Did the Big Bang Have A Cause

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Big bang cosmology is based on four sets of equations. The most fundamental are the field equations of the general theory of relativity (GTR), which relate the metric tensor g_ab and its derivatives to the energy-momentum tensor T_ab. The formula summarizing the ten field equations is

R_ab is the Ricci tensor of the metric g_ab, R is the Ricci scalar, lambda is the cosmological constant (probably zero), c is the velocity of light and G is Newton’s constant of gravitation. The field equations are solved for the universe if the relevant observational values are introduced. Since our universe is (approximately) homogeneous and isotropic, it is described by the Robertson-Walker metric, which is determined by the radius of the universe at a given time and the curvature of space-time. The metric is

where ds is the space-time interval between two events, a the scale factor representing the radius of the universe at a given time, and do is the line element of a space with constant curvature. The application of this metric to the field equations provides us with the Friedmann’s solutions, which are the heart of big bang cosmology. With the cosmological constant omitted, these solutions read:

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In these equations a as before is the scale factor representing the radius of the universe at a given time. da/dt is the first derivative of a with respect to time; it measures the rate of change of a with time (the rate at which the universe expands or contracts). D²a/dt² is the second derivative of a with respect to time; it measures the rate of change of da/dt (the acceleration of the expansion or the deceleration of the contraction). P is the pressure of matter and p its density. k is a constant which takes one of three values: 0 for a flat Euclidian space, -1 for a hyperbolic space, or +1 for a spherical space.

If Friedmann’s solutions are supplemented by the Hawking—Penrose singularity theorems, and the theorems are satisfied, it follows that our Friedmann universe began to exist with a big bang singularity. The theorems state that a singularity is inevitable given the following five conditions:

(b) There are no closed timelike curves (the principle of causality is not violated).

(c) Gravity is always attractive; that is, for any timelike vector V^a, the energy momentum tensor of matter satisfies the inequality

(d) The spacetime manifold is not too symmetric, such that every space- time path of a particle or light ray encounters some matter or

randomly oriented curvature. That is, any timelike or null geodesic contains some point at which

(e) There is some point p such that all the past directed spacetime paths from p start converging again. This condition implies that there is enough matter present in the universe to focus every past directed spacetime path from some point p, such that the matter causes a time-reversed closed trapped surface.

If these conditions hold in our universe, it follows that there is a big bang singularity which constitutes the beginning of the universe and the earliest time. (Actually, as Weingard ([1979] p. 199) emphasizes, the theorems merely tell us that there is a nonspacelike geodesic that is only finitely extendible into the past, which is consistent with the singularity being a timelike singularity from which only some world lines originate. However, as Hawking argues, these are special cases and are unstable; in the general case there will be a curvature singularity that will intersect every world line. Thus general relativity predicts a beginning of time’. (Hawking [1980] p. 149).)

The above-mentioned four sets of equations (the field equations, the Robertson—Walker line element, the Freidmann solutions and the singularity theorems) entail that the universe began without a cause. This is true in a threefold sense. (1) Since there is nothing earlier than the big bang singularity, nothing earlier than the singularity can cause it. (2) Since nothing different than the singularity exists simultaneously with the singularity, nothing

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simultaneous with the singularity can cause it. (3) Since (by condition (b) of the singularity theorems) there are no closed timelike curves, there is nothing later than the singularity that can cause it.

Distinct from big bang cosmology are philosophies of big bang cosmology, which interpret the theses of big bang cosmology in the light of certain philosophical theories or principles. Now it is prima facie possible that there is a sound philosophy of big bang cosmology that entails or renders it probable that the big bang singularity has a cause. This would be the case, for example, if there were a sound philosophical argument that it is a synthetic a priori truth that everything that begins to exist has cause. Alternatively, a philosopher might argue strictly on a posteriori grounds and claim that cosmologists misused inductive logic and drew false conclusions about the uncaused character of the singularity from the observational evidence.

In this paper I discuss whether there are sound philosophical arguments for the thesis that the singularity has a cause. My conclusion shall be negative. In Sections 2 and 3, I discuss whether or not there is a sound a posteriori argument for such a cause. Weingard [1979] has discussed some of the analogies between vacuum black hole singularities and the big bang singularity, and in Section 3 I use some of his ideas to show how interesting arguments from analogy can be constructed for the conclusion that the big bang has a cause. I shall conclude, however, consistently with Weingard’s remarks, that there is no convincing observational evidence that there are vacuum black hole singularities and hence, that there is no reason to think the analogical arguments I construct are sound. In Section 4, I discuss William Lane Craig’s [1979], [1986], [1989] argument that there is an a priori argument for a cause of the big bang based on the synthetic a priori principle that everything that begins to exist has a cause. I will criticize Craig’s Kant- based argument for this principle and conclude that this principle does not belong to the class of synthetic a priori truths.

