Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T14:59:47.126Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical relativity: challenges for computational science

Published online by Cambridge University Press:  07 November 2008

Gregory B. Cook  and
Saul A. Teukolsky
Gregory B. Cook
Affiliation:
Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA E-mail: [email protected], [email protected]
Saul A. Teukolsky
Affiliation:
Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA E-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the burgeoning field of numerical relativity, which aims to solve Einstein's equations of general relativity numerically. The field presents many questions that may interest numerical analysts, especially problems related to nonlinear partial differential equations: elliptic systems, hyperbolic systems, and mixed systems. There are many novel features, such as dealing with boundaries when black holes are excised from the computational domain, or how even to pose the problem computationally when the coordinates must be determined during the evolution from initial data. The most important unsolved problem is that there is no known general 3-dimensional algorithm that can evolve Einstein's equations with black holes that is stable. This review is meant to be an introduction that will enable numerical analysts and other computational scientists to enter the field. No previous knowledge of special or general relativity is assumed.

Type
Research Article
Information
Acta Numerica , Volume 8 , January 1999 , pp. 1 - 45
Copyright
Copyright © Cambridge University Press 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abrahams, A. M., Bernstein, D., Hobill, D., Seidel, E. and Smarr, L. (1992), ‘Numerically generated black-hole spacetimes: Interaction with gravitational waves’, Phys. Rev. D 45, 3544–3558.CrossRefGoogle ScholarPubMed
Abrahams, A. M., Cook, G. B., Shapiro, S. L. and Teukolsky, S. A. (1994 a), ‘Solving Einstein's equations for rotating spacetimes: Evolution of relativistic star clusters’, Phys. Rev. D 49, 5153–5164.CrossRefGoogle ScholarPubMed
Abrahams, A. M., Shapiro, S. L. and Teukolsky, S. A. (1994 b), ‘Calculation of gravitational wave forms from black hole collisions and disk collapse: Applying perturbation theory to numerical spacetimes’, Phys. Rev. D 51, 4295–4301.CrossRefGoogle Scholar
Abrahams, A. M. et al. , The Binary Black Hole Alliance (1998), ‘Gravitational wave extraction and outer boundary conditions by perturbative matching’, Phys. Rev. Lett. 80, 1812–1815.CrossRefGoogle Scholar
Abramovici, A., Althouse, W. E., Drever, R. W. P., Gürsel, Y., Kawamura, S., Raab, F. J., Shoemaker, D., Sievers, L., Spero, R. E., Thorne, K. S., Vogt, R. E., Weiss, R., Whitcomb, S. E. and Zucker, M. E. (1992), ‘LIGO: The laser interferometer gravitational-wave observatory’, Science 256, 325–333.CrossRefGoogle ScholarPubMed
Alcubierre, M. and Schutz, B. F. (1994), ‘Time-symmetric ADI and causal reconnection: Stable numerical techniques for hyperbolic systems on moving grids’, J. Comput. Phys. 112, 44–77.CrossRefGoogle Scholar
Anderson, A., Choquet-Bruhat, Y. and York, J. W. Jr (1997), ‘Einstein-Bianchi hyperbolic system for general relativity’, Topological Methods in Nonlinear Analysis 10, 353–373.CrossRefGoogle Scholar
Anninos, P., Camarda, K., Massó, J., Seidel, E., Suen, W.-M. and Towns, J. (1995 a), ‘Three dimensional numerical relativity: The evolution of black holes’, Phys. Rev. D 52, 2059–2082.CrossRefGoogle ScholarPubMed
Anninos, P., Daues, G., Massó, J., Seidel, E. and Suen, W.-M. (1995 b), ‘Horizon boundary condition for black hole spacetimes’, Phys. Rev. D 51, 5562–5578.CrossRefGoogle ScholarPubMed
