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Maximal extension of conformally flat globally hyperbolic space–times

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Abstract

The notion of maximal extension of a globally hyperbolic space–time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein’s equations of general relativity. Choquet-Bruhat and Geroch proved (Commun Math Phys 14:329–335, 1969) that if the Cauchy problem has a local solution, this solution has a unique maximal extension. Since the causal structure of a space–time is invariant under conformal changes of metrics we may generalize this notion of maximality to the conformal setting. In this article we focus on conformally flat space–times of dimension greater or equal than 3. In this case, by a Lorentzian version of Liouville’s theorem, these space–times are locally modeled on the Einstein space–time. In the first part of the article we use this fact to prove the existence and uniqueness of the maximum extension for globally hyperbolic conformally flat space–times. In the second part we give a causal characterization of globally hyperbolic conformally flat maximal space–times whose developing map is a global diffeomorphism.

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Notes

  1. Our convention is to consider the zero vector as a spacelike vector. In particular, a causal or lightlike vector is non zero.

  2. In fact, as explained in [21], the definition of limit curve has been fitted in order to have this convergence property for globally hyperbolic space–times.

  3. The set \(A=\{x\in \mathbb {R}^{1,n} \ :\ \Vert x\Vert _{1,n}=-1\}\) is a closed achronal edgeless subset of \(\mathbb {R}^{1,n}\). But it is not a Cauchy hypersurface, because no lightlike straight lines going through the origin intersect \(A\).

  4. For intellectual satisfaction we should also mention that this condition is also necessary for strongly causal space–times. More precisely: in Hawking et al. [15] have shown that if \(g\) and \(g'\) define the same causal structure over \(M\) and if this structure is strongly causal, then \(g\) and \(g'\) are conformally equivalent. The hard part of their proof is to show that every homeomorphism which preserves the causal structure of a strongly causal space–time is differentiable. Starting from that it is easy to show that the identity is a conformal map.

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Acknowledgments

I want to thanks my Ph.D. advisor Thierry Barbot, from University of Avignon, and the professor Virginie Charette, from University of Sherbrooke, for the suggestions and the helpful re-reading of this paper.

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  1. Département de Mathématiques, Université Paris-Sud, 91405, Orsay Cedex, France

    Clara Rossi Salvemini

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  1. Clara Rossi Salvemini
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Salvemini, C.R. Maximal extension of conformally flat globally hyperbolic space–times. Geom Dedicata 174, 235–260 (2015). https://doi.org/10.1007/s10711-014-0015-y

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