A Finite Universe with no Absolute Beginning
Introduction
In this essay, I present two arguments that classical relativity denies the universe a beginning. The first argument, primarily based on Adolf Grünbaum’s work, states that the Robertson-Walker metric is naturally past-finite at the singularity yet spacetime remains beginningless, a point confirmed by experts like Jean-Marc Levy-Leblond and Brian Pitts. The second argument states that superior methods of measuring time at the singularity, developed by physicists like Levy-Leblond and Charles Misner, imply an infinite past despite its finitude in ordinary time. Together, these arguments constitute a strong case that relativity rules out an absolute beginning.
Finite and Boundless
As Adolf Grünbaum, who held a degree in physics and specialized in the philosophy of physics, repeatedly argued in many of his papers, general relativity theory (GTR) excludes an initial instant t=0 from the set of real physical instants. The cosmic time-interval of the Robertson-Walker spacetime metric is past-open, but does not extend before t=0. That is, there is no temporally first physical event of the spacetime. The past on this open interval is ordinally unbounded, and yet the age of the universe in the given metric (say, in years) is finite. And there is no initial instant in GTR because the Big Bang singularity does not meet the requirements for being a real physical event. Instead, there is a hole in the spacetime manifold at t=0, such that at unboundedly ever earlier and briefer moments of time the spacetime metric becomes degenerate, and the curvature as well as the density approach infinity.
The term “Big Bang singularity”, in GTR, is a shorthand way of speaking about this degenerate mathematical behavior of the metric and scalar curvature at regressively earlier times. This behavior robs the Big Bang singularity of its event-status in GTR. Grünbaum cites John Stachel’s finding that points of the theoretical manifold first acquire the physical significance of being events when they stand in the spatio-temporal relations specified by the metric of the manifold. In GTR, there is no structure on the differentiable manifold that is both independent of the metric tensor and able to serve as an individuating field, that is, to turn the ideal points of the manifold into points of physical spacetime. Thus, in GTR, it turns out that the notion of an event makes physical sense only when both manifold and metric structure are well defined around it. But the Big Bang singularity does not qualify as a physical point-event of the spacetime to which one could assign three spatial coordinates, and one time coordinate. Therefore, instead of having a first physical event, the past cosmic time-interval is open or unbounded, rather than closed or bounded by a first moment or instant, although its metrical duration in years is only finite. And hence the past of the Big Bang universe did not include a first physical event or state at which it could be said to have begun. As the astrophysicist Jean-Marc Levy-Leblond explained:
My purpose [in this paper] is to dissent from the common but mistaken idea that Big Bang cosmologies imply “that there was an instant at which time literally began and, so, by extension, an instant before which there was no time”. In truth, the so-called ‘initial’ instant t=0 corresponds to a singularity of the spacetime model (as expressed by the Robertson-Walker metric) which has to be taken at face value: because the model is not defined for t=0, this value does not belong to its physical time domain. The range of physical time consists only of the open interval. The initial time t=0 is not a moment in the life of the Universe, and does not belong to its past. As such, this out-of-reach instant may be said to be infinitely remote, irrespective of its finite numerical value on a conventional timescale.
Physicist Brian Pitts clarified this issue further in one of his papers (relevant parts here). Dr. Pitts wrote:
Within the Robertson-Walker cosmologies, spacetime for t > 0 (which is to say, always), one can ‘explain’ (in the fashion of Laplace’s demon) each moment in terms of an earlier one. Thus, there is no beginning required as far as physics can tell. … It appears, then, that whether one is tolerant or intolerant toward singularities, it turns out that there is no first moment, because every moment is preceded by earlier moments. Thus… there is no beginning implied by physics… In order for the Big Bang singularity to [be an absolute beginning], the singularity must be well enough behaved to be a real and intelligible part of spacetime, and badly enough behaved that it cannot have a past. Satisfying both conditions seems difficult and unlikely to be achieved.
The obvious implication of this, as professor John Earman explained, is that “the standard big bang models are not compatible in any obvious way with the idea that the universe has a physically uncaused beginning [since] these models imply that for every time t there is a prior time t’ and that state at t’ is a cause (in the sense of causal determinism) of a state at t.” (p.208) Dr. Quentin Smith added that in this scenario “the first interval… is ‘past-open,’ which means that there is no instant t that is the first instant of each earliest interval of any length, be the interval an hour, minute or second, etc. Before any instant in an earliest hour, minute, second, etc., there is an infinite number of other instants.” Elsewhere, Dr. Smith also noted that “there is no first instant and there are an infinite number of briefer and briefer first intervals of a given length.” (p.190) Chris Smeenk, who has a Master of Science (M.S.) degree in Physics and Astronomy, agreed with this conclusion, stating that “there is no ‘first instant’ of time in the [Big Bang] picture any more than there is a ‘first point’ in an open interval of the real line... yet the universe does not have an infinite past.”
