Abstract
There exists a growing literature on the so-called physical Church–Turing thesis in a relativistic spacetime setting. The physical Church–Turing thesis is the conjecture that no computing device that is physically realizable (even in principle) can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, Németi and Dávid (Appl Math Comput 178:118–142, 2006) have shown how an appreciation of the physical Church–Turing thesis necessitates the confluence of mathematical, computational, physical, and indeed cosmological ideas. In this essay, I will honour István’s seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in Németi and Dávid (Appl Math Comput 178:118–142, 2006) might be complemented by a quantum-information-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo–Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church–Turing thesis.
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Notes
And hence, the argument presented in this essay is not touched by the complaints by Shagrir and Pitowsky (2003, p. 94).
Cf. also Németi and Dávid (2006, §3).
A Cauchy surface is a spacelike hypersurface \(\Sigma \) such that every timelike or null curve without endpoints intersects \(\Sigma \) exactly once.
Wald (1984), Theorem 8.3.14.
For a more detailed treatment of Kerr–Newman spacetime in the context of relativistic computation, cf. Németi and Dávid (2006, §2); for an accessible entry to the main ideas, cf. Németi and Andréka (2006). For systematic treatments of Kerr spacetimes in GR, cf. O’Neill (1995) and Wiltshire et al. (2009).
Cf. Earman (1995, §4.8) and Németi and Dávid (2006, §5) for an assessment of these potential obstacles. Cf. also the earlier Earman and Norton (1993), and the recent Manchak (2010). I concur with Manchak’s assessment that the physical reasonableness of Malament–Hogarth spacetimes is yet to be settled. As to whether this means that a relativistic computer is physically reasonable, cf. Piccinini (Piccinini (2011), §4.2). Piccinini accepts that relativistic computers are in principle possible (although computations in our Kerr–Newman computer are not repeatable), but nevertheless concludes that “for now and the foreseeable future, relativistic hypercomputers do not falsify [the physical Church–Turing thesis].” (759) On the question of repeatability, cf. also Andréka et al. (2009, §5).
This is a technical term describing the radial stretching and the tangential squeezing acting on our traveller.
Cf. Wüthrich (1999, §3.4) for a precise calculation of the size of the tidal forces encountered by such a traveller.
Németi and Dávid consider these ideas to solve distinct though related aspects of the problem. I will somewhat carelessly run these together. For further ideas of how to solve this problem, cf. Andréka et al. (2009, §4.1).
In keeping with my earlier gender choices, I invert the nomenclature of Xian-Hui and You-Gen (2005).
Cf. Nielsen and Chuang (2000, §1.3.7) for an excellent recapitulation of quantum teleportation.
Cf. Peres and Terno (2004, §2.E) for a more rigorous statement of the results and a commented list of references for the specific results.
As was asserted by a referee.
One might worry at this point, as did a referee, that the usual conceptualization of entanglement between states in algebraic QFT required that their algebras commute and hence that the notion of timelike entanglement is meaningless for non-commuting algebras in timelike-separated spacetime regions, or—worse—that such entanglement may not exist. I cannot easily allay this concern, but only remark that if the concept of entanglement presupposes that the algebras predicated of the respective spacetime regions commute, then it would be a conceptual truth that entanglement cannot be used for signalling between these regions, quite regardless of how they are causally related. It is clear that this could not be justified on the basis of special-relativistic prohibitions of superluminal signalling alone. In what follows, I will simply assume—perhaps against better advice—that the possibility of entanglement-based ‘timelike signalling’ remains open.
Cf. the excellent survey in Berkovitz (2008), particularly §7.
Cf. §7.2 and §7.3 of Berkovitz (2008), respectively.
Cf. Butterfield et al. (1993).
Cf. also Peacock (2009).
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I am grateful to István Németi and Hajnal Andréka for discussions on the topic and for their willingness to share old material from their personal archive. I thank Gergely Székely for his great patience with my procrastinating, Joseph Berkovitz and Kent Peacock for correspondence, and the audience at the István-Fest, as well as John Dougherty and the referees for this journal for comments.
To István Németi on the occasion of his 70th birthday.
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Wüthrich, C. A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer. Synthese 192, 1989–2008 (2015). https://doi.org/10.1007/s11229-014-0502-6
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DOI: https://doi.org/10.1007/s11229-014-0502-6