e-mail: shibata@provence.c.u-tokyo.ac.jp 2005 1. 40 % 1 4 1) 1 PSRB1913 16 30 2) 3) X g 1990 g 4) g g 2 g 2. 1990 2000 3) 10 1 Page 1
5) % 1 g g 3. 1 3 1 6) 3 S S S n m (1/a, b k /a) a b k 1 1 3 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) 1 3 6 (2), (3) g ij 1 2 g ij (4) g ij m G mn 0 m mn 3 g ij K 6) ij 3 1 1 5.1 1 2 Vol. 61, No. 5, 2006 Page 2
4. 2 0) 1) 3 2) 3) 1) 3) 2 3 5.1 (2), (3) 7 1 10 2 8 5.3 5.2 5. 5.1 3 1 3 3 1 11) 3 1 1987 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
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) 5.2 3 2 S S S S 3 2 1971 1984 (W. Unruh) 9) 10) 6 5.3 2 1 1 15 m 2Gm/c 2 0.14Gm/c 2 5Gm/c 2 M 50 70GM/c 2 M 2m 1 600, 3 600 3 1,500 3 200 1 8 300 4 4 Vol. 61, No. 5, 2006 Page 4
1 1 FACOM-VPP5000 720 460 6. 2005 2005 6.1 2 5) 1) 3 2) 2 g 14) 15) 16) 35 km 3 1 1 10 1 13) LIGO 3) 300 Mpc 10 100 13) 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 1 0.2 ms (t 2.04 ms) 2 99.9% 5 Page 5
6 50 Mpc 45 5 4 1.3M 2.06 ms (0.0005M ) 2.9M, a M J cj/gm 2 0.75 5.2 1.3M Hypermassive neutron star; HMNS 1.5M 2.6M 2.18M HMNS HMNS T/W 0.14 18) 4ms 6.2 6 6 2 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 0.75 19) HMNS 6 t 3 ms 3 3.5 khz t 10 ms 10 ms HMNS HMNS 50 ms ms 6 Vol. 61, No. 5, 2006 Page 6
7 (HMNS) x z HMNS P c P c 0.2 ms 2 3kHz 100 200 3kHz D h e# 5 10 21 50 Mpc (6) D LIGO HMNS 20) 6.3 HMNS 1) HMNS 2) HMNS 50 ms 21) 7 HMNS HMNS x z 5 10 12 10 ms 7 2 HMNS3 7ms 3, 4 1,000 K (n)(n ) nn g HMNS g 22) 6.4 1 7 Page 7
5.2 5.3 (F. Pretorius) 23) 8 24, 25) 3 1 3 1 8 23) M 0 GM 0/c 3 Y 4 2 t/(gm 0/c 3 ) 200 8 5) 7. 1) 2) 3) 1 1 1 2 8 Vol. 61, No. 5, 2006 Page 8
1) I. H. Stairs: Science 304 (2004) 547. 2) J. H. Taylor: Phil. Trans. R. Soc. London 341 (1992) 117. 3) LIGO, TAMA300, GEO600, VIRGO http://www.ligo.caltech.edu/, http://tamago.mtk.nao.ac.jp/, http://www.geo600.unihannover.de/, http://www.virgo.infn.it/ 4) 60 (2005) 271. 5) 53 (1998) 581. 6) J. W. York, Jr.: in Sources of Gravitational Radiation, ed. L. Smarr (Cambridge Univ. Press, 1979) p. 83. 7) G. B. Cook: Living Reviews in Relativity 3 (2000) 5, http://relativity. livingreviews.org/articles/. 8) J. A. Font: Living Review Relativity 3 (2000) 2, http://relativity. 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) 104006. 11) 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) 024007; M. Alcubierre, et al.: Phys. Rev. D 61 (2000) 041501; H. Shinkai and G. Yoneda: Phys. Rev. D 66 (2002) 124003. 13) V. Kalogera, et al.: Astrophys. J. 601 (2004) L179; 614 (2004) L137. 14) M. Shibata and K. Uryū: Phys. Rev. D 61 (2001) 064001; Prog. Theor. Phys. 107 (2002) 265: M. Shibata, K. Taniguchi and K. Uryū: Phys. Rev. D 68 (2003) 084020; 71 (2005) 084021. 15) M. Shibata and K. Taniguchi: Phys. Rev. D 73 (2006) 064027. 16) A. Akmal, V. R. Pandharipande and D. G. Ravenhall: Phys. Rev. C 58 (1998) 1804; F. Douchin and P. Haensel: Astron. Astrophys. 380 (2001) 151. 17) K. Uryū, M. Shibata and Y. Eriguchi: Phys. Rev. D 62 (2000) 104015; K. Taniguchi and E. Gourgoulhon: Phys. Rev. D 66 (2002) 104019. 18) S. Chandrasekhar: Ellipsoidal Figures of Equilibrium (Yale Univ. Press, New Haven, 1969). 19) E. W. Leaver: Proc. R. Soc. London A 402 (1985) 285. 20) M. Shibata: Phys. Rev. Lett. 94 (2005) 201101. 21) M. D. Duez, Y. T. Liu, S. L. Shapiro and B. C. Stephens: Phys. Rev. D 72 (2005) 024028; M. Shibata and Y.-I. Sekiguchi: Phys. Rev. D 72 (2005) 044014; M. D. Duez, et al.: Phys. Rev. Lett. 96 (2006) 031101. 22) M. Shibata, et al.: Phys. Rev. Lett. 96 (2006) 031102. 23) F. Pretorius: Phys. Rev. Lett. 95 (2005) 121101. 24) M. Campanelli, et al.: Phys. Rev. Lett. 96 (2006) 111101. J. G. Baker, et al.: Phys. Rev. Lett. 96 (2006) 111102. F. Herrmann, et al.: gr-qc/ 0601026: 25) P. Diener, et al.: gr-qc/0512108. Latest Progress in Numerical Relativity 2006 2 13 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 2005. 9 Page 9