Abstract
This is a technical chapter to prepare the following ones. We motivate and perform a conformal decomposition of the 3-metric on each hypersurface of a 3+1 slicing. To avoid dealing with tensor densities, we introduce a background flat 3-metric. The link between the connections of the physical 3-metric and the conformal one is exhibited, leading to the computation of the Ricci tensor of the conformal 3-metric. Two associated decompositions of the extrinsic curvature are presented, with two different conformal rescalings of the traceless part. The 3+1 Einstein equations are then rewritten in terms of the conformal quantities. Finally, we discuss the Isenberg-Wilson-Mathews approximation to general relativity, which amounts to assuming that the conformal 3-metric is flat and that the slicing is maximal.
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Notes
- 1.
See also Ref. [2] which is freely accessible on the web.
- 2.
The \(C^k_{\;\,ij}\) are not to be confused with the components of the Cotton tensor discussed in Sect. 7.1 Since we shall no longer make use of the latter, no confusion may arise.
- 3.
Notice that we are using a hat, instead of a tilde, to distinguish this quantity from that defined by (7.63).
- 4.
To be discussed in Sect. 10.2.2.
References
Lichnerowicz, A.: L’intégration des équations de la gravitation relativiste et le problème des n corps, J. Math. Pures Appl. 23, 37 (1944). Reprinted in A. Lichnerowicz : Choix d’oeuvres mathématiques, Hermann, Paris (1982), p. 4
Lichnerowicz, A.: Sur les équations relativistes de la gravitation, Bulletin de la S.M.F. 80, 237 (1952). Available at http://www.numdam.org/item?id=BSMF_1952__80__237_0
York, J.W.: Gravitational degrees of freedom and the initial-value problem. Phys. Rev. Lett. 26, 1656 (1971)
York, J.W.: Role of conformal three-geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082 (1972)
Cotton, E.: Sur les variétés à à trois dimensions, Annales de la faculté des sciences de Toulouse Sér. 2, 1, 385 (1899). Available at http://www.numdam.org/item?id=AFST_1899_2_1_4_385_0
Damour, T.: Advanced General Relativity, lectures at Institut Henri Poincaré, Paris (2006). Available at http://www.luth.obspm.fr/IHP06/
Blanchet, L.: Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Relativity 9, 4 (2006). http://www.livingreviews.org/lrr-2006-4
Blanchet, L.: Theory of Gravitational Wave Emission, lectures at Institut Henri Poincaré, Paris (2006). Available at http://www.luth.obspm.fr/IHP06/
Shibata, M., Nakamura, T.: Evolution of three-dimensional gravitational waves: harmonic slicing case. Phys. Rev. D 52, 5428 (1995)
Baumgarte, T.W., Shapiro, S.L.: Numerical integration of Einstein’s field equations. Phys. Rev. D 59, 024007 (1999)
Bonazzola, S., Gourgoulhon, E., Grandclément, P., Novak, J.: Constrained scheme for the Einstein equations based on the Dirac gauge and spherical coordinates. Phys. Rev. D 70, 104007 (2004)
Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, New York (1973)
Nakamura, T.: 3D Numerical Relativity. In: Sasaki M. (eds) Relativistic Cosmology, Proceedings of the 8th Nishinomiya-Yukawa Memorial Symposium. Universal Academy Press, Tokyo, (1994) pp. 155
Isenberg, J.A.: Waveless Approximation Theories of Gravity, preprint University of Maryland (1978). Published in Int. J. Mod. Phys. D 17, 265 (2008). Available as http://arxiv.org/abs/gr-qc/0702113 an abridged version can be found in Ref. [16].
Wilson, J.R., Mathews, G.J.: Relativistic hydrodynamics. In: Evans, C.R., Finn, L.S., Hobill, D.W. (eds) Frontiers in numerical relativity., pp. 306. Cambridge Univ. Press, Cambridge (1989)
Isenberg, J., Nester, J.: Canonical Gravity. In: Held, A. (eds) General Relativity and Gravitation, one hundred Years after the Birth of Albert Einstein, pp. 23. Plenum Press, New York (1980)
Friedman, J.L., Uryu, K. and Shibata, M.: Thermodynamics of binary black holes and neutron stars, Phys. Rev. D 65, 064035 (2002), Erratum in Phys. Rev. D 70, 129904(E) (2004)
Cordero-Carrión, I., Ibáñez, J.M., Morales-Lladosa, J.A.: Maximal slicings in spherical symmetry: local existence and construction. J. Math. Phys. 52, 112501 (2011)
Cook, G.B., Shapiro, S.L., Teukolsky, S.A.: Testing a simplified version of Einstein’s equations for numerical relativity. Phys. Rev. D 53, 5533 (1996)
Mathews, G.J., Wilson, J.R.: Revised relativistic hydrodynamical model for neutron-star binaries. Phys. Rev. D 61, 127304 (2000)
Faber, J.A., Grandclément, P., Rasio, F.A.: Mergers of irrotational neutron star binaries in conformally flat gravity. Phys. Rev. D 69, 124036 (2004)
Oechslin, R., Uryu, K., Poghosyan, G., Thielemann, F.K.: The Influence of Quark Matter at High Densities on Binary Neutron Star Mergers. Mon. Not. Roy. Astron. Soc. 349, 1469 (2004)
Dimmelmeier, H., Font, J.A., Müller, E.: Relativistic simulations of rotational core collapse I. Methods, initial models, and code tests. Astron. Astrophys 388, 917 (2002)
Dimmelmeier, H., Font, J.A., Müller, E.: Relativistic simulations of rotational core collapse II. Collapse dynamics and gravitational radiation. Astron. Astrophys 393, 523 (2002)
Dimmelmeier, H., Novak, J., Font, J.A., Ibáñez, J.M., Müller, E.: Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations. Phys. Rev. D 71, 064023 (2005)
Saijo, M.: The collapse of differentially rotating supermassive stars: conformally flat simulations. Astrophys. J. 615, 866 (2004)
Saijo, M.: Dynamical bar instability in a relativistic rotational core collapse. Phys. Rev. D 71, 104038 (2005)
Cordero-Carrión, I., Cerdá-Durán, P., Dimmelmeier, H., Jaramillo, J.L., Novak, J., Gourgoulhon, E.: Improved constrained scheme for the Einstein equations: an approach to the uniqueness issue. Phys. Rev. D 79, 024017 (2009)
Shibata, M., Uryu, K.: Merger of black hole-neutron star binaries: nonspinning black hole case. Phys. Rev. D 74, 121503(R) (2006)
Gourgoulhon, E.: Constrained schemes for evolving the 3+1 Einstein equations, presentation at the CoCoNuT Meeting 2009 (Valencia, Spain, 4–6 November 2009). Available at http://www.mpa-garching.mpg.de/hydro/COCONUT/valencia2009/intro.php
Bucciantini, N., Del Zanna, L.: General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: the X-ECHO code. Astron. Astrophys. 528, A101 (2011)
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Gourgoulhon, É. (2012). Conformal Decomposition. In: 3+1 Formalism in General Relativity. Lecture Notes in Physics, vol 846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24525-1_7
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