My concern in this paper is with classical big bang cosmology or the so- called ‘standard hot big bang model’. I shall not discuss the new ideas introduced by string theory (see, for example, Weingard [1989] and Green et al [1987] the Hartle-Hawking [1983] theory, Guth’s [1981] original inflationary theory, Linde’s [1982] Albrecht’s and Steinhardt’s [1982] new inflationary theory, Linde’s [1984] chaotic inflationary theory, Tryon’s [1973] Gott’s [1982] Zeldovich’s [1982] and others vacuum fluctuation theories, and Dewitt’s [1973] grafting of Everett’s many-worlds interpretation of quantum mechanics onto big bang cosmology. These theories are certainly worth discussing on their own merits (some are discussed in Smith [1985], [1986], [1988], [1990] but discussing them in the present paper would take us too far afield from our restricted topic. My concern is only with whether there is a sound philosophy of classical big bang cosmology that entails or renders probable that the singularity has a cause. But I should

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emphasize that classical big bang cosmology will probably be replaced some day by a mature version of one of the aforementioned theories, or by some other cosmology based on a quantum theory of gravity, and consequently the conclusions about the universe drawn in this paper that are based on the classical theory should be understood as having a correspondingly provisional character.

An assumption I am making in this paper is that the initial singularity is real rather than a theoretical fiction. Although some (Swinburne [1981] p. 254; Craig [1979] p. 116-7) have claimed the singularity is a ‘physically impossible’ state, I believe there are no good arguments for its physical impossibility and that there are plausible arguments for its reality (see Smith [1991], [1992]). If the initial singularity is treated as a theoretical fiction, then the earliest temporal interval of each length is open in the earlier direction. This entails there is no earliest instant and is compatible with the principle that every event has a cause. In this sense, it may be argued that classical big bang cosmology is not committed to the thesis that the universe has a beginning and that this beginning is uncaused. However, my question in this paper, ‘Did the Big Bang have a Cause?’ is intended to mean ‘Did the Big Bang Singularity have a Cause?’, where the singularity is treated as real.[1]

2 SOME PRIMA FACIE PROBLEMS WITH A POSTERIORI ARGUMENTS FOR A CAUSE OF THE BIG BANG SINGULARITY

It is by no means obvious that there is a logically coherent a posteriori philosophical argument for a cause of the big bang singularity, let alone a sound argument. After all, how could an a posteriori argument possibly reach a different conclusion than big bang cosmology, which is (‘by definition’) the theory that results from empirical investigation of the large scale structure and the beginning and ending (or beginninglessness and endlessness) of the universe? Since the philosopher does not have in his study a more powerful telescope than the cosmologists, he cannot develop an argument from new or additional observational evidence than that possessed by the cosmologists. Rather, there seems only one route open to the philosopher, namely, an argument that the cosmologists have not correctly reasoned from the available observational evidence, that they have not adequately followed the principles of inductive logic or the rules of application of this logic. It might be suggested

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that this is the line of thinking behind the a posteriori argument of W. H. Newton-Smith that the big bang singularity has a cause. He writes [1980] p. 111:

It appears that the prospects for ever having evidence of a genuine first event are remote. For, 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 emerged from some prior cosmological going-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. So the prospects for ever being warranted in positing a beginning of time are dim.

However, it is not obvious that this passage expresses a philosophy of big bang cosmology or a philosophical re-interpretation of the implications of the cosmological data as distinct from a misunderstanding of big bang cosmology by a philosopher. Specifically, it is not clear whether Newton-Smith simply misuriderstands the concept of a singularity, as this concept is defined in big bang cosmology, or whether he is advancing an argument that is intended to qualify or override the conclusions reached in big bang cosmology. According to the standard cosmological definition, a state of the universe is a singularity if and only if it is a boundary of space-time, such that space-time cannot be extended beyond it. The quoted passage suggests that Newton-Smith thinks that infinite density is the defining property of a singularity, such that it is essential to a singularity to be infinitely dense but accidental to be a boundary of space-time. If this is in fact Newton-Smith’s understanding, it is mistaken, since infinite density is an accidental property of a singularity and being a boundary of space-time is the essential property. (Of course physical singularities, not coordinate singularities, are here at issue.) Consider, for example, the following definition of a singularity by Misner, Thorne and Wheeler (which is based on Schmidt’s [1971] now standard definition). They ask us to suppose that a space-time path terminates at a certain point.

Suppose, further, that it is impossible to extend the spacetime manifold beyond that termination point—e.g. because of infinite curvature there. Then that termination point, together with all adjacent termination points, is called a ‘singularity’. (Misner et al. [ p. 934).

