*To the editors*:

Alasdair Richmond’s essay comprises an interesting discussion, both physical and philosophical, of the homogeneous rotating universe model discovered in 1949 by Kurt Gödel. This exact solution of Einstein’s general relativity equations contains closed timelike curves and therefore allows for a trip into the past, accompanied by all the paradoxes that arise from a possible violation of the principle of causality.

At the time when Gödel published his article, time travel was already a staple of science fiction, and some of its associated paradoxes had already been discussed in several issues of the 1917 edition of *Amazing Stories*. 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. The French writer Rene Barjavel used the same idea in the final scene of his famous novel *The Reckless Traveler*, which was published in 1944.^{1} 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. 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 cat, at once half-dead and half-alive, as well as the many-worlds theory proposed by Hugh Everett in 1957.

As Richmond points out, some writers have suggested that Everett’s theory offers a possible solution to the grandfather paradox: by killing his ancestor, the traveler would bring about a change in the future, resulting in a bifurcation of space-time along several different causal lines. Such a solution is unsatisfactory because it is based on a misunderstanding of the many-worlds hypothesis. Everett proposed an interpretation of the measurement process in quantum physics, according to which all states in superposition continue to exist after measurement, but in disjoint universes—contrary to the standard Copenhagen interpretation, where the measurement sets the system in a unique classical state. But the murder of an ancestor is not a quantum process. Moreover, even assuming a bifurcation of the universe engendered by the elimination of some ancestor, the paradox would be logically resolved only if different causal lines interacted in a very particular way, which is also contrary to Everett’s theory.

George Gamow had entertained the concept of a universe in global rotation in 1946; he thought it legitimate to generalize to the universe as a whole the observed rotation of local cosmic masses such as planets, stars, and galaxies.^{2} Gamow did not derive an exact analytic solution from the equations of general relativity to justify his idea, nor did he introduce astronomical observation of the large-scale universe that could support it. The first solution of the Einstein field equations describing the field generated by a rotating body was discovered in 1937 by Willem van Stockum.^{3} This was also the first example in which there was a connection between time travel and the existence of a rotation associated with space-time. The work of van Stockum and Gamow certainly encouraged Gödel to search for an exact relativistic solution describing a rigid rotating universe containing closed timelike curves. In addition to the technical feat, which impressed Einstein himself, Gödel sought, above all, to find support for his philosophical point of view on the illusory nature of time. According to Gödel, from the moment when an observer starting from a point in space-time, *p*, travels along a trajectory allowed by the principle of relativity—that is to say, confined inside light cones—to his chronological past, J^{–}(*p*), time exists relative only to the observer and loses all independent reality. As Richmond rightly points out, “Gödel thought the unreality of time was a physical … discovery,” and not just a metaphysical option. From his point of view as a logician, Gödel’s model of rotating universe, whether realistic or not, was indeed a universe possible to the extent that it was an exact mathematical solution of Einstein’s field equations. And for Gödel, if time was an illusion in a possible universe, this must also be true in the real universe.

The philosophical arguments developed by Gödel to deny the physical nature of time have been analyzed by many authors.^{4} The most thorough study was made by Pierre Cassou-Noguès, who had access to thousands of pages and unpublished notes that are preserved at the Institute for Advanced Studies of Princeton.^{5} In Gödel’s papers, Cassou-Noguès saw evidence of a mental system that while logically consistent also represented the mind of a scientific genius molded by a bizarre psychology, mixing irrational phobias, superstitions, hypochondria, and paranoia.

The rotating universe of Gödel is not expanding, which is in blatant contradiction with observations of the redshifts of distant galaxies and the cosmic background radiation. More and more drastic experimental constraints have been placed on the possible large-scale vorticity of the universe,^{6} such that the possibility of a cosmological model in global rotation, although it has been carefully studied, has been almost universally rejected.^{7} It is fascinating to note that the philosophical option advocated by Gödel, namely the illusory nature of time, features prominently on the agenda of contemporary physics, with some approaches attempting to reconcile general relativity and quantum field theory. The fundamental equations of loop quantum gravity do not contain a time variable. The theory describes processes in which changes take place not under the action of an identifiable temporal variable, but as the result of a non-commutative sequence of spatial operators whose ordered classification simulates an irreversible flow of time. As in Gödel’s view, but in a very different physico-mathematical context, the experience of passing time would then be relative to the particular conditions in which the observer finds himself. It would also depend on the structure of his cognitive system. Our own perception of time is certainly different from that of other animals. We should not look for a manifestation of our own experience of time in the realm of fundamental physics.^{8} On this view, time appears as an emergent quality, underpinned by strata mixing spatial transformations and cognitive processes.

