Energy norms and the stability of the Einstein evolution equations

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 66, Issue 8, 2002, Pages

  Lindblom, Lee ^a   Scheel, Mark A ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYTIC METHOD; ARTICLE; CALCULATION; ENERGY; GROWTH RATE; MATHEMATICAL ANALYSIS; MOLECULAR STABILITY; PHYSICAL PARAMETERS;

EID: 3843053453     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.66.084014     Document Type: Article

References (72)

1. R. Arnowitt, S. Deser, and C.W. Misner, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962), pp. 227–265.
2. J.W. York, Jr., in Sources of Gravitational Radiation, edited by L. L. Smarr (Cambridge University Press, Cambridge, England, 1979), pp. 83–126.
3. M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428 (1995).
4. T.W. Baumgarte and S.L. Shapiro, Phys. Rev. D 59, 024007 (1999).
5. A.E. Fischer and J.E. Marsden, in General Relativity: An Einstein Centenary Survey, edited by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1979), pp. 138–211.
6. C. Bona and J. Massó, Phys. Rev. Lett. 68, 1097 (1992).
7. S. Frittelli and O. Reula, Commun. Math. Phys. 166, 221 (1994).
8. C. Bona, J. Massó, E. Seidel, and J. Stela, Phys. Rev. Lett. 75, 600 (1995).
9. Y. Choquet-Bruhat and J.W. York, Jr., C. R. Acad. Sci., Ser. I: Math. 321, 1089 (1995).
10. A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.W. York, Jr., Phys. Rev. Lett. 75, 3377 (1995).
11. M.H.P.M. van Putten and D.M. Eardley, Phys. Rev. D 53, 3056 (1996).
12. S. Frittelli and O.A. Reula, Phys. Rev. Lett. 76, 4667 (1996).
13. H. Friedrich, Class. Quantum Grav. 13, 1451 (1996).
14. F.B. Estabrook, R.S. Robinson, and H.D. Wahlquist, Class. Quantum Grav. 14, 1237 (1997).
15. M.S. Iriondo, E.O. Leguizamon, and O.A. Reula, Phys. Rev. Lett. 79, 4732 (1997).
16. A. Anderson, Y. Choquet-Bruhat, and J.W. York, Jr., Topol. Methods Nonlinear Anal. 10, 353 (1997).
17. M.A.G. Bonilla, Class. Quantum Grav. 15, 2001 (1998).
18. G. Yoneda and H. Shinkai, Phys. Rev. Lett. 82, 263 (1999).
19. A. Anderson and J.W. York, Jr., Phys. Rev. Lett. 82, 4384 (1999).
20. S. Frittelli and O.A. Reula, J. Math. Phys. 40, 5143 (1999).
21. M. Alcubierre, B. Brügmann, M. Miller, and W.-M. Suen, Phys. Rev. D 60, 064017 (1999).
22. S.D. Hern, Ph.D. thesis, University of Cambridge, 1999, gr-qc/0004036.
23. H. Friedrich and A. Rendall, in Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics, edited by B. G. Schmidt (Springer-Verlag, Berlin, 2000), pp. 127–223.
24. G. Yoneda and H. Shinkai, Int. J. Mod. Phys. D 9, 13 (2000).
25. H. Shinkai and G. Yoneda, Class. Quantum Grav. 17, 4799 (2000).
26. L.E. Kidder, M.A. Scheel, and S.A. Teukolsky, Phys. Rev. D 64, 064017 (2001).
27. G. Yoneda and H. Shinkai, Class. Quantum Grav. 18, 441 (2001).
28. H. Shinkai and G. Yoneda, Class. Quantum Grav. 19, 124019 (2002).
29. K. Alvi, “First-order symmetrizable hyperbolic formulations of Einstein’s equations including lapse and shift as dynamical fields,” gr-qc/0204068.
30. O. Sarbach and M. Tiglio, Phys. Rev. D 66, 064023 (2002).
31. G. Yoneda and H. Shinkai, “Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system,” gr-qc/0204002.
32. P. Laguna and D. Shoemaker, Class. Quantum Grav. 19, 3679 (2002).
33. By unconstrained evolution we mean that the evolution equations are used to advance the fields in time, but the constraint equations are used only to set initial data and to check the accuracy of the results.
34. M.A. Scheel, T.W. Baumgarte, G.B. Cook, S.L. Shapiro, and S.A. Teukolsky, Phys. Rev. D 58, 044020 (1998).
35. M. Alcubierre, B. Brügmann, T. Dramlitsch, J.A. Font, P. Papadopoulos, E. Seidel, N. Stergioulas, and R. Takahashi, Phys. Rev. D 62, 044034 (2000).
36. G. Calabrese, J. Pullin, O. Sarbach, and M. Tiglio, Phys. Rev. D 66, 064011 (2002).
