Abstract
Inspired by Raychaudhuri’s work, and using the equation named after him as a basic ingredient, a new singularity theorem is proved. Open non-rotating Universes, expanding everywhere with a non-vanishing spatial average of the matter variables, show severe geodesic incompletness in the past. Another way of stating the result is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies.
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The first instance of a non-singular cosmological solution, compatible with the energy and causality conditions and the Raychaudhuri equation, was given by Senovilla. This generated an intense interest in AKR on the nature and behaviour of such solutions. His initial proposal of the vanishing of the space-time average of the energy density (and other physical and kinematical quantities) as a condition for their existence got refined through many discussions with Dadhich and Senovilla. It evolved finally to the realization that the vanishing of the spatial average of the energy density is what distinguishes all these non-singular solutions, as stated in this theorem proved by Senovilla. In our opinion, this theorem, to which both Raychaudhuri and Senovilla have contributed, provides a succinct and complete characterization of the nature of all non-singular cosmological models — Editors.
In memory of Amal Kumar Raychaudhuri (1924–2005).
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Senovilla, J.M.M. A singularity theorem based on spatial averages. Pramana - J Phys 69, 31–47 (2007). https://doi.org/10.1007/s12043-007-0109-2
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DOI: https://doi.org/10.1007/s12043-007-0109-2