Abstract
In this paper we consider a Lorentzian manifold which is globally hyperbolic with a non-compact Cauchy surface. We show that continuous representation of the spacetime is possible by causally admissible systems of its Cauchy surfaces. For that purpose we use Vietoris topology. One application is also included. The work is in the line with research on causality in relativistic spacetimes.
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References
Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243, 461 (2003)
Bernal, A.N., Sánchez, M.: Smoothness ot time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43 (2005)
Esty, N.C.: \(CL(\mathbb{R})\) is simply connected under Vietoris topology, applied general. Topology 8(2), 259–265 (2007)
Filippov, V.V.: Basic topological structures of ordinary differential equations. Kluwer Academic Publishers, Netherlands (1998)
Geroch, R.: Domain of dependence. J. Math. Phys. 11, 437 (1970)
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-time. Cambridge University Press, Cambridge (1973)
Kim, D.-H.: A note on non-compact Cauchy surfaces. Class. Quantum Gravity 25, 238002 (2008)
Kim, D.-H.: An Imbedding of Lorentzian manifolds. Class. Quantum Gravity 26, 075004 (2009)
Penrose, R.: Techniques of Differential Topology in Relativity. SIAM, Philadelphia (1972)
Vatandoost, M., Bahrampour, Y.: Some necessary and sufficient conditions for admitting a continuous sphere order representation of two-dimensional space-times. J. Math. Phys. 53, 122501 (2012)
Wald, M.: General Relativity. University of Chicago Press, Chicago (1984)
Minguzzi, E., Sanchez, M.: The causal hierarchy of spacetimes. ESI lect. Math. Phys. 299–358 (2008)
Acknowledgments
We are thankful to Indian Institute of Engineering Science and Technology for the support. We are also grateful to the learned referees for their valuable suggestions.
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Choudhury, B.S., Mondal, H.S. Continuous representation of a globally hyperbolic spacetime with non-compact Cauchy surfaces. Anal.Math.Phys. 5, 183–191 (2015). https://doi.org/10.1007/s13324-014-0093-x
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DOI: https://doi.org/10.1007/s13324-014-0093-x
Keywords
- Lorentzian geometry
- Spacetime
- Causality
- Strong causality
- Global hyperbolicity
- Cauchy surface
- Vietoris topology
- Causally admissible system