Notice that Misner et al are not making the claim, ‘If the point has infinite curvature [or density], it (together with adjacent points of infinite curvature) is a singularity’, but the claim, ‘if space-time is inextendible beyond the point, for whatever reason, it (together with adjacent termination points) is a singularity’. It may be, as a matter of fact, that all singularities are infinitely dense, but this is not a defining condition of singularities but merely explains why (in the actual cases) that certain points are singular. If Newton-Smith’s scenario were true, i.e. if the state of infinite density that obtained about 15 billion years ago

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was caused by an earlier state of the universe, the infinitely dense state would not be a singularity, for space-time would be extended beyond the point(s) and the point(s) would thereby not comprise a boundary of space-time. But if this were the case classical big bang cosmology would need to be revised or subsumed under some broader theory of the universe.

This last remark suggests a second reading of Newton-Smith’s passage, namely, that Newton-Smith is suggesting that (1) the big bang singularity is the initial boundary of the universe as the universe is represented in big bang cosmology, but that (2) there are reasons (based on Newton-Smith’s more careful use of inductive principles in reasoning from the observational data) to think that the singularity is not in fact an initial boundary. But it is not obvious that this second reading enables Newton-Smith’s argument to be reconstructed in a logically coherent way. One difficulty centers on the sense of ‘the universe’. Classical big bang cosmology uses a definition of ‘the universe’ that is based on the following definitions of a space-time and cosmological model.

D1: S is a space-time=df. S is an inextendible, connected, differential manifold that has a smooth, nondegenerate pseudo-Riemannian metric of Lorentz signature (+, —, . . . , —) and that has a continuous, nonvanishing vector field which assigns a timelike vector to every point.

It is often added that S has a global time function all of whose time slices are Cauchy surfaces, but I shall not assume this since this assumption is inconsistent with the thesis that there are white holes and we cannot rule out white holes without begging the question against a certain argument that the big bang singularity has a cause, as shall appear in Section 3. It perhaps goes without mentioning that GTR allows other sorts of space-time than that defined in D1, but D1 is the definition thought to be actually instantiated and is the one standardly employed in big bang cosmological discussions.

In order to define ‘the universe’, we also need a definition of a cosmological model:

D2: C is a cosmological model = df. C is a pair whose first element is a space-time S and whose second element is a symmetric tensor field of type (2,0).

D3: U is the universe =U is a cosmological model C that is described by Friedmann’s solutions to the field equations and that satisfies the Hawking—Penrose singularity theorems.

(Note that ‘cosmological model’ is here used to refer to the physical reality described by a theory rather than to the theory itself.) The big bang singularities and other singularities (e.g. black hole singularities) are attached to spacetime S as its boundaries. The big bang singularity is defined as the

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boundary of the universe in the earlier direction, the beginning-point of the universe.

Given these definitions, if we read Newton-Smith’s statement that some prior state of the universe led to the production of this particular singularity’ as involving a use of the word ‘universe’ in the sense of D3 we get an implicit contradiction, since the big bang singularity is defined as the boundary of the universe in the earlier direction. Newton-Smith’s statement would reduce, by substitution of synonyms for synonyms, to the explicit contradiction: ‘the boundary of the universe in its earlier direction is produced by a state of the universe earlier than this boundary’. Thus, if Newton-Smith’s passage is to be given a coherent interpretation, he cannot be taken as using ‘universe’ in the sense of D3. If there are physical processes earlier than the big bang singularity, then these processes are parts of ‘the universe’ in some novel sense. Now we need not insist that the believer in such earlier processes produce a specific physical theory of these processes or of the spatio-temporal structure they occupy, since all that is necessary for the causal argument to go through is a general definition that encompasses both what went before and what went after the big bang singularity. We may use the definition (capitalizing ‘Universe’ to distinguish it from D3 and capitalizing ‘Space-time’ to distinguish it from D1):

D4: U’ is a Universe = U’ is a Space-time T occupied by physical processes, such that there is no Space-time T’ that is both continuous with T and contains T as a proper part.

Here ‘Space-time’ is taken as undefined, with the only conditions upon it being that space-time in the sense defined in D1 is a proper part of it and that its other proper part—the part not defined by D1—is earlier than its proper part that is defined by D1. In order to have a distinct expression for the proper part not defined by D1, I shall use the term ‘spatio-temporal structure’.

It seems logically possible that there is a Universe U’, such that the Friedmann universe U that begins with the big bang singularity and the spatio temporal structure that precedes the singularity are both parts of U’. About this earlier part of the Universe U’, we need not say anything definite at all, except that it is a part of U’, is a spatio-temporal structure, is earlier than the singularity, and contains physical processes that causes the singularity. But note that ‘inextendible’ in D1 should be taken as implying merely that S is inextendible to a larger Space-time S’ that obeys the equations of GTR, such that Dl allows that it is logically possible that S is extendible to a Space-time T that contains as its other proper part a spatio-temporal structure that does not obey GTR.

