⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀A Finite Universe with no Absolute Beginning

Introduction

In this essay, two arguments are presented to show that classical (relativistic) Big Bang cosmology does not require or support an absolute beginning of the universe. The first argument states that even if spacetime is past-finite, it probably still lacks a beginning due to the nature of the singularity and time itself. The second (supplementary) argument states that an alternative way of measuring time close to the initial singularity can assign infinite duration to the past despite its finitude in cosmic time. Together, these arguments constitute a strong case against the necessity of a beginning.

Finite But Unbounded

Science popularizers often assert that the Big Bang theory proves the universe had an absolute beginning in a singularity; a moment when time, space, and matter came into existence from nothing. But the mathematics of the Robertson-Walker models tells a different story.

As Dr. Adolf Grünbaum (M.S. in physics from Yale University) repeatedly explained in many of his papers, the general theory of relativity (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 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. 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.

The French astrophysicist Jean-Marc Lévy-Leblond (known for the Lévy-Leblond equation) explained this issue very well in his paper The Unbegun Big Bang in the prestigious journal Nature:

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 ]t=0, ∞[; 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.

The obvious implication of this, as Dr. John Earman (expert in general relativity) 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 of a state at t.” (p.208) Dr. Quentin Smith (specialist in quantum cosmology and time) 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. Before any instant in an earliest hour, minute or second, there is an infinite number of other instants.” Dr. Brian Pitts (PhD in physics) elaborated further, saying that “if time is continuous, then the world’s being finitely old does not entail its having a first moment. Within the Robertson-Walker cosmologies, spacetime for t > 0 (i.e., always), one can ‘explain’ each moment in terms of an earlier one. Thus, there is no beginning required as far as physics can tell.” Dr. Chris Smeenk (M.S. in physics and astronomy) agreed with this conclusion, stating that “there is no ‘first instant’ of time in this 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.”

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This is very similar to what Dr. Roberto Torretti (specialist in relativity theory) 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.” Similarly, Dr. Daniel Linford (B.S. in physics) stated that “the singular FLRW models traditionally used to describe the Big Bang do not include a first moment. By analogy, consider that there is no immediate next real number after 0. Hence, there is no first element of the set of real numbers greater than 0. Singular FLRW models have a similar feature; while they include a finite past, they do not include a first moment.” Likewise, Dr. Craig Callender (specialist in the physics of time) pointed out that “the geometrical modeling used in the big bang model [implies] the past is finite even though there is no first moment, just as there is no first positive real number.”

The technical way of explaining why the first interval is past-open is that the singularity cannot be treated as a boundary point of spacetime. As Dr. Smeenk noted: “It would be inconsistent to treat singular points as points in spacetime, although one can study singularities by adding boundary points. One could add a boundary consisting of the point –T on each geodesic, much like adding boundary points to make an open interval closed. But the resulting spacetime with boundary is not taken to represent physical spacetime.” Dr. Earman made the same point: “There are ways of doctoring standard relativistic cosmological models to represent singularities as boundary points attached to the spacetime manifold. In this way one could hope to reclaim a ‘first instant’ for time. But… the boundary points are ideal elements… [and] ideal points are not points of spacetime.” (p.209) Dr. Torretti likewise observed 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.”

Adequate Singular Timescales

Physicists usually talk about the age of the universe in terms of Einstein’s proper time (i.e., as measured by a clock in gravitational freefall). That idea runs into problems with relativity, though. Special relativity doesn’t require us to apply proper time to the whole cosmos, and while general relativity can give ‘absolute’ times (viz., cosmic time, in which observers anywhere could use the local density of matter as a basis for a universal clock), the choice of coordinates is arbitrary. More importantly, near the singularity the heat and gravity would destroy every known physical clock, from a metal watch down to atomic nuclei. Proper and cosmic time therefore lose meaning in a place where no clock made of matter could survive, which is why an alternative timescale is perfectly adequate in that environment. (For further reading, I highly recommend Joe Rosen’s short paper When did the Universe Begin? and Silvia Bianchi’s Time and Cosmology)

