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There is sufficient evidence at present to justify the belief that the universe began to exist without being caused to do so. This evidence includes the Hawking-Penrose singularity theorems that are based on Einstein’s General Theory of Relativity, and the recently introduced Quantum Cosmological Models of the early universe. The singularity theorems lead to an explication of the beginning of the universe that involves the notion of a Big Bang singularity, and the Quantum Cosmological Models represent the beginning largely in terms of the notion of a vacuum fluctuation. Theories that represent the universe as infinitely old or as caused to begin are shown to be at odds with or at least unsupported by these and other current cosmological notions.

My purpose in this paper is to argue that there is sufficient evidence at present to warrant the conclusion that the universe probably began to exist over ten billion years ago, and that it began to exist without being caused to do so. I believe accordingly that the positions held by many if not most contemporary philosophers concerning this issue are unjustified, for their beliefs typically fall into one of three mutually exclusive categories, (1) the universe is probably infinitely old, (2) the universe began to exist and its beginning was caused by God, and (3) insufficient evidence is available to enable us to decide about whether the universe began to exist or is infinitely old.

1. The Prediction of a Space-time Singularity in our Past. Most philosophers today are aware that the Big Bang cosmological theory has superseded the Steady State theory, but a great number of these philosophers erroneously believe either that there is probably an infinite number of cycles of expansion and contraction of the universe, or that there is insufficient evidence to decide between the infinitely oscillating model and the theory that there was an earliest or single expansion, or that there is a first expansion that needs to be explained by introducing divine causality. That these beliefs are unfounded becomes apparent once the evidence for the prediction of a singularity in our past by the Big Bang cosmological model is adequately clarified.

The most important but by no means the only observational evidence for the Big Bang theory is the redshift of the light from distant galactic clusters, first discovered by Slipher and Hubble, which indicates the universe to be expanding uniformly in all directions.¹ This suggests that there is some time in the past when all the galactic clusters, or all the materials in these clusters, were arbitrarily close together, and that this time represents the beginning of the universe.

This is more exactly understood in terms of the models of the universe provided by the so-called Friedmann solutions of the field equations of Einstein’s General Theory of Relativity (GTR). The field equations show that the metric of space-time is dependent upon the matter present in that space-time.² The field equations can be solved for the universe as a whole if figures reflecting the observed values of the universe are introduced. Since the universe is isotropic (the same in all directions) and homogeneous (the matter is evenly distributed), it is described by the Robertson-Walker metric,³ which applied to the field equations enable them to be reduced to (with the cosmological constant / omitted):

-3d²a/dt² = 4*pi*G (p + 3P/c²)a

3(da)²/dt = 8*pi*Gpa² – 3kc²

a is the scale factor representing the radius of the universe at a given time. da/dt is the rate of change of a with time; it is the rate at which the universe expands or contracts. d²a/dt² is the rate of change of da/dt, that is, the acceleration of the expansion or the deceleration of the contraction. G is the gravitational constant and c the speed of light. P is the pressure of matter and p its density. k is a constant which takes one of three values: 0 for a flat Euclidean space (in which case the universe is open, that is, expands forever), -1 for a hyperbolic space (in which case the universe is also open), or +1 for a spherical space (in which case the universe is closed, that is, will contract).

What is important to note about these Friedmann equations is that if p, the density of matter in the universe, is positive, then the right side of the first equation is positive, and this entails that d²a/dt², the acceleration of the expansion or the deceleration of the contraction, cannot be zero. d²a/dt² must be negative, which means that the acceleration of the expansion is decreasing or that the acceleration of the contraction is increasing. In a word, if there is matter present in the universe, then the universe must be either expanding or contracting with a varying acceleration.

It is the case of expansion that especially interests us, for the universe is now expanding. If the acceleration of the expansion is decreasing, this implies that the further we go into the past the greater the increase of the acceleration and the smaller the scale factor a of the radius of the universe, until a time t₀ is reached when a = 0. As d²a/dt² increases and a decreases, the density of matter p increases, until at t₀ the value of p is infinite. At this time the entire universe is squeezed into at least one point of infinite density, infinite temperature, and infinite curvature. We have reached a space-time singularity.

I shall argue in the next sections that these considerations support the idea that there is an uncaused beginning of the universe. In the remainder of this section I shall discuss the issue of whether or not the singularity is real.

