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The KalÄm Cosmological Argument: âScienceâ Excerpt
William Lane Craig and James D. Sinclair
2.3 Scientific confirmation
The sort of philosophical problems with the infinity of the past which have been the object of our discussion are now being recognized in scientific papers by leading cosmologists and philosophers of science.1 For example, Ellis, Kirchner, and Stoeger ask, âCan there be an infinite set of really existing universes? We suggest that, on the basis of well-known philosophical arguments, the answer is Noâ (Ellis et al. 2003, p. 14, our emphasis). Similarly, noting that an actual infinite is not constructible and therefore not actualizable, they assert, âThis is precisely why a realized past infinity in time is not considered possible from this standpointâsince it involves an infinite set of completed events or momentsâ (Ellis et al. 2003, p. 14). These misgivings represent endorsements of both the kalÄm arguments defended above. Ellis and his colleagues conclude, âThe arguments against an infinite past time are strongâitâs simply not constructible in terms of events or instants of time, besides being conceptually indefiniteâ (Ellis et al. 2003, p. 14).
Apart from these philosophical arguments, there has emerged during the course of the twentieth century provocative empirical evidence that the universe is not past eternal. This physical evidence for the beginning of the universe comes from what is undoubtedly one of the most exciting and rapidly developing fields of science today: astronomy and astrophysics. Prior to the 1920s, scientists had always assumed that the universe was stationary and eternal. Tremors of the impending earthquake that would topple this traditional cosmology were first felt in 1917, when Albert Einstein made a cosmological application of his newly discovered gravitational theory, the General Theory of Relativity (Einstein 1917, pp. 177â88). In so doing he assumed that the universe is homogeneous and isotropic and that it exists in a steady state, with a constant mean mass density and a constant curvature of space. To his chagrin, however, he found that General Relativity (GR) would not permit such a model of the universe unless he introduced into his gravitational field equations a certain âfudge factorâ Î in order to counterbalance the gravitational effect of matter and so ensure a static universe. Einsteinâs universe was balanced on a razorâs edge, however, and the least perturbationâeven the transport of matter from one part of the universe to anotherâwould cause the universe either to implode or to expand. By taking this feature of Einsteinâs model seriously, the Russian mathematician Alexander Friedmann and the Belgian astronomer Georges LemaĂźtre were able to formulate independently in the 1920s solutions to the field equations which predicted an expanding universe (Friedmann 1922).
Friedmannâs first equation is:
where H = Hubble parameter, a = scale factor, G = gravitational constant, Ï = mass density of universe, Î = cosmological constant, K = curvature parameter, c = speed of light. By way of explanation, the scale factor, âa,â of the universe is a global multiplier to universe size. Imagine the universe as a perfectly laid out city with streets that travel only north-south and east-west. Streets are spaced at equal distance intervals. Street intersections then define perfectly symmetric city blocks. One could go further and think of buildings in the city as analogous with galaxies in the universe.
The distance from one city block to another is a function of two values: the originally laid out distance (call that the ânormalizedâ distance) and the scale factor multiplier âa.â Note that, as in Fig. 1.1, when one multiplies by a scale factor of 1/2, one still has precisely the same city with the same number of city blocks. The only thing that has changed is the distance interval between the city blocks.
Now consider buildings within the city. If the city block distance were reduced to the size of buildings, clearly something must give. The buildings would be squeezed together and destroyed. This is analogous to what happens with matter in the real universe. The sizes of non-elementary mass structures such as protons, neutrons, atomic nuclei, and so on, are fixed; they do not change with the scale factor. Other physical structures, such as massless particles, do adjust with the scale factor. The wavelength of radiation adjusts and hence would gain (in the case of contraction) or lose (in the case of expansion) energy as a result.2 When one crowds these particles in upon themselves, one will see a transition to different physics.
Figure 1.1 Analogy of the universe as a city laid out in a grid.
Recall that the full city is always present regardless of the value of the scale factor. So now consider two additional situations. First, suppose that the scale factor were to shrink to zero. Space (and time) would disappear. Any structure that could not transform to zero size would be destroyed. If there were no physical process that would allow such a thing to happen, we should seem to have a paradox. Either there must be an undiscovered physical process or the scale factor cannot, in reality, assume a null value.
Second, imagine that the city is of infinite size. Conceptually, there is no problem with extending the streets north-south and east-west to infinity in each direction. What does it mean to scale the universeâs size in such a situation? No matter what scale factor one adopts, the size of the universe remains infinite. Nevertheless, the idea of scaling still retains coherency in that we can apply a multiplier to the finite distance between city blocks. Yet what would be the meaning of applying a zero scale factor in this situation? Now it would appear that the size of the full universe is âzero times infinityâ which, in general, can be any finite number (Barrow 2005, p. 160). What does this mean, given that the distance between any spot in the universe to any other spot must still be zero? GR simply breaks down at zero scale factor.
Whether or not the full universe is of infinite or finite size is given in the Friedmann equation by the curvature parameter âK.â A positive K indicates that the universe, much like the surface of the Earth, is unbounded yet of finite size. Going back to the analogy, imagine that the city is laid out over Earthâs entire curved surface. A traveler on 1st street would never come to the end; rather he would eventually come back to the location where he started. A positive K yields positive curvature and a closed universe. This is one type of âcompact metricâ within GR.
