Causal relation between regions I and IV of the Kruskal extension

Transcription

1 Causal elation between egions I and IV of the Kuskal extension Yi-Ping Qin 1,2,3 1 Yunnan Obsevatoy, Chinese Academy of Sciences, Kunming, Yunnan , P. R. China 2 National Astonomical Obsevatoies, Chinese Academy of Sciences 3 Chinese Academy of Science-Peking Univesity joint Beijing Astophysical Cente axiv:g-qc/ v1 27 Ap 2000 Summay By extending the exteio Schwazschild spacetime in two opposite diections with the Kuskal method, we get an extension which has the same T X spacetime diagam as has the conventional Kuskal extension, while allowing its egions I and IV to coespond to diffeent diections of the oiginal spacetime. We futhe extend the exteio Schwazschild spacetime in all diections and get a 4-dimensional fom of the Kuskal extension. The new fom of extension includes the conventional one as a pat of itself. Fom the point of view of the 4-dimensional fom, egion IV of the conventional extension does not belong to anothe univese but is a potion of the same exteio Schwazschild spacetime that contains egion I. The two egions ae causally elated: paticles can move fom one to the othe. It is known that the conventional Kuskal extension contains fou pats of spacetime, a black hole, a white hole, and two egions of the exteio Schwazschild spacetime [1]. The two egions wee believed to belong to diffeent univeses, and messages can not be exchanged between them. In the following we show that this intepetation is not coect. The standad metic fom of the exteio Schwazschild spacetime is [2] ds 2 = (1 2M 1 )dt M d (dθ 2 + sin 2 θdφ 2 ) ( > 2M). (1) Let t = t x = sin θ cosφ y = sin θ sin φ z = cosθ ( > 2M). (2) 1

2 Then (1) can be witten as 2My 2 ds 2 = (1 2M )dt2 2Mx + [ 2 2 ( 2M) + 1]dx2 + [ 2 ( 2M) + 1]dy2 2Mz + [ 2 2 ( 2M) + 1]dz2 (3) + 2 ( 2M) (xydxdy + yzdydz + zxdzdx) ( > 2M), = x 2 + y 2 + z 2. (4) Now we conside the extension of the exteio Schwazschild spacetime in two opposite diections. We notice fom the tansfomation (2) that when θ = π/2 and φ = 0, π then y = z = 0 and x = ± ( > 2M), (5) x = + coesponds to φ = 0 and x = coesponds to φ = π. That is, while x > 0 coesponds to one diection, x < 0 coesponds to the opposite one. In this situation, fom (3) and (4) one has ds 2 = (1 2M x x )dt2 + x 2M dx2 (t, x, y, z) x + x, (6) x + {(t, x, y, z) > t >, > x > 2M, y = 0, z = 0} is a potion of the exteio Schwazschild spacetime in one diection and x {(t, x, y, z) > t >, 2M > x >, y = 0, z = 0} is anothe potion of the spacetime in the opposite diection. It can be veified that the tansfomation, T = ( x 2M 1)1/2 e x / sh t X = ( x 2M 1)1/2 e x / ch t y = y z = z (t, x, y, z) x + (7) and will ewite (6) in the fom T = ( x 2M 1)1/2 e x / sh t X = ( x 2M 1)1/2 e x / ch t y = y z = z (t, x, y, z) x, (8) T and X satisfy ds 2 = 32M3 e x /2M ( dt 2 + dx 2 ) (T, X, y, z) x X + X, (9) X 2 T 2 = ( x 2M 1)e x /2M ( x > 2M), (10) 2