At first it was thought that the singularity predicted by the Friedmann equations was fictitious, since its prediction depended upon the assumption that the universe is exactly homogeneous and isotropic, whereas in reality it is only approximately so. Consider an inexactly symmetrical contracting universe: as the radius of the universe approaches zero the convergence of particles, due to small perturbations, would not focus upon a single point; rather the particles would rebound off one another and result in a “bounce” of the universe and a new phase of expansion. In the words of E. M. Lifshitz and I. M. Khalatnikov, who developed one of the more recent arguments for this scenario, the fluctuations “exclude the possibility of the existence of a singularity in the future of the contracting universe and imply that the contraction of the universe (if this must in general occur) must finally be turned into an expansion” (Lifschitz and Khalatnikov 1963, p. 207). This is the basis of the idea of an oscillating universe, according to which the universe runs through successive cycles of expansion and contraction. The present phase of the expansion, accordingly, can be understood as a result of a prior phase of contraction.

Before I explain how it can be proven that the above argument is mistaken and that a singularity must occur even if the universe is inexactly symmetrical, I shall show first that the assumption that the singularity is fictitious and that the universe oscillates does not render probable the idea that the universe is infinitely old.

Models of an oscillating universe usually predict that with each new cycle there is an increase in the size of the radius of the universe, amount of radiation present, and entropy.⁴ Radiation from previous cycles accumulates in each new cycle, and the accompanying increase in pressure causes the new cycle to be longer than the last one; the universe expands to a greater radius and takes a longer time to complete the cycle. This disallows an infinite regress into the past, for a regress will eventually arrive at a cycle that is infinitely short and a radius that is infinitely small; this cycle, or the beginning of some cycle with values approaching the values of this cycle, will count as the beginning of the oscillating universe.

The inference to a finite past can also be made from a measure of the amount of radiation present in the universe; if there were an infinite number of previous cycles, an infinite amount of radiation would be present in the current cycle, but the amount measured is finite. Joseph Silk calculates that the amount of radiation observed in the present expansion allows there to be “about 100 previous expansion and collapse cycles of the universe” (Silk 1980, p. 311).

The conclusion that the past is finite also follows from facts about entropy; if an infinite number of previous cycles have elapsed, each with increasing entropy, then the present cycle would be in a state of maximum entropy-but in fact it is in a state of relatively low entropy.

John Wheeler sweeps away these objections to an infinitely oscillating universe by supposing that at the end of each contracting phase all the constants and laws of that cycle disappear and the universe is “reprocessed probabilistically” (Misner, Thorne and Wheeler 1973, p. 1214) so as to acquire new constants and laws in the next cycle. No information about a previous cycle is passed on to the next cycle. Accordingly, no inference to a finite past can be made on the basis of present observations and the laws and constants that hold in the current cycle.

Now there is no reason to think that such a universe is logically impossible, but that is not germane to our present concern, which is to establish probabilist grounds for a belief in the finitude or infinitude of the universe’s past. It is logically possible that at the point of onset of each new cycle all laws and constants are transformed, but since these occurrences cannot be predicted according to any known physical law, there is no reason to think that these transformations occur.

Indeed, there is a theoretical reason to prefer the finite oscillatory models to Wheeler’s model (supposing that we must choose among oscillating models). The finite models, through being constructed in accordance with the known physical laws and constants, obey a principle related to the principle of induction; the related principle is that physical laws and constants originally inductively established for one domain of physical events should be applied to other domains of physical events if there is no observational evidence that events in these other domains differ in the relevant respects from those in the original domain. In the present context, the domains are cycles; since there is no observational evidence that events in past cycles differ relevantly from those in our cycle, we are not justified in supposing that the laws and constants inductively established in our cycle do not apply to the events in previous cycles.

The issue of whether oscillating universes are finite or infinite in respect of the past lost much of its urgency in the middle and late 1960s, with the development of the Hawking-Penrose singularity theorems (Penrose 1965; Hawking 1965, 1966, 1970), which entailed that an inexactly homogeneous and isotropic universe must have a singularity. A space-time contains a singularity if (1) the space-time satisfies the equations of GTR, (2) time-travel into one’s own past is impossible and the principle of causality is not violated (there are no closed timelike curves), (3) the mass density and pressure of matter never becomes negative,⁵ (4) the universe is closed and/or there is enough matter present to create a trapped surface, and (5) the space-time manifold is not too highly symmetric.⁶

It is reasonable to assume that all of these conditions, except perhaps (4), apply to our universe. Condition (4) might seem to be open to question if the universe is not closed and the condition of a trapped surface must obtain. A trapped surface is one from which light and matter cannot escape due to the intensity of the gravitational forces, such that under the influences of these forces the space-time paths of radiation and matter within the trapped surface converge towards a singularity. If the singularity is in the past, the geodesics of the rays and particles stem from the singularity; if it is in the future, the geodesics aim towards it. In the case of our universe, there is a singularity in its past (be the universe open or closed) if there is enough matter present to create a trapped surface. And there is enough matter:

Recent observations of the microwave background indicate that the universe contains enough matter to cause a time-reversed closed trapped surface. This implies the existence of a singularity in the past.⁷

2. The Beginning of the Universe Defined. In order to show how the foregoing considerations render probably true the idea that the universe spontaneously began, a precise definition of the beginning of the universe must be developed. That is the task of the present section.

