×

zbMATH — the first resource for mathematics

Initial data for numerical relativity. (English) Zbl 1024.83001
Summary: Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein’s equations constrain our choices of these initial data. We examine several of the formalisms used for specifying Cauchy initial data in the \(3+1\) decomposition of Einstein’s equations. We then explore how these formalisms have been used in constructing initial data for spacetimes containing black holes and neutron stars. In the topics discussed, emphasis is placed on those issues that are important for obtaining astrophysically realistic initial data for compact binary coalescence.

MSC:
83-08 Computational methods for problems pertaining to relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83B05 Observational and experimental questions in relativity and gravitational theory
PDF BibTeX XML Cite
Full Text: DOI Link EuDML
References:
[1] Abrahams, AM; Bernstein, D.; Hobill, D.; Seidel, E.; Smarr, L., “Numerically generated black-hole spacetimes: Interaction with gravitational waves”, Phys. Rev. D, 45, 3544-3558, (1992)
[2] Abrahams, AM; Evans, CR, “Critical Behavior and Scaling in Vacuum Axisymmetric Gravitational Collapse”, Phys. Rev. Lett., 70, 2980-2983, (1993)
[3] Abrahams, AM; Price, RH, “Black-hole collisions from Brill-Lindquist initial data: Predictions of perturbation theory”, Phys. Rev. D, 53, 1972-1976, (1996)
[4] Anderson, A.; York, JW, “Hamiltonian Time Evolution for General Relativity”, Phys. Rev. Lett., 81, 1154-1157, (1998) · Zbl 0949.83010
[5] Arbona, A.; Bona, C.; Carot, J.; Mas, L.; Massò, J.; Stella, J., “Stuffed black holes”, Phys. Rev. D, 57, 2397-2402, (1998)
[6] Arnowitt, R.; Deser, S.; Misner, CW; Witten, L. (ed.), “The dynamics of general relativity”, 227-265, (1962), New York
[7] Asada, H., “Formulation for the internal motion of quasiequilibrium configurations in general relativity”, Phys. Rev. D, 57, 7292-7298, (1998)
[8] Bardeen, JM, “A Variational Principle for Rotating Stars in General Relativity”, Astrophys. J., 162, 71-95, (1970)
[9] Bardeen, JM; Wagoner, RV, “Relativistic Disks. I. Uniform Rotation”, Astrophys. J., 167, 359-423, (1971)
[10] Baumgarte, TW, “Innermost stable circular orbit of binary black holes”, Phys. Rev. D, 62, 024018/1-8, (2000)
[11] Baumgarte, TW; Cook, GB; Scheel, MA; Shapiro, SL; Teukol-sky, SA, “Binary Neutron Stars in General Relativity: Quasi-Equilibrium Models”, Phys. Rev. Lett., 79, 1182-1185, (1997)
[12] Baumgarte, TW; Cook, GB; Scheel, MA; Shapiro, SL; Teukolsky, SA, “General Relativistic Models of Binary Neutron Stars in Quasiequilibrium”, Phys. Rev. D, 57, 7299-7311, (1998)
[13] Baumgarte, TW; Cook, GB; Scheel, MA; Shapiro, SL; Teukolsky, SA, “The Stability of Relativistic Neutron Stars in Binary Orbit”, Phys. Rev. D, 57, 6181-6184, (1998)
[14] Bildsen, L.; Cutler, C., “Tidal Interaction of Inspiralling Compact Binaries”, Astrophys. J., 400, 175-180, (1992)
[15] Bishop, NT; Isaacson, R.; Maharaj, M.; Winicour, J., “Black hole data via a Kerr-Schild approach”, Phys. Rev. D, 57, 6113-6118, (1998)
