Ultimate Explanations of the Universe (6 page)

The solution to Einstein’s equations found by Gödel, and subsequently many other solutions with similar properties, have proved irrefutably that worlds with closed time are within the realm of possibilities afforded by the general theory of relativity. But it is still an open question whether such solutions are physically realistic, or whether they are a purely theoretical possibility; or, to put it more precisely, whether there exist any laws of physics which would make the closing up of time-like curves impossible. Such laws would serve as selection rules admitting only of solutions with unclosed time. Do selection rules of this kind exist?

As we have already said, other “strange behaviours” apart from time-loops may occur in space-time, giving rise to a variety of causal pathologies. For instance, almost closed time-like curves may occur in space-time. These do not give rise to any problems with causality in themselves, but a slight disturbance in the gravitational field – and this can never be ruled out – may bring about their closure, which would in turn entail a “causal disaster.” Brandon Carter
¹⁹
has compiled an immense hierarchy of such instances of pathological behaviour. The question arises whether a general rule can be formulated for the elimination of all of them. It turns out that it can. But before we formulate it, we shall have to recall a few basic concepts from the geometry of space-time.

As is well-known, in the theory of relativity time-like curves represent the histories of particles which have a rest mass (which we usually call material particles). But apart from time-like curves there are also null curves (also known as light curves), which represent particles with zero rest mass, i.e. photons. Time-like and null curves together are referred to as causal curves. The occurrence of closed null curves would also give rise to a variety of pathologies, e.g. by using a photon we would be able to send a message back into the past. So in order to exclude all the causal pathologies we should also take into account the behaviour of the null curves, in other words speak of the causal curves.

Let’s assume that we have a space-time in which there are no closed causal curves, but we also want to protect it against any causal pathologies which might threaten it. The rule is fairly obvious. What we have to do is impose a requirement that there should be no disturbance of the gravitational field, however small,
²⁰
capable of bringing about the occurrence of closed causal curves. Space-times in which this rule is observed are known as
stably causal
space-times. As we have said, this rule seems fairly obvious, but the proof that in stably causal space-times there are no unwanted pathologies is not at all straightforward.

We should expect that if there are no problems over causality in stably causal space-times, then there should be no trouble, either, with the time record for the history of the universe. To verify this assumption we first have to have a “theoretical clock” available to measure the universe’s time.

Let’s take a look at any clock that we use in everyday life. It will be a device which assigns a real number to each particular moment. For example, as I am writing these words my wristwatch is showing an assignation of the numbers 9 and 36/60 in the conventional units known as hours and minutes. If I now recall that I and my watch are marking out a time-like curve in space-time, then I may designate any function continuously
²¹
increasing along a time-like curve as a clock.
²²
If such clocks exist along every time-like curve in the given space-time, then there exists a global time in that space-time. Or strictly speaking, we say that a global time (also known as cosmic time) exists in the given space-time if in that space-time there exists a continuous function with real number values continuously increasing along every causal curve.

Our attention to these somewhat pedantic definitions is rewarded with an elegant theorem:
²³
in a space-time there exists a global time if and only if the given space-time is stably causal.

We now know how to rule out causal pathologies and along with them lay the ghost of closed time: the requirement is that the space-time must be stably causal. But are there any additional physical reasons to justify this postulate? Again the answer is yes, but in order to appreciate it we have to consider the problem of measurements in physics.

As is well-known, every measurement is subject to error; there are no perfect measurements. Let’s assume that a theory of physics
T
predicts that a certain specific reading for a physical parameter
q
is to give a value
q
₀
. To prove or disprove the theory, we carry out the measurement, but we may not expect to obtain a reading of exactly
q
₀
. We shall consider the measuring experiment to have verified theory
T
if the result we have obtained for the reading lies in the range [
q
₀
– Δ
q
₀
,
q
₀
+ Δ
q
₀
], where Δ
q
₀
is appropriately small. If we obtain such a reading, we are entitled to say that the experiment has confirmed the theory to a good approximation.

