The Journal of Geometric Analysis

, Volume 26, Issue 2, pp 791–836 | Cite as

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

  • Erasmo Caponio
  • Anna Valeria Germinario
  • Miguel Sánchez
Article

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.

Keywords

Stationary spacetime Finsler manifold Randers metric  Convex boundary Timelike and lightlike geodesics  Gravitational lensing Asymptotic flatness 

Mathematics Subject Classification

53C50 53C60 53C22 58E10 83C30 

Notes

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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Copyright information

© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  • Erasmo Caponio
    • 1
  • Anna Valeria Germinario
    • 2
  • Miguel Sánchez
    • 3
  1. 1.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly
  3. 3.Departamento de Geometría y Topología, Facultad de CienciasUniversidad de GranadaGranadaSpain

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