3 and X + {(T, X, y, z) > T >, > X > 0, y = 0, z = 0; X 2 T 2 > 0}, X {(T, X, y, z) > T >, 0 > X >, y = 0, z = 0; X 2 T 2 > 0}. The above tansfomation, denoted f x, is a 1-1 map,with f x ( x + ) = X +, f x ( x ) = X and f x ( x + x ) = X + X. Note that, if elation (7) is applied in the tansfomation fo both x + and x, then the map will no longe be 1-1 (even though the metic fom of (9) is maintained). Keeping the metic fom of (9) but allowing X 2 T 2 > 1 instead of X 2 T 2 > 0 (these coespond to x > 0 and x > 2M espectively), we have an extension of the pat, x + x, of an exteio Schwazschild spacetime, confined in two opposite diections, in a manifold X, X {(T, X, y, z) > T >, > X >, y = 0, z = 0} which is an R 2 space. The new extension is 2-dimensional and has the same T X spacetime diagam as has the conventional Kuskal extension. (Note that, hee x = accoding to (4) and (5).) Howeve, fo the new extension, while egion I of the diagam (i.e. X+ ) coesponds to a potion of the exteio Schwazschild spacetime in one diection, x +, egion IV (i.e. X ) coesponds to none othe than a potion in the opposite diection, x. This esult led us futhe to conside the extension of the exteio Schwazschild spacetime in an equatoial plane. In the spacetime θ = π 2, (11) (1) becomes ds 2 = (1 2M 1 )dt M d dφ 2 ( > 2M, θ = π ). (12) 2 The following tansfomation T = ( 2M 1)1/2 e / sh t R = ( 2M 1)1/2 e / ch t θ = θ φ = φ (t,, θ, φ) π/2, (13) π/2 {(t,, θ, φ) > t >, > > 2M, θ = π 2, 2π φ 0}, ewites (12) in the fom ds 2 = 32M3 e /2M ( dt 2 + dr 2 ) + 2 dφ 2 (T, R, θ, φ) R π/2, (14) T, R and ae elated by R 2 T 2 = ( 2M 1)e/2M ( > 2M), (15) 3

4 and R π/2 {(T, R, θ, φ) > T >, > R > 0, θ = π 2, 2π φ 0; R2 T 2 > 0}. The above tansfomation, denoted f, is a 1-1 map, with f ( π/2 ) = R π/2, π/2 is an R 3 space with an infinite length hollow column and R π/2 is an R 3 with two head-to-head opposite hollow cones extending to infinity. In the same way, the extension of R π/2 is ealized by keeping the metic fom of (14) but allowing R 2 T 2 > 1 instead of R 2 T 2 > 0 (they coespond to > 0 and > 2M espectively). In this extension we find that a black hole is inside one of the two cones and a white hole is inside the othe cone. This extension is 3-dimensional and can be ealized by otating the conventional Kuskal extension aound its T axis. With the same method, the whole exteio Schwazschild spacetime (in all diections) can be extended by making the following tansfomation T = ( 2M 1)1/2 e / sh t R = ( 2M 1)1/2 e / ch t θ = θ φ = φ (t,, θ, φ), (16) {(t,, θ, φ) > t >, > > 2M, π θ 0, 2π φ 0}, which ewites (1) in the fom ds 2 = 32M3 e /2M ( dt 2 + dr 2 ) + 2 (dθ 2 + sin 2 θdφ 2 ) (T, R, θ, φ) R, (17) R 2 T 2 = ( 2M 1)e/2M ( > 2M), (18) and R {(T, R, θ, φ) > T >, > R > 0, π θ 0, 2π φ 0; R 2 T 2 > 0}. Then, keeping the metic fom of (17), we extend the ange of R 2 T 2 > 0 to R 2 T 2 > 1 (they coespond to > 2M and > 0 espectively). This yields ds 2 = 32M3 e /2M ( dt 2 + dr 2 ) + 2 (dθ 2 + sin 2 θdφ 2 ) (T, R, θ, φ) K, (19) R 2 T 2 = ( 2M 1)e/2M ( > 0), (20) and K {(T, R, θ, φ) > T >, > R > 0, π θ 0, 2π φ 0; R 2 T 2 > 1}. This extension is 4-dimensional, and contains the extension of the equatoial plane as a 3- dimensional section, and the conventional extension as a 2-dimensional section. Fom the point of view of the 4-dimensional extension, egion IV of the conventional extension does not belong 4

5 to a diffeent univese but is a potion of the same exteio Schwazschild spacetime that contains egion I. Paticles can move fom one to the othe. Fo example, the wold line of a paticle in a cicula obit outside the Schwazschild black hole (with = constant > 2M ) can be a line aound the T axis and towads the diection of T fom egion I ( θ = π 2, φ = 0) though the egions of (θ = π 2, 0 < φ < π) to egion IV ( θ = π 2, φ = π), shown in the spacetime diagam of the 3-dimensional head-to-head double-cone extension. Acknowledgements It is my pleasue to thank D. Tao Kiang fo helpful discussion. Thanks ae also given to Pofessos G. Z. Xie, Xue-Tang Zheng and Shi-Min Wu fo thei help. This wok was suppoted by the United Laboatoy of Optical Astonomy, CAS, the Natual Science Foundation of China, and the Natual Science Foundation of Yunnan. Refeences 1. M. D. Kuskal, Phys. Rev. 119, 1743 (1960). 2. R. M. Wald, Geneal Relativity (The Univesity of Chicago Pess, Chicago, 1984). 5