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1 % 1 4 1) 1 PSRB ) 3) X g 1990 g 4) g g 2 g ) 10 1 Page 1
2 5) % 1 g g ) 3 S S S n m (1/a, b k /a) a b k S n m, a, b k S b k 2 4 adt a S(t dt) 4 mn S g mn mn n m n n (1) g mn n n 0 1 g mn 6 10 mn a, b k, g ij 10 n m, g m n G mn kt mn 3 G mn n m n n kt mn n m n n, (2) G mn n m g n k kt mn n m g n k, (3) G mn g m j g n k kt mn g m j g n k. (4) G mn T mn G c k 8pGc 4 (2) (4) (2), (3) g ij 1 2 g ij (4) g ij m G mn 0 m mn 3 g ij K 6) ij Vol. 61, No. 5, 2006 Page 2
3 4. 2 0) 1) 3 2) 3) 1) 3) (2), (3) ) F i d jk ( k g ij, G d ij g ij, K K ij g ij (5) 5 d jk ( k g ij d ij g ij F i G 3 Page 3
4 g ij G G d ij g ij (2), (3) 5 9 (T. W. Baumgarte), (M. Alcubierre), (B. Brügmann), 12) BSSN (Baumgarte Shapiro Shibata Nakamura) S S S S (W. Unruh) 9) 10) m 2Gm/c Gm/c 2 5Gm/c 2 M 50 70GM/c 2 M 2m 1 600, , Vol. 61, No. 5, 2006 Page 4
5 1 1 FACOM-VPP ) 1) 3 2) 2 g 14) 15) 16) 35 km ) LIGO 3) 300 Mpc ) 4 1.5M (log a) (ms) 1.84 ms 2 (E. Gourgoulhon), 17) 35 km 1 4, 5 15) ln a; 4 1.5M M 5 1.3M (Pandharipande) 16) 2.18M 1.5M ms (t 2.04 ms) % 5 Page 5
6 6 50 Mpc M 2.06 ms (0.0005M ) 2.9M, a M J cj/gm M Hypermassive neutron star; HMNS 1.5M 2.6M 2.18M HMNS HMNS T/W ) 4ms ms 4, 5 2 A t f d A exp( t/t) sin(2pft d) 5 t 2.35 ms f 6.5 khz, t 0.18 ms M 2.9M, a ) HMNS 6 t 3 ms khz t 10 ms 10 ms HMNS HMNS 50 ms ms 6 Vol. 61, No. 5, 2006 Page 6
7 7 (HMNS) x z HMNS P c P c 0.2 ms 2 3kHz kHz D h e# Mpc (6) D LIGO HMNS 20) 6.3 HMNS 1) HMNS 2) HMNS 50 ms 21) 7 HMNS HMNS x z ms 7 2 HMNS3 7ms 3, 4 1,000 K (n)(n ) nn g HMNS g 22) Page 7
8 (F. Pretorius) 23) 8 24, 25) ) M 0 GM 0/c 3 Y 4 2 t/(gm 0/c 3 ) ) 7. 1) 2) 3) Vol. 61, No. 5, 2006 Page 8
9 1) I. H. Stairs: Science 304 (2004) ) J. H. Taylor: Phil. Trans. R. Soc. London 341 (1992) ) LIGO, TAMA300, GEO600, VIRGO ) 60 (2005) ) 53 (1998) ) J. W. York, Jr.: in Sources of Gravitational Radiation, ed. L. Smarr (Cambridge Univ. Press, 1979) p ) G. B. Cook: Living Reviews in Relativity 3 (2000) 5, livingreviews.org/articles/. 8) J. A. Font: Living Review Relativity 3 (2000) 2, livingreviews.org/articles/. 9) S. W. Hawking and G. F. R. Ellis: Large Scale Structure of Spacetimes (Cambridge Univ. Press, 1973). 10) M. Alcubierre and B. Brügmann: Phys. Rev. D 63 (2001) ) T. Nakamura, K. Oohara and Y. Kojima: Prog. Theor. Phys. Supplement 90 (1987) 1. 12) M. Shibata and T. Nakamura: Phys. Rev. D 52 (1995) 5428; M. Shibata: Prog. Theor. Phys. 101 (1999) 251; T. W. Baumgarte and S. L. Shapiro: Phys. Rev. D 59 (1999) ; M. Alcubierre, et al.: Phys. Rev. D 61 (2000) ; H. Shinkai and G. Yoneda: Phys. Rev. D 66 (2002) ) V. Kalogera, et al.: Astrophys. J. 601 (2004) L179; 614 (2004) L ) M. Shibata and K. Uryū: Phys. Rev. D 61 (2001) ; Prog. Theor. Phys. 107 (2002) 265: M. Shibata, K. Taniguchi and K. Uryū: Phys. Rev. D 68 (2003) ; 71 (2005) ) M. Shibata and K. Taniguchi: Phys. Rev. D 73 (2006) ) A. Akmal, V. R. Pandharipande and D. G. Ravenhall: Phys. Rev. C 58 (1998) 1804; F. Douchin and P. Haensel: Astron. Astrophys. 380 (2001) ) K. Uryū, M. Shibata and Y. Eriguchi: Phys. Rev. D 62 (2000) ; K. Taniguchi and E. Gourgoulhon: Phys. Rev. D 66 (2002) ) S. Chandrasekhar: Ellipsoidal Figures of Equilibrium (Yale Univ. Press, New Haven, 1969). 19) E. W. Leaver: Proc. R. Soc. London A 402 (1985) ) M. Shibata: Phys. Rev. Lett. 94 (2005) ) M. D. Duez, Y. T. Liu, S. L. Shapiro and B. C. Stephens: Phys. Rev. D 72 (2005) ; M. Shibata and Y.-I. Sekiguchi: Phys. Rev. D 72 (2005) ; M. D. Duez, et al.: Phys. Rev. Lett. 96 (2006) ) M. Shibata, et al.: Phys. Rev. Lett. 96 (2006) ) F. Pretorius: Phys. Rev. Lett. 95 (2005) ) M. Campanelli, et al.: Phys. Rev. Lett. 96 (2006) J. G. Baker, et al.: Phys. Rev. Lett. 96 (2006) F. Herrmann, et al.: gr-qc/ : 25) P. Diener, et al.: gr-qc/ Latest Progress in Numerical Relativity Masaru Shibata abstract: Einstein s equations are nonlinear, partial di#erential, and simultaneous. It is impossible to derive an analytic solution which describes a dynamical phemomemon in nature. However, it has been required to develop a method for the solutions to clarify general relativistic astrophysical phenomena which can be observed by large telescopes and detectors. The field which develops methods for numerically solving Einstein s equations is referred to as Numerical relativity. In this article, I first explain the reason that the numerical relativity is needed and its outline. Then, the latest progress is reviewed focusing particularly on the results obtained in Page 9
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