There are at least three possible definitions that might seem to be consistent with the ideas of the last section. Either the universe began (i) at the singularity, (ii) after the singularity, or (iii) neither at nor after the singularity.

(i) If the universe were closed and perfectly homogeneous and isotropic, then the definition of the beginning of the universe “at the singularity” would be relatively simple. At the first time, t₀, when a = 0, there exists a single point in which the entire universe is compressed, and the existence of this point counts as the beginning of the universe. This point exists for one instant before exploding in the Big Bang. However, since the universe is imperfectly homogeneous and isotropic, a more complicated definition is necessary. The imperfect symmetry implies that the universe began nonsimultaneously at a series of points.⁸ Moreover, it is necessary that these points be infinite in number if the universe is open and space is infinite, for in one point there can be compressed only a finite volume of space. Given these factors, the appropriate definition of the beginning of the universe is that it began at the earliest singularity, this singularity consisting of the existence at t₀ of the first point(s) to explode in a Big Bang. The universe began at this time in the sense that this is the earliest time at which some part of the universe exists.

There is a serious problem with this definition. The universe is standardly defined as the set of events, each event being a point in a 4-dimensional space-time continuum, such that each event is characterized by four coordinates (x₁,x₂,x₃,t), the first three being spatial and the fourth temporal. But the singularity at t₀ is not in a 3-d space; it is in a space either of 0 dimensions (if it is just one point), 1 dimension (if it is a series of points constituting a line or line segment) or 2 dimensions (if it is a series of points comprising a surface-like space). Accordingly, the singularity at t₀ is not a part of the universe and a fortiori not the earliest part of the universe. Rather it is a source of the universe. The universe itself began at some time after t₀, which leads us to the second definition of the beginning of the universe.⁹

(ii) On this definition the beginning of the universe is the explosion of 4-dimensional space-time out of the earliest singularity, the singularity at t₀. In other words, the beginning of the universe is the Big Bang. The Big Bang is the first state of the universe.

The Big Bang occurs at t > t₀. However there is not some instant at which the Big Bang occurs, for (assuming that time is dense or continuous) there is no earliest instant after the first instant t₀; for every instant t_a > t₀ there is another instant t_b < t_a. Accordingly, if the phrase “the Big Bang” is to be used unequivocally, it must be used to designate a state occupying an interval that is the first interval of some length to elapse after t₀. Although on a priori grounds there is no nonarbitrary basis for selecting this length, there are empirical reasons for identifying the first post-t₀ interval of length 10^-43 second as the time of the Big Bang. The earliest state of the universe that cosmologists have determined to be unpreceded by a state of a different kind is the state constitutive of the Planck era, which occupies the first post-t₀ interval of length 10^-43 second. A cosmological state of some kind K is the only state at which all and only the types of particles and forces present in that state exist. It is speculated by many cosmologists that during the Planck era and only during the Planck era there existed only one type of force, the superforce, and one type of particle, the superparticle; the superforce is the force from which the gravitational, strong, weak and electromagnetic forces subsequently separated due to symmetry breaking, and the superparticle likewise became differentiated into the various types of bosons and fermions. Following the Planck era there is the GUT era from 10^-43 to 10^-35 second (the inflationary expansion occurs at 10^-35 second at the end of the GUT era), the electroweak era from 10^-35 second to 10^-10 second, the free quark era from 10^-10 second to 10^-4 second, and so on up to the present.

In order to eliminate possible confusion about the identity of this Big Bang, it must be noted that it is the explosion of 4-dimensional space-time out of the point(s) that exist(s) at t₀; there are other explosions from points that exist at instants later than t₀. The Big Bang that explodes from the singularity at t₀ is the first Big Bang, and can be designated as “the Big Bang₁“. It is the Big Bang₁ that is the beginning of the universe.

The Big Bang₁ is “the first state of the universe” in two senses. In the first sense, the Big Bang₁ begins and ends before every other state of a different kind begins; in this sense, the first state of the universe is a nonoverlapped state. In the second sense, the Big Bang₁ begins before every other Big Bang begins; in this sense the first state of the universe is a partially overlapped state, for it is likely that other Big Bangs begin before the Big Bang₁ ends.

(iii) A third possible definition is that the universe began with the Big Bang₁ at the earliest interval of 10^-43 second, but that it did not begin at or after the earliest singularity. This does not entail that the universe began before this singularity, for it is possible for the universe to begin neither before, at nor after this singularity; this possibility is actual if there is no time at which the singularity exists. The concept of a singularity, on this view, is a limiting concept that refers to nothing existent. The prediction by the Friedmann equations of a time t₀ when a = 0 is interpreted as a prediction of a limit to time and to the radius of the universe. “t₀” does not refer to a time but expresses a concept of an ideal limit that past times can approach with arbitrary closeness but can never reach; every actual time t is such that t >t₀. The same holds for the concept of a = 0, and the concepts of infinite density, temperature and curvature.

