Global Hyperbolicity in Space-Time Manifold

Mohajan, Haradhan (2016): Global Hyperbolicity in Space-Time Manifold. Published in: International Journal of Professional Studies , Vol. 1, No. 1 (30 June 2016): pp. 14-30.

Abstract

Global hyperbolicity is the most important condition on causal structure space-time, which is involved in problems as cosmic censorship, predictability etc. An open set O is said to be globally hyperbolic if, i) for every pair of points x and y in O the intersection of the future of x and the past of y has compact closure i.e., a space-time is said to be globally hyperbolic if the sets are compact for all (i.e., no naked singularity can exist in space-time topology), and ii) strong causality holds on O i.e., there are no closed or almost closed time like curves contained in O. Here is causal future and is the causal past of an event x. If a space-time is timelike or null geodesically incomplete but cannot be embedded in a larger space-time then we say that it has a singularity. An attempt is taken here to discuss global hyperbolicity and space-time singularity by introducing definitions, propositions and displaying diagrams appropriately.

• Item Type: MPRA Paper
• Language: English
• Keywords: Cauchy surface, causality, global hyperbolicity, space-time manifold, space-time singularities.
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C30 - General

References

Bernal, A.N. and Sánchez, M. (2003), On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem, Communications in Mathematical Physics, 243(3): 461–470.

Bernal, A.N. and Sánchez, M. (2005), Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes, arXiv:gr-qc/0401112v3 15.

Clarke, C.J.S. (1986), Singularities: Global and Local Aspects in Topological Properties and Global Structure of Space-time (ed. P.G. Bergmann and V. de. Sabbata), Plenum Press, New York.

Geroch, R.P. (1970), Domain of Dependence, Journal of Mathematical Physics, 11: 437–449.

Geroch, R.P.; Kronheimer, E.H. and Penrose, R. (GKP) (1972), Ideal Points in Space-time, Proceedings of the Royal Society, London, A 237: 545–567.

Hawking, S.W. (1966a), The Occurrence of Singularities in Cosmology I, Proceedings of the Royal Society, London, A294: 511–521.

Hawking, S.W. (1966b), The Occurrence of Singularities in Cosmology II, Proceedings of the Royal Society, London, A295: 490–493.

Hawking, S.W. (1994), Classical Theory, arXiv:hep-th/9409195v1 30 Sep 1994.

Hawking, S.W. and Ellis, G.F.R. (1973), The Large Scale Structure of Space-time, Cambridge University Press, Cambridge.

Hawking, S.W. and Penrose, R. (1970), The Singularities of Gravitational Collapse and Cosmology, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 314: 529–548.

Hawking, S.W. and Sachs, R.K. (1974), Causally Continuous Spacetimes, Communications in Mathematical Physics, 35: 287–296.

Joshi, P.S. (1996), Global Aspects in Gravitation and Cosmology, 2nd Edition, Clarendon Press, Oxford.

Kruskal, M.D. (1960), Maximal Extension of Schwarzschild Metric, Physical Review, 119(5): 1743–1745.

Leray, J. (1953), Hyperbolic Differential Equations, The Institute for Advanced Study, Princeton, N. J.

Minguzzi, E. and Sánchez, M. (2008), The Causal Hierarchy of Spacetimes, arXiv:gr-qc/0609119v3.

Mohajan, H.K. (2013a), Friedmann, Robertson-Walker (FRW) Models in Cosmology, Journal of Environmental Treatment Techniques, 1(3): 158–164.

Mohajan, H.K. (2013b), Schwarzschild Geometry of Exact Solution of Einstein Equation in Cosmology, Journal of Environmental Treatment Techniques, 1(2): 69–75.

Mohajan, H.K. (2013c), Singularity Theorems in General Relativity, M. Phil. Dissertation, Lambert Academic Publishing, Germany.

Mohajan, H.K. (2015), Basic Concepts of Differential Geometry and Fibre Bundles, ABC Journal of Advanced Research, 4(1): 57–73.

Penrose, R. (1972), Techniques of Differential Topology in Relativity, A.M.S. Colloquium Publications, SIAM, Philadelphia.

Sachs, R.K. and Wu, H. (1977), General Relativity and Cosmology, Bulletin of the American Mathematical Society, 83(6): 1101–1164.

Sánchez, M. (2010), Recent Progress on the Notion of Global Hyperbolicity, arXiv:0712.1933v2 [gr-qc] 25 Feb 2010.

Szekeres, P. (1960), On the Singularities of a Riemannian Manifold, Publicationes Mathematicae, Debrecen, 7: 285–301.

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