Given these definitions, we may ask if there is a possible reconstruction of Newton-Smith’s argument that is a viable argument for the claim that there is some Universe U’ containing both the Friedmann universe U and a spatio-

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temporal structure earlier than the singularity that is occupied by physical processes that cause the singularity. If we interpret the passage earlier quoted from Newton-Smith in a sufficiently general fashion, we may extract an argument that avoids the above-mentioned contradiction and has a true observational premise, namely (1). The argument goes

But this argument is sound only if the suppressed minor premise, ‘the big bang singularity is a macroscopic event’ is true, which it is not. So far from being macroscopic, the singularity is so ‘small’ it is less than 4D. If the universe is finite, the singularity has zero spatial dimensions and zero temporal dimensions; it is a point of infinitely compressed matter that exists for an instant only. If the universe is infinite, then any given finite volume of space is compressed to a single point at the singularity; in this case, there are an infinite number of points constituting the singularity and it may be considered as 1D, as a ‘line’ of sorts. Thus, Newton-Smith does not give us a sound a posteriori argument for a cause of the big bang singularity. These considerations suggest that a different and more cosmologically sophisticated approach must be taken if we are to have any hope of constructing a sound a posteriori philosophical argument for a cause of the singularity.

3 SOME SOPHISTICATED A POSTERIORI ARGUMENTS FOR A CAUSE OF THE BIG BANG SINGULARITY

A minimal condition of a sophisticated a posteriori philosophical argument will be that it refer the big bang singularity to a class to which it actually belongs and which is relevant to the singularity’s possession or non-possession of the property of having a cause. This reference class is obviously not the class of macroscopic events since the singularity is not a macroscopic event, but it is not the class of microscopic events either, since the singularity is not a microscopic event. Instead, the singularity is a boundary of the manifold of events and as such the reference class should consist only of singularities. One way to construct an a posteriori argument based on the class of singularities is an argument from analogy, which involves at least four formal concepts:

(F) The reference property F, which is the property that is possessed by all and only members of a reference class to which x belongs.

(G) The sample property C. which is the property whose possession or non-possession by x we are endeavoring to discover.

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(S) The sample class S, which is the class of all and only those members of the reference class R of which we know whether or

(R-S) The remainder class, which is the class of the members of the reference class R that is left over when the sample class S is subtracted from R. The remainder class consists at least of x and possibly only of x.

Now for an argument from analogy to be sound, it must not only have true premises and be logically correct but also must satisfy the rules of application of inductive logic, most notably the requirement of total relevant evidence. In arguments from analogy, the total relevant evidence includes all the relevant similarities and dissimilarities between the entities in the sample class S and the individual x. An argument from analogy is sound if it has true premises, is logically correct and satisfies this rule of application:

(3) The members of the sample class S are more relevantly similar to than different from the individual x.

Suppose we let the individual x be the big bang singularity, the reference class R the class of all singularities, the sample property G the property of having a cause, the sample class S the class consisting of all the singularities of which we know whether or not they have a cause, and the remainder class K—S the class consisting of all the singularities of which we do not know whether or not they have a cause. Consider this simplified argument from analogy.

(4) Every member of the class of singularities of which we know whether or not it has a cause, has a cause.

The class of singularities, it might be argued, includes at least (i) big bang singularities, (ii) big crunch singularities, (iii) Schwarzschild gravitational collapse black hole singularities (iv) Schwarzschild vacuum black hole singularities, (v) Reissner-Nordstrom gravitational collapse black hole singularities, (vi) Reissner-Nordstrom vacuum black hole singularities (vii) Kerr gravitational collapse black hole singularities, and (viii) Kerr vacuum black hole singularities. It might be argued that we know of each of these simularities, except for (i), that it has a cause, and since each of these singularities is more similar to than different from the big bang singularity, it follows that it is more probable than not that the big bang singularity possesses the sample property, having a cause.

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hope of getting off the ground. To begin with, we need to make explicit something I have been assuming, namely, that our talk of a singularity as having a cause requires that the concept of causality used in GTR be replaced by a broader conception of causality, for GTR recognizes as causes and effects only events that are parts of the 4D space-time manifold and singularities are boundaries rather than parts of the manifold.[2] Secondly, we need to narrow our reference class R since as it stands the rule of application (3) is not satisfied; that is, it is not the case that the members of the sample class S are more similar to than different from the individual x, the big bang singularity. Our sample class S includes big crunch and gravitational collapse black hole singularities and these singularities are more different from than similar to the big bang singularity, as will become implicitly clear in the following. We need to exclude these singularities from our reference class. The narrowing of our reference class requires a more detailed exploration of black hole singularities of the Schwarzschild, Reissner-Nordstrom and Kerr types.

where M is the mass of the body measured in geometric units, r the radial coordinate and theta and phi the angular coordinates. This is the geometry of a spherical, non-rotating and uncharged gravitating body, which collapses to a singularity when r = 0. But this Schwarzschild black hole singularity should not be included in our reference class, since this singularity is art endpoint of spacetime curves in the Friedmann universe U and therefore is not sufficiently similar to the big bang singularity to give us a sound argument from analogy. This black hole singularity possesses the properties

(J) being an effect of a body’s B gravitational collapse, such that B is a part of U,

Accordingly, the Schwarzschild gravitational collapse singularity is not sufficiently similar to the big bang singularity to Support a causal argument from analogy.