An alternate explication of the definition of the universe as beginning “neither before, at nor after the singularity” is implied by some remarks of Richard Swinburne. According to this explication, there is no time at which the singularity exists, but there is time, empty time, prior to the Big Bang₁. If t₀ is the time at which the first singularity would have existed had it existed, then the Friedmann equations can be taken as predicting that “the Universe must have come into being after t₀” (Swinburne 1981, p. 254). Paul Fitzgerald interprets Swinburne to mean something that Fitzgerald takes to be absurd, that “the universe popped into being, preceded by a finite empty lapse of time!” (Fitzgerald 1976, p. 635). But this is a misinterpretation of Swinburne, for Swinburne argues that it is logically necessary for past time to be infinite (Swinburne l981, pp. 172-173), and therefore that if the universe began at t > t₀ there must be an infinite amount of empty time prior to the Big Bang₁ or to t₀.

I have elsewhere shown that Swinburne’s and others’ putative proofs of the necessary infinitude of time are fallacious, so Swinburne’s characterization of the beginning of the universe need not be accepted (Smith 1985c). But that is not to say that it is logically impossible for the Big Bang₁ to have occurred after an infinite (or finite) period of empty time, as J. G. Whitrow (1980, p. 32) and others have argued it to be; this is logically possible.¹⁰ What is pertinent to my present investigation is that this is improbable, given that the empirically established cosmological theories that predict a beginning of the universe do so by predicting a beginning of time (this will be proven in the next section).

If we reject the “empty time” explication of the third definition of the beginning of the universe, that leaves us with two seemingly viable definitions, the second and the third as originally explicated. I shall assume the second definition to be the correct one, as it treats the singularity as real and thus complies with the Hawking-Penrose singularity theorems. The few objections that have been made to the reality of the singularity since the development of the singularity theorems have been based for the most part on philosophical grounds, and do not seem very convincing.¹¹ In any case, I will show that the conclusion that the universe spontaneously began follows no less if the third definition is used.

3. Arguments that the First Singularity and the Big Bang₁ are Uncaused. The idea that the Friedmann equations and the Hawking-Penrose singularity theorems predict an uncaused beginning of the universe is resisted by many philosophers. W. H. Newton-Smith writes:

. . . supposing that the Big Bang emerged from a singularity of infinite density, it is hard to see what would constitute a reason for denying that that singularity itself emerge from some prior cosmological goings-on. And as we have reasons for supposing that macroscopic events have causal origins, we have reason to suppose that some prior state of the universe led to the production of this particular singularity. (1980, p. 111)

This argument fails on several accounts. Note first that

(1) We have reason for supposing that macroscopic events have causal origins

entails

(2) We have reason for supposing that the cosmological singularity has a causal origin

only given the additional premise

(3) The cosmological singularity is a macroscopic event

which is false, for the singularity, far from being a macroscopic event, is infinitely smaller than the smallest microscopic event that physicists have yet detected. Moreover, the singularity is not even an event, that is, a point in 4-dimensional space-time; it is not a part of but a boundary or edge of the 4-d-space-time continuum.

Furthermore, it belongs analytically to the concept of the cosmological singularity that it is not the effect of prior physical events. The definition of a singularity that is employed in the singularity theorems entails that it is impossible to extend the space-time manifold beyond the singularity. The definition in question is based on the concept of inextendible curves, a concept that has been most completely and precisely explicated by B. G. Schmidt (1971). In a space-time manifold there are timelike geodesics (paths of freely falling particles), spacelike geodesics (paths of tachyons), null geodesics (paths of photons), and timelike curves with bounded acceleration (paths along which it is possible for observers to move). If one of these curves terminates after a finite proper length (or finite affine parameter in the case of null geodesics), and it is impossible to extend the space-time manifold beyond that point (for example, because of infinite curvature), then that point, along with all adjacent terminating points, is a singularity. Accordingly, if there is some point p beyond which it is possible to extend the space-time manifold, beyond which geodesics or timelike curves can be extended, then p by definition is not a singularity.

This effectively rules out the idea that the singularity is an effect of some prior natural process. A more difficult question is whether or not the singularity or the Big Bang probably is an effect of a supernatural cause, God. I will consider first the question of whether the Big Bang₁ is probably supernaturally caused. This fits in with W. L. Craig’s argument for a divine causality of the beginning of the universe, for Craig rejects the singularity as unreal and treats the Big Bang as the first physical state (Craig does not distinguish among the several Big Bangs) (Craig 1979). Craig’s argument includes the steps

(4) We have reason to believe that all events have a cause

(5) The Big Bang is an event (or a set of events)

(6) Therefore, we have reason to believe that the Big Bang has a cause

Additional steps are introduced to show that the cause of the Big Bang probably is a personal Creator.

