Cuvatue singulaity We wish to show that thee is a cuvatue singulaity at 0 of the Schwazschild solution. We cannot use eithe of the invaiantsr o R ab R ab since both the Ricci tenso and the Ricci scala vanish. The next siplest invaiant is R abcd R abcd The cuvatues ae given by whee R 1 010 1 2 ν,11e ν λ 1 4 ν,1ν,1 e ν λ 1 4 ν,1λ,1 e ν λ R 1 212 1 2 λ,1e λ R 1 313 2 λ,1e λ sin 2 θ R 2 020 1 2 ν,1e ν λ R 2 323 1 e λ sin 2 θ R 3 030 1 2 ν,1e ν λ Theefoe, e ν e λ ν,1 1 λ,1 R010 1 1 2 ν,11 ν,1 ν,1 e ν λ 1 4 2 4 3 1 4 3 1 4 2 4 R 1 212 R 1 313 sin2 θ R 2 020 3 R 2 323 sin2 θ R 3 030 3 Check R 1 010 1 2 ν,11e ν λ 1 4 ν,1ν,1 e ν λ 1 4 ν,1λ,1 e ν λ 1 2 ν,11 ν,1 ν,1 e ν λ 1
Loweing the uppe index, 1 4 2 4 3 1 4 2 4 1 2 4 3 1 R 1010 3 R 1212 R 1313 1 sin 2 θ R 2020 1 R 2323 sin 2 θ R 3030 sin 2 θ and aising all fou indices, R 1010 3 R 1212 5 1 R 1313 5 sin 2 1 θ R 2020 5 R 2323 7 sin 2 θ R 3030 5 sin 2 θ To copute the invaiant, we will have sus including all eaangeents of the indices of the nonvanishing tes, fo exaple, R 1212 R 1212 R 2112 R 2112 R 1221 R 1221 R 2121 R 2121 So each te ust be counted fou ties. Theefoe R abcd R abcd 4 3 3 4 1 5 4 1 sin 2 θ 5 sin 2 θ 2
4 1 5 8 sin 2 θ 7 sin 2 θ 4 1 sin 2 θ 5 sin 2 θ so siplifying, R abcd R abcd 162 6 162 6 42 6 42 6 which diveges stongly at 0, but is egula at. Regulaity at 42 6 162 6 42 6 It tuns out that the Schwazschild etic has only a coodinate singulaity at. A change of coodinates eoves the singula facto,. Null coodinates, u and v, such that holding eithe u o v constant gives a null geodesic, tun out to not only eove the singulaity, but also give a cleae pictue of the causal elationships nea the sta. Fist, we find the null geodesics in the t-plane. We have aleady shown that and the line eleent then tells us that u 0 k 0 k2 u 0 u 1 u 1 2 so that Taking the quotient, and integating, u 1 ±k d dt u1 u 0 ± ˆ d ±t ˆ d ˆ ˆ d d ±t c 3
Theefoe, defining u t v t we see that u constant gives an ingoing null geodesic and v constant gives an outgoing null geodesic. To wite the line eleent in tes of u and v, copute thei diffeentials, du dt d 1 dt d dv dt d Theefoe, This is still singula at, but now define Then Substituting into the line eleent, we have dudv dt 2 d 2 1 2 ds 2 1 dt 2 d2 1 2 dω 2 1 dudv 2 dω 2 U e u 4 V e v 4 dudv 1 u v dudve 4 162 1 16 2 dudve 16 2 e dudv ds 2 162 e dudv 2 dω 2 whee U, V. Thee is clealy no poble at, which is called the event hoizon. Radial infall We now conside a paticle falling into the black hole. Since the neighbohood of is now established to be egula, thee can be nothing to keep a paticle fo falling acoss the hoizon. We theefoe conside a paticle falling fo just inside the hoizon towad the singulaity. In this egion, becoes the tielike coodinate. The geodesic equation fo u 0 ay be witten as 1 u 0 k 4
with initial value at 0 given by 1 u 0 0 k 0 Since is now the tielike coodinate, we ay choose u 0 0 0, and theefoe, k 0, and u 0 0 along the entie geodesic. Then u 1 is given by Integating, 1 1 1 u 1 τ u 1 Letting y, and theefoe, y 2, ˆ 1 dτ 0 d 0 d 1 Now let y sin θ so that τ 2 τ 1 2 0 cos 2 θdθ 1 cos 2θ dθ 0 θ 12 0 sin 2θ ˆ y2 dy 2 0 2 θ sin θ cos θ 0 0 acsin 1 2 2 π 2 acsin 1 0 We ay take the initial position to be 0, which gives τ π 4 1 0 0 0 1 0 which is, in paticula, finite. The infall doesn t take long fo stella sized black holes. Fo a black hole with 10 ties the ass of the sun, we have τ GMπ 4c 2 6.67 10 11 1.99 10 31 π 4 9 10 16 3690 sec 5
o just ove an hou. Fo a black hole at the cente of a galaxy, which ay have a ass a illion ties as geat, the infall will take just shot of two yeas. The escape of light Finally, conside light which stats nea the event hoizon at t 0. How long does it take to escape fo the egion of the black hole? We conside an outgoing null geodesic, which has v constant, Suppose light leaves 0 δ at tie t 0. Then and the light eaches adius at tie c t c δ δ ct δ δ Copae this tie to the tie it would take in fee space fo light to tavel fo δ to, given by We find t 0 δ c t δ δ t 0 δ δ 1 δ 1 δ δ Conside a black hole with 10 ties the ass of the sun, so that 2GM c 2 2950 The tie to each the obit of Eath 1.5 10 11 fo nea the hoizon would noally be about 500 seconds. This is inceased by t 500 δ δ 500 2950 1.5 1011 1.5 1011 δ 9.83 10 6 1.5 1011 δ Fo a distance of δ 1c above the hoizon, the tie delay is t 9.83 10 6 1.5 10 13 2.98 10 4 sec so a collapsing sta will appea to settle to its Schwazschild adius exteely quickly. 6