1. Introduction

The cosmological model given by Gödel (Reference Gödel1949a) is an exact solution of Einstein’s equation in which matter takes the form of a pressure-free perfect fluid. Its peculiar causal properties (e.g., a global time function fails to exist) have been of considerable interest to philosophers of time since the properties seem to imply the nonexistence of an objective time lapse. But it is not clear how the peculiar features of the Gödel model bear on the nature of time in our own universe. This thought Gödel explicitly considered. He writes (Reference Gödel and Schilpp1949b, 561–62): “It might, however, be asked: Of what use is it if such conditions prevail in certain possible worlds? Does that mean anything for the question interesting us whether in our world there exists an objective lapse of time?” Gödel offers two remarks in response to the questions (562):

Concerning remark 1, Gödel has been somewhat neglected (Yourgrau Reference Yourgrau1991; Savitt Reference Savitt1994; Earman Reference Earman1995; Dorato Reference Dorato and Callender2002; Belot Reference Belot2005; Smeenk and Wüthrich Reference Smeenk, Wüthrich and Callender2011). Concerning remark 2, Gödel has been heavily criticized (Savitt Reference Savitt1994; Earman Reference Earman1995; Dorato Reference Dorato and Callender2002; Belot Reference Belot2005; Smeenk and Wüthrich Reference Smeenk, Wüthrich and Callender2011). What follows is intended to be a straightforward defense of remark 1 and charitable reconstruction of remark 2 that, together, serve to clarify the significance of Gödel’s work for the nature of time in our world.

2. Preliminaries

We begin with a few preliminaries concerning the relevant background formalism of general relativity.Footnote ¹ An n-dimensional, relativistic space-time (for $n\ge 2$) is a pair of mathematical objects $(M,g_{ab})$, where M is a connected n-dimensional manifold (without boundary) that is smooth (infinitely differentiable). Here, $g_{ab}$ is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentz signature $(-,+,\dots ,+)$ defined on M. Two space-times $(M,g_{ab})$ and $(M^{\prime },{g^{\prime }}_{ab})$ are isometric if there is a diffeomorphism $ɸ:M\to M^{\prime }$ such that ${ɸ}_{*}\left(g_{ab}\right)={g^{\prime }}_{ab}$.

For each point $p\in M$, the metric assigns a cone structure to the tangent space $M_p$. Any tangent vector ${\xi }^a$ in $M_p$ will be time-like (if $g_{ab}{\xi }^a{\xi }^b<0$), null (if $g_{ab}{\xi }^a{\xi }^b=0$), or space-like (if $g_{ab}{\xi }^a{\xi }^b>0$). Null vectors create the cone structure; time-like vectors are inside the cone, while space-like vectors are outside. A time orientable space-time is one that has a continuous time-like vector field on M. A time orientable space-time allows us to distinguish between the future and past lobes of the light cone. In what follows, it is assumed that space-times are time orientable.

For some interval $I\subseteq ℝ$, a smooth curve $\gamma :I\to M$ is time-like if the tangent vector ${\xi }^a$ at each point in $\gamma \left[I\right]$ is time-like. Similarly, a curve is null (respectively, space-like) if its tangent vector at each point is null (respectively, space-like). A curve is causal if its tangent vector at each point is either null or time-like. A causal curve is future directed if its tangent vector at each point falls in or on the future lobe of the light cone. A time-like curve $\gamma :I\to M$ is closed if there are two distinct points $s_1,s_2\in I$ such that $\gamma \left(s_1\right)=\gamma \left(s_2\right)$.

For any two points $p,q\in M$, we write $p\ll q$ if there exists a future-directed time-like curve from p to q. We write $p<q$ if there exists a future-directed causal curve from p to q. These relations allow us to define the time-like and causal pasts and futures of a point p: $I^{-}\left(p\right)=\{q:q\ll p\}$, $I^{+}\left(p\right)=\{q:p\ll q\}$, $J^{-}\left(p\right)=\{q:q<p\}$, and $J^{+}\left(p\right)=\{q:p<q\}$. We say a space-time $(M,g_{ab})$ admits a global time function if there is a smooth function $t:M\to ℝ$ such that, for any distinct points $p,q\in M$, if $p\in J^{+}\left(q\right)$, then $t\left(p\right)>t\left(q\right)$. The function assigns a “time” to every point in M such that it increases along every (nontrivial) future-directed causal curve. Following the literature (e.g., Earman Reference Earman1995; Dorato Reference Dorato and Callender2002), we take the existence of a global time function to be a necessary (but not sufficient) condition for the objective lapse of time.