An argument of this sort avoids the problems in Newton-Smith’s argument, for it does not argue from macroscopic events to something that is neither macroscopic nor an event, but argues from events in general to another event or set of events, the Big Bang. Furthermore, it does not violate the singularity theorems in supposing that the space-time manifold is extended beyond a singularity.

Nevertheless, the argument fails because its first premise, (4), is false. Craig writes of (4):

Constantly verified and never falsified, the causal proposition may be taken as an empirical generalization enjoying the strongest support experience affords. (1979, p. 145)

However, quantum mechanical considerations show that the causal proposition is limited in its application, if applicable at all, and consequently that a probabilistic argument for a cause of the Big Bang cannot go through. It is not relevant to the demonstration of this fact whether the causal relation be analyzed in terms of physical necessity or in terms of the regular but nonnecessary conjunction of events of a certain kind. Either analysis may be assumed. It is sufficient to understand causality in terms of a law enabling single predictions to be deduced, precise predictions of individual events or states. That there are uncaused events in this sense follows from Heisenberg’s uncertainty principle, which states that for conjugate magnitudes such as the position q and momentum p of a particle, it is impossible in principle to measure both simultaneously with precision. If p lies within a certain interval of length delta-p, and q lies within a certain interval of length delta-q, then if delta-p is made very small (measured exactly), delta-q cannot at the same time be made very small (measured exactly). Exactly put, the product of delta-p and delta-q cannot be made smaller than Planck’s constant h divided by 4*pi, so that

delta-p • delta-q >= h/(4*pi)                     (7)

Now if the initial conditions such as p and q of a particle x cannot all be known precisely at time t₁, then the subsequent conditions of x at time t₂ cannot be precisely predicted. The prediction of the conditions of the conjugate magnitudes of x at t₂ must be statistical and indeterministic. For example the position of x at t₂ is represented in terms of various possible positions each with a different probability value, such that none of these values is able to arbitrarily approach 1. These predictions are effected by means of the Schrodinger wave function psi; the square of the amplitude of psi at each point of psi determines the probability distribution of condition q of x at t₂. If by d(q, t₂₎ we mean the probability distribution of q at t₂ , this can be calculated as

d(q,t₂) = |(q,t₂)|²                     (8)

Equations (7) and (8) at most tend to show that acausal laws govern the change of condition of particles, such as the change of particle x’s position from q₁ to q₂. They state nothing about the causality or acausality of absolute beginnings, of beginnings of the existence of particles. Consequently, supposing that with suitable additional premises we can draw inferences from (7) and (8) to the whole universe, the only relevant argument that we could show to be unsuccessful is

(9) We have reason to believe that all changes of condition are caused

(10) Therefore, we have reason to believe that all changes of condition of the universe as a whole are caused

and the failure of this argument does not entail the failure of

(11) We have reason to believe that all beginnings of existence are caused

(12) Therefore, we have reason to believe that the beginning of the universe’s existence is caused.

Thus if the latter argument is to be refuted, it is necessary to find premises more relevant to absolute beginnings than (7) and (8).

Such premises can be obtained on the basis of Heisenberg’s uncertainty relation

delta-E • delta-t >= h/(4*pi)                     (13)

where E [is equivalent to] energy, t [is equivalent to] time and h [is equivalent to] Planck’s constant. This relation implies that if the energy of a particle is measured precisely, so that delta-E is made very small, the time at which the particle possesses this energy can be known only imprecisely, so that delta-t is very large. Now if delta-t is small enough, delta-E becomes so large that it becomes impossible in principle to determine if the law of energy conservation is violated. During the interval of time

delta-t [is congruent to] (h/(4*pi)*delta-E                     (14)

this law is inapplicable and consequently an amount of energy delta-E can spontaneously come into existence and then (before the interval has elapsed) cease to exist. There is observational evidence, albeit indirect, that this uncaused emergence of energy or particles (notably virtual particles) frequently occurs. It appears, then, that the argument (11)-(12) is unsuccessful and that the crucial step in the argument to a supernatural cause of the Big Bang, or more exactly, of the Big Bang₁, is faulty.

It might be objected that quantum acausality applies only on the microscopic level and not to macroscopic cosmological states or beginnings. This could be granted without detriment to my argument, for the physical processes constitutive of the Big Bang₁ (which I have defined to occur during the Planck era) are one and all microscopic and occur at dimensions where quantum mechanical principles unquestionably apply.

