Hausdorff closed limits and the causal boundary of globally hyperbolic spacetimes with timelike boundary
We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopte...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
06.11.2018
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Subjects | |
Online Access | Get full text |
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Summary: | We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally
hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a
metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff
himself in the 1950's in set theory, can be advantageously adopted on the
Geroch-Kronheimer-Penrose causal completion of M, retaining essentially all the
good properties of previous topologies in this ambient. In particular, we show
that if the globally hyperbolic spacetime $M$ admits a conformal boundary,
defined in such broad terms as to include all the standard examples in the
literature, then the latter is homeomorphic to the causal boundary endowed with
the CLT. We also discuss how our recent proposal arXiv:1807.00152 for a
definition of null infinity using only causal boundaries can be translated when
using the CLT, simplifying a number of technical aspects in the pertinent
definitions. In a more technical vein, in the appendix we discuss the
relationship of the CLT with the more generally applicable (but not always
Hausdorff) chronological topology, and show that they coincide exactly in those
cases when the latter happens to be Hausdorff. |
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DOI: | 10.48550/arxiv.1811.02670 |