[16] Bocquet, M.; Bonazzola, S.; Gourgoulhon, E.; Novak, J., “Rotating Neutron Star Models With Magnetic Field”, Astron. Astrophys., 301, 757-775, (1995)
[17] Bona, C.; Massò, J., “Harmonic synchronizations of spacetime”, Phys. Rev. D, 38, 2419-2422, (1988)
[18] Bonazzola, S.; Gourgoulhon, E.; Marck, J-A, “A relativistic formalism to compute quasi-equilibrium configurations of non-synchronized neutron star binaries”, Phys. Rev. D, 56, 7740-7749, (1997)
[19] Bonazzola, S.; Gourgoulhon, E.; Marck, J-A, “Numerical approach for high precision 3-D relativistic star models”, Phys. Rev. D, 58, 104020/1-14, (1998)
[20] Bonazzola, S.; Gourgoulhon, E.; Marck, J-A, “Numerical models of irrotational binary neutron stars in general relativity”, Phys. Rev. Lett., 82, 892-895, (1999) · Zbl 0961.85002
[21] Bonazzola, S.; Gourgoulhon, E.; Salgado, M.; Marck, J-A, “Axisym-metric rotating relativistic bodies: a new numerical approach for ‘exact’ solutions”, Astron. Astrophys., 278, 421-443, (1993)
[22] Bonazzola, S.; Schneider, J., “An Exact Study of Rigidly and Rapidly Rotating Stars in General Relativity with Application to the Crab Pulsar”, Astrophys. J., 191, 273-286, (1974)
[23] Bowen, JM, “General form for the longitudinal momentum of a spherically symmetric source”, Gen. Relativ. Gravit., 11, 227-231, (1979) · Zbl 0439.35069
[24] Bowen, JM, “General solution for flat-space longitudinal momentum”, Gen. Relativ. Gravit., 14, 1183-1191, (1982) · Zbl 0497.53037
[25] Bowen, JM; York, JW, “Time-asymmetric initial data for black holes and black-hole collisions”, Phys. Rev. D, 21, 2047-2056, (1980)
[26] Bowen, JW; Rauber, J.; York, JW, “Two black holes with axisymmetric parallel spins: Initial data”, Class. Quantum Grav., 1, 591-610, (1984)
[27] Brandt, S.; Brügmann, B., “A simple construction of initial data for multiple black holes”, Phys. Rev. Lett., 78, 3606-3609, (1997)
[28] Brandt, SR; Seidel, E., “Evolution of distorted rotating black holes. I: Methods and tests”, Phys. Rev. D, 52, 856-869, (1995)
[29] Brandt, SR; Seidel, E., “Evolution of distorted rotating black holes. II: Dynamics and analysis”, Phys. Rev. D, 52, 870-886, (1995)
[30] Brandt, SR; Seidel, E., “Evolution of distorted rotating black holes. III: Initial data”, Phys. Rev. D, 54, 1403-1416, (1996)
[31] Brill, DR; Lindquist, RW, “Interaction energy in geometrostatics”, Phys. Rev., 131, 471-476, (1963) · Zbl 0117.23604
[32] Butterworth, EM, “On the Structure and Stability of Rapidly Rotating Fluid Bodies in General Relativity. II. The Structure of Uniformly Rotating Pseudopolytropes”, Astrophys. J., 204, 561-572, (1976)
[33] Butterworth, EM, “On the Structure and Stability of Rapidly Rotating Fluid Bodies in General Relativity. III. Beyond the Angular Velocity Peak”, Astrophys. J., 231, 219-223, (1979)
[34] Butterworth, EM; Ipser, JR, “Rapidly Rotating Fluid Bodies in General Relativity”, Astrophys. J. Lett., 200, l103-l106, (1975)
[35] Butterworth, EM; Ipser, JR, “On the Structure and Stability of Rapidly Rotating Fluid Bodies in General Relativity. I. The Numerical Method for Computing Structure and Its Application to Uniformly Rotating Homogeneous Bodies”, Astrophys. J., 204, 200-233, (1976)
