Abstract
What is the analogous notion of Gromov–Hausdorff convergence for sequences of spacetimes? Since a Lorentzian manifold is not inherently a metric space, one cannot simply use the traditional definition. One approach offered by Sormani and Vega (Class Quant Gravity, 33:085001, 2016) is to define a metric space structure on a spacetime by means of the null distance. Then one can define convergence of spacetimes using the usual definition of Gromov–Hausdorff convergence. In this paper we explore this approach by giving many examples of sequences of warped product spacetimes with the null distance converging in the Gromov–Hausdorff sense. In addition, we give an optimal convergence theorem which shows that under natural geometric hypotheses a sequence of warped product spacetimes converge to a specific limiting warped product spacetime. The examples given further serve to show that the hypotheses of this convergence theorem are optimal.
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Acknowledgements
The author would like to thank Piotr T. Chruściel, Melanie Graf, Michael Kunzinger, Ettore Minguzzi, and Roland Steinbauer, the organizers of the Non-regular Spacetime Geometry workshop at the Erwin Schrödinger Institute, for the invitation to participate and speak at this wonderful workshop. While at the workshop the author was reminded of the desire to explore a VADB type theorem for the null distance and this paper would not have happened without it.
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Allen, B. Null distance and Gromov–Hausdorff convergence of warped product spacetimes. Gen Relativ Gravit 55, 118 (2023). https://doi.org/10.1007/s10714-023-03167-8
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DOI: https://doi.org/10.1007/s10714-023-03167-8