It might then be objected that the Big Bang₁ is the beginning of the existence of 4-d space-time itself, and that the uncaused events I have specified involve merely the beginnings of existence within 4-d space-time. Surely “There are some uncaused beginnings of existence within space-time” is irrelevant to and thus cannot increase the probability value of “The beginning of the existence of space-time itself is uncaused”.

My response is that if this is so (and I will provide reasons for doubting that this is the case in Section 4 with my discussion of the vacuum fluctuations models of the universe), then the same holds for the parallel argument for a supernatural cause of 4-d space-time; for “There are some caused beginnings of existence within space-time” or even “All beginnings of existence within space-time are caused” would by the same token be irrelevant to and thus fail to increase the probability of “The beginning of the existence of 4-d space-time is caused”.

I conclude that quantum mechanical considerations show that the argument to a divine cause of the Big Bang₁ based on the causal principle (4) is unsuccessful.

But this does not end the matter, for it is still open to a defender of the theistic argument to claim that I have no right to introduce quantum mechanical acausality into the discussion, since the Big Bang cosmological model is based on GTR and GTR presupposes a causal determinism to operate in the domain of its application.

I shall not decide whether this objection is valid, but will instead show that even if it is valid it still can be proven that we have no reason to think the Big Bang₁ is caused, supernaturally or otherwise. This can be demonstrated solely on the basis of the GTR Big Bang cosmological model itself.

Let us begin by assuming what I have already maintained to be the case, that the singularity at t₀ is real. Given that, we can note that the classical notions of space and time and all known laws of physics (since they are formulated on a classical space-time background) break down at the singularity, and consequently it is impossible to predict what will emerge from the singularity. This impossibility is not due to our ignorance of the correct theory but is a limitation upon possible knowledge that is similar but additional to the limitation entailed by the quantum mechanical uncertainty principle. The former limitation is due to the causal structure of space-time that is postulated by GTR; the interaction region postulated by GTR can be bounded not only by an initial surface on which data are given and a final surface on which measurements are made but also by a hidden surface. A hidden surface is one about which any possible observer can have only limited information, such as (in the case of black holes) the mass, angular momentum and charge. This surface “emits with equal probability all configurations of particles compatible with the observers limited knowledge” (Hawking 1976, p. 2460). A surface close to the Big Bang₁ singularity, a surface at the Planck time 10^-43 second, is a hidden surface. The singularity hidden by this surface “would thus emit all configurations of particles with equal probability” (Hawking 1976, p. 2463). If we assume with Craig that the singularity is unreal, and that the first physical state is the Big Bang₁, then the hidden surface is not taken to be subsequent to the singularity; instead of the particles being regarded as randomly and spontaneously being emitted from the singularity, they are regarded as randomly and spontaneously being emitted from nothing at all. This means, precisely put, that if the Big Bang₁ is the first physical state, then every configuration of particles that does constitute or might have constituted this first state is as likely on a priori grounds to constitute it as every other configuration of particles. In either case, the constitution of the Big Bang₁ is impossible in principle to predict and thus is uncaused (for “uncaused” minimally means “in principle unpredictable”).

The singularity itself is also regarded in the GTR-based Big Bang theory as uncaused, although for a different reason. It is defined as a point beyond which space-time curves cannot be extended, and thus which cannot have causal antecedents.

In sum, then, we may say that although the GTR-based Big Bang theory does suppose causality to operate in its domain of application, it also supposes that there is a limit to its operation; it represents causality as breaking down at the initial physical states, the singularity and the Big Bang. Consequently, this theory cannot be used to support the thesis that the initial physical states are probably caused and that this cause is God.

4. Quantum Gravity and the Uncaused Beginning of the Universe. There is a serious lacuna in my foregoing account of the beginning of the universe; I have been presuming that the GTR-based Big Bang theory has an unlimited application and therefore applies to the extreme conditions of the universe during the Big Bang₁. In fact, GTR fails to apply when quantum mechanical interactions predominate, and these predominate when the temperature is at or above 10³²K, when the density is at or above 10⁹⁴ gm cm^-3, and when the radius of curvature becomes of the order of 10^-33 cm. Since these conditions obtain during the Planck era at the first 10^-43 second after the singularity, the GTR-based Big Bang theory cannot be used as a reliable guide in reconstructing the physical processes that occurred during this time and a fortiori cannot be used as a reliable basis for predicting that the density, temperature and curvature reached infinite values prior to this time. Accordingly it seems that the foregoing probabilistic argument to an uncaused beginning of the universe is in jeopardy.

