Injectivity Radius of Lorentzian Manifolds |
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Authors: | Bing-Long Chen Philippe G LeFloch |
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Institution: | (1) Department of Mathematics, Sun Yat-Sen University, 510275 Guang-Zhou, People’s Republic of China;(2) Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université de Paris VI, 4 Place Jussieu, 75252 Paris, France |
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Abstract: | Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural
curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of
a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically
associate with a given observer (p, T) –where p is a point of the manifold and T is a future-oriented time-like unit vector prescribed at p only. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the
reference Riemannian metric, and then express them in terms of the Lorentzian metric. In the context of general relativity,
our estimate on the injectivity radius of an observer should be useful to investigate the regularity of spacetimes satisfying
Einstein field equations. |
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Keywords: | |
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