Anninos, P., Hobill, D., Seidel, E., Smarr, L. and Suen, W.-M. (1993), ‘Collision of two black holes’, Phys. Rev. Lett. 71, 2851–2854.CrossRefGoogle ScholarPubMed
Anninos, P., Hobill, D. W., Seidel, E., Smarr, L. and Suen, W.-M. (1995 c), ‘Head-on collision of two equal mass black holes’, Phys. Rev. D 52, 2044–2058.CrossRefGoogle ScholarPubMed
Arnold, D. N., Mukherjee, A. and Pouly, L. (1998), Adaptive finite elements and colliding black holes, in Numerical Analysis 1997: Proceedings of the 17th Dundee Biennial Conference (Griffiths, D. F., Higham, D. J. and Watson, G. A., eds), Addison Wesley Longman, Harlow, England, pp. 1–15.Google Scholar
Banyuls, F., Font, J. A., Ibáñez, J. M., Martí, J. M. and Miralles, J. A. (1997), ‘Numerical {3 + 1} general relativistic hydrodynamics: A local characteristic approach’, Astrophys. J. 476, 221–231.CrossRefGoogle Scholar
Barrett, J. W., Galassi, M., Miller, W. A., Sorkin, R. D., Tuckey, P. A. and Williams, R. M. (1997), ‘A parallelizable implicit evolution scheme for Regge calculus’, Int. J. Theor. Phys. 36, 815–839.CrossRefGoogle Scholar
Baumgarte, T. W. and Shapiro, S. L. (1999), ‘On the integration of Einstein's field equations’, Phys. Rev. D 59, 024007.CrossRefGoogle Scholar
Baumgarte, T. W., Cook, G. B., Scheel, M. A., Shapiro, S. L. and Teukolsky, S. A. (1996), ‘Implementing an apparent-horizon finder in three dimensions’, Phys. Rev. D 54, 4849–4857.CrossRefGoogle ScholarPubMed
Baumgarte, T. W., Cook, G. B., Scheel, M. A., Shapiro, S. L. and Teukolsky, S. A. (1998), ‘General relativistic models of binary neutron stars in quasiequilibrium’, Phys. Rev. D 57, 7299–7311.CrossRefGoogle Scholar
Bernstein, D., Hobill, D. W., Seidel, E., Smarr, L. and Towns, J. (1994), ‘Numerically generated axisymmetric black hole spacetimes: Numerical methods and code tests’, Phys. Rev. D 50, 5000–5024.CrossRefGoogle ScholarPubMed
Bishop, N. T., Gómez, R., Holvorcem, P. R., Matzner, R. A. and Winicour, P. P. J. (1997a), ‘Cauchy-characteristic evolution and waveforms’, J. Comput. Phys. 136, 140–167.CrossRefGoogle Scholar
Bishop, N. T., Gómez, R., Lehner, L. and Winicour, J. (1996), ‘Cauchy-characteristic extraction in numerical relativity’, Phys. Rev. D 54, 6153–6165.CrossRefGoogle ScholarPubMed
Bishop, N. T., Gómez, R., Lehner, L., Maharaj, M. and Winicour, J. (1997 b), ‘High-powered gravitational news’, Phys. Rev. D 56, 6298–6309.Google Scholar
Bona, C. and Massó, J. (1992), ‘A hyperbolic evolution system for numerical relativity’, Phys. Rev. Lett. 68, 1097–1099.CrossRefGoogle ScholarPubMed
Bona, C., Massó, J. and Stela, J. (1995 a), ‘Numerical black holes: A moving grid approach’, Phys. Rev. D 51, 1639–1639.CrossRefGoogle ScholarPubMed
Bona, C., Massó, J., Seidel, E. and Stela, J. (1995b), ‘New formalism for numerical relativity’, Phys. Rev. Lett. 75, 600–603.CrossRefGoogle ScholarPubMed
Bonazzola, S., Gourgoulhon, E. and Marck, J.-A. (1997), ‘A relativistic formalism to compute quasi-equilibrium configurations of non-synchronized neutron star binaries’, Phys. Rev. D 56, 7740–7749.CrossRefGoogle Scholar
Bonazzola, S., Gourgoulhon, E. and Marck, J.-A. (1998), ‘Numerical approach for high precision 3-d relativistic star models’, Phys. Rev. D 58, 104020.CrossRefGoogle Scholar
Bonazzola, S., Gourgoulhon, E. and Marck, J.-A. (1999), ‘Numerical models of irrotational binary neutron stars in general relativity’, Phys. Rev. Lett. 82, 892–895.CrossRefGoogle Scholar
Bonazzola, S., Gourgoulhon, E., Salgado, M. and Marck, J.-A. (1993), ‘Axisymmetric rotating relativistic bodies: A new numerical approach for “exact” solutions’, Astron. Astrophys. 278, 421–443.Google Scholar