I believe, however, that there are three reasons for a continued support of the idea that the universe spontaneously began to exist. To comprehend these reasons, we must first observe that the reason why GTR is inapplicable during the Planck era is that the theory of gravity in GTR is unable to account for the quantum mechanical behavior of gravity during this era. A new quantum theory of gravity is needed. Although such a theory has not yet been developed, there are some general indications of what it may predict. It is in terms of these indications that our three reasons are to be understood.

First, it is thought that a quantum theory of gravity may show gravity to be repulsive rather than attractive under conditions that obtain during the Planck era. During this time regions of negative energy density may be created by the forces and particles present, and these regions lead to a gravitational repulsion. This suggests that any given finite set of past-directed timelike or null geodesics will not converge in a single point but will be pushed apart, as it were, by the repulsive gravitational force. This possibility is consistent with an oscillating universe, for as each contracting phase ends gravity becomes repulsive and prevents converging geodesics from terminating in a point; gravity repels them so that they enter a new expanding phase.

But this way of avoiding the singularity predicted by the Hawking-Penrose theorems does not give us a universe that is infinitely old. For-and this is the first of the three reasons I want to mention-this oscillating quantum-gravitational universe would still be subject to the same problems that were discussed in Section One, namely, increase in radius, length of cycle, radiation and entropy with each new cycle. Consequently, this theory does no more than push the cosmological singularity further into the past, at a time just before (or at) the beginning of the first cycle when the radius of the universe is zero (or near zero).

The second reason is that there is a way in which the Hawking-Penrose theorems’ prediction of a singularity at the beginning of the present expansion can be made consistent with a quantum theory of repulsive gravity. These theorems do not define a singularity as that wherein curvature, density and temperature are infinite and the radius is zero. A singularity is defined as a point or series of points beyond which the space-time manifold cannot be extended. Consequently, if the effects of quantum gravity prevent a build up of temperature, density and curvature to infinite values, and a decrease of radius to zero, this need not mean there is no singularity at the beginning of the present expansion. The singularity could occur with finite and nonzero values.

The third reason is that the most theoretically developed attempts to account for the past of the universe on the basis of specifically quantum mechanical principles have represented the universe as spontaneously beginning at the onset of the present expansion. These theories are collectively known as the “vacuum fluctuation models of the universe”. The models developed by Tryon (1973), Brout, Englert and Gunzig (1978), Grishchak and Zeldovich (1982), Atkatz and Pagels (1982), and Gott (1982) picture the universe as emerging spontaneously from an empty background space, and the model of Vilenkin (1982) depicts it as emerging without cause from nothing at all.

The first vacuum fluctuation model was developed by Edward Tryon in 1973. A vacuum fluctuation is an uncaused emergence of energy out of empty space that is governed by the uncertainty relation delta-E • delta-t >= h/(4*pi), and which thus has zero net value for conserved quantities. Tryon argues that the universe is able to be a fluctuation from a vacuum in a larger space in which the universe is embedded since it does have a zero net value for its conserved quantities. Observational evidence (Tryon claims) supports or is consistent with the fact that the positive mass-energy of the universe is cancelled by its negative gravitational potential energy, and that the amount of matter created is equal to the amount of antimatter. (But this last point is inconsistent with current Grand Unified Theories.)

A disadvantage of Tryon’s theory, and of other theories that postulate a background space from which the universe fluctuates, is that they explain the existence of the universe but only at the price of introducing another unexplained given, namely, the background space. This problem is absent from Vilenkin’s theory, which represents the universe as emerging without a cause “from literally nothing” (1982, p. 26). The universe appears in a quantum tunneling from nothing at all to de Sitter space. Quantum tunneling is normally understood in terms of processes within space-time; an electron, for example, tunnels through some barrier if the electron lacks sufficient energy to cross it but nevertheless still does cross it. This is possible because the abovementioned uncertainty relation allows the electron to spontaneously acquire the additional energy for the short period of time required for it to tunnel through the barrier. Vilenkin applies this concept to space-time itself; in this case, there is not a state of the system before the tunneling, for the state of tunneling is the first state that exists. The state of tunneling thus is the analogue of the Big Bang₁ in the third definition of the beginning of the universe offered in Section 2, for it is the first state of the universe and there is no time before this state. The equation describing this state is a quantum tunneling equation, specifically the bounce solution of the Euclidean version of the evolutionary equation of a universe with a closed Robertson-Walker metric.¹² The universe emerged from the tunneling with a finite size (a = H^-1) and with a zero rate of expansion or contraction (da/dt = 0). It emerged in a symmetric vacuum state, which then decays and the inflationary era begins; after this era ends, the universe evolves according to the standard Big Bang model.

