Existence of a Cauchy surface and compactness of causally convex hulls in a spacetime

Dwaipayan Mishra; Himadri Shekhar Mondal

doi:10.5269/bspm.65472

Dwaipayan Mishra Research Scholar at Midnapore College(Autonomous)
Himadri Shekhar Mondal Vidyasagar University

Resumo

In this paper it is shown that, if a spacetime contains a Cauchy surface then the causally convex hulls of compact sets are compact. The converse is not true in general, however if the spacetime is causal then the compactness of causally convex hulls of compact sets of the spacetime ascertain the existence of a Cauchy surface. Also, the convex hull operator is denoted and some results regarding it has been explored.

Downloads

Não há dados estatísticos.

Biografia do Autor

Dwaipayan Mishra, Research Scholar at Midnapore College(Autonomous)

Department of Mathematics

Himadri Shekhar Mondal, Vidyasagar University

Department of Mathematics

Referências

Hounnonkpe, R. A., Minguzzi, E.: Globally hyperbolic spacetimes can be defined without the causal condition. Classical and Quantum Gravity 36 (19) (2019)

Minguzzi, E.: Lorentzian causality theory . Living Rev. Relativ 22 (3) (2019)

Hawking, S. W., Ellis, G. F. R.: The Large Scale Structures of Space-Time. Cambridge University Press, Cambridge (1973).

Bernal, A. N., Sanchez, M.: On smooth Cauchy hypersurfaces and Gerochs splitting theorem. Commun. Math. Phys. 243, 461-470 (2003)

Bernal, A. N., Sanchez, M.: Globally hyperbolic definition can be defined as causal instead of strongly causal. Class. Quantum Grav. 42, 745-749 (2007)

Leray, J.: Hyperbolic Differential Equations. IAS, Princeton (1952)

Beem, J. K., Ehrlich, P. E., Easley, K. L.: Global Lorentzian Geometry. New York: Marcel Dekker Inc. (1996)

Penrose, R.: Techniques of Differential Topology in Relativity, CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia: Siam (1972)

Choudhury, B. S., Mondal, H. S.: Continuous representation of a globally hyperbolic spacetime with non-compact Cauchy surfaces. Anal. Math. Phys. (2014)

Minguzzi, E.: Causality theory for closed cone structures with applications. Rev. Math. Phys 31:1930001 (2019)

Minguzzi, E.: Charecterization of some causality conditions through the continuity of the Lorentzian distance. J. Geom. Phys. 59(7), 827-833 (2009)

Geroch, R.: Domain of dependence. J. Math. Phys 11, 437-449 (1970)

Fathi, A., Siconolfi,: A. On smooth time functions. Math. Proc. Camb. Phil. Soc.152, 303-339 (2012)

Chrusciel, P. T., Grant,J. D. E., Minguzzi, E.: On differentiabilioty of volume time functions. Ann. Henri Poincare 17, 2801-2824 (2016)

Bernard, P., Suhr, S.: Lyapounov functions of closed cone fields: from Conley theory to time functions. Commun. Math. Phys. 359, 467-498 (2018)

Samann, C.: Global hyperbolicity for spacetimes with continuous metrics. Ann. Henri Poincare 17(6), 1429-1455 (2016)

Minguzzi, E., Sanchez, M.: The causal hierarchy of spacetimes. Recent developments in pseudo-Riemannian geometry. ESI Lect. Math. Phys.,EMS Pub. House, Zurich, ed. By H. Baum and D. Alekseevsky299-358 (2008)

Penrose, R.: Techniques of Differential Topology in Relativity. CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia: SIAM(1972)

Galloway, G. J.: Notes on Lorentzian causality. ESI-EMS-IAMP Summer School on Mathematical Relativity (2014)

Wald, R. M.: General Relativity.The University of Chicago Press, Chicago (1984)

Kian, M.: Epigraph of operator functions. Quaestiones Mathematicae, 39(5), 587-594, (2016).