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Our best hope is to follow the clue offered by Robert Weingard [1979] about the relation between the Schwarzschild black hole singularities and the initial big bang singularity and the final big crunch singularity:

Now it can be pointed out that while in the case of the Robertson—Walker final singularity, there is a corresponding Schwarzschild singularity that is connected with the collapse of a gravitating body, the Schwarzschild singularity corresponding to the Robertson-Walker-initial singularity is within a vacuum black hole—one that does not arise from gravitational collapse. (Weingard [1979] p. 198, my italics)

Thus, it is to the Schwarzschild vacuum black hole that we should look if we want to discover an analogy with the initial singularity. The vacuum black hole is described by taking the limit of the above-stated line element in which the radius of the gravitating body is set to 0. This black hole may be described in terms of two distinct asymptotically fiat regions of a space-time, each of which contains a singularity. These singularities connect and form a wormhole that expands to maximum radius of 4(pie)M and maximum surface area 16(pie)M² and then contracts and pinches off, leaving two disconnected singularities again. Woridlines enter the space-time regions from the original singularities and worldlines from these regions crash into the singularities that exist after the pinching off of the wormhole. The singularities that exist prior to the pinching off of the wormhole are white hole singularities, singularities that (like the big bang singularity) emit rather than absorb particles; they have the property K but not I. These white hole singularities are further analogous to the big bang singularity in that the particle creation associated with them occurs in the highly curved space time in the region of the singularity. (For a development of this point, see Weingard [1979] p. 202-6.) These white holes singularities may be included in a reference class that is narrower than the one mentioned in (5) and thus may provide a stronger analogical argument for (6).

Suppose, then, that our reference class includes only these Schwarzschild white hole singularities and the big bang singularity. Our analogical argument will succeed if (i) the sample class includes all and only the Schwarzschild white hole singularities, (iii) we know of each of these singularities that it has a cause, and (iii) the set of the other relevant properties of the white hole singularities is more similar than different to the set of relevant properties of the big bang singularity. These three conditions would justify the conclusion that it is more probable than not that the big bang singularity has a cause.

But there is an apparent difficulty in the way of such an argument. The Schwarzschild white holes do not arise from a gravitational collapse but are built into the initial conditions of the universe. Misner, Thorne and Wheeler ([1973] p. 842-3) observe that the Schwarzschild vacuum black hole singularities ‘can exist only if the expanding universe. . . was “born” with the necessary initial conditions—with “r=0” Schwarzschild singularities ready and waiting to blossom forth into wormholes’. The Schwarzschild white hole

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singularities are retarded pieces of the big bang singularity and abide in a quiescent state until an unpredictable time when wormholes arise from them. The problem this poses is that the Schwarzschild white hole singularities, qua (retarded) parts of the big bang singularity, are no more effects of a causal process than the big bang singularity itself. We will have reason to believe the white singularities have causes only if we already have reason to believe the big bang singularity has a cause, which is inimical to the prospects of a sound analogical argument. An independent reason to reject this argument is that the sample class is empty; there is no observational evidence whatsoever that there are Schwarzschild white hole singularities.

However, there are two other main classes of black holes, Reissner-Nordstrom black holes (which are electrically charged) and Kerr black holes (which have angular momentum) and these may provide us with a suitable sample class. Reissner-Nordstrom black holes have an electrically charged, nonrotating and spherically symmetric gravitating body which occupy a spacetime structure with the geometry (where Q is the electric charge):

If this describes a gravitating body, then we will have only a singularity that is an endpoint of curves and not a beginningpoint. But let us take the limit of this solution in which the radius of the charged gravitating body is set to 0. This gives us a vacuum black hole, which has a wormhole to another universe through which a particle can travel (unlike the case with the Schwarzschild wormhole) and also a timelike singularity that connects the two universes. For an observer emerging into one of these universe from the wormhole, the singularity will appear as a white hole singularity, a beginningpoint of curves, and thus will possess the property K also possessed by the big bang singularity and the Schwarzschild white hole singularity.

But Reissner-Nordstrom white hole singularities are less analogous to the big bang singularity than the Schwarzschild white hole singularity since the Reissner-Nordstrom singularities are timelike singularities and the big bang and Schwarzschild singularities are spacelike singularities. This would provide a very weak causal argument from analogy. More importantly, there is simply no observational evidence that there are any Reissner-Nordstrom white hole singularities and so once again we are left with an empty sample class.

