Two New Conditions for the Occurrence of Krolak Strong Curvature
Ben Whale
(Australian National University)
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It is sometimes stated that the only reason for believing that singularities occur in gravitational collapse is the singularity theorems [1, 2, 4]. Given this, investigating the physical consequences of singularity theorems is an important part of understanding gravitational collapse and singularity formation. Tipler and Krolak's strong curvature conditions provide a link between the sometimes very geometric assumptions of singularity theorems and physical behaviour. Unfortunately the Krolak [3] and Tipler [5] conditions make statements about geodesic divergence at every point along a geodesic, while the singularity theorems make a statement only about the geodesic divergence at a single point on a geodesic. If one wishes to use the Krolak and Tipler conditions to describe physical behaviour in singularity formation we need to overcome this difference. We give two necessary and sufficient conditions for a timelike geodesic to satisfy the Krolak strong curvature condition in terms of the geodesic divergence from a single point. These results are the first step towards providing a concrete path to linking the formation of singularities in gravitational collapse to curvature behaviour. 1. Bizon, P., E. Malec, and N. O. Murchadha, Trapped surfaces due to concentration of matter in spherically symmetric geometries. Classical and Quantum Gravity, 1989. 6(7), p. 961-976.
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