Now, in order for the timescale to match the unbounded description of the past, Dr. Leblond suggested that we refine our concepts of time accordingly (despite Grünbaum’s insistence that this additional step is unnecessary). He proposed “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 timescale that assigns infinite duration to 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 new timescale corresponds to a clock geared to the expansion of the universe, where each unit of linear time corresponds to a specific change in the radius of space itself. Thus, Leblond concludes: “Naturally, on the linear scale, the lifespan of the Universe is infinite, so that never did the Big Bang begin.” (For details, see Dr. Leblond’s paper Did the Big Bang Begin?)

Objections Considered

Zeno Reloaded?

The most common objection presented by folks who first hear about this reasoning is that it is just a clever application of Zeno’s Dichotomy paradox, that is, to travel any distance, you must first traverse infinite halves, never arriving at your destination. But if we are allowed to use this paradox in the cosmic interval, why not apply it to every other interval since the Big Bang? All intervals are infinitely divisible, but we easily complete later ones. So why can’t we complete the first one and reach t=0?

According to Dr. Quentin Smith, the relevant difference between later intervals and the first one lies in the nature of the singularity. Unlike regular intervals, where both endpoints (instants) are well-defined physical events (e.g., noon and 1 PM), the initial interval has no closing instant in the earlier direction because t=0 is not a physical event, i.e., it is not a real spacetime point due to the breakdown of the metric. This means there is no “there” to reach. Therefore, the singularity’s nature makes it an unreachable boundary (or an “unattainable boundary point” as Dr. Leblond calls it) rather than a traversable point. As Dr. Smith explained:

Some philosophers have argued that if the first instant of the first hour after the Big Bang can be “deleted” (i.e., regarded as a nonexistent), then the first instant of any hour can be deleted. ... Their mistake is failing to realize that the first hour after the Big Bang lacks a first instant because of a unique circumstance, that there is a cosmic singularity. There is no cosmic singularity at the present hour or at the various hours they mention and [so] these hours or hour-long processes must have a first instant.

Non-Sequitur Fallacy?

Perhaps a follow-up objection to Dr. Smith’s reply is that, just because the singular t=0 cannot qualify as the first instant, it doesn’t logically follow that there can’t be a first instant, for the first instant could simply be one where (or when) the spacetime metric is well-defined. Therefore, this argument commits the non-sequitur fallacy.

In response to this, Dr. Smith argued that, since time is continuous, it follows that “between any two instants, there are an infinite number of other instants.” But if that’s the case, then “we will not find a certain instant that immediately comes after [the hypothetical] t=0. For example, the instant y corresponding to the number 0.5 cannot be the first instant, since between the number 0 and the number 0.5 there is the number 0.25 and some instant z corresponding to 0.25. The same holds for any other number in the interval 0 > x < or = 1.” Similarly, Dr. Pitts pointed out that “there is no earliest instant at which the space-time metric is well-defined. It is well-defined, according to Big Bang cosmology, for all moments after 0” (personal correspondence).

Planck Against Continuity?

Another very common objection is that time is not actually continuous at all. Quantum mechanics entails that time is discrete, such that there is a limit to how small an interval can be, and this limit is called the Planck time. The Planck time is the shortest measurable time unit in theory, representing the duration for a photon to travel one Planck length. However, if the first interval is discrete rather than infinitely divisible, then it must have a first instant.

The problem with this objection is that quantum mechanics does not require time to be discrete at all, and the Planck scale is a theoretical limit for measurement instead of a fundamental constraint. As physicist Alex Klotz explained, the Planck length (and by extension, Planck time) is not a fundamental pixel of the universe but a scale where quantum gravity effects become significant: “There is nothing in established physics that says this is the case, nothing in general relativity or quantum mechanics pointing to it” and “just because units are ‘natural’ it doesn’t mean they are ‘fundamental’.” The fact is that, as the astrophysicist Ethan Siegel pointed out, in order for spacetime to be truly physically discrete, it would have to violate the CPT theorem, Lorentz invariance, and the principle of relativity, all of which are well established.