[36] Cantor, M.; Kulkarni, AD, “Physical distinctions between normalized solutions of the two-body problem of general relativity”, Phys. Rev. D, 25, 2521-2526, (1982)
[37] Choquet-Bruhat, Y.; York, JW, “The Cauchy problem”, General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein, 1, 99-172, (1980)
[38] Cook, GB, “Initial data for axisymmetric black-hole collisions”, Phys. Rev. D, 44, 2983-3000, (1991)
[39] Cook, GB, “Three-dimensional initial data for the collision of two black holes II: Quasicircular orbits for equal mass black holes”, Phys. Rev. D, 508, 5025-5032, (1994)
[40] Cook, GB; Abrahams, AM, “Horizon structure of initial-data sets for axisymmetric two-black-hole collisions”, Phys. Rev. D, 46, 702-713, (1992)
[41] Cook, GB; Choptuik, MW; Dubal, MR; Klasky, S.; Matzner, RA; Oliveira, SR, “Three-dimensional initial data for the collision of two black holes”, Phys. Rev. D, 47, 1471-1490, (1993)
[42] Cook, GB; Scheel, MA, “Well-behaved harmonic time slices of a charged, rotating, boosted black hole”, Phys. Rev. D, 56, 4775-4781, (1997)
[43] Cook, GB; Shapiro, SL; Teukolsky, SA, “Spin-up of a rapidly rotating star by angular momentum loss: Effects of general relativity”, Astrophys. J., 398, 203-223, (1992)
[44] Cook, GB; Shapiro, SL; Teukolsky, SA, “Rapidly rotating neutron stars in general relativity: Realistic equations of state”, Astrophys. J., 424, 823-845, (1994)
[45] Cook, GB; Shapiro, SL; Teukolsky, SA, “Rapidly rotating polytropes in general relativity”, Astrophys. J., 422, 227-242, (1994)
[46] Cook, GB; Shapiro, SL; Teukolsky, SA, “Testing a Simplified Version of Einstein’s Equations for Numerical Relativity”, Phys. Rev. D, 53, 5533-5540, (1996)
[47] Damour, T.; Jaranowski, P.; Schäfer, G., “Determination of the last stable orbit for circular general relativistic binaries at the third post-Newtonian approximation”, Phys. Rev. D, 62, 084011/1-21, (2000)
[48] Doran, C., “A new form of the Kerr solution”, Phys. Rev. D, 61, 1-4, (2000)
[49] Flanagan, EE, “Possible Explanation for Star-Crushing Effect in Binary Neutron Star Simulations”, Phys. Rev. Lett., 82, 1354-1357, (1999)
[50] Friedman, JL; Ipser, JR, “Errata: “Rapidly rotating relativistic stars”, Philos. Trans. R. Soc. London, Ser. A, 341, 561, (1992) · Zbl 0770.76074
[51] Friedman, JL; Ipser, JR, “Rapidly rotating relativistic stars”, Philos. Trans. R. Soc. London, Ser. A, 340, 391-422, (1992) · Zbl 0770.76074
[52] Friedman, JL; Ipser, JR; Parker, L., “Rapidly Rotating Neutron Star Models”, Astrophys. J., 304, 115-139, (1986)
[53] Friedman, JL; Ipser, JR; Sorkin, RD, “Turning-Point Method for Axisymmetric Stability of Rotating Relativistic Stars”, Astrophys. J., 325, 722-274, (1988)
[54] Garat, A.; Price, RH, “Nonexistence of conformally flat slices of the Kerr spacetime”, Phys. Rev. D, 61, 124011/1-4, (2000)
[55] Gourgoulhon, E., “Relations between three formalisms for irrotational binary neutron stars in general relativity”, (April, 1998), [Online Los Alamos Archive Preprint]: cited on 18 July 2000, http://xxx.lanl.gov/abs/gr-qc/9804054. 15
[56] Gourgoulhon, E.; Bonazzola, S., “Noncircular axisymmetric stationary spacetimes”, Phys. Rev. D, 48, 2635-2652, (1993)
