely smooth at b r D 2m; but the Schwarzschild coordinate chart is unable to cover b r D 2m: (Recall that the spherical polar coordinate chart of Example 1.1.2 cannot cover the North or the South pole.) Eddington [82], Painlev´e [203], and Gullstrand [119] extended the Schwarzschild chart on and beyond the Schwarzschild radius. Eddington employed one null coordinate in his construction. (Finkelstein [100] also similarly extended the same with one null coordinate.) Lemaˆıtre [163] used the geodesic normal time coordinate (or comoving coordinates) for the extension whereas Synge [241] devised a doubly null coordinate chart to extend all the previous charts for the maximal extension of the original Schwarzschild chart. A modification of Synge’s chart was constructed independently by Kruskal [153] and Szekeres [245]. It is the commonly used chart now for the maximal extension of the Schwarzschild chart.
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical Exposition, DOI 10.1007/978-1-4614-3658-4 5, © Springer Science+Business Media New York 2012
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5 Black Holes
We shall start our investigations on black holes with the Lemaˆıtre metric of (3.105), with a change of notation, given by i h 2m .dr/2 C ŒY .r; t/2 .d/2 C sin2 .d'/2 .dt/2 ; Y .r; t/ h i2=3 p Y .r; t/ W D .3=2/ 2m .r t/ > 0;
ds 2 D
D W D f.r; ; '; t/ W 0 < r t; 0 < < ; < ' < g :
(5.2)
The above metric and the orthonormal components of the curvature tensor diverge in the limit .r t/ ! 0C : The transformation from a subset of the Lemaˆıtre chart to the Schwarzschild chart is furnished by b r D Y .r; t/; b ;b ' D .; '/; ˇ ˇp ˇ Y .r; t/ C p2m ˇ p p ˇ ˇ b t D t 2 2m Y .r; t/ C .2m/ ln ˇ p p ˇ; ˇ Y .r; t/ 2m ˇ Ds W D f.r; ; '; t/ W .4m=3/ < r t; 0 < < ; < ' < g ; n o bs W D b r; b ; b ' ;b t W 2m < b r; 0 < b < ; < b ' < ; b t2R ; D r @ b r ;b t 2m D > 0: @.r; t/ b r
(5.3)
From the sub-Jacobian above (which has the same value as the Jacobian), it is clear that the mappin
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