3. Concerning Remark 1

Gödel’s first consideration relating to the ideality of time and our universe is clear: despite the empirical data collected by cosmologists (e.g., data suggesting an expanding universe), there remains the epistemic possibility that our universe is one in which absolute time cannot be defined. Concerning 1, Yourgrau (Reference Yourgrau1991), Savitt (Reference Savitt1994), and Dorato (Reference Dorato and Callender2002) are silent. Earman (Reference Earman1995) and Belot (Reference Belot2005) do consider Gödel’s claim but swiftly find it unconvincing.

Belot states that Gödel himself grants that the more adequate models of our cosmos support an absolute time (Reference Belot2005, 270). But the text certainly does not lead us to this conclusion. And we know that, even late in his life, Gödel had still not given up on the possibility that we inhabit a Gödel-type model. Indeed, he would remain intensely interested in the collection of all astronomical data relevant to this possibility (Bernstein Reference Bernstein1991). Earman (Reference Earman1995, 199) claims that “we have all sorts of … experiences which lend strong support to the inference that we do not inhabit a Gödel type universe but rather a universe that fulfills all of the necessary conditions for an objective lapse of time.” However, Earman does leave open the possibility that Gödel’s remark 1 can be defended. To do so, it is sufficient to show that “there are cosmological models that (i) lack the features necessary for an objective time lapse, but (ii) reproduce the redshift, etc., so that they are effectively observationally indistinguishable from models that fit current astronomical data and have the spatiotemporal structure needed to ground an objective lapse of time” (199).

Earman strongly doubts that one can find “models which allow for time travel and which are observationally indistinguishable from non-time travel models” (Reference Earman1995, 200). But this remark is puzzling since one need not find models that are so causally misbehaved as to allow for time travel—it would suffice to find models that lack some feature or other necessary for an objective time lapse. And it has already been shown by Malament (Reference Malament1977, 79–80) that the relation of observational indistinguishability as introduced by Glymour (Reference Glymour1977) and used by Earman (Reference Earman1995) does not always preserve some properties necessary for an objective time lapse (in particular, the existence of a global time function). Also, the relation of observational indistinguishability as introduced by Glymour is symmetric and allows observers to live eternally. But, these conditions can be justifiably softened. Indeed, Malament (Reference Malament1977) has introduced the weaker relation: the space-time $(M,g_{ab})$ is weakly observationally indistinguishable from the space-time $(M^{\prime },{g^{\prime }}_{ab})$ if for every point $p\in M$ there is a point $p^{\prime }\in M^{\prime }$ such that $I^{-}\left(p\right)$ and $I^{-}\left(p^{\prime }\right)$ are isometric.

So, one might now wonder whether there are cosmological models that fit current astronomical data that are weakly observationally indistinguishable from models that lack features necessary for an objective time lapse (in particular, the existence of a global time function). And it turns out that there are. In fact, one can show that every cosmological model is weakly observationally indistinguishable from some model that lacks features necessary for an objective time lapse.

It should be clear that the proposition (a proof is given in the appendix) provides support for Gödel’s remark 1. Indeed, it remains an epistemic possibility, just as Gödel claimed it was, that we inhabit a world that has no objective time lapse. Further, even in the face of any (as yet uncollected) astronomical data, this epistemic possibility remains. One final comment concerning the proposition: it makes precise the heavily criticized statement made elsewhere by Gödel that “the experience of the lapse of time can exist without an objective lapse of time” (Reference Gödel and Schilpp1949b, 561).

4. Concerning Remark 2

Gödel’s second remark is sometimes interpreted to be an argument that time in our universe is ideal (Savitt Reference Savitt1994; Earman Reference Earman1995; Dorato Reference Dorato and Callender2002). But this reading seems to be a bit strong. Gödel only states that “the mere compatibility with the laws of nature of worlds in which there is no distinguished absolute time … throws some light on the meaning of time in those worlds in which an absolute time can be defined.” In other words, the existence of Gödel-type solutions simply have implications regarding the nature of time for all cosmological models. However, it seems “there is a consensus that even this modest conclusion is not warranted” (Smeenk and Wüthrich Reference Smeenk, Wüthrich and Callender2011, 597).