[57] Gourgoulhon, E., Grandclèment, P., Taniguchi, K., Marck, J.-A., and Bonazzola, S., “Quasiequilibrium sequences of synchronized and irrota-tional binary neutron stars in general relativity. I. Method and tests”, (July, 2000), [Online Los Alamos Archive Preprint]: cited on 18 July 2000, http://xxx.lanl.gov/abs/gr-qc/0007028. Submitted to Phys. Rev. D. 15
[58] Grandclèment, P., Bonazzola, S., Gourgoulhon, E., and Marck, J.-A., “A multi-domain spectral method for scalar and vectorial Poisson equations with non-compact sources”, (March, 2000), [Online Los Alamos Archive Preprint]: cited on 25 October 2000, http://xxx.lanl.gov/abs/gr-qc/0003072. Submitted to J. Comp. Phys. 15
[59] Gullstrand, A., No article title, Arkiv. Mat. Astron. Fys., 16, 1, (1922)
[60] Kley, W.; Schäfer, G., “Relativistic dust disks and the Wilson-Mathews approach”, Phys. Rev. D, 60, 027501/1-4, (1999)
[61] Kochanek, CS, “Coalescing Binary Neutron Stars”, Astrophys. J., 398, 234-247, (1992)
[62] Komatsu, H.; Eriguchi, Y.; Hachisu, I., “Rapidly rotating general relativistic stars—I. Numerical method and its application to uniformly rotating polytropes”, Mon. Not. R. Astron. Soc., 237, 355-379, (1989) · Zbl 0671.76064
[63] Komatsu, H.; Eriguchi, Y.; Hachisu, I., “Rapidly rotating general relativistic stars—II. Differentially rotating polytropes”, Mon. Not. R. Astron. Soc., 239, 153-171, (1989) · Zbl 0701.76050
[64] Kraus, P.; Wilczek, F., “Some applications of a simple stationary line element for the Schwarzschild geometry”, Mod. Phys. Lett. A, 9, 37133719, (1994) · Zbl 1015.83501
[65] Kulkarni, AD, “Time-asymmetric initial data for the \(N\) black hole problem in general relativity”, J. Math. Phys., 25, 1028-1034, (1984)
[66] Kulkarni, AD; Shepley, LC; York, JW, “Initial data for \(N\) black holes”, Phys. Lett. A, 96, 228-230, (1983)
[67] Lake, K., “A class of quasi-stationary regular line elements for the Schwarzschild geometry”, (July, 1994), [Online Los Alamos Archive Preprint]: cited on 18 July 2000, http://xxx.lanl.gov/abs/gr-qc/9407005. 3.3.3
[68] Lichnerowicz, A., “L’integration des équations de la gravitation relativiste et le problème des \(n\) corps”, J. Math. Pures Appl., 23, 37-63, (1944). 2.2 · Zbl 0060.44410
[69] Lindquist, RW, “Initial-Value Problem on Einstein-Rosen Manifolds”, J. Math. Phys., 4, 938-950, (1963) · Zbl 0118.23201
[70] Lousto, CO; Price, RH, “Improved initial data for black hole collisions”, Phys. Rev. D, 57, 1073-1083, (1998)
[71] Marronetti, P.; Mathews, GJ; Wilson, JR, “Binary neutron-star systems: From the Newtonian regime to the last stable orbit”, Phys. Rev. D, 58, 107503/1-4, (1998)
[72] Marronetti, P.; Mathews, GJ; Wilson, JR, “Irrotational binary neutron stars in quasiequilibrium”, Phys. Rev. D, 60, 087301/1-4, (1999)
[73] Marsa, RL; Choptuik, MW, “Black-hole-scalar-field interactions in spherical symmetry”, Phys. Rev. D, 54, 4929-4943, (1996)
[74] Martel, K.; Poisson, E., “Regular coordinate systems for Schwarzschild and other spherical spacetimes”, Am. J. Phys., 69, 476480, (2001)