This is very similar to what Roberto Torretti argued in one of his papers: “The Big Bang is often described as the beginning of the world… [but] this description is misleading. Big Bang universes have their duration bounded below, but do not have a proper beginning. There is in them no first instant that might tempt us to ask what went on before it. For every given instant along a causal line there is another instant preceding it, at which something was already going on.” In another paper, Torretti defended this conclusion, arguing that “mathematicians are conversant with methods for adding boundary points to Riemannian manifolds. Resorting to one of these methods, the open-ended butt on the past of every Friedmann timelike geodesic can be closed by fiat. But such closure is not required by theory, nor backed by experiment or observation.” (p.80)
Adequate Singular Metrics
Now, in order for the metric to match this unbounded description of the past, some prominent physicists suggested that we refine our concepts of time accordingly (despite Grünbaum’s insistence that this additional step is unnecessary). For instance, Leblond suggested “to send back the birth of the Universe [metrically] to (minus) infinity where, or rather when it seems to [ordinally] belong.” To achieve that, he introduced a new linear time metric, which confers an infinite duration on the ordinally unbounded past of the universe. Importantly, he notes that “no new physics is involved in the introduction and use of linear time rather than cosmic time.” This metric corresponds to a clock geared to the expansion of the universe. Thus, Leblond concludes: “Naturally, on the linear scale, the lifespan of the Universe is infinite, so that never did the Big Bang begin.” (For more on this, see Leblond’s paper Did the Big Bang Begin?)
Physicists Midy and Petit from the Universite Paris-XI confirmed Leblond's findings, saying that “in physical (i.e. conformal) time, the origin of the universe is shifted to −∞, corroborating the ideas developed by Levy-Leblond… and this is linked with the logarithmic dependence of ϑ on t as results from (51). This removes the question of knowing what existed before the big bang.” (Midy & Petit, 1999; p.18)
Leblond wasn’t the first to introduce a more adequate notion of time to describe the metric at the singularity. As Charles Misner, John Wheeler and Nobel laureate Kip Thorne explained (Gravitation; pp. 813–814, and other works), another way to describe an infinite metric at the singularity is to measure the oscillations of the curvature of space itself. These physicists, armed with strong evidence, claim that in the most general forms of a relativistic universe, an infinite number of curvature oscillations occur at the singularity – the spatial geometry turns into a chaotic mess, stretching and squashing wildly in different directions, making it highly anisotropic. Consequently, when measured by the curvature clock, it takes an infinite amount of time to reach the supposed t=0. Thus, Misner concludes: “An infinite regress should be a part of our picture of the cosmological singularity” and “infinite density becomes a formal abstraction never realized in the course of evolution.” (For further reading, see John Barrow’s book World Within the World; pp. 234–236, 317–321)
Following the same line of reasoning, physicist Gianluca Calcagni pointed out that if we “assume [that] we cannot get rid of the big bang” singularity, what we “would see if we went towards the singularity” is an “oscillatory [behavior] in the scale factors” which implies that “at any time there is always one scale factor increasing backwards in time. This process happens infinitely many times before reaching the singularity.” (Lectures on Classical and Quantum Cosmology; p.32)
It also worth pointing out that Dr. Pitts has made other arguments to support the idea that an initial singularity entails a beginningless past. For example, in the paper “Nonsingularity of Flat Robertson-Walker Models”, Dr. Pitts and his collaborator W. C. Schieve considered what one should make of singularities if one supposes a background space-time geometry that provides an independent notion of spacetime 'distance' that could differ from the observable rods-and-clocks distance used in General Relativity. They argued that one could and should locate the Big Bang infinitely far away in terms of the background geometry:
The Big Bang singularity for the flat Robertson-Walker cosmological spacetimes is dissolved by exile to the infinite past. (This idea formally resembles a suggestion made by Misner, Agnese, Wataghin, and Levy-Leblond in general relativity, but with a different justification.)
In a more recent paper (2018), they reiterated their findings, explaining that their approach has more physical motivations than the Leblonde-Misner's approach:
This problem can likely be resolved by stretching the Robertson-Walker solution so that the singularity occurs at unrenormalized past infinity. This move resembles one by Misner, Agnese, Wataghin, and Levy-Leblond, but our introduction of a flat background metric gives a compelling physical motivation, which they perhaps lacked.
Topology to the Rescue?
Finally, Dr. Pitts mentioned that some have attempted to avoid this implication by ignoring metrics altogether and appealing to topology in order to define a first moment. But even topology cannot avoid this issue:
Given that neither existence nor uniqueness of a metric (for timelike curves) holds necessarily, the natural move is to adopt a topological rather than metrical notion of beginning. However, to adopt a first moment as the criterion for a beginning is to admit defeat because the two plausible moves relative to contemporary cosmology (viz., taking space-time to contain only points ‘after’ the singularity or invoking some perhaps presently unknown theory that resolves the singularity) both lack a first moment.
Conclusion
Classical relativity, more specifically the Robertson-Walker metric and refined time measurements, negates the notion of an absolute beginning for the universe – Adolf Grünbaum’s argument shows the past-finite cosmic time-interval is ordinally unbounded, lacking a physical t=0 event as the spacetime metric degenerates at the singularity, a finding backed by Leblond and Pitts, while superior metrics from Leblond and Misner, based on curvature oscillations and linear scales, stretch the past to infinity despite its finite measure in ordinary time, with every instant preceded by another in a structure that has no first cause or starting point. Finally, Dr. Pitts demonstrates that attempts to impose a first moment through topology fail.