Quantum Mechanics Strikes Again?

Here’s another way critics can appeal to quantum theory to attempt to refute the Leblond thesis. They might claim that it doesn’t work anymore because it entirely depends on general relativity. Quantum theory becomes important at the Big Bang and many argue it eliminates the initial singularity altogether. Thus, this thesis has no relevance to current cosmology. And, indeed, Dr. Leblond acknowledged this issue, saying that he chose “to deal with perfectly conventional (big-bang) and classical (nonquantum) cosmological scenarios [since] we do not wish to complicate with quantum effects... The crucial point is that for t=tᵢ, there is a true singularity of the model.”

However, the usual opponents of the Leblond thesis cannot appeal to quantum theory to attack it. Why not? Because they defend the proposition that the cosmos had an absolute beginning, and they make their case by appealing to theorems (e.g., Aron Wall’s Quantum Singularity theorem) that supposedly demonstrate the existence of a singularity in spite of quantum effects. But as long as spacetime has a singularity, the Leblond thesis applies. If, on the other hand, one rejects the singularity due to quantum effects, then one must accept that the past is probably infinite. Either way the cosmos has no beginning. As Dr. Brian Pitts explained:

The question therefore arises whether to take the singularity seriously as a feature of the real world, or to dismiss it as an artifact of incomplete physical understanding. As one sees all the time in papers on quantum gravity, most people who work on quantum gravity take for granted that the Big Bang singularity is an artifact of incomplete physical understanding and expect or hope that uniting gravity with quantum mechanics in some kind of quantum gravity will resolve the singularity into some well-defined situation that admits extrapolation to still earlier times, ad infinitum. … 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.

Inflating the Cosmic Interval?

Critics may argue that the Leblond thesis exclusively assumes the hot Big Bang model, which was later modified by inflation. In inflationary cosmology, the hot Big Bang is no longer the earliest phase; it is preceded by an exponential expansion driven by false vacuum decay. Dr. Grünbaum himself noted the change, saying that according to inflationary theory, “the Big Bang is no longer held to comprise all of the earliest instants; instead, the inflationary expansion is preceded [and driven] by the so-called vacuum states in a metrically finite past.” However, if the open-interval was replaced by an inflationary phase and a vacuum state in which spacetime is well-defined, then the Leblond thesis no longer applies.

This objection fails for the same reason that the quantum criticism fails. Either the inflationary models retain the singularity or they do not. If inflation eliminates the singularity, then the past must be infinite. But the usual opponents of the Leblond thesis insist (on the basis of “proofs” like the Borde-Guth-Vilenkin theorem) that inflationary spacetimes have singularities (or geodesic incompleteness) as well. However, if they retain it, the Robertson-Walker metric still degenerates at the supposed boundary, and no first moment exists. In that case, the vacuum state and the inflationary phase simply occur after the open interval. Therefore, in neither case is there an absolute beginning.

Violation of Relativity?

According to philosopher Robin Collins, the idea that there is an infinite regress of previous states presupposes that it is physically intelligible to talk of the instantaneous state of the whole universe in some absolute sense. However, Collins says, Einstein’s relativity theory denies that this is possible, implying instead that any set of instantaneous states will have to be defined relative to a specific inertial frame of reference.

This objection fails because causation can still work very well without Newton’s universal “now”. Although each state is being caused by a previous state in its own frame of reference (or relative clock), it is still caused nonetheless. In other words, every state of the universe, in every frame, is caused; the causal chain doesn’t care about one’s clock. Thus, in the open interval, each state is being caused by a previous state in its own relative clock. Additionally, Collins’ objection is incorrectly presupposing special relativity, as Dr. Smith explained:

[Leblond’s thesis] assumes Friedmann’s cosmology, not special relativity. Friedmann’s cosmology is the solution of Einstein’s theory of general relativity that applies to our universe; special relativity is the “vacuum solution” of the equations of general relativity. This point about special and general relativity, which is often not recognized by philosophers, is crucial here. For Friedmann’s cosmology does admit of an absolute frame, or a “privileged frame,” as is often said.

