Myers and Hawking Theorems: Geometry for the Limits of the Universe
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It is known that the celebrated theorem by Hawking which assures the existence of a Big-Bang under physically motivated hypotheses, uses geometric ideas inspired in classical Myers theorem. Our aim here is to go a step further: first, a result which can be interpreted as the exact analogy in pure Riemannian geometry to Hawking theorem will be proven and, then, the isomorphic role of the hypotheses in both theorems will be analyzed. This will provide some interesting links between Riemannian and Lorentzian geometries, as well as an introduction to the latter.
The reader interested only in Riemannian Geometry can regard this new result as a simple application of Myers theorem combined with the properties of focal points. However, readers with broader perspectives will learn that when a geometer thinks in our space as a complete Riemannian manifold, a relativist may think in our spacetime as predictable, or that suitable bounds on the Ricci tensor will force geodesics either to converge in the space or to join in the time. Moreover, the limitation of the distance from any point to a hypersurface in a Riemannian manifold, may turn out into a catastrophic relativistic limit for the duration of our physical Universe.
Mathematics Subject ClassificationPrimary 53C50 53C20 Secondary 53C21 83C75
KeywordsFocal points expanding hypersurface positive Ricci curvature singularity theorems separating and Cauchy hypersurfaces timelike convergence
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- 1.Aazami A., Javaloyes M.A. (2014): Penrose’s singularity theorem in a Finsler spacetime. Preprint, arxiv:1410.7595.
- 2.Bailleul I. (2011): A probabilistic view on singularities. J. Math. Phys., 52, 023520.Google Scholar
- 3.Beem J.K., Ehrlich P.E., Easley K.L. (1996): Global Lorentzian geometry. Vol. 202 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, second edition.Google Scholar
- 4.Bernal A.N., Sánchez M. (2003): On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Comm. Math. Phys., 243, pp. 461–470.Google Scholar
- 5.Bernal A.N., Sánchez M. (2005): Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys., 257, pp. 43–50.Google Scholar
- 6.Bernal A.N., Sánchez M. (2007): Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’. Classical Quantum Gravity, 24, pp. 745–749.Google Scholar
- 9.Do Carmo M.P. (1992): Riemannian geometry. Mathematics: Theory and Applications, Birkhäuser Boston Inc., Boston, MA.Google Scholar
- 10.Galloway G. J., Senovilla J. M. M. (2010): Singularity theorems based on trapped submanifolds of arbitrary co-dimension, Classical Quantum Gravity, 27, no. 15, 152002.Google Scholar
- 11.Geroch R. (1970): Domain of dependence. J. Mathematical Phys., 11, pp. 437–449.Google Scholar
- 12.Hawking S.W., Ellis G.F.R. (1973): The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London - New York.Google Scholar
- 13.Javaloyes M.A., Sánchez M. (2010): An introduction to Lorentzian geometry and its applications. Editorial Universidad de Sao Paulo, Sao Paulo, Brasil. ISBN: 978-85-7656- 1.Google Scholar
- 14.Minguzzi E., Sánchez M. (2008): The causal hierarchy of spacetimes. Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, pp. 299–358.Google Scholar
- 15.Müller O., Sánchez M. (2014): An invitation to Lorentzian Geometry. Jahresbericht der Deutschen Mathematiker-Vereinigung 115 No 3–4: 153–183.Google Scholar
- 17.O’Neill B. (1983): Semi-Riemannian geometry with applications to Relativity. Academic Press Inc., New York.Google Scholar
- 18.Senovilla, J.M.M. (1997): Singularity Theorems and Their Consequences. Gen. Relat. Grav. 29, No. 5 701-848.Google Scholar
- 19.Senovilla J.M.M., Garflinke D. (2015): The 1965 Penrose singularity theorem. Class. Quantum Grav. 32, 124008, 45pp.Google Scholar