But a certain condition must be satisfied for the entire procedure to make sense. Let’s assume that a small disturbance of the conditions in which the measurement is taken gives rise to very diverse results. The box of errors [
q
₀
– Δ
q
₀
,
q
₀
+ Δ
q
₀
] would then contain very many values of
q
, and we would not be able to say whether the experimental measurements had confirmed theory
T
, which had predicted the value
q
₀
, or some other theory for which the predicted value would lie inside the box of errors. We must therefore assume that a small perturbation of the conditions in which the measurement is made gives rise to small changes in the results of measurement. This assumption is known as the assumption of
structural stability
of measurement, and is a (generally tacitly) adopted assumption of the experimental method. Without this assumption the experimental method would be groundless.

Measurements of time and space, viz. readings for time intervals and lengths in space, are some of the most important measurements in physics. The measuring-rod and the clock belong to the physicist’s fundamental set of instruments. Hence measurements of time and space should also be characterised by structural stability. But in the theory of relativity space and time are only aspects, observed from a certain reference frame, of space-time, which is independent of the choice of a frame of reference. Therefore, in compliance with the postulate of structural stability, a small perturbation in the structure of space-time may yield only small changes in the results of space-time measurements. But the structure of space-time is determined by the gravitational field, which in the general theory of relativity is the curvature of space-time,
²⁴
in other words small perturbations in the gravitational field may give rise only to small changes in the results of space-time measurements.

Let us now consider a space–time in which there are no closed time-like curves. Perturbations of the gravitational field that produce closed time-like curves may not be called small. The postulate excluding such an occurrence, we recall, is known as the postulate of causal stability and is a necessary and sufficient condition for the existence of a global time in the universe. Therefore there is a strict correlation between the existence of global time and the possibility of measurements of space and time to be conducted, in other words the very possibility of physics as an experimental science. If the principle of stable causality were not valid in the universe, there would be no global time, small perturbations in the gravitational field could give rise to large changes in the structure of space-time, the principle of structural stability would not hold, and the experimental method employed in physics would be in jeopardy.

Does all this mean that in the real universe time-loops cannot occur, since if they did, it would be impossible to conduct experimental physics? In the macrocosm we have been engaged in the practise of physics for quite a long time, and our level of success shows that the experimental method applied in physics is working very well, which in turn is a strong argument for the stability of the properties of the universe, in other words for the existence of global time. Global time operates at the macrocosmic level; while at the Planck level, until we obtain a complete theory of time, we shall have to be ready to admit a variety of possibilities. For instance, according to a fairly popular hypothesis, the closer we come to the Planck level, the more contorted space-time becomes, until at the Planck level itself it turns into a jumble of all the possible geometrical options. Various configurations of curves, including closed time-like curves, may occur in such a “space-time foam.” Thus time also participates in the jumble of geometrical forms. Only as it proceeds to the higher levels does space-time gradually smooth out and a time measuring out cosmic history emerges.

However we should remember that the concept of a “space-time foam” is highly hypothetical, and the possibility of time-loops occurring in such a chaos of diverse configurations does not smack of an ultimate explanation.