The only other type of black hole singularity besides the Schwarzschild and Reissner-Nordstrom singularities is the Kerr black hole singularity. The Kerr metric is

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A Kerr black hole is rotating and has a singularity at r=0. If we apply the metric to a vacuum black hole, we will have at r=0 a disc singularity (alternately, one can interpret the Kerr black hole in terms of closed timelike curves, as Weingard [1979] p. 207-12 [1979a] p. 331-2 explains, but we shall not take this route here). The edge of the disc is a singularity where curvature is infinite and the rest of the disc is an extendible edge of spacetime, i.e., an end of worldlines that can be extended to a larger spacetime. This will violate cosmic completeness (the principle that a universe has an inextendible spacetime) but this violation will occur only with the horizon of the Kerr black hole (by the cosmic censorship principle). The white hole singularity will be the edge of the disc as considered from the viewpoint of the new region of spacetime into which the worldlines can be extended. Thus, we again have a possible class of caused singularities that have the property K and are partially analogous to the big bang singularity.

But the problem once again is that there simply are no Kerr white hole singularities to give us a nonempty sample class. But the situation is not as dismal as with the Schwarzschild and Reissner—Nordstrom cases, since there are gravitational collapse Kerr black hole singularities (in fact, all actually extant gravitational collapse black hole singularities are of the Kerr type) and it is possible that the Kerr vacuum solution partially describes the Kerr gravitational collapse black holes. Specifically, it is possible that the vacuum solutions apply to the extent that they allow the edge of the disc formed from the collapsed star to be a white hole singularity as seen from the perspective of the spacetime into which the worldlines could be extended. This would give us a nonempty sample class of white hole singularities. Unfortunately, however, this is not the actual situation. Whereas the Kerr gravitational collapse solutions describe the exterior geometry of all actual gravitational collapse black holes, the interior geometry (which I was suggesting might be partially described by the Kerr vacuum solutions) is not in fact described by the Kerr vacuum solutions but is more accurately described by the Schwarzschild gravitational collapse solutions, where the infalling matter hits a spacelike singularity and is there crushed out of existence (with no white holes formed). (For more details, see WaId [1977] p. 78-91.)

The upshot of these various considerations is that the prospects are exceedingly dim for an posteriori argument for a cause of the big bang singularity. In order to obtain a sample class sufficiently analogous to the big bang singularity, we needed to exclude all singularities but white hole singularities. But whereas there are a variety of types of white holes that may be similar enough to the big bang singularity to warrant a causal argument from analogy, the sad fact is that there are no instances of these types (as far as current observational evidence suggests). Consequently, in direct opposition to Newton-Smith’s conclusion (it is hard to see what would constitute a reason for denying that [the big bang] singularity itself emerged from some prior

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cosmological goings-on’), we must draw the inference that it is hard to see what would constitute a reason for affirming that the big bang singularity itself emerged from some prior cosmological goings-on.

The obvious candidate for an a priori argument for a cause of the big bang would be based on a premise that a relevant principle of causality is a necessary truth, specifically, a synthetically necessary a priori truth. The leading proponent of such an argument is William Lane Craig;[3] he maintains that it is a ‘synthetic a priori proposition’ [1979] p. 147 that ‘everything that begins to exist has a cause of its existence’ [1979] p. 145. This provides the premise

(8) The big bang singularity is the beginning of the existence of the universe U,

But what reason is there to think that (7) is a synthetic a priori truth? Some philosophers reject the category of synthetic a priori truths, but I shall assume here there are some instances of this category. It is arguable, for example, that the proposition no body is both orange all over at t and blue all over at t is a synthetic a priori truth. This proposition is synthetic in that the concept of blue is not a proper part of the concept of orange (which it would have to be if the concept of not-blue is a proper part of the concept of orange), and the concept of orange is not a proper part of the concept of blue. This proposition is a priori in that it can be known to be necessarily true simply by ratiocination by anybody who possesses the concepts that are parts of this proposition. (A more precise definition of a priori ratiocination and related concepts is given in [Smith forthcoming]. but the following account suffices for our present purposes.).

The ratiocination involved in a priori knowledge may be immediate or mediate. If immediate, the proposition is epistemically necessary; that is, it cannot be conceived to be possibly false—anybody who understands it immediately understands that it must be true. Our proposition about blueness and orangeness is of this epistemically necessary type. But epistemic necessity is not the same thing as ‘intuitive obviousness’. For example, it is intuitively obvious to me that I now have two hands, but this is not epistemically necessary since I can conceive this proposition to be possibly false. I can conceive the

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possibility that I am now hallucinating my hands or that no physical objects correspond to the series of visual and tactile sense data that I take as evidence of my hands.

