[The following post is based on a seminar given at the Scuola Normale Superiore (Pisa) in June 2011. Here you can find the complete text of the talk (in Italian).]
In 1946, Paul Artur Schilpp asked Kurt Gödel to write a paper to be included in the collective volume on Albert Einstein for the “Library of Living Philosophers” collection (edited by Schilpp himself). As shown by a letter from Schilpp to Gödel dated July 10th 1946, Gödel himself should have had to propose an argument for the paper, but he never replied. Schilpp, then, suggested he write an article on the topic: “The realistic standpoint in physics and mathematics”. But Gödel was hesitant: he thought himself to be insufficiently expert on the topic to worthily contribute. At the most, he considered himself able to contribute some considerations on the notion of time as resulting from Einstein’s theory of relativity and on the relationship between this and the idealistic thesis of the non-existence of objective time. Schilpp was enthusiastic and thus, three years later (in 1949), a short article was published under the title “A remark about the relationship between relativity theory and idealistic philosophy”1
The article turned out to be very interesting, not least because it is the only place in Gödel’s published works where he takes a stand on a general philosophical problem not strictly related to mathematics. Gödel argues that,
Following up the consequences [of the relativity theory, particularly of the general one] […] one obtains an unequivocal proof for the view of those philosophers who, like Parmenides, Kant, and the modern idealists, deny the objectivity of change and consider change as an illusion or an appearance due to our special mode of perception. (p. 202)
Relativity theory and idealism
The argument is actually quite simple. The theory of relativity upsets all our traditional intuitions concerning time: we are used thinking of time as an objective succession of instants and this succession is intuitively considered as the same for all the observers; but the (special) theory of relativity made us aware that two events A and B, that are simultaneous for a certain observer x, can be non-simultaneous for a different observer y. It can even happen that A follows B according to x and that B follows A according to y. So, we must admit, simultaneity is not absolute, but rather relative (to the observer). But if relativity theory leads us to the conclusion that simultaneity is relative, then we can no longer think of reality as composed of an objective succession of temporal instants. And if temporal instants do not objectively exist, then we must conclude that even the change cannot exist, since it can take place only within them. However, this argument needs some clarifications in order to be convincing — and here is where Gödel’s genius comes to the fore.
The first problem is that saying that temporal instants (or intervals) are relative does not necessarily mean that they are not objective. For example, the relation “to be to the left of” is relative, since its validity depends on the position of the observer; but it still expresses an objective disposition in the world. However, what we usually call “time” is something very different from what comes out from this relativization. For we usually think that what exists, objectively exists within temporal intervals that are objective and exact as well. To relativize these time intervals would mean to relativize what they contain.
A relative lapse of time […], if any meaning at all can be given to this phrase, would certainly be something entirely different from the lapse of time in the ordinary sense, which means a change in the existing. The concept of existence, however, cannot be relativized without destroying its meaning completely. (p. 203n)
Moreover, this argument actually shows that time flows differently according to the observers, but one could still say that this flow of time is nonetheless an objective property of reality. However, Gödel replies,
A lapse of time […] which is not a lapse in some definite way seems to me as absurd as a colored object which has no definite colors. But, even if such a thing were conceivable, it would again be something totally different from the intuitive idea of the lapse of time to which the idealistic assertion refers. (p. 203n)
A more serious objection is the following. The complete equivalence of all the observers moving at different (but uniform) velocities is valid only within the abstract spatio-temporal framework of the special relativity theory. De facto, the existence of matter and the spatio-temporal curvature produced by it destroy the equivalence of all the observers by telling apart some of them apart, i.e. those who follow, in their movement, the average movement of the matter. In all the cosmological solutions known until 1949 the local times of these “privileged” observers could be composed together into a unique global time. Thus, one might consider this global time as the absolute time. This absolute time flows objectively, and all the discrepancies between it and the observers’ relative times are to be traced to the effect of the motion relative to the average state of motion of the matter on measuring processes and on physical processes in general.2
Gödel’s reply to this objection represents the original contribution of the author to the debate. The reply is based indeed on Gödel’s discovery of a new cosmological solution to the equations of the general theory of relativity, according to which it is impossible to define an absolute time in the way we have just seen. For, in the universes resulting from this cosmological solution, the local times of the “privileged” observers cannot be composed together into a unique, global, absolute time. And this is not all. It is also impossible to define any other procedure able to define an absolute time reasonably similar to our intuitive notion of what such an absolute time should be.
What makes this operation impossible in such universes is the fact that
the compass of inertia in them everywhere rotates [in the same direction] relative to matter, which in our world would mean that it rotates relative to the totality of galactic systems. (p. 204n)
From this comes the name “rotating universes” by which we usually refer to them. If we impose on such universes the characteristic of being static and spatially homogeneous, and if we bestow a value >0 on the cosmological constant, then we obtain a universe in which time lines are circular and closed. As a consequence of this, we have it that
by making a round trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present, and future, and back again, exactly as it is possible in other worlds to travel to distant parts of space. (p. 205)
Thus, we have that in these universes, for any possible definition of global absolute time, one could travel in regions of the universe belonging to the past of that definition — and to the future of the observer. But, Gödel concludes,
if the experience of the lapse of time can exist without any objective lapse of time, no reason can be given why an obctive lapse of time should be assumed at all. (p. 206)
We subjectively experience time flowing (and hence a change within it), but to this subjective experience does not (cannot, given the general relativity) correspond any objective temporal order.