Beginning With No Starting Point?

In response to the Leblond thesis, philosopher William Craig argued that a beginning for time does not require that time have a first instant but only that time be finite in the past, viz., time’s beginning can be reached if it is traversed by intervals of equal length (e.g., seconds) rather than instants. So for Craig “beginning” should be understood metrically (by the finitude of the past) rather than topologically (by the existence of a first instant), so that the universe still has a beginning.

Dr. Brian Pitts replied that a metrical criterion for a beginning is inadequate because there are viable extensions or underdetermined alternatives to general relativity which supply neither a metric (as in Bach-Weyl theory) nor a unique metric (as in scalar-tensor theories); in the latter, the past is infinite in some metrics and finite in others. Nevertheless, Dr. Pitts replaced the demand for a first instant with the weaker condition of lacking maximal past extension, i.e., history could have contained everything it does contain plus some earlier events; it could have had “more of a past.” However, once “beginning” is understood this way, the singularity no longer counts as a beginning. How so? Dr. Pitts says the reason is that “the universe has as much past as it possibly could have, because there couldn’t have been more history prior to the Big Bang singularity ‘added’ onto the Big Bang singularity story” (personal correspondence).

For further reading on whether past-finitude qualifies as a beginning, see Graham Oppy’s  Arguing About Gods (pages 147-150) and Bede Rundle’s Why is There Something Rather Than Nothing? (pages 122-124).

Do ‘Instants’ Even Exist?

It was suggested to me that another possible way to avoid the implications of the Leblond thesis is to posit that instants – i.e., temporal points with no duration, analogous to movie frames – do not even exist in the real world. In that case, it is meaningless to state that the cosmic interval has infinitely many instants.

Now, the Leblond thesis works even if instants aren’t real because even if there are no actual instants of zero duration, shorter and shorter sub-intervals – that constitute the cosmic interval – exist, and there is an infinite number of them. Or, as Dr. Quentin Smith says, “there is no first instant and there are an infinite number of briefer and briefer first intervals of a given length.” Elsewhere, Dr. Smith made this point more explicitly:

The shortest time is the zero duration instant. But that really doesn’t matter, because I suppose that none of these [infinitely long line of events at the first state of the universe] are zero duration instants.

Conclusion

General relativity does not require an absolute beginning of the universe. Dr. Grünbaum’s argument demonstrates that the cosmic interval is ordinally unbounded because the spacetime metric degenerates at the singularity. Dr. Leblond’s linear time stretches the past to infinity despite its finite measure in cosmic time, so that every instant is preceded by another and no starting point exists.

References:

Callender, Craig, and Casey David McCoy. “Time in cosmology.” The Routledge Companion to Philosophy of Physics (2021): 707-718.

De Bianchi, Silvia. “Time and Cosmology.” The Routledge Companion to Philosophy of Time. Routledge, 2025. 364-373.

Earman, John. “Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes”. Oxford University Press, 1995.

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Grünbaum, Adolf. “Pseudo-Creation of the ‘Big Bang,’” Free Inquiry 13(1) (Winter 1992/1993): 14–15.

Grünbaum, Adolf. “Creation as a Pseudo-Explanation in Current Physical Cosmology.” In Erkenntnis Orientated: A Centennial Volume for Rudolf Carnap and Hans Reichenbach, 233–254. Dordrecht: Springer Netherlands, 1991.

Grünbaum, Adolf. “Narlikar’s ‘Creation’ of the Big Bang Universe was a Mere Origination.” Philosophy of Science 60.4 (1993): 638–646.

Grünbaum, Adolf. “Some Comments on William Craig’s ‘Creation and Big Bang Cosmology.’” Philosophia Naturalis 31.2 (1994).

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Klotz, Alex. “What Planck Length Is and Its Common Misconceptions.” Physics Forums, 9 September 2015.

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Lévy-Leblond, Jean-Marc. “Did the Big Bang Begin?” American Journal of Physics 58.2 (1990): 156–160.

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