[75] Mathews, GJ; Marronetti, P.; Wilson, JR, “Relativistic hydrodynamics in close binary systems: Analysis of neutron-star collapse”, Phys. Rev. D, 58, 043003/1-13, (1998)
[76] Mathews, GJ; Wilson, JR, “Revised relativistic hydrodynamical model for neutron-star binaries”, Phys. Rev. D, 61, 127304/14, (2000)
[77] Matzner, RA; Huq, MF; Shoemaker, D., “Initial Data and Coordinates for Multiple Black Hole Systems”, Phys. Rev. D, 59, 024015/1-6, (1999)
[78] Misner, CW, “The Method of Images in Geometrostatics”, Ann. Phys. (N. Y.), 24, 102-117, (1963) · Zbl 0112.44202
[79] Misner, C.W., Thorne, K.S., and Wheeler, J.A., Gravitation, (W. H. Freeman and Company, New York, New York, 1973). 1.1, 3.3.1
[80] Ó Murchadha, N.; York, JW, “Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds”, J. Math. Phys., 14, 1551-1557, (1973) · Zbl 0281.53031
[81] Ó Murchadha, N.; York, JW, “Initial-value problem of general relativity. I. General formulation and physical interpretation”, Phys. Rev. D, 10, 428-436, (1974)
[82] Ó Murchadha, N.; York, JW, “Initial-value problem of general relativity. II. Stability of solution of the initial-value equations”, Phys. Rev. D, 10, 437-446, (1974)
[83] Ó Murchadha, N.; York, JW, “Gravitational Potentials: A Constructive Approach to General Relativity”, Gen. Relativ. Gravit., 7, 257-261, (1976)
[84] Oppenheimer, JR; Volkoff, G., “On massive neutron cores”, Phys. Rev., 55, 374-381, (1939) · Zbl 0020.28501
[85] Painlevé, P., No article title, C. R. Acad. Sci., 173, 677, (1921)
[86] Pfeiffer, H.P., Teukolsky, S.A., and Cook, G.B., “Quasi-circular Orbits for Spinning Binary Black Holes”, (June, 2000), [Online Los Alamos Archive Preprint]: cited on 18 July 2000, http://xxx.lanl.gov/abs/gr-qc/0006084. 3.2.3, 3.4, 3.4
[87] Rieth, R.; Królak, A. (ed.), “On the validity of Wilson’s approach to general relativity”, Proceedings of the Workshop on Mathematical Aspects of Theories of Gravitation, Warsaw, February 29-March 30, 1996, Warsaw · Zbl 0900.83004
[88] Robertson, H.P., and Noonan, T.W., Relativity and Cosmology, (Saunders, London, 1968). 3.3.3 · Zbl 0181.28505
[89] Rowan, S., and Hough, J., “Gravitational Wave Detection by Interfer-ometry (Ground and Space)”, (June, 2000), [Article in Online Journal Living Reviews in Relativity]: cited on 31 October 2000, http://www.livingreviews.org/Articles/Volume3/2000-3hough. 1 · Zbl 0944.83005
[90] Shapiro, SL; Teukolsky, SA, “Gravitational Collapse to Neutron Stars and Black Holes: Computer Generation of Spherical Spacetimes”, Astrophys. J., 235, 199-215, (1980)
[91] Shapiro, SL; Teukolsky, SA, “Gravitational collapse of rotating spheroids and the formation of naked singularities”, Phys. Rev. D, 45, 2006-2012, (1992)
[92] Shibata, M., “A relativistic formalism for computation of irrotational binary stars in quasiequilibrium states”, Phys. Rev. D, 58, 024012/1-5, (1998)
[93] Smarr, L.; Čadež, A.; DeWitt, B.; Eppley, K., “Collision of two black holes: Theoretical framework”, Phys. Rev. D, 14, 2443-2452, (1976)
[94] Smarr, L.; York, JW, “Kinematical conditions in the construction of spacetime”, Phys. Rev. D, 17, 2529-2551, (1978)
[95] Sorkin, RD, “A Criterion for the Onset of Instabilities at a Turning Point”, Astrophys. J., 249, 254-257, (1981)
