Abstract
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.
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References
Aazami A., Javaloyes M.A. (2014): Penrose’s singularity theorem in a Finsler spacetime. Preprint, arxiv:1410.7595.
Bailleul I. (2011): A probabilistic view on singularities. J. Math. Phys., 52, 023520.
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.
Bernal A.N., Sánchez M. (2003): On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Comm. Math. Phys., 243, pp. 461–470.
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.
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.
Bray H. (2003) Black holes, geometric flows, and the Penrose inequality in general relativity. Not. Am. Math. Soc. 49: 1372–1381
Chrusciel P., Galloway G.J., Pollack D. (2010) Mathematical general relativity: a sampler. Bull. Am. Math. Soc. (N.S.) 47(4): 567–638
Do Carmo M.P. (1992): Riemannian geometry. Mathematics: Theory and Applications, Birkhäuser Boston Inc., Boston, MA.
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.
Geroch R. (1970): Domain of dependence. J. Mathematical Phys., 11, pp. 437–449.
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.
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.
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.
Müller O., Sánchez M. (2014): An invitation to Lorentzian Geometry. Jahresbericht der Deutschen Mathematiker-Vereinigung 115 No 3–4: 153–183.
Myers S.B. (1941) Riemannian manifolds with positive mean curvature. Duke Mathematical Journal 8(2): 401–404
O’Neill B. (1983): Semi-Riemannian geometry with applications to Relativity. Academic Press Inc., New York.
Senovilla, J.M.M. (1997): Singularity Theorems and Their Consequences. Gen. Relat. Grav. 29, No. 5 701-848.
Senovilla J.M.M., Garflinke D. (2015): The 1965 Penrose singularity theorem. Class. Quantum Grav. 32, 124008, 45pp.
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Morales Álvarez, P., Sánchez, M. Myers and Hawking Theorems: Geometry for the Limits of the Universe. Milan J. Math. 83, 295–311 (2015). https://doi.org/10.1007/s00032-015-0241-2
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DOI: https://doi.org/10.1007/s00032-015-0241-2