Abstract
Consider a set M equipped with a structure ∗. We call a natural topology T ∗, on (M, ∗), the topology induced by ∗. For example, a natural topology for a metric space (X, d) is a topology T d induced by the metric d, and for a linearly ordered set (X, <), a natural topology should be the topology T < that is induced by the order < . This fundamental property, for a topology to be called “natural,” has been largely ignored while studying topological properties of spacetime manifolds (M, g), where g is the Lorentz “metric,” and the manifold topology T M has been used as a natural topology, ignoring the spacetime “metric” g. In this survey, we review critically candidate topologies for a relativistic spacetime manifold, and we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Göbel, Hawking-King-McCarthy and others, and we examine what should be meant by the term “natural topology” for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Gao–Wald type of “time dilation” are such examples.
“Time stays long enough for anyone who will use it.”—Leonardo da Vinci
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Notes
- 1.
The term metric, for the Lorentz tensor field, is an abuse of language, as was also pointed by Zeeman in [29], but it is so widely used that we will put it in quotes, in order to distinguish from the Riemannian metric.
- 2.
- 3.
For example, Nada, Agarwal, Shrivastava, Dossena and Williams; for a complete list of names and articles, see [26].
- 4.
“Better” in a topological sense: that is, topologies easier to work with and rich in topological properties.
- 5.
For a short survey, see Section 5, from [13].
- 6.
We refer, again, to [29] for a rigorous proof.
- 7.
For more details, and Thorne’s arguments, read Section 3, from [27].
- 8.
For a critical survey on this discussion, we refer to [18].
- 9.
Choros stands for space, in Greek, like chronos stands for time.
- 10.
Here, the word “cone” is used in a generalized sense, i.e. it is a cone on \(I \times \mathbb {S}^{n-2}\) in Minkowski space \(\mathcal {M}^n\).
- 11.
Indeed, there are solutions of the Einstein’s field equation in general relativity, which imply an extreme tilt of the light cones that lead, for example, to CTCs: independently of whether there exists a chronology protection mechanism in a more general frame, something that was conjectured by Hawking, or if such solutions are once accepted (see [27]), we should underline that our discussion lies within the scope of general relativity and not where the theory collapses within a singularity.
- 12.
See [17] for an introductory discussion on this particular problem.
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Acknowledgements
The author wishes to thank Rolf Suabedissen, from Oxford, for being kind to reply to our topological questions, even if they were elementary; the author is grateful for his valuable time and for the collegiality. Infinite thanks to Andreas Boukas for sharing thoughts on quantum gravity, some of which are incorporated in the last section.
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Papadopoulos, K. (2021). Natural vs. Artificial Topologies on a Relativistic Spacetime. In: Rassias, T.M., Pardalos, P.M. (eds) Nonlinear Analysis and Global Optimization. Springer Optimization and Its Applications, vol 167. Springer, Cham. https://doi.org/10.1007/978-3-030-61732-5_18
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