One of man’s ancient dreams is to build a
perpetuum mobile
– a machine which would work without the need to take in energy from without. The discovery of the second law of thermodynamics brought an end to such dreams: in an isolated system, although the total amount of energy is conserved (by the first law of thermodynamics), nevertheless it is dissipated and the machine’s potential to perform useful work decreases all the time. But does the universe as a whole not fulfil the dream of the
perpetuum mobile
? Does it not ultimately provide some kind of explanation for its own existence? That it has always existed and will continue to exist forever. The early attempts in cosmology to accomplish this ideal failed. Contrary to his intentions, Einstein did not succeed in constructing a static model of an eternal universe. The universe is dynamic; it does not seem likely for changeability to last interminably; thus there looms a “ghost of the beginning.” A solution was to come in the form of a perpetually oscillating cosmos, but this, too, turned out to be problematic in view of thermodynamics. The idea of a closed time is an alternative to a cosmology with a beginning, but one that replaces the latter with problems with causality, and even logic. All the indications are that if we wish to have a model of an eternal cosmos, we shall have to fit it out with some additional regenerative mechanisms. The first brave attempt of this sort was the cosmology of the steady state put forward in 1948 by Hermann Bondi, Thomas Gold, and Fred Hoyle. It expressed their reaction to the undeniable theoretical and experimental problems (the age of the universe) challenging the young relativistic cosmology; but from its outset it was also inspired by philosophical considerations and worldview. Its three designers found it particularly hard to accept the existence of a singularity at the beginning of “relativistic evolution.” The theory’s predictability broke down at the singularity – it was impossible to determine what happened before the beginning – and the capacity to predict is the fundamental feature required of any theory in physics.
¹
Moreover, the singularity was too uncomfortably reminiscent of the concept of a creation of the universe, which all three scientists rejected on grounds of worldview.

There is a persistent habit of mind which suggests that an eternal universe must be static and unchanging. But must it? Could not a dynamic state coexist with eternity? It could, but the continuous dissipation of energy would have to be counterbalanced with some sort of “regenerating” mechanism. After Hubble’s observations it was impossible to return to the idea of a static world. If the universe was to be eternal, it had to be a stationary system, viz. notwithstanding its variability it must always look the same. The density of matter decreases with increasing distance separating the galaxies moving away from each other; there was thus a need for a mechanism which would continuously restore the losses.

The theory of the universe in a steady state appeared in two versions: Bondi and Gold’s, and Hoyle’s. Initially Gold’s idea of a continuous creation of matter was quite vague, but developed a more definite shape in the course of the three scientists’ discussions. Gradually, however, they went their separate ways. Hoyle followed a more mathematical reasoning and tried to reconcile the concept of the creation of matter with the formalism of the general theory of relativity; while Bondi and Gold took an approach in opposition to relativistic cosmology, building up their model from scratch. Emphatically, neither Hoyle nor Bondi and Gold rejected the general theory of relativity as a theory in physics; they were only against its application in cosmology which they said was an unwarranted extrapolation. In effect two independent papers were produced, and despite the rivalry between their authors, by a strange coincidence both were published in the same issue of
The Monthly Notices of the Royal Astronomical Society
.

Hoyle was faster than his competitors and his article was ready much earlier, but it was turned down by the editors of two scientific journals. The prestigious British
Proceedings of the Royal Physical Society
gave the postwar shortage of paper as the grounds for its rejection; while the editors of the American
Physical Review
wanted Hoyle to shorten the article, which he refused to do. In outcome it appeared in the
Monthly Notices of the Royal Astronomical Society
.
²
Originally Hoyle did not want to publish in this journal, which was edited by British astronomers, since he was apprehensive of their reaction to his unconventional ideas. But in fact quite the opposite happened, as the secretary of the British Astronomical Society at the time, who decided as to what was to be published, was William McCrea, a supporter of the hypothesis of the continuous creation of matter. The same man had already earmarked Bondi and Gold’s article for publication.
³
They had not been trying their luck with other journals.
⁴
Thereby a great controversy was launched between the cosmology of the steady state and relativistic cosmology, and for the next two decades was to dominate developments in the science of the universe.

The principal assumption in the model of the steady state is that “the universe viewed globally does not change.” The authors of this model call this the perfect cosmological principle. It differs from the (ordinary) cosmological principle applied in the Friedman-Lemaître cosmology in that it assumes that the picture of the universe is independent not only of the observer’s position in space (as in the ordinary principle), but also of the point in time of his observation. A large part of Bondi and Gold’s argument boils down to propaganda on behalf of the perfect cosmological principle.

Copