Essay Abstract

The existence of a global time is often taken for granted but should instead be considered as a matter of investigation. By using the tools of global Lorentzian geometry I prove that, under physically reasonable conditions, the impossibility of finding a global time implies the singularity of spacetime.

Author Bio

Ettore Minguzzi is researcher of mathematical physics at the University of Florence, Italy. He has earned his PhD from Milano University in 2002. His main research interests are in general relativity and applied gauge theories. In the last years he has contributed to global Lorentzian geometry and causality theory in particular with the study of the "causal ladder of spacetimes". He is a member of SIGRAV "Società Italiana di Relatività Generale e Fisica della Gravitazione", SEGRE "Spanish Society on Relativity and Gravitation" and ISGRG "International Society on General Relativity and Gravitation".

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I inform the readers that recently, the technical papers on which this essay is based, namely reference [13] "Chronological spacetimes without lightlike lines are stably

causal" and [14] "K-causality coincides with stable causality" have received positive referee reports from Communications in Mathematical Physics (minor changes have been asked).

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Dear Dr. Minguzzi,

I read your essay, which I consider impressive, as it shows that in a Lorentz manifold the existence of singularities can be guaranteed under more relaxed circumstances than usual. I find interesting your main result concerning the existence of singularities in the absence of a global time without closed causal curves.

Best wishes,

Cristi Stoica

“Flowing with a Frozen River”,

http://fqxi.org/community/forum/topic/322

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Thank you for the appreciation Cristi, it is nice to receive some feedback. Ettore

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References [13] and [14] have now been accepted for publication in CMP.

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Dear Dr. Minguzzi,

Reading your Essay is a joy. You have this utterly professional skills to elaborate on the most intricate issues in GR -- the obvious, and deeply puzzling, fact that we observe mass with one "charge" only.

May I ask a question. You wrote: "Instead, only massless particles make sense, and thus the concept of non-singularity must be expressed from the ‘point of view’ of massless particles."

I wonder how you would explain the absence of Closed Timelike Curves (CTCs) from the ‘point of view’ of massless particles. CTCs are exactly what we should expect from GR (cf. W.B. Bonnor, Closed timelike curves in general relativity, Int. J. Mod. Phys. D12 (2003) 1705-1708; gr-qc/0211051 v1). It seems to me that CTCs (and timelike naked singularities, cf. R. Goswami et al., gr-qc/0410041v1) are just like the ultraviolet catastrophe from 1905, which too has never happened.

Thank you for your (global) time.

Dimi Chakalov

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Dear Dr. Chakalov,

thank you for the appreciation and feedback.

My opinion is that spacetimes with chronology violating regions (CTC) are worth studying and should not be dismissed a priori. Nevertheless, in this work I assume that the absence of CTC could be physically justified at least in some limit. In reference [13] I considered the case in which the spacetime admits CTC. Thanks to a result due to Kriele, the main conclusion of my work does not change if the chronology violating region is of limited size (i.e. does not extend to infinity). It is sometimes speculated that future civilizations could perhaps produce chronology violating regions but in this case these regions would probably be limited (time machines). Thus in this case the consequences of the main theorem (i.e. the existence of a time function under null geodesic completeness) would still hold.

As for the presence of timelike naked singularity, their possible presence does not change the main result of this work since the cosmic censorship conjecture is not assumed. Indeed, assuming it for instance in the strong form would be equivalent to the assumption of global hyperbolicity which implies stable causality and hence the existence of a time function. Clearly, in this case it would be trivial to deduce the existence of a time function.

Best wishes,

E. Minguzzi

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Regarding CTCs: I believe it was proven that any locally reflecting spacetime with a CTC is 'totally vicious', i.e. one with a CTC through every point (Jong-Chul Kim and Jin-Hwan Kim, Totally vicious space-times, J Math Phys 34 (1993) 2435-2439). A CTC is worse than the Ebola virus, as it is interwoven in the possible models of spacetimes permitted in GR. Yet there is no trace from any CTCs in the Cosmic Microwave Background Radiation. No trace from any timelike naked singularity either. Ergo, we might be facing the kind of "ultraviolet catastrophe" I mentioned before, and need some really huge breakthrough to explain why all these nasty things have never happened.

I wish you best of luck with your work.

Dimi Chakalov

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