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Closed Timelike Curves, Singularities and Causality: a Survey from Gödel to Chronological Protection

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universe Review Closed Timelike Curves, Singularities and Causality: A Survey from Gödel to Chronological Protection Jean-Pierre Luminet Review Observatoire de Paris (LUTH), Centre de Physique Théorique (CPT), Laboratoire d’Astrophysique de Closed Timelike Curves,Marseille (LAM), Singularities Aix-Marseille Universit andé, CNRS, Causality: LAM, 38 rue F. A Joliot-Curie, Survey 13013 Marseille, France; from Gödel to [email protected] Protection Abstract: I give a historical survey of the discussions about the existence of closed timelike curves in Jean-Pierre Luminet general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying Observatoire de Paris (LUTH), Centre de Physique Théorique (CPT), Laboratoire d’Astrophysique de Marseille (LAM),into Aix-Marseille the past Université, is intimately CNRS, LAM, linked 38 rue to F. Joliot-Curie, spacetime 13013 models Marseille, devoid France; of timelike singularities. Since such [email protected] arise as an inevitable consequence of the equations of general relativity given physically Abstract:reasonable I give a historical assumptions, survey of time the discussions travel in theabout past the becomesexistence of possible closed timelike only curves when one or another of these in generalassumptions relativistic ismodels violated. of the Ituniverse, is the caseopening with the wormhole-typephysical possibility solutions.of time travel S. in Hawking the and other authors past, haveas first triedrecognized to “save” by K. Gödel the paradoxical in his rotating consequences universe model of 1949. time I travelemphasize in thethat pastjour- by advocating physical neyingmechanisms into the past is ofintimately chronological linked to spacet protection;ime models however, devoid of such timelike mechanisms singularities. remainSince presently unknown, such singularities arise as an inevitable consequence of the equations of general relativity given physicallyeven reasonable when quantum assumptions, fluctuations time travel nearin the horizonspast becomes are possible taken only into when account. one or I an- close the survey by a brief otherand of these pedestrian assumptions discussion is violated. ofIt is Causal the case Dynamicalwith wormhole-type Triangulations, solutions. S. anHawking approach and to quantum gravity in otherwhich authors causalityhave tried playsto “save” a seminalthe paradoxical role. consequences of time travel in the past by ad- vocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close Keywords: time travel; closed timelike curves; singularities; wormholes; Gödel’s universe; chrono- the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantumlogical gravity protection; in which causal causality dynamical plays a seminal triangulations role. Keywords: time travel; closed timelike curves; singularities; wormholes; Gödel’s universe; chron- ological protection; causal dynamical triangulations Citation: Luminet, J.-P. Closed 1. Introduction TimelikeCitation: Curves, Luminet, Singularities J.-P. Closed and In 1949, the mathematician and logician Kurt Gödel, who had previously demon- Causality:Timelike ACurves, Survey Singularities from Gödel and to 1. Introduction Causality: A Survey from Gödel to strated the incompleteness theorems that broke ground in logic, mathematics, and philoso- Chronological Protection. Universe In 1949, the mathematician and logician Kurt Gödel, who had previously demon- Chronological Protection. Universe phy, became interested in the theory of general relativity of Albert Einstein, of which he 2021, 7, 12. https://doi.org/ strated the incompleteness theorems that broke ground in logic, mathematics, and phi- 2021, 7, x. became a close colleague at the Institute for Advanced Study at Princeton. He then discov- 10.3390/universe7010012https://doi.org/10.3390/xxxxx losophy, became interested in the theory of general relativity of Albert Einstein, of which he becameered ana close exact colleague solution at the of Institute the gravitational for Advanced field Study equations at Princeton. describing He then a rotating universe Received: 1 December 2020 discovered an exact solution of the gravitational field equations describing a rotating Received: 1 December 2020 model [1]. Accepted: 8 January 2021 universe model [1]. Accepted: 8 January 2021 Gödel’s universe is infinite, non-expanding, and filled with an idealized, homogenous Published: 12 January 2021 Gödel’s universe is infinite, non-expanding, and filled with an idealized, homoge- Published: 12 January 2021 nousperfect perfect fluid. fluid. It Itrotates rotates to stay to staybalanced balanced against againstgravitational gravitational collapse, the collapse, angular the angular velocity Publisher’s Note: MDPI stays velocityoutward outward juxtaposed juxtaposed to to the the gravitational gravitational pull inward. pull inward. Gödel explained Gödel explainedthat mat- that matter rotates neutral with regard to jurisdictional ½ 1 Publisher’s Note: MDPI stays neu- ter rotatesrelative relative to theto the compass compass of of inertia inertia with with the angular the angular velocity velocity 2(πGρ) , where 2(πG ρ) is2 , where is the mean claims in published maps and the mean density of matter, and G, Newton’s gravitational constant. However, unlike a tral with regard to jurisdictional clai- density of matter, and G, Newton’s gravitational constant. However, unlike a spinning institutional affiliations. spinning top, Gödel’s universe does not rotate about a privileged geometrical axis. The ms in published maps and institutio- localtop, inertial Gödel’s frames rotate universe with respect does to not a distant rotate frame about defined a privilegedby faraway galaxies geometrical so axis. The local nal affiliations. that inertialevery observer frames sees rotate himself with at the respect center of to rotation, a distant hardly frame an unusual defined situation. by faraway galaxies so that Copyright: © 2021 by the authors. Moreevery extraordinarily, observer in seesGödel’s himself rotating at universe, the center there of are rotation, space-time hardly trajectories an unusualthat situation. More Licensee MDPI, Basel, Switzerland. returnextraordinarily, to their starting point, in Gödel’s namely rotatingclosed timelike universe, curves there (CTCs). are Such space-time curves remain trajectories that return to confined locally to their future light cones, and an object traveling along a CTC never This article is an open access article their starting point, namely closed timelike curves (CTCs). Such curves remain confined Copyright:distributed© under 2021 the by terms the and author. Li- moves faster than the local speed of light, so that CTCs do not all violate the laws of censeeconditions MDPI, of the Basel, Creative Switzerland.Commons speciallocally relativity, to their while future representing light cones,possible andpaths an for object material traveling objects. This along is a avery CTC never moves faster Attribution (CC BY) license strange result because a traveler could journey into the future but arrive in the past, This article is an open access article than the local speed of light, so that CTCs do not all violate the laws of special relativity, (http://creativecommons.org/licenses accompanied by all the paradoxes that arise from a possible violation of the principle of distributed under the terms and con- while representing possible paths for material objects. This is a very strange result because /by/4.0/). causality. ditions of the Creative Commons At- a traveler could journey into the future but arrive in the past, accompanied by all the tribution (CC BY) license (https:// paradoxes that arise from a possible violation of the principle of causality. creativecommons.org/licenses/by/Universe 2021, 7, x. https://doi.org/10.3390/xxxxx At the time when Gödel published his article,www.mdpi.com/journal/universe time travel in the past was already a 4.0/). staple of science fiction, and some of its associated paradoxes had already been discussed Universe 2021, 7, 12. https://doi.org/10.3390/universe7010012 https://www.mdpi.com/journal/universe Universe 2021, 7, 12 2 of 11 in several issues of the 1917 edition of Electrical Experimenter (which would become later the infamous Amazing Stories magazine). The best known of the paradoxes is the grandfather paradox. As far as I know, it was first formulated in 1936 by Catherine Moore in a short story entitled “Tryst in Time,” in which a time traveler tries to kill his own grandfather [2]. The French writer René Barjavel used the same idea in the final scene of his famous novel The Reckless Traveler, published in 1944 (for an English translation, see [3]). Eager to change the course of history, the hero travels back in time to the Napoleonic era and inadvertently kills a man who is, in fact, one of his ancestors. Would he survive the encounter if he broke the causal chain leading to his own existence? The 1958 edition also includes a postscript entitled “To Be and Not to Be,” in which the novelist specifies the nature of the time travel paradox. It seems that Barjavel, an avid reader of popular science literature, was aware of Schrödinger’s thought experiment involving a cat at once half-dead
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