Abstract
Most approaches to quantum gravity suggest that relativistic spacetime is not fundamental, but instead emerges from some non-spatiotemporal structure. This paper investigates the implications of this suggestion for the possibility of time travel in the sense of the existence of closed timelike curves in some relativistic spacetimes. In short, will quantum gravity reverse or strengthen general relativity’s verdict that time travel is possible?
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Notes
- 1.
For a recent review, see [30].
- 2.
- 3.
It certainly appeared so to me when Smeenk and I wrote Smeenk and Wüthrich ([30], Sect. 8).
- 4.
This expectation is confirmed, e.g., for physical fields in curved spacetimes, which propagate in accordance to hyperbolic wave equations ([33], Chap. 10).
- 5.
- 6.
For a recent—and optimistic—review, see [31].
- 7.
These boundaries are so-called ‘future Cauchy horizons’, i.e., boundaries of future domains of dependence of global time slides, where these domains are defined as those regions such that every past inextendible causal or timelike through any event in the region intersects the global time slice.
- 8.
- 9.
This is a paraphrase of a theorem due to [21]. More precisely, the theorem states that for any two ‘distinguishing’ (and temporally oriented) spacetimes \(\langle M, g_{ab}\rangle \) and \(\langle M', g'_{ab}\rangle \), a causal isomorphism \(\phi : M \rightarrow M'\) is a smooth conformal isometry. A bijection \(\phi : M \rightarrow M'\) is a causal isomorphism just in case for all \(p, q \in M\), p is in the chronological past of q if and only if \(\phi (p)\) is in the chronological past of \(\phi (q)\). A spacetime \(\langle M, g_{ab}\rangle \) is distinguishing just in case for all \(p, q \in M\), if the chronological past of p is identical to the chronological past of q, then \(p=q\), and if the chronological future of p is identical to the chronological future of q, then \(p=q\). A causal isomorphism \(\phi \) is a conformal isometry just in case it is a diffeomorphism and there exists a conformal factor \(\varOmega : M' \rightarrow \mathbb {R}\) such that \(\phi _*(g_{ab}) = \varOmega ^2 g'_{ab}\) with \(\varOmega \ne 0\).
- 10.
A binary relation R on a domain D is antisymmetric just in case for all \(x, y \in D\), if Rxy and Ryx, then \(x=y\).
- 11.
- 12.
An accessible textbook for both approaches to canonical quantum gravity described in this section is [7].
- 13.
See Gambini and Pullin ([7], Chap. 8).
- 14.
For a further discussion concerning the physical interpretation of these spin networks, see Wüthrich ([39], Sect. 2.1).
- 15.
Huggett et al. ([19], Sect. 2) offers a more detailed explanation of the problem and brief survey of reactions to this ‘problem of time’.
- 16.
Though they are philosophically rich in other ways [17].
- 17.
See Smeenk and Wüthrich ([30], 623) for more details.
- 18.
Strictly speaking, it is not even target space, or at least not the metric g in it, is fundamental; rather, given a general metric in the action of a theory, one obtains a quantum theory of perturbations around a coherent state, which corresponds to the classical relativistic metric ([16], Sect. 3).
- 19.
See [18].
- 20.
To articulate precisely what asymptotic flatness amounts to, and, connectedly, what it is for a system to be isolated in a background-independent theory such as GR is far from trivial and requires some unpacking, as it is offered, e.g., in Wald ([33], Sect. 11).
- 21.
For a much more detailed account of the emergence of spacetime in causal set theory, see Huggett and Wüthrich ([18], Chaps. 3, 4).
- 22.
For a more precise formulation, see Huggett and Wüthrich ([18], Chap. 4).
- 23.
See [39] for a more detailed sketch of the current state of the art.
- 24.
The latter is of course not really a ‘local’ phenomenon as it concerns the early stages of the whole cosmos; however, since the description is really of a very small universe during the first few ‘Planck times’, the description would be only of what is really a very small part of spacetime. This is indeed the remit of ‘quantum cosmology’, which thus becomes an ‘astrophysical’ theory under the present use of the term.
- 25.
[18] consider the state of the art regarding the relationship between quantum theories of gravity and GR much more fully.
- 26.
I thank the anonymous referee for pressing this conclusion. I agree that this is an important upshot of my discussion.
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Acknowledgements
I am grateful to the editors for their kind invitation and to Hajnal Andréka, Stefano Furlan, Niels Linnemann, István Németi and an anonymous referee for their comments on earlier versions of this paper and for discussions. I am also grateful to Hajnal Andréka and István Németi for their collaboration on earlier projects. But most of all, I am honoured by their friendship.
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Wüthrich, C. (2021). Time Travelling in Emergent Spacetime. In: Madarász, J., Székely, G. (eds) Hajnal Andréka and István Németi on Unity of Science. Outstanding Contributions to Logic, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-030-64187-0_19
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