Now it seems evident that the proposition (7) is not a priori in the sense that it can be known to be necessarily true by immediate ratiocination by anybody who possesses the concepts in (7). For (7) is not epistemically necessary; I can conceive of something beginning to exist without a cause. A case in point is the universe. I can conceive of the universe existing at a certain time t (say at the time of the big bang singularity), such that (a) there is no time earlier than t, (b) nothing else exists at t, (c) nothing timelessly exists that causes the universe, and (d) there are no closed timelike curves whereby ‘later’ parts of the universe cause the universe to exist at t. Indeed, Craig himself admits that an uncaused beginning can be conceived: ‘We can in our mind’s eye picture the universe springing into existence uncaused’ [1979] p.145. But Craig insists that the imaginability or conceivability of this does not entail it is really possible (as distinct from being merely epistemically possible):

We can in our mind’s eye picture the universe springing into existence uncaused, but the fact that we can construct and label such a mental picture does not mean the origin of our universe could have really come about in this way. .. . Just because we can imagine something’s beginning to exist without a cause it does not mean this could ever occur in reality. ([1979] p. 145)

Now Craig cannot be making the point that Kripke and Putnam make about certain scientific statements that they allege to be necessarily true, such as that the atomic number of gold is 79. Kripke and Putnam argue that the fact that such propositions can be conceived to be false does not entail they are not necessarily true; they claim that certain empirical investigations are required to determine the truth value of a posteriori necessities and that this is the reason we can conceive them to be false. In a word, the Kripke-Putnam examples are a posteriori necessary truths. But Craig’s causal proposition (7) is purported to be an a priori necessary truth: Craig is not suggesting that we need to look through a microscope or telescope to determine the truth value of (7). Accordingly, Craig is best interpreted as suggesting that only a priori ratiocination is required to know the truth of the causal proposition (7), but that this ratiocination is mediate. That is, (7) is not an epistemically necessary proposition that cannot be conceived to be possibly false, but instead is a proposition whose necessary truth needs to be proved. (Although Craig is not entirely clear on this point, for he also claims this proposition is ‘intuitively obvious’ [1979] p. 148: perhaps we can grant that this proposition is or seems intuitively obvious to Craig but, as I suggested above, intuitive obviousness is not the same thing as epistemic necessity.) In any case, since (7) is not epistemically necessary, the only route left open to Craig is to

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appeal to some argument or proof that it is synthetic and a priori. To his credit, Craig does appeal to some argument, specifically, to a Kantian transcendental argument. According to Craig, the Kantian or neo-Kantian philosopher ‘defends the validity of the causal proposition as the expression of the operation of a mental a priori category of causality which the mind brings to experience’ [1979] p. 145-6.

But if Kant’s theory is to be introduced to defend the synthetic a priori status of the law of causality, Kant’s theory must first be rectified in several respects, for it is unacceptable as it stands. For one thing, his theory implies the contradiction that the category of causality is inapplicable to the noumenal causes of phenomena. Second. Kant’s causal principle is that every alteration of substance has a cause, whereas Craig’s principle (7) is something Kant rejected outright. Kant claimed that it is impossible for anything to begin to exist and therefore that it is impossible for something to be caused to begin to exist: for Kant. every change is an alteration in the state of some permanent substance (Kant [A182/B225 – A189/B232]). Thus, our second modification must be that the Kantian-type arguments be used to support (7) rather than Kant’s own principle. Third, Kant assumes that the a priori causal law is determinist and this assumption does not comport well with quantum mechanics. Thus, unless we are to sacrifice quantum mechanics on the alter of the Critique, we must suppose the causal law is probabilistic. Fourth, Kant’s idealism is inconsistent with scientific realism (i.e., with the view that there exist some imperceptible physical entities that are not dependent on the human mind), which is the better justified position (in my judgement: see Smith [1986a] Chapter VI). I believe all these required modifications to Kant’s theory would be acceptable to Craig (indeed, he makes some of them himself). But if Kant’s theory is modified to deal with these problems, can it still be used to demonstrate that (7) is a synthetic a priori proposition? That is, can we soundly argue in a Kantian fashion that ‘the causal principle (7) is a condition of the possibility of experience and therefore a synthetic a priori proposition’?

In order to answer these questions, we must first make more precise what is meant by ‘experience’ in a Kantian context. By ‘experience’ Kant means an objectifying interpretation of our representations in which ‘we posit an object for our representations’, such that our representations are apprehended ‘not in so far as they (as represented) are objects, but only insofar as they stand for an object’ [A190/B235]. Without this objectifying interpretation, ‘we should then have only a play of representations, relating to no object’ [A194/B239]. Let us call any synthetic proposition that is presupposed by our objectifying interpretations of our representations a transcendental proposition, such that the following definition holds:

D5 P is a transcendental proposition = df. P is synthetic and it is necessarily true that for any person x, if x objectively interprets his representations, x is committed (at least implicitly) to P.