Beyond these few supplementary details, which are essentially bibliographic in nature, let us turn to the main purpose of this letter, which is to complete Richmond’s analysis by showing that journeying into the past is intimately linked to space-time models devoid of singularities. From physically reasonable assumptions, singularities arise as an inevitable consequence of the equations of general relativity. Time travel to the past becomes possible only when one or another of these assumptions is violated. This point, which seems essential to me, is usually omitted in most of the literature on the subject. Let’s take a closer look.

In the context of the metrics of space-time solutions to the equations of general relativity, the possibility of travel to the past is related to the existence of closed timelike curves. Gödel’s model is a particular example, but it is far from being unique. As Richmond rightly points out, the causal structure of space-time is described by the field of light cones at each point. In special relativity, free from gravitation, all light cones passing through all events are parallel to one another, somewhat akin to a field of wheat in which all the ears are vertical; a closed timelike curve would require a departure from the light cones, which is impossible. In general relativity the situation is much more interesting. By virtue of the principle of equivalence, which stipulates the influence of gravitation on all forms of energy, light cones incline and deform according to their local curvature. In the same way, the identification between points of space allowed by a multi-connected topology deforms the field of light cones.^{9} We can, therefore, conceive of a space-time with strong curvature, or non-trivial global topology, in which the light cones are sufficiently inclined along particular temporal trajectories to allow them to close in a loop.

In Gödel’s rotating universe there is a privileged axis of symmetry. As one moves away from this axis and the speed of rotation increases, the field of light cones reaches a point allowing for the formation of closed timelike loops. Van Stockum’s cylinder, and its generalization, which was discovered in 1974 by Frank Tipler, allow for the existence of closed time curves in the same manner. Other exact solutions containing closed time curves, but not cited by Richmond, are the Taub-NUT empty space,^{10} the Misner space obtained by introducing a multiconnected toric typology into Minkowski space-time,^{11} or the Gott space representing two parallel cosmic strings of infinite length that intersect at relativistic speed.^{12} But the most famous solution is Kerr’s space-time describing a rotating black hole.^{13}

These space-times include chronical regions devoid of closed timelike curves, and one or more achronal regions that contain them; their common boundary is termed the chronological horizon.^{14} The Kerr solution has two event horizons denoted *r*_{+} and *r*_{–} respectively. The regions *r* > *r*_{+} and *r*_{+}> *r* > *r*_{–} are chronical, while the region r < *r*_{–} is achronal; the horizon of internal events *r*_{–} plays the role of chronological boundary. A temporal curve from a point *p* of the chronical region outside the black hole can successively cross the two horizons *r*_{+} and *r*_{–}, cross the achronal region r < *r*_{–} and emerge from the black hole to arrive in the chronological past *J*^{–}(*p*), thus achieving a closed time curve of the type referred to as a wormhole.

The first wormhole solution was published in 1916 by Ludwig Flamm.^{15} It was studied in 1935 by Einstein and Nathan Rosen as part of Schwarzschild’s solution describing a static spherical black hole, and was referred to as an Einstein-Rosen Bridge.^{16} The term “wormhole” first appeared in a famous article published in 1957 by Charles Misner and John Wheeler.^{17} The introduction of the Kruskal coordinate system in the 1960s made it possible fully to describe the geometric and topological structure of Schwarzschild’s space-time, and to show that the existence of a temporal singularity at *r* = 0 in the middle of Einstein-Rosen Bridge renders a wormhole untraversable.

The question then arose whether there were more general solutions that allowed for closed timelike curves in association with traversable wormholes. Answers in the affirmative could be found from Kerr’s solutions for a rotating black hole, the Reissner-Nordström metric for an electrically charged black hole, and so-called Lorentzian wormholes lined with exotic material, that is to say, negative energy density.^{18} If these space-times have many singularities, they are not timelike but spacelike. Kerr’s black holes have a ring-shaped singularity lying in the equatorial plane, and perfectly lawful particle paths can either fly over it or pass through it without touching it. The models of Gödel, van Stockum, Tipler, Misner and Gott are devoid of any singularity. The existence of closed timelike curves is, in fact, contingent on the absence of time-type singularities.

The crucial question of the occurrence of singularities in general relativity was dealt with in the 1960s by Roger Penrose and Stephen Hawking in the context of the gravitational collapse of stars (a singularity of the future), and its analogue, reached by reversing time in big bang cosmological solutions (a singularity of the past). Using differential topology, they succeeded in proving that the appearance of time-type singularities was generically inevitable, given some simple physical hypotheses. These are grouped into two major classes. The first places certain conditions on space-time: the existence of a Cauchy surface on which it is possible to define initial conditions whose development unequivocally determines the past and the future, or a limit on the rotation speed of the universe or the bodies it contains. The second consists of various constraints on the positivity of energy—so-called strong, weak or dominant conditions.

