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Revista mexicana de física
versión impresa ISSN 0035-001X
Rev. mex. fis. vol.53 supl.2 México feb. 2007
Introducción a la relatividad numérica
M. Alcubierre
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70543, México D.F. 04510, México
Recibido el 18 de julio de 2005
Aceptado el 14 de marzo de 2005
Resumen
Se presenta una introducción a los conceptos básicos de la relatividad numérica. Partiendo de una breve discusión de la relatividad general, se presenta la formulación 3+1, que es la más utilizada en cálculos numéricos. Se introducen los conceptos de foliación, métrica espacial, funciones de norma y curvatura extrínseca y se discute la separación de las ecuaciones de Einstein que dan lugar a las ecuaciones de ArnowittDeserMisner (ADM). Se menciona también la existencia de formulaciones alternativas de las ecuaciones de evolución y se discute el concepto de hiperbolicidad. Finalmente se presentan las ideas principales de las aproximaciones en diferencias finitas, que son el método mas comúnmente utilizado en relatividad numérica.
Descriptores: Relatividad numérica.
Abstract
I present an introduction to the basic concepts of numerical relativity. Starting from a brief discussion of general relativity, I present the 3+1 formulation, which is the most widely used for numerical calculations in relativity. I introduce the concepts of foliation, spatial metric, gauge functions and extrinsic curvature, and discuss the splitting of Einstein's equations that result in the ArnowittDeserMisner (ADM) equations. I also mention the existence of alternative formulations of the evolution equations and discuss the concept of hyperbolicity. Finally, I discuss the main ideas of finite difference approximations, which are the most common method used in numerical relativity.
Keywords: Numerical relativity.
PACS: 04.20.Ex; 04.25.Dm; 95.30.Sf
Referencias
1. S.G. Hahn y R.W. Lindquist, Ann. Phys., 29 (1964) 304. [ Links ]
2. L. Smarr, A. Cadez, B. DeWitt, y K. Eppley, Phys. Rev. D, 14 N. 10 (1976) 2443. [ Links ]
3. K. Eppley, The Numerical Evolution of the Collision of Two Black Holes. PhD thesis, Princeton University, Princeton, New Jersey, (1975). [ Links ]
4. K. Eppley, Phys. Rev. D, 16 N 6 (1977) 1609. [ Links ]
5. M. W. Choptuik, "Universality and scaling in gravitational collapse of massless scalar field," Phys. Rev. Lett, 70 (1993) 9. [ Links ]
6. C. Gundlach, "Critical phenomena in gravitational collapse," Living Reviews in Relativity, 2 (1999) 4. [ Links ]
7. L. Lehner, "Numerical relativity: A review," Class. Quantum Grav. 18 (2001) R25R86. [ Links ]
8. LIGO http://www.ligo.caltech.edu/. [ Links ]
9. http://www.virgo.infn.it/. [ Links ]
10. GEO600 http://www.geo600.unihannover.de/. [ Links ]
11. TAMA http://tamago.mtk.nao.ac.jp [ Links ]
12. A. Einstein, "Die feldgleichungen der gravitation," Preuss. Akad. Wiss. Berlin, Sitzber, (1915) 844847. [ Links ]
13. A. Einstein, "Zur algemeinen relativitäts theorie," Preuss. Akad. Wiss. Berlin, Sitzber, (1915) 778786. [ Links ]
14. J. A. Wheeler, A journey into gravity and spacetime. distributed by W. H. Freeman, (New York, U.S.A.: Scientific American Library, 1990). [ Links ]
15. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. (San Francisco: W. H. Freeman, 1973). [ Links ]
16. B. Schutz, A First Course in General Relativity. (Cambridge University Press, 1985). [ Links ]
17. R. M. Wald, General Relativity. (Chicago: The University of Chicago Press, 1984). [ Links ]
18. J. Winicour, Living Reviews in Relativity, 1 (1998). [ Links ]
19. H. Friedrich, Proc. Roy. Soc. London, 378 (1981) 401. [ Links ]
20. J. W. York, Jr., Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation L. L. Smarr, ed., (Cambridge, UK: Cambridge University Press, 1979) p83. [ Links ]
21. R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research L. Witten, ed., (New York: John Wiley, 1962) p227. [ Links ]
22. A. Lichnerowicz, J. Math Pures et Appl., 23 (1944) 37. [ Links ]
23. J. W. York, Phys. Rev. Lett., 82 (1999) 1350. [ Links ]
24. G. B. Cook, Living Reviews in Relativity, 2000 (200) 5. [ Links ]
25. L. Smarr and J. York, Phys. Rev. D, 17 (1978) 2529. [ Links ]
26. C. Bona, J. Masso, E. Seidel, and J. Stela, Phys. Rev. Lett., 75 (1995) 600. [ Links ]
27. A. Abrahams, D. Bernstein, D. Hobill, E. Seidel, and L. Smarr, Phys. Rev. D, 45 (1992) 3544. [ Links ]
28. P. Anninos, D. Hobill, E. Seidel, L. Smarr, and W.M. Suen, Phys. Rev. Lett., 71 N. 18 (1993) 2851. [ Links ]
29. M. Alcubierre, Class. Quantum Grav., 20 N. 4 (2003) 607. [ Links ]
30. L. Smarr and J. York, Phys. Rev. D, 17 (1978) 1945. [ Links ]
31. M. Alcubierre, B. Brugmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D, 67 (2003) 084023. [ Links ]
32. T. W. Baumgarte and S. L. Shapiro, Physical Review D, 59 (1999) 024007. [ Links ]
33. M. Shibata and T. Nakamura, Phys. Rev. D, 52 (1995) 5428. [ Links ]
34. M. Alcubierre, G. Allen, B. Brugmann, E. Seidel, and W.M. Suen, Phys. Rev. D, 62 (2000) 124011. [ Links ]
35. H.O. Kreiss and J. Lorenz, InitialBoundary Value Problems and the NavierStokes Equations. (New York: Academic Press, 1989). [ Links ]
36. C. Bona, I Masso, E. Seidel, and J. Stela, Phys. Rev. D, 56 (1997) 3405. [ Links ]
37. S. Frittelli and O. Reula, Phys. Rev. Lett., 76 (1996) 4667. grqc/9605005. [ Links ]
38. L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. D, 64 (2001) 064017. grqc/0105031. [ Links ]
39. O. Sarbach, G. Calabrese, J. Pullin, and M. Tiglio, Phys. Rev. D, 66 (2002) 064002. [ Links ]
40. R. D. Richtmyer and K. Morton, Difference Methods for Initial Value Problems. (New York: Interscience Publishers, 1967). [ Links ]
41. A. R. Mitchell, The finite element method in partial differential equations. (U.S.A.: J. Wiley and Sons, 1977). [ Links ]
42. S. Bonazzola and J.A. Marck, in Frontiers in Numerical Relativity C. Evans, L. Finn, and D. Hobill, eds., (Cambridge, England: Cambridge University Press, 1989) p. 239. [ Links ]
43. L. E. Kidder and L. S. Finn, Phys. Rev. D, 62 (2000) 084026.grqc/9911014. [ Links ]
44. L. E. Kidder, M. A. Scheel, S. A. Teukolsky, E. D. Carlson, and G. B. Cook, Phys. Rev. D, 62 (2000) 084032. grqc/0005056. [ Links ]
45. B. Szilagyi, R. Gómez, N. T. Bishop, and J. Winicour, Phys. Rev. D, 62 (2000). grqc/9912030. [ Links ]
46. B. Szilagyi, B. Schmidt, and J. Winicour, Phys. Rev. D, 65 (2002). grqc/0106026 [ Links ]