A zero value for âKâ yields a âflatâ universe. 1st street is unbounded and of infin ite length (in both directions). A similar situation obtains for a negative K value. Here one has negative, or âsaddle-shapedâ curvature. Two travelers moving east and side-by-side up 1st and 2nd streets would actually get laterally farther apart as the curvature of the surface causes the streets to diverge from each other. The latter case gives an infinitely sized âopenâ universe.
The components of the universe (all the energy, keeping in mind that E = mc2) determine what type of curvature the universe possesses. The âstrengthâ of gravity, included in the equation via the parameter âG,â affects the magnitude of the curvature.
The parameters Ï and Î indicate the type and magnitude of the different types of energy that cause the curvature. The parameter âÏâ represents the density (that is, the energy per unit volume) of the two types of âordinaryâ energy: matter and radiation. It is âordinaryâ in the sense that we are familiar with it in daily life and it is of a form that makes gravity an attractive force. Î represents an exotic type of energy density which can transform gravity from an attractive to a repulsive force.
Friedmannâs first equation tells us how the scale factor changes as time elapses. Mathematically, this is the first derivative of the scale factor âa,â known as âa-dot,â or Ă„. One can see that the increase (or decrease) in the scale factor is strongly a function of the universeâs energy content. Now the âordinaryâ energy density Ï will become smaller as the universe expands, since one has the same amount of energy spread out over a greater volume. So its causal impact on the expansion will progressively diminish at ever later times (this works in reverse for contraction). By contrast, Î, which represents the dark energy density, is constant. The dark energy does not become more dilute during expansion or concentrated during contraction. Hence, early in the life of an expanding universe Î is unimportant compared to Ï. But its impact âsnowballsâ as time goes on. As long as the impact of Ï in the early universe is not enough to overturn an expansion and begin a contraction, the effect of Î will eventually lead to a runaway expansion of the universe. There will appear a moment in the history of the universe when the dark energy will begin to dominate the ordinary energy, and the universeâs expansion will begin to accelerate. Recent observations, in fact, seem to show precisely this effect in our own universe, with a transition age at 9 billion years (Overbye 2006).
Friedmannâs second equation gives the rate of change of the expansion rate:
It determines whether the expansion itself is slowing or accelerating. This acceleration is referred to as âa-double-dot,â or Ă€. A new term âpâ appears in the equation. This is the pressure (similar to the pressure of a gas inside a balloon). Pressure itself can produce gravitational force. Pressure is normally negligible for ordinary matter, although it can play a role in radiation dominated universes. Pressure, however, can have a tremendous impact given a universe dominated by dark energy. As Friedmannâs second equation shows, the rate at which the expansion of the universe accelerates is proportional to: (- ” - 3p). But the pressure in the vacuum is just equal to the negative of the energy density; this is called the equation of state. Hence, overall, the acceleration is positive (which will produce expansion) and proportional to twice the energy density.
Ordinary matter will exert positive pressure (which will keep a balloon inflated, for example). This type of pressure will produce an attractive gravitational force, which supplements the attractive gravity that accrues from mass. Dark energy has the bizarre property that it generates negative pressure. But dark energy, while it has a positive energy density (which contributes to attractive gravity) will, on net, produce a repulsive gravitational effect. Looking at Friedmannâs second equation, one sees that an attractive gravitational contribution tends to slow down expansion (or accelerate contraction), while repulsive gravity will do the opposite.
The monumental significance of the Friedmann-LemaĂźtre model lay in its historization of the universe. As one commentator has remarked, up to this time the idea of the expansion of the universe âwas absolutely beyond comprehension. Throughout all of human history the universe was regarded as fixed and immutable and the idea that it might actually be changing was inconceivableâ (Naber 1988,pp. 126â27). But if the Friedmann-LemaĂźtre model is correct, the universe can no longer be adequately treated as a static entity existing, in effect, timelessly. Rather the universe has a history, and time will not be a matter of indifference for our investigation of the cosmos.
In 1929 the American astronomer Edwin Hubble showed that the red-shift in the optical spectra of light from distant galaxies was a common feature of all measured galaxies and was proportional to their distance from us (Hubble 1929, pp. 168â73). This red-shift, first observed by Vesto Slipher at the Lowell Observatory,3 was taken to be a Doppler effect indicative of the recessional motion of the light source in the line of sight. Incredibly, what Hubble had discovered was the isotropic expansion of the universe predicted by Friedmann and LemaĂźtre on the basis of Einsteinâs GR. It was a veritable turning point in the history of science. âOf all the great predictions that science has ever made over the centuries,â exclaims John Wheeler, âwas there ever one greater than this, to predict, and predict correctly, and predict against all expectation a phenomenon so fantastic as the expansion of the universe?â (Wheeler 1980, p. 354).
2.31 The standard hot Big Bang model
According to the Friedmann-LemaĂźtre model, as time proceeds, the distances separating the ideal particles of the cosmological fluid constituted by the matter and energy of the universe become greater. It is important to appreciate that as a GR-based theory, the model does not describe the expansion of the material content of the universe into a pre-existing, empty, Newtonian space, but rather the expansion of space itself. The ideal particles of the cosmological fluid are conceived to be at rest with respect to space but to recede progressively from...