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Distance to a focal point and singularity theorems with weak timelike convergence condition

Published online by Cambridge University Press:  01 March 2009

SEONG-HUN PAENG
SEONG-HUN PAENG*
Affiliation:
Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul 143-701, Korea. e-mail: [email protected]
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Abstract

In classical general relativity, the timelike convergence condition is Ric(v, v) ≥ 0 for every timelike vector v. But recent astronomical observations show that the timelike convergence condition does not hold. In this paper, we obtain an explicit upper bound of the volume form and the distance to a focal point (such as Hubble distance) along timelike geodesic with integrals of the negative part of Ricci curvature, which entails singularity theorems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 475 - 487
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Baez, J. C. and Bunn, E.The meaning of Einstein's equation. arxiv-gr-qc/0103044.Google Scholar
[2]Beem, J. K., Ehrlich, P. and Easley, K. L.Global Lorentzian Geometry. (Marcel Dekker, Inc., 1996).Google Scholar
[3]Galloway, G. J.Some results on occurence of compact minimal submanifolds. Manuscripta math. 35 (1981), 209–219.CrossRefGoogle Scholar
[4]Hawking, S. W. and Penrose, R.The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. London A314 (1970), 529–548.Google Scholar
[5]O'Neil, B.Semi-Riemannian Geometry with Applications to Relativity. (Academic Press, 1983).Google Scholar
[6]Paeng, S.-H.Volume expansion rate of Lorentzian manifold based on integral Ricci curvature over a timelike geodesic. J. Geom. Phys. 57 (2007), 1499–1503.CrossRefGoogle Scholar
[7]Petersen, P. and Wei, G.Relative volume comparison with integral curvature bounds. GAFA 7 (1997), 1031–1045.Google Scholar
[8]Riess, A. G. et al. The farthest known supernova, Astrophys. J. 560 (2001), 49–71.CrossRefGoogle Scholar
[9]Tipler, F. J.Energy conditions and spacetime singularities. Phys. Rev. D 17 (1978), 2521–2528.CrossRefGoogle Scholar
[10]Tipler, F. J.General relativity and conjugate ordinary differential equations. J. Diff. Eq. 30 (1978), 165–174.CrossRefGoogle Scholar
[11]Townsend, P. K. and Wohlfarth, M. N. R.Accelerating cosmologies from compactification. Phys. Rev. Lett. 91 (2003), 061302.CrossRefGoogle ScholarPubMed
[12]Yun, J.-G. Singularity theorems in general relativity with an averaged curvature condition, preprint.Google Scholar