37. M. Alcubierre, G. Allen, B. Brügmann, E. Seidel, and W.-M. Suen, Phys. Rev. D 62, 124011 (2000).
38. A.M. Knapp, E.J. Walker, and T.W. Baumgarte, Phys. Rev. D 65, 064031 (2002).
39. R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
40. P. Painlevé, Acad. Sci., Paris, C. R. 173, 677 (1921).
41. A. Gullstrand, Ark. Mat., Astron. Fys. 16, 1 (1922).
42. K. Martel and E. Poisson, Am. J. Phys. 69, 476 (2001).
43. H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations (Academic, San Diego, 1989).
44. We will typically consider (Formula presented) that also satisfy the initial data constraints, but this is not essential.
45. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II (Wiley, New York, 1962).
46. B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods, Pure and Applied Mathematics (Wiley, New York, 1995).
47. KST System III 26 with (Formula presented) (or (Formula presented) and (Formula presented)
48. The actual evolution equations studied here involve not only these physical constraints, but a large number of additional constraints as well. We illustrate in Sec. IV A that all the constraints grow at the same rate.
49. This projection operator is not unique, however, in the cases that we study here there is an obvious simple choice.
50. Our (Formula presented) differs from the quantity (Formula presented) used by KST by a factor of two: (Formula presented)
51. We note that this representation in terms of (Formula presented) and (Formula presented) encompasses both cases of the representation in terms of (Formula presented) and (Formula presented) given by KST in their Eqs. (2.38) and (2.39) 26.
52. While this assumption is not essential, such a symmetrizer must exist if there is any positive definite symmetrizer that depends only on the dynamical fields (Formula presented) In a neighborhood of flat space only (Formula presented) is nonzero so any symmetrizer in this neighborhood must depend on (Formula presented)
53. The common belief that strongly hyperbolic systems are also symmetric hyperbolic is true only for equations on spacetimes with a single spatial dimension.
54. We do not reproduce here the rather complicated expressions for the eigenvalues of (Formula presented) in the Rindler geometry, but we would be happy to supply them to anyone who has an interest in them.
55. L.E. Kidder, M.A. Scheel, S.A. Teukolsky, E.D. Carlson, and G.B. Cook, Phys. Rev. D 62, 084032 (2000).
56. R.J. Leveque, Numerical Methods for Conservation Laws (Birkhauser Verlag, Basel, 1992).
57. J.P. Boyd, Chebyshev and Fourier Spectral Methods (Springer-Verlag, Berlin, 1989).
58. C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, Berlin, 1988).
59. L.E. Kidder, M.A. Scheel, H.P. Pfeiffer, and S.A. Teukolsky (in preparation).
60. E. Seidel and W.-M. Suen, Phys. Rev. Lett. 69, 1845 (1992).
61. M. Alcubierre and B.F. Schutz, J. Comput. Phys. 112, 44 (1994).
62. P. Anninos, G. Daues, J. Massó, E. Seidel, and W.-M. Suen, Phys. Rev. D 51, 5562 (1995).
63. C. Gundlach and P. Walker, Class. Quantum Grav. 16, 991 (1999).
64. M.A. Scheel, T.W. Baumgarte, G.B. Cook, S.L. Shapiro, and S.A. Teukolsky, Phys. Rev. D 56, 6320 (1997).
65. M.A. Scheel, S.L. Shapiro, and S.A. Teukolsky, Phys. Rev. D 51, 4208 (1995).
66. M.A. Scheel, S.L. Shapiro, and S.A. Teukolsky, Phys. Rev. D 51, 4236 (1995).
67. G.B. Cook et al., Phys. Rev. Lett. 80, 2512 (1998).
68. S. Brandt et al., Phys. Rev. Lett. 85, 5496 (2000).
69. M. Alcubierre and B. Brügmann, Phys. Rev. D 63, 104006 (2001).
70. M. Alcubierre, B. Brügmann, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D 64, 061501(R) (2001).
71. B. Szilágyi and J. Winicour, “Well-posed initial-boundary evolution in general relativity,” gr-qc/0205044.
72. G. Calabrese, L. Lehner, and M. Tiglio, Phys. Rev. D 65, 104031 (2001).