This reply refers to one possible cosmological solution. We are not sure that this solution really describes how our universe is. So, one might reply, it is true that in those rotating universes it is not possible to define an absolute time, but this does not imply that it is not possible to do so in our universe, since our universe could be a non-rotating universe. However, Gödel notices that
The mere compatibility with the laws of nature of worlds in which there is no distinguished absolute time, and [[in which]], therefore, no objective lapse of time can exist, throws some light on the meaning of time also in those worlds in which an absolute time can be defined. For, if someone asserts that that this absolute time is lapsing, he accepts as a consequence that whether or not an objective lapse of time exists (i.e., whether or not a time in the ordinary sense of the word exists) depends on the particular way in which matter and its motion are arranged in the world. (p. 207)
In other words, if something like an absolute time existed (and if we admit that such a concept must preserve something of the original intuition we have of it), it should be valid in all the possible universes. But if it does not happen, then a philosophical view should account for such an anomaly — and this does not seem to be very easy to achieve.3
There is a second objection to Gödel’s reply. We previously assumed that these rotating universes are static. Now, a static solution hardly can be considered a proper description of our universe, since in these universes it seems impossible to account for the so-called red shift. However, rotating solutions can be found for expanding (non-static) universes too, and in these solutions it can be impossible to define a notion of absolute time. But this needs a clarification of what Gödel means by “absolute time”. This clarification is offered by Gödel in a footnote: in such universes absolute time could even not exist,
At least if it required that successive experiences of one observer should never be simultaneous in the absolute time or (which is equivalent) that the absolute time should agree in direction with the times of all possible observers. Without this requirement an absolute time always exists in an expanding (and homogeneous) world. (p. 206n)
On the occasion of the second German edition (1955), Gödel clarifies further the point, by adding that
By an “absolute time” I understand a world time that can be defined without reference to particular objects and that satisfies the requirement formulated at the beginning of this footnote. More precisely, this should be called a “possible absolute time”, since several can exist within one world, even though that is only exceptionally the case in spatially homogeneous universes. (p. 206n)
Thus, summing up: relativity theory forces us to abandon the idea of time flowing equally for all observers. Different observers, different times. However, it still seems to be possible to accept a notion of absolute time as something that flows in the same direction for all observers. On the contrary, Gödel’s discovery of the rotating universes shows that in some universes (that could coincide with our universe) it is at least possible to halt the time flowing (or even invert it). If we admit that the notion of absolute time cannot undergo such a twisting, then we must abandon such a notion, and accept the idealistic idea according to which time is just a product of our subjectivity.
The (theoretical) possibility, in Gödel’s rotating universes, to travel through time stimulated several responses. Many authors focused on the theme, some of them criticizing Gödel for having credited such an absurd idea, some of them believing that Gödel’s discovery could have interesting consequences for our theory of time. However, it must be noted that Gödel himself excluded, in the article we are considering, the possibility of time travel, on the basis of its physical impossibility. In a footnote he writes:
Basing the calculation on a mean density of matter equal to that observed in our world, and assuming one were able to transform matter completely into energy, the weight of the “fuel” of the rocket ship, in order to complete the voyage in years (as measured by the travellers), would have to be of the order of magnitude of times the weight of the ship (if stopping, too, is effected by recoil). This estimate applies to . Irrespective of the value of , the velocity of the ship must be at least of the velocity of light. (p. 205n)
Time travels would be contradictory (and hence would exclude the plausibility of such universes) only if a time travel were practically feasible. But since at the moment it seems to be impossible (and what today seems to be just a physical impossibility could be a theoretical impossibility tomorrow), such an objection cannot exclude, a priori, that the space-time structure of our universe is actually of this kind.
Therefore, Gödel’s argument is not founded, in any sense, on the possibility of time travel, but only on the impossibility of defining, in such rotating universes, an absolute time; and this only depends on the existence, in these rotating universes, of closed time-lines.
Moreover, as correctly pointed out by Yourgrau,4 one cannot claim, at the same time, that time travel is (even only theoretically) possible and that time is nothing but an illusion!
What must be admitted, of course, — Yourgrau writes — is that Gödel believes he has shown the compatibility with the GTR of universes permitting time travel […]. But it is this very fact that Gödel takes to indicate that , the standard variable for time, should not be read here as standing for genuine, successive time. But if there is no genuine time, there can be no genuine time travel. […] Gödel describes the R-universe as permitting time travel, but only if we do not read “time” as denoting a relativistic formal simulacrum of the real thing. (pp. 3-4)
 ⇑ The article can be found in Kurt Gödel, Collective Works, vol. 2: Publications 1938-1974, Oxford University Press, New York-Oxford, 1990; pp. 199-207. The shortness of the article doesn’t do justice to the complexity of its drafting. Actually, five different manuscripts were written before the final version. Two of these five manuscripts can be found in the third volume of the Collective Works.
 ⇑ It was just by reasoning on these considerations that the physicists James Jeans concluded that there is no reason to abandon the intuitive idea of an absolute time which flows objectively.
 ⇑ One could exclude the rotating solutions just because they don’t permit us to define an absolute time, but this would be an arbitrary and ad hoc solution.
 ⇑ P. Yourgrau, The Disappearance of Time. Kurt Gödel and the Idealistic Tradition in Philosophy, Cambridge University Press, Cambridge 1991.