These quantum mechanical models of the beginning of the universe are explanatorily superior in one respect to the standard GTR-based Big Bang models; they do not postulate initial states at which the laws of physics break down but explain the beginning of the universe in accordance with the laws of physics. The GTR-based theory predicts a beginning of the universe by predicting initial states at which the laws of the theory that are used to predict these states break down. The singularity and the explosion of 4-d space-time from the singularity obey none of the laws of GTR that are obeyed by states within the universe or subsequent states of the universe. In contrast, the quantum mechanical theories represent the universe as coming into existence via the same laws that processes within the universe obey. Instead of an exploding singularity, there is a quantum fluctuation or tunneling that is analogous to the fluctuations or tunnelings within the universe and that obeys the same acausal laws as the latter fluctuations or tunnelings.¹³

This review of the role of quantum mechanics in accounts of the beginning of the universe strongly suggests that the probabilistic argument to an uncaused beginning of the universe, although more complicated than we had been supposing in Sections 1-3, still goes through. Its conclusion is summarized in this disjunctive statement: it is probably true that EITHER the universe began without cause at the beginning of this expansion (a) subsequent to a singularity of infinite density, temperature and curvature, and zero radius, or (b) at a singularity with finite and nonzero values, or (c) in a vacuum fluctuation from a larger space or a tunneling from nothing, OR the universe spontaneously began to exist at the beginning of some prior expansion phase under conditions described in (a), (b) or (c).

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1. See G. Stromberg’s summary of V. M. Slipher’s measurements in Stromberg 1925; also see Hubble 1929. Other observational evidence that supports Big Bang cosmology includes the background microwave radiation of 2.7 K, which is a remnant of the intense heat generated at an early stage of the expanding universe. This radiation was first discovered (and initially measured to be 3.5 K) by A. Penzias and R. Wilson in Penzias and Wilson 1965.

A third major set of data supporting the Big Bang cosmology is the abundance of helium 4, deuterium, helium 3, and lithium 7, the formation of which is predicted to occur in the first minutes of the Big Bang.

2. According to John Wheeler (1973, p. 220), the simplest expression of the Einstein equations is

(curvature of space time) = 8*pi (density of the mass-energy present in that space-time).

More completely, it can be said that the field equations relate the metric tensor g_µv and its derivatives, which describe the geometry of space-time, to the energy-momentum tensor T_µv, which is determined by the distribution of the mass and energy in that space-time. These equations enable paths in space-time (specifically, geodesic paths) to be calculated. The formula summarizing the ten field equations is

R_µv – (1/2)Rg_µv + lamda*g_µv = -(8*pi*G/c²)T_µv

The terms on the left-hand side are composed of g_µv and its derivatives, and also of the constant lamda. G is the constant of gravitation and c the velocity of light.

3. The Robertson-Walker metric is determined by a, the radius of the universe at a certain time, and by the curvature of space-time. The metric of a homogeneous and isotropic universe is

ds² = dt² – (1/c²)a²*(d-sigma)²

where ds is the space-time interval between two events, d-sigma [derivative of sigma] is the line element of a space of constant curvature. and c the velocity of light.

4. The most widely discussed models have been developed in Tolman (1934, pp. 440 ff) and in Landsberg and Park (1975).

5. That is, the stress-energy tensor satisfies

( T_alpha-beta – 1/2*g_alpha-beta*T )u^alpha*u^beta >= 0

6. That is, the space-time is such that

^t(a^Rb)cd(e^tf)^{(t^c)*(t^d)} [is not equal to] 0

holds at some point along each timelike or null geodesic. t^a is the tangent vector.

7. Hawking and Ellis (1973, p. 3). The proof that the trapped surface created by this matter implies a singularity in our past (rather than in our future) is given on pages 356-359.

8. The less dense parts of the universe exploded from points first, followed by the more dense parts. See Barrow and Silk (1983, p. 42).

It should also be noted that if the universe is not sufficiently isotropic and homogeneous, some past-directed timelike geodesics will not end in singularities. Observational evidence, however, suggests that the universe is sufficiently symmetric so that all do end in singularities. See Hawking and Ellis (1973, pp. 358-359).

9. Although the space of the singularity is standardly defined as less than 3-d, Roger Penrose has proposed a definition of the cosmological singularity as a 3-d spacelike surface, in which case it could count as a part of the universe and thus as its beginning. See Penrose (1974).

10. I have shown that it is logically possible for there to be empty time before the Big Bang₁ in Smith (1985a).

11. A frequent objection is that singularities involve infinite values and that infinities cannot be real. See for example Craig (1979, pp. 116-117). I have rebutted Craig’s and others’ arguments against infinite realities in Smith (1987).

12. The bounce solution is a(t) = H^-1cos(Ht). See Vilenkin (1982, p. 26).

13. For further discussion of the vacuum fluctuation theories of the beginning of the universe, see Smith (1986b, esp. pp. 81-84). Other pertinent cosmological discussions can be found in Smith (1985b) and Smith (1986a, chapter Vl).