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D5 is not stated by Kant but I think it captures a clear and coherent part of his theory and is consistent with the above-mentioned changes made to his theory. Let us suppose, for the sake of argument, that the causal proposition (7) is a transcendental proposition. Does it follow that (7) is synthetic and a priori? I think it is clear that if synthetic a priori propositions are a species of necessarily true propositions (propositions true in all possible worlds), it follows that ‘P is a transcendental proposition’ does not entail ‘P is a synthetic a priori proposition’. For some transcendental propositions are false in some possible worlds, For example, the proposition that there are objectifying interpretations is a transcendental proposition, since no person can objectively interpret his representations without being committed, at least implicitly, to this proposition. Yet this proposition is not necessarily true, since it is false in all the possible worlds that include only matter or only representations that are not objectively interpreted. It follows, therefore, that if (7) is transcendentally true, that does not show that the big bang singularity cannot occur uncaused.

However, if (7) is merely a transcendental proposition, it still can be used to block the argument that the big bang singularity is actually uncaused. If (7) is transcendental, the proponent of an uncaused big bang is committed to (7), since this proponent objectively interprets his representations. Accordingly, if somebody maintains that the big bang is uncaused, he is contradicting himself, since he is also committed to (7), which implies the big bang has a cause. Consequently, it is encumbent upon a proponent of the big bang cosmological theory (which entails the big bang is actually uncaused) to show that (7) is not a transcendental proposition.

Kant’s argument that his causal principle is transcendental is that it is required to distinguish the subjective succession of representations from the objective succession of events in the world. His basic assumption is that

(10) Our representations must be objectively interpreted in such a way that every event is understood as have a definite and determinate position in an objective temporal order.

Kant believes this interpretation involves viewing events as following from one another in accordance with a rule. If we assume the rule is probabilist (an event of kind y follows with some degree of probability upon an event of kind x) and that it pertains to events that are beginnings of existence as well as to events that are changes of state (since both kinds of events make up the objective succession in the world), we may state the following general principle (which entails that (7) is a transcendental causal proposition):

(11) Necessarily, for any person x, if x objectively interprets his representations, x is committed (at least implicitly) to the proposition that every event has a probabilist cause.

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representations can be distinguished from the objective succession of events, consistently with assumption (10). if one allows there to be a first uncaused event. Consider this Kantian argument (the lettering is mine):

(a) Let us suppose there is nothing antecedent to an event, upon which it must follow according to a rule. (b) All succession of perception would then be only in the apprehension, that is, would be merely subjective, and would never enable us to determine objectively which perceptions are those that really precede and which are those that follow [A194/B239].

We need not question the determinism implied by the ‘must follow’ or the idealism implied by the designation of objective events as ‘perceptions’ in order to see that this argument does not establish that there can be no uncaused first event. The inference from (a) to (b) is valid only if ‘an event’ in (a) means any event. If ’an event’ in (a) means some event, the inference is invalid, for if there is an earliest uncaused event, and all later events are caused, then the succession of perceptions that relates to the later events would not be merely subjective. The later events would succeed one another in accordance with a causal rule and thereby their succession would be objective. The perception of the first event would also not be purely subjective, for this first event would be related by a causal rule to the second event and thereby would belong to the objective order. The first event x might be related to the second event y by the rule an event of kind y follows from an event of kind x. The fact that x is a cause of a later event suffices to bring it into the objective succession: it need not in addition to this be an effect of something else. Therefore (11) is false, and (7) is not a transcendental proposition.

The considerations advanced in this section suggest that there is no sound Kantian-style argument that the big bang singularity has a cause. I am not aware of any other even remotely plausible a priori argument that the big bang has a cause, and so I think it is reasonable to conclude that the prospects for such arguments are poor. If we combine this result with the results of Sections 2 and 3, we may conclude that there is no good reason to believe that there is a sound a posteriori or a priori argument for a cause of the big bang. Thus we reach a general conclusion: there is no philosophy of big bang cosmology that makes it reasonable to reject the fundamental thesis of big bang cosmology: that the universe began to exist without a cause.[4]

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[1] It should be added that some attempts have been made within the context of classical big bang cosmology to avoid the prediction of a singularity altogether (whether it be treated as real or fictitious). For example. J. D. Bekenstein [1975] has argued that the stress energy tensor of a classical massive scalar field can be negative and prevent the closed universe singularities. (In this regard. also see Smith and Weingard [1990].) Ellis [1978] has also developed a static universe model that does not have an initial singularity.

[2] Accordingly. I am not using the definitions of cause’ and ‘effect’ employed in Smith 119901. Furthermore. I depart from Smith [1990] by adopting a broader definition of ‘time’ that allows me to say that the singularities are temporally related to the events in the spacetime manifold. There is no inconsistency between the two papers, only a use of different definitions. On this point, see Smith [1991] p. 38.

(Note that the word ‘best’ in line two of the definition D3 in Smith [1991] p. 27 is a misprint

for ‘set’.)

[3] See Craig [1979], [1986], [1989], [1990]. Craig also argues for the further thesis that the cause is God, hut I shall not address this theistic issue here. I have discussed the connection between theism and big hang cosmology in Smith [1991], [1992S].

[4] I should like to thank a referee for helpful comments art an earlier version of this paper.