Now we have already shown that the nonexistence of a global time function in certain models does have epistemic implications for all models. But Gödel seems to have something more in mind concerning remark 2. Indeed, of great importance seems to be that fact that “whether or not an objective lapse of time exists … depends on the particular way in which matter and its motion are arranged in the world.” And it is unclear how this statement leads to any general implications concerning all models. Here, we provide one way to spell out some of the details.

Consider an arbitrary model $(M,g_{ab})$ that has all the geometric properties necessary for absolute time. In particular, assume it admits a global time function. Following Gödel, consider an observer at some point $p\in M$ who “asserts that this absolute time is lapsing.” Since time is lapsing objectively, this means that at p—the event of the assertion—all events in the set $I^{+}\left(p\right)$ have yet to “come into existence” (Gödel Reference Gödel and Schilpp1949b, 558). In other words, at p, if one takes the idea of an objective time lapse seriously, one is led to consider as fixed not the space-time $(M,g_{ab})$ but rather merely a portion of it. It is then natural to wonder whether there is a sense in which this leaves open from the perspective of the observer making the assertion the nomological possibility that, after the time of the assertion, matter and its motions might be (re)arranged in such a way that a global time function can no longer be defined. This question whether a cosmological model can “start out” with well-behaved causal structure but not “end up” that way was, in a sense, asked some time ago by Stein (Reference Stein1970, 594). “Consider either an arbitrary given cosmological model, or a model having the structure of one of the sorts assumed to hold in the real world. Then: is it ((a) ever, (b) always) possible to introduce into such a model a continuous deformation of the structure, leading through intermediate states, all compatible with Einstein’s theory, to a state in which Gödel-type relationships occur?”

Here is one way to formulate this question precisely. Let $(M,g_{ab})$ be any space-time that admits a global time function. For any point $p\in M$, is there is a space-time $(M,{g^{\prime }}_{ab})$ that (i) fails to admit a global time function and (ii) is such that ${g^{\prime }}_{ab}=g_{ab}$ on the region $M-I^{+}\left(p\right)$? When $(M,g_{ab})$ is at least three-dimensional, yes.

It should be clear how the proposition (proof given in the appendix) can be used to understand Gödel’s remark 2. It is philosophically unsatisfying (although not a contradiction) for one to assert that time is objectively lapsing in one’s universe when from the perspective of the observer making the assertion there remains the nomological possibility that, after the time of the assertion, matter and its motions might be smoothly (re)arranged in such a way so as to prohibit an objective time lapse.

5. Conclusion

Despite the above propositions, one might insist that “we do not inhabit a Gödel type universe but rather a universe that fulfills all of the necessary conditions for an objective lapse of time” (Earman Reference Earman1995, 199). We close with one final word of caution. It seems reasonable that “in order to be physically significant, a property of space-time ought to have some form of stability, that is to say, it should also be a property of ‘nearby’ space-times” (Hawking and Ellis Reference Hawking and Ellis1973, 197). There are a number of different ways to understand the notion of nearby space-times—none entirely satisfactory (Geroch Reference Geroch1971; Fletcher Reference Fletcher2015). Here we simply note one sense in which causally well-behaved space-times can be “close” to space-times that are not.

Consider the one-parameter family of space-times $(M,g_{ab}(\lambda \left)\right)$ where $\lambda \in [0,1]$, $M={ℝ}^4$, and $g_{ab}\left(\lambda \right)=-{\nabla }_{a}t{\nabla }_{b}t+{\nabla }_{a}x{\nabla }_{b}x-(1/2)\exp \left(2\lambda x\right){\nabla }_{a}y{\nabla }_{b}y-2\exp \left(\lambda x\right){\nabla }_{(a}t{\nabla }_{b)}y+{\nabla }_{a}z{\nabla }_{b}z$. One can easily verify that $(M,g_{ab}(\lambda \left)\right)$ is Gödelian for all $\lambda \in (0,1]$. But what about $(M,g_{ab}(0\left)\right)$? Surprisingly, one finds it is a Minkowski space-time.Footnote ² Thus, there is sense in which a model satisfying all of the necessary conditions for an objective lapse of time is “close” to a set of models that do not satisfy these conditions. Should this fact not give us pause?