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The space of light rays: Causality and L–boundary

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Abstract

The space of light rays \({\mathcal {N}}\) of a conformal Lorentz manifold \((M,{\mathcal {C}})\) is, under some topological conditions, a manifold whose basic elements are unparametrized null geodesics. This manifold \({\mathcal {N}}\), strongly inspired on R. Penrose’s twistor theory, keeps all information of M and it could be used as a space complementing the spacetime model. In the present review, the geometry and related structures of \({\mathcal {N}}\), such as the space of skies \(\varSigma \) and the contact structure \({\mathcal {H}}\), are introduced. The causal structure of M is characterized as part of the geometry of \({\mathcal {N}}\). A new causal boundary for spacetimes M prompted by R. Low, the L-boundary, is constructed in the case of 3–dimensional manifolds M and proposed as a model of its construction for general dimension. Its definition only depends on the geometry of \({\mathcal {N}}\) and not on the geometry of the spacetime M. The properties satisfied by the L–boundary \(\partial M\) permit to characterize the obtained extension \({\overline{M}}=M\cup \partial M\) and this characterization is also proposed for general dimension.

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Notes

  1. The expression of the equation of Jacobi fields depends on the sign of definition of the Riemann curvature tensor \({\mathbf {R}}\) and on the order of the arguments. So,

    $$\begin{aligned} J'' - {\mathbf {R}}\left( J,\gamma '\right) \gamma ' = 0, J'' + {\mathbf {R}}\left( \gamma ',J\right) \gamma ' = 0, J'' - {\mathbf {R}}\left( \gamma ',J\right) \gamma ' = 0 \end{aligned}$$

    are other expressions we can find in the literature.

  2. This is not an automatic property for any \({\mathcal {U}}\in {\mathcal {N}}\) and \(\overline{{\mathcal {U}}}\in \overline{{\mathcal {N}}}\). If \(U=\varSigma ({\mathcal {V}}) \) and \({\overline{U}}=\varSigma (\overline{{\mathcal {U}}})\) such that \(\varPhi \left( U\right) \subset {\overline{U}}\), then we can choose \({\mathcal {U}}=\phi ^{-1}\left( \phi \left( {\mathcal {V}}\right) \cap \overline{{\mathcal {U}}}\right) \subset {\mathcal {V}}\). Now, if \(X\subset {\mathcal {V}}\), since \(\varPhi \left( U\right) \subset {\overline{U}}\) then \(\phi \left( X\right) \subset \phi \left( {\mathcal {V}}\right) \cap \overline{{\mathcal {U}}}\) and since \(\phi \) is a diffeomorphism, then \(X\in {\mathcal {U}}\). So we have \(U=\varSigma \left( {\mathcal {U}}\right) = \varSigma \left( {\mathcal {V}}\right) \) and \(\phi \left( {\mathcal {U}}\right) \subset \overline{{\mathcal {U}}}\).

  3. This can be shown if we notice that, since U is globally hyperbolic, by [51, Thm. 3.78], there exists a diffeomorphism \(h:C\times \mathbb {R}\rightarrow U\) such that \(C_{\lambda }=h\left( C\times \{\lambda \}\right) \) is a smooth spacelike Cauchy surface in U. Without any lack of generality, we can assume that \(C=C_0\) and \(C_{-}=C_c\) for some \(c\in \mathbb {R}\). Observe that any light ray can be parametrized by \(\gamma (s)=\mathrm {exp}_{\gamma (0)}\left( s\cdot \gamma '(0)\right) \) then, if \(p_2:C\times \mathbb {R}\rightarrow \mathbb {R}\) is the canonical projection, then the equation \(p_2 \circ h^{-1}\left( \gamma (s)\right) =c\) can be written in coordinates by an equation such that \(F(x,y,\theta ,s)=c\). Then, the existence of \(s_{\gamma }=s_{\gamma }(x,y,\theta )\) follows from the Theorem of implicit function.

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Acknowledgements

The authors wish to thank two anonymous referees who provided useful comments and suggestions to improve the quality of this paper and A. Bautista would like to thank the organizing committee of the meeting Singularity theorems, causality, and all that. A tribute to Roger Penrose for its kind invitation.

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Authors and Affiliations

  1. Dpto. de Análisis Económico: Economía Cuantitativa, Univ. Autónoma de Madrid, C/ Francisco Tomás y Valiente, 5, 28049, Madrid, Spain

    A. Bautista

  2. Instituto de Ciencias Matemáticas CSIC - UAM - UC3M - UCM (ICMAT) and Dpto. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganés, Madrid, Spain

    A. Ibort

  3. Dpto. de Geometría y Topología and Instituto de Matemáticas Interdisciplinar (IMI), Univ. Complutense de Madrid, Avda. Complutense s/n, 28040, Madrid, Spain

    J. Lafuente

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Bautista, A., Ibort, A. & Lafuente, J. The space of light rays: Causality and L–boundary. Gen Relativ Gravit 54, 59 (2022). https://doi.org/10.1007/s10714-022-02942-3

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