Bowen, J. M. (1979), ‘General form for the longitudinal momentum of a spherically symmetric source’, Gen. Relativ. Gravit. 11, 227–231.CrossRefGoogle Scholar
Bowen, J. M. and York, J. W. Jr (1980), ‘Time-asymmetric initial data for black holes and black-hole collisions’, Phys. Rev. D 21, 2047–2056.CrossRefGoogle Scholar
Brandt, S. and Brügmann, B. (1997), ‘A simple construction of initial data for multiple black holes’, Phys. Rev. Lett. 78, 3606–3609.CrossRefGoogle Scholar
Brandt, S. R. and Seidel, E. (1995), ‘The evolution of distorted rotating black holes I: Methods and tests’, Phys. Rev. D 52, 856–869.CrossRefGoogle ScholarPubMed
Brodbeck, O., Frittelli, S., Hübner, P. and Ruela, O. A. (1999), ‘Einstein's equations with asymptotically stable constraint propagation’, J. Math. Phys. 40, 909–923.CrossRefGoogle Scholar
Brügmann, B. (1996), ‘Adaptive mesh and geodesically sliced Schwarzschild space-time in 3+1 dimensions’, Phys. Rev. D 54, 7361–7372.CrossRefGoogle Scholar
Butterworth, E. M. and Ipser, J. R. (1976), ‘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.CrossRefGoogle Scholar
Choptuik, M. W. (1991), ‘Consistency of finite-difference solutions of Einstein's equations’, Phys. Rev. D 44, 3124–3135.CrossRefGoogle ScholarPubMed
Choptuik, M. W. (1993), ‘Universality and scaling in gravitational collapse of a massless scalar field’, Phys. Rev. Lett. 70, 9–12.CrossRefGoogle ScholarPubMed
Choquet-Bruhat, Y. and York, J. W. Jr (1980), The Cauchy problem, in General relativity and gravitation. One hundred years after the birth of Albert Einstein (Held, A., ed.), Vol. 1, Plenum, New York, pp. 99–172.Google Scholar
Choquet-Bruhat, Y. and York, J. W. Jr (1995), ‘Geometrical well posed systems for the Einstein equations’, C. R. Acad. Sci. Paris A321, 1089–1095.Google Scholar
Christodoulou, D. and Murchadha, N. Ó (1981), ‘The boost problem in general relativity’, Commun. Math. Phys. 80, 271–300.CrossRefGoogle Scholar
Cook, G. B. (1991), ‘Initial data for axisymmetric black-hole collisions’, Phys. Rev. D 44, 2983–3000.CrossRefGoogle ScholarPubMed
Cook, G. B. (1994), ‘Three-dimensional initial data for the collision of two black holes II: Quasicircular orbits for equal mass black holes’, Phys. Rev. D 50, 5025–5032.CrossRefGoogle ScholarPubMed
Cook, G. B. and York, J. W. Jr (1990), ‘Apparent horizons for boosted or spinning black holes’, Phys. Rev. D 41, 1077–1085.CrossRefGoogle ScholarPubMed
Cook, G. B., Choptuik, M. W., Dubal, M. R., Klasky, S., Matzner, R. A. and Oliveira, S. R. (1993), ‘Three-dimensional initial data for the collision of two black holes’, Phys. Rev. D 47, 1471–1490.CrossRefGoogle ScholarPubMed
Cook, G. B., Shapiro, S. L. and Teukolsky, S. A. (1994), ‘Rapidly rotating neutron stars in general relativity: Realistic equations of state’, Astrophys. J. 424, 823–845.CrossRefGoogle Scholar
Cook, G. B. et al. , The Binary Black Hole Alliance (1998), ‘Boosted three-dimensional black-hole evolutions with singularity excision’, Phys. Rev. Lett. 80, 2512–2516.CrossRefGoogle Scholar
Dubal, M. R., d'Inverno, R. A. and Clarke, C. J. S. (1995), ‘Combining Cauchy and characteristic codes II: The interface problem for vacuum cylindrical symmetry’, Phys. Rev. D 52, 6868–6881.CrossRefGoogle ScholarPubMed
Einstein, A. and Rosen, N. (1935), ‘The particle problem in the general theory of relativity’, Phys. Rev. 48, 73–77.CrossRefGoogle Scholar
Evans, C. R. (1984), ‘A method for numerical relativity: Simulation of axisymmetric gravitational collapse and gravitational radiation generation’, PhD thesis, University of Texas at Austin.Google Scholar
