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Convex Regions of Stationary Spacetimes and Randers Spaces. Applications to Lensing and Asymptotic Flatness

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Abstract

By using stationary-to-Randers correspondence (SRC, see Caponio et al. in Rev Mat Iberoamericana 27:919–952, 2011), a characterization of light and time-convexity of the boundary of a region of a standard stationary \((n+1)\)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler \(n\) or \((n+1)\)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of hypersurfaces \(S^{n-1}(r)\times \mathbb {R} \), where \(S^{n-1}(r)\) is a sphere of large radius in a spacelike section of an end, as well as the characterization of their time-convexity with natural physical interpretations. The lens effect of both light rays and freely falling massive particles with a finite lifetime, (i.e., the multiplicity of such connecting curves) is characterized in terms of the focalization of the geodesics in the underlying Randers manifolds.

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Notes

  1. Recall that these techniques were initiated in [11] with the introduction of time and light convexity in the static case. This case becomes quite simpler, as it is related to Riemannian instead of properly Finslerian metrics, see [6]. The techniques also apply to periodic trajectories and other Lorentzian variational problems on convex domains (see the subtleties in [7] and references therein).

  2. It is worth pointing out that a second natural notion of forward and backward Cauchy sequence can be given, see [28, Section 3.2.2]. This notion is not equivalent to that stated above, but it yields equivalent (forward and backward) Cauchy completions and, thus, equivalent notions of completeness.

  3. Usually, one has to assume that the hypersurface is also non-degenerate but, since our definition (recall Remark 2.1) does not involve the second fundamental form of \(H\), the non-degeneracy assumption can be dropped (cf. [17, Remark 3]). Clearly, in the non-degenerate case, one recovers the usual condition about the sign of the second fundamental form.

  4. Notice that, for lightlike geodesics, the construction of the Fermat metric was conformally invariant and, so, the elements \(h, \omega \) where normalized so that \(\beta \) could be regarded as an overall conformal factor, eventually equal to 1. However, this conformal invariance does not hold for timelike geodesics, and it is emphasized by means of the subscript \(\beta \).

  5. Recall that, for example, in order to have a well defined, unique and non-necessarily vanishing ADM mass of a \(3\) dimensional Riemannian manifold \((S,h)\), the decay rate of \(h\)—which involves also the Ricci tensor—must be of order not less than \(1/2\) and not greater than \(1\), [2, Theorems 4.2 and 4.3]).

  6. Even though this excludes the presence of electromagnetic fields, it is justified as it allows one to make simpler statements, leaving to the reader the task to determine precise fall-off hypotheses for the energy-momentum tensor (compare, e.g., with [33, Section 2.3]) whenever the vacuum assumption is dropped.

  7. See, about the difficulty of this problem, the claim on its over-determinacy in [48], as well as the construction of a solution involving Kerr–Newman spacetime [47] (recall that the latter is not vacuum around \(\mathcal {J}^\pm \)).

  8. The result follows even if we allow here and in (46) \(q'=0\).

  9. Following the detailed discussion and nomenclature about Kerr in [56, p. 209], such geodesics are either (ordinary) bounded orbits or (exceptional) crash-crash ones. Recall that the ordinary orbits of Kerr geodesics are bounded (those with two \(r\)-turning points, one of them a maximum), flyby (one \(r\)-turning point, necessarily a minimum) or transit (no \(r\)-turning points). The exceptional orbits are spherical ones (i.e., \(r(s)\equiv r_{0}\)), asymptotic orbits to a spherical one, crash-escape orbits and crash-crash ones. None of these have a \(r\)-turning point \(r_{0}\) except the crash-crash one (as in the example of the pebble), where \(r_{0}\) is a maximum.

  10. Recall that [3, Theorem 1.3] is stated under \(C^{2,1}_\mathrm{loc}\) regularity, but the result holds also for a \(C^2\) domain (see [17]).

  11. One could try to improve this hypothesis by replacing it with some condition intrinsic to \(D\) (i.e., independent of if it is regarded as a domain of a bigger manifold) or relaxing the hypotheses of smoothness on \(\partial D\). Nevertheless, such conditions are rather technical even in the Riemannian case [5] (and, thus, even in the simple static case). Moreover, the non-trivial relation between the symmetrized distance \(d_{s}\) and the generalized distance \(d\) in a Finsler manifold, complicates more the situation—recall, for example, that \(d_{s}\) may not be a length metric, which must be taken into account in the picture of the intrinsic Cauchy boundary, see [28, Remark 3.23].

  12. We use standard terminology as in [23].

  13. However, one could still find a result of multiplicity in purely causal terms by using timelike homotopy classes as in [62], which allows some sharp conclusions on the behavior of the geodesics.

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Acknowledgments

We would like to thank R. Bartolo for several discussions on a preliminary version of this work. We also thank the referee for her/his interesting comments and questions. EC and AVG are partially supported by PRIN2009 “Metodi variazionali ed applicazioni allo studio di equazioni differenziali nonlineari”. MS is partially supported by Spanish MTM2010-18099 (MICINN) and P09-FQM-4496 (Junta de Andalucía) grants, both with FEDER funds. This research is a result of the activity developed within the Spanish-Italian Acción Integrada HI2008.0106/Azione Integrata Italia-Spagna IT09L719F1.

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Authors and Affiliations

  1. Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy

    Erasmo Caponio

  2. Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125, Bari, Italy

    Anna Valeria Germinario

  3. Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071, Granada, Spain

    Miguel Sánchez

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  1. Erasmo Caponio
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Caponio, E., Germinario, A.V. & Sánchez, M. Convex Regions of Stationary Spacetimes and Randers Spaces. Applications to Lensing and Asymptotic Flatness. J Geom Anal 26, 791–836 (2016). https://doi.org/10.1007/s12220-015-9572-z

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