[96] Sorkin, RD, “A Stability Criterion for Many-Parameter Equilibrium Families”, Astrophys. J., 257, 847-854, (1982)
[97] Stergioulas, N., “Rotating Stars in Relativity”, (June, 1998), [Article in Online Journal Living Reviews in Relativity]: cited on 18 July 2000, http://www.livingreviews.org/Articles/Volume1/1998-8stergio. 4.2 · Zbl 1023.83014
[98] Stergioulas, N.; Friedman, JL, “Comparing Models of Rapidly Rotating Relativistic Stars Constructed by Two Numerical Methods”, Astrophys. J., 444, 306-311, (1995)
[99] Teukolsky, SA, “Irrotational Binary Neutron Stars in Quasiequilibrium in General Relativity”, Astrophys. J., 504, 442-449, (1998)
[100] Thornburg, J., “Coordinate and boundary conditions for the general relativistic initial data problem”, Class. Quantum Grav., 4, 1119-1131, (1987)
[101] Uryū, K.; Eriguchi, Y., “New numerical method for constructing quasiequilibrium sequences of irrotational binary neutron stars in general relativity”, Phys. Rev. D, 61, 124023/1-19, (2000)
[102] Uryū, K.; Shibata, M.; Eriguchi, Y., “Properties of general relativistic, irrotational binary neutron stars in close quasiequilibrium orbits: Polytropic equations of state”, Phys. Rev. D, 62, 104015/115, (2000)
[103] Wilson, JR, “Models of Differentially Rotating Stars”, Astrophys. J., 176, 273-286, (1972)
[104] Wilson, JR; Mathews, GJ; Evans, CR (ed.); Finn, LS (ed.); Hobill, DW (ed.), “Relativistic Hydrodynamics”, 306-314, (1989), Cambridge, England
[105] Wilson, JR; Mathews, GJ, “Instabilities in close neutron star binaries”, Phys. Rev. Lett., 75, 4161-4164, (1995)
[106] Wilson, JR; Mathews, GJ; Marronetti, P., “Relativistic Numerical Method for Close Neutron Star Binaries”, Phys. Rev. D, 54, 13171331, (1996)
[107] York, JW, “Gravitational degrees of freedom and the initial-value problem”, Phys. Rev. Lett., 26, 1656-1658, (1971)
[108] York, JW, “Role of conformal three-geometry in the dynamics of gravitation”, Phys. Rev. Lett., 28, 1082-1085, (1972)
[109] York, JW, “Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity”, J. Math. Phys., 14, 456-464, (1973) · Zbl 0259.53014
[110] York, JW, “Covariant decompositions of symmetric tensors in the theory of gravitation”, Ann. Inst. Henri Poincare, A, 21, 319-332, (1974) · Zbl 0308.53018
[111] York, JW; Smarr, LL (ed.), “Kinematics and Dynamics of General Relativity”, 83-126, (1979), Cambridge, England
[112] York, JW; Tipler, FJ (ed.), “Energy and Momentum of the Gravitational Field”, 39-58, (1980), New York
[113] York, JW, “Initial data for \(N\) black holes”, Physica (Utrecht), 124A, 629-637, (1984)
[114] York, JW; Evans, CR (ed.); Finn, LS (ed.); Hobill, DW (ed.), “Initial Data for Collisions of Black Holes and Other Gravitational Miscellany”, 89-109, (1989), Cambridge, England
[115] York, JW, “Conformal ‘thin-sandwich’ data for the initial-value problem of general relativity”, Phys. Rev. Lett., 82, 1350-1353, (1999) · Zbl 0949.83011
[116] York, JW; Piran, T.; Matzner, RA (ed.); Shepley, LC (ed.), “The Initial Value Problem and Beyond”, 147-176, (1982), Austin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.