Fischer, A. E. and Marsden, J. E. (1972), ‘The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I’, Commun. Math. Phys. 28, 1–38.CrossRefGoogle Scholar
Fourès-Bruhat, Y. (1952), ‘Théorèm d'existence pour certain systèms d'équations aux dérivées partielles non linéaires’, Acta. Math. 88, 141–225.CrossRefGoogle Scholar
Friedman, J. L., Ipser, J. R. and Parker, L. (1986), ‘Rapidly rotating neutron star models’, Astrophys. J. 304, 115–139.CrossRefGoogle Scholar
Friedrich, H. (1985), ‘On the hyperbolicity of Einstein's and other gauge field equations’, Commun. Math. Phys. 100, 525–543.CrossRefGoogle Scholar
Friedrich, H. (1996), ‘Hyperbolic reductions for Einstein's equations’, Class. Quantum Gravit. 13, 1451–1469.CrossRefGoogle Scholar
Frittelli, S. (1997), ‘Note on the propagation of the constraints in standard 3 + 1 general relativity’, Phys. Rev. D 55, 5992–5996.CrossRefGoogle Scholar
Frittelli, S. and Reula, O. A. (1996), ‘First-order symmetric hyperbolic Einstein equations with arbitrary fixed gauge’, Phys. Rev. Lett. 76, 4667–4670.CrossRefGoogle ScholarPubMed
Gentle, A. P. and Miller, W. A. (1998), ‘A fully (3+l)-d Regge calculus model of the Kasner cosmology’, Class. Quantum Gravit. 15, 389–405.CrossRefGoogle Scholar
Gundlach, C. (1998 a), ‘Critical phenomena in gravitational collapse’, Adv. Theor. Math. Phys. 2, 1–49.CrossRefGoogle Scholar
Gundlach, C. (1998 b), ‘Pseudo-spectral apparent horizon finders: An efficient new algorithm’, Phys. Rev. D 57, 863–875.CrossRefGoogle Scholar
Hawking, S. W. and Ellis, G. F. R. (1973), The Large Scale Structure of Spacetime, Cambridge University Press, Cambridge, England.CrossRefGoogle Scholar
Komatsu, H., Eriguchi, Y. and Hachisu, I. (1989), ‘Rapidly rotating general relativistic stars I: Numerical method and its application to uniformly rotating polytropes’, Mon. Not. R. Astr. Soc. 237, 355–379.CrossRefGoogle Scholar
Kulkarni, A. D. (1984), ‘Time-asymmetric initial data for the N black hole problem in general relativity’, J. Math. Phys. 25, 1028–1034.CrossRefGoogle Scholar
Kulkarni, A. D., Shepley, L. C. and York, J. W. Jr (1983), ‘Initial data for N black holes’, Phys. Lett. 96A, 228–230.CrossRefGoogle Scholar
Marsa, R. L. and Choptuik, M. W. (1996), ‘Black-hole-scalar-field interactions in spherical symmetry’, Phys. Rev. D 54, 4929–4943.CrossRefGoogle ScholarPubMed
Matzner, R. A., Huq, M. F. and Shoemaker, D. (1999), ‘Initial data and coordinates for multiple black hole systems’, Phys. Rev. D 59, 024015.CrossRefGoogle Scholar
Misner, C. W. (1963), ‘The method of images in geometrostatics’, Ann. Phys. 24, 102–117.CrossRefGoogle Scholar
Misner, C. W., Thorne, K. S. and Wheeler, J. A. (1973), Gravitation, Freeman, New York.Google Scholar
Murchadha, N. Ó and York, J. W. Jr (1974), ‘Initial-value problem of general relativity. I. General formulation and physical interpretation’, Phys. Rev. D 10, 428–436.CrossRefGoogle Scholar
Parashar, M. and Brown, J. C. (1995), Distributed dynamical data-structures for parallel adaptive mesh-refinement, in Proceedings of the International Conference for High Performance Computing (Sahni, S., Prasanna, V. K. and Bhatkar, V. P., eds), Tata McGraw-Hill, New Delhi, India. See also www.ticam.utexas.edu/∼parashar/public_html/DAGH.Google Scholar
Pons, J. A., Font, J. A., Ibáñez, J. M., Martí, J. M. and Miralles, J. A. (1998), ‘General relativistic hydrodynamics with special relativistic Riemann solvers’, Astro, and Astroph. 339, 638–642.Google Scholar
Regge, T. (1961), ‘General relativity without coordinates’, Nuovo Cimento 19, 558–571.CrossRefGoogle Scholar
Rezzolla, L., Abrahams, A. M., Matzner, R. A., Rupright, M. E. and Shapiro, S. L. (1999), ‘Cauchy-perturbative matching and outer boundary conditions: Computational studies’. Phys. Rev. D 59, 064001.CrossRefGoogle Scholar
Sachs, R. K. and Wu, H. (1977), General Relativity for Mathematicians, Springer-Verlag, New York.CrossRefGoogle Scholar
Scheel, M. A., Baumgarte, T. W., Cook, G. B., Shapiro, S. L. and Teukolsky, S. A. (1997), ‘Numerical evolution of black holes with a hyperbolic formulation of general relativity’, Phys. Rev. D 56, 6320–6335.CrossRefGoogle Scholar
Scheel, M. A., Baumgarte, T. W., Cook, G. B., Shapiro, S. L. and Teukolsky, S. A. (1998), ‘Treating instabilities in a hyperbolic formulation of Einstein's equations’, Phys. Rev. D 58, 044020.CrossRefGoogle Scholar
Scheel, M. A., Shapiro, S. L. and Teukolsky, S. A. (1995 a), ‘Collapse to black holes in Brans–Dicke theory I: Horizon boundary conditions for dynamical space-times’, Phys. Rev. D 51, 4208–4235.CrossRefGoogle Scholar
Scheel, M. A., Shapiro, S. L. and Teukolsky, S. A. (1995 b), ‘Collapse to black holes in Brans–Dicke theory II: Comparison with general relativity’, Phys. Rev. D 51, 4236–4249.CrossRefGoogle ScholarPubMed
Seidel, E. and Suen, W.-M. (1992), ‘Towards a singularity-proof scheme in numerical relativity’, Phys. Rev. Lett. 69, 1845–1848.CrossRefGoogle ScholarPubMed
Shapiro, S. L. and Teukolsky, S. A. (1992), ‘Collisions of relativistic clusters and the formation of black holes’, Phys. Rev. D 45, 2739–2750.CrossRefGoogle ScholarPubMed
Shibata, M. (1998), ‘A relativistic formalism for computation of irrotational binary stars in quasi-equilibrium states’, Phys. Rev. D 58, 024012.CrossRefGoogle Scholar
Smarr, L. L. and York, J. W. Jr (1978), ‘Kinematical conditions in the construction of spacetime’, Phys. Rev. D 17, 2529–2551.CrossRefGoogle Scholar
Sorkin, R. D. (1982), ‘A stability criterion for many-parameter equilibrium families’, Astrophys. J. 257, 847–854.CrossRefGoogle Scholar
Stark, R. F. and Piran, T. (1987), ‘A general relativistic code for rotating axisymmetric configurations and gravitational radiation: Numerical methods and tests’, Computer Physics Reports 5, 221–264.CrossRefGoogle Scholar
Taylor, M. E. (1996), Partial Differential Equations III: Nonlinear Equations, Springer, New York.CrossRefGoogle Scholar
Teukolsky, S. A. (1998), ‘Irrotational binary neutron stars in quasiequilibrium in general relativity’, Astrophys. J. 504, 442–449.CrossRefGoogle Scholar
Thornburg, J. (1987), ‘Coordinate and boundary conditions for the general relativistic initial data problem’, Class. Quantum Gravit. 4, 1119–1131.CrossRefGoogle Scholar
van Putten, M. H. P. M. and Eardley, D. M. (1996), ‘Hyperbolic reductions for Einstein's equations’, Phys. Rev. D 53, 3056–3063.CrossRefGoogle Scholar
Wald, R. M. (1984), General Relativity, The University of Chicago Press, Chicago.CrossRefGoogle Scholar
Williams, R. M. and Tuckey, P. A. (1992), ‘Regge calculus: A brief review and bibliography’, Class. Quantum Gravit. 9, 1409–1422.CrossRefGoogle Scholar
Wilson, J. R. (1979), A numerical method for relativistic hydrodynamics, in Sources of Gravitational Radiation (Smarr, L. L., ed.), Cambridge University Press, Cambridge, England, pp. 423–445.Google Scholar
Wilson, J. R., Mathews, G. J. and Marronetti, P. (1996), ‘Relativistic numerical method for close neutron star binaries’, Phys. Rev. D 54, 1317–1331.CrossRefGoogle Scholar
York, J. W. Jr (1979), Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation (Smarr, L. L., ed.), Cambridge University Press, Cambridge, England, pp. 83–126.Google Scholar