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3+1 Formalism in General Relativity

Part of the book series: Lecture Notes in Physics ((LNP,volume 846))

Abstract

The 3+1 formalism is an approach to general relativity that relies on the slicing of the four-dimensional spacetime by three-dimensional surfaces (hypersurfaces). These hypersurfaces have to be spacelike, so that the metric induced on them by the Lorentzian spacetime metric [signature \((-,+,+,+)\)] is Riemannian [signature \((+,+,+)\)]. From the mathematical point of view, this procedure allows to formulate the problem of resolution of Einstein equations as a Cauchy problem with constraints. From the pedestrian point of view, it amounts to a decomposition of spacetime into “space” + “time”, so that one manipulates only time-varying tensor fields in some “ordinary” three-dimensional space, where the scalar product is Riemannian. One should stress that this space + time splitting is not some a priori structure of general relativity but relies on the somewhat arbitrary choice of a time coordinate. The 3 + 1 formalism should not be confused with the 1+3 formalism (cf. e.g. Ref. [1]), where the basic structure is a congruence of one-dimensional curves (mostly timelike curves, i.e. worldlines), instead of a family of three-dimensional surfaces.

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Notes

  1. 1.

    These three persons have some direct affiliation: Georges Darmois was the thesis adviser of André Lichnerowicz, who was himself the thesis adviser of Yvonne Choquet-Bruhat.

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Authors and Affiliations

  1. Lab. Univers et Théories (LUTH) UMR 8102 du CNRS, Observatoire de Paris, Université Paris Diderot, place Jules Janssen 5, 92190, Meudon, France

    Éric Gourgoulhon

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  1. Éric Gourgoulhon
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Gourgoulhon, É. (2012). Introduction. 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_1

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  • DOI: https://doi.org/10.1007/978-3-642-24525-1_1

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  • Print ISBN: 978-3-642-24524-4

  • Online ISBN: 978-3-642-24525-1

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