These important singularity theorems are similar to theorems governing rays in geometrical optics.^{19} A singularity is similar to a point of convergence in an optical system. The event horizon of a black hole is like a convergent lens: all the material and luminous trajectories that cross it are focused on a single point, which corresponds to a central singularity. In the same way, the appearance of a singularity in big bang cosmology is established from the convergence of all temporal trajectories as one goes back in the past.

By marking an interruption of time for material and luminous trajectories, singularities prevent these trajectories from closing in a loop and violating causality. In classical general relativity, i.e. without quantum corrections, the possibility of traveling in the past is therefore necessarily related to a violation of one or another of the hypotheses concerning the occurrence of time-type singularities.

According to general relativity, a wide variety of solutions give rise to closed timelike curves. But in each case, one or another of the hypotheses governing the occurrence of singularities is not satisfied. The cylinders of infinite length in relativistic rotation are not physically relevant, our universe does not rotate as fast as the Gödel solution requires, cosmic strings of infinite length are unrealistic, traversable wormholes violate the positivity energy condition, and so on. The debate might simply end with a decree that nature forbids the formation of closed timelike curves. Nonetheless, one can well imagine a technologically advanced civilization that is able to artificially create closed timelike curves in order to travel in time. This is the scenario depicted in Carl Sagan’s novel *Contact*. An ancient extraterrestrial race transmits a radio message that contains the plans for a time machine. When travelling from star to star, the machine uses a space-time shortcut in the form of a wormhole, an approach that violates the singularity theorems. Sagan asked his colleague Kip Thorne to find a workaround. Thorne, a renowned specialist in relativistic astrophysics,^{20} thus embarked upon a parallel career as consultant and scientific advisor for novelists and filmmakers.^{21} He replied to Sagan that it was sufficient to violate the positive energy conditions guaranteeing the appearance of singularities by introducing a form of exotic energy with negative pressure. All the usual forms of matter obey positive energy conditions. No body made of classical material in gravitational implosion, such as a black hole, or even the big bang, can generate an achronal region. Quantum field theory offers a means of escape. Quantum vacuum energy fluctuations are described as bubbling pairs of particles and antiparticles of all types and masses (electrons, quarks, photons, mini black holes, and so on), constantly appearing and disappearing. These are felt, in particular, as part of the effect predicted by Hendrik Casimir in 1948, and verified experimentally by Steve Lamoreaux in 1997. The introduction into the quantum vacuum of two weakly separated conductive plates modifies the energy value of a vacuum, which then behaves as if it possesses a negative energy.

In 1988, Thorne et al. showed that not only was it theoretically possible to open a passage between two points of space and to travel almost instantaneously between the stars, but that we must also be able to travel in time.^{22} Having opened and stabilized a traversable wormhole, one can imagine leaving one of its mouths on earth, while the second is carried inside a space ship in relativistic flight, making a round trip of, say, ten thousand years for an observer remaining behind on earth. If the journey is undertaken at speeds close to that of light, the passage of time onboard with be only a few days or weeks. A time lag will then arise between the two mouths of the wormhole. This would allow for time travel up to ten thousand years back into the past for those onboard the ship, or up to ten thousand years into the future for those left behind on earth.

To counter this physically embarrassing eventuality, although based on semi-classical general relativity calculations, Hawking published a chronological protection conjecture in 1992.^{23} There exists, according to Hawking’s conjecture, a physical mechanism capable of preventing closed timelike curves from forming in any conceivable circumstances, whether natural or artificial. The achronal regions of the Misner, Taub-NUT, and Kerr spaces are classically unstable; particles and fields falling in a Kerr black hole, or traveling at relativistic speed in Taub-NUT and Misner spaces, see their spectral shift diverge toward blue as the chronological horizon approaches, and it seems reasonable to think that the associated energy density, which is also divergent, has sufficient feedback over space-time to prevent the formation of closed timelike curves. The conjecture remains unproven. Hence it may be the case that chronology is not always protected at macroscopic scales, as suggested by the model of traversable wormholes. And even if it were, quantum gravity could give rise to nonzero probability amplitudes allowing closed time curve formation on a microscopic scale.

One possible method to find a universal protection mechanism can be found in the quantum instability of time horizons. This can be described as a stacking of the quantum fluctuations of the vacuum in the vicinity of the chronological horizon, so that the fluctuations have a non-zero renormalized energy density that diverges as the horizon approaches. In turn, semi-classical Einstein equations suggest that this energy should distort the space-time geometry in order to protect the timeline. Such a feedback mechanism is the quantum analogue of the well-known Larsen effect in acoustics.^{24}

A tentative approach to these questions is appropriate because no rigorous proof of chronological protection has ever been formulated that is valid in all the spaces that could conceivably host closed timelike curves. From our current knowledge of physics, journeys to the past seem absolutely impossible. Time machines seem destined to remain forever in the realm of cinema and science fiction. Still, we must be careful not to prejudge the surprises that the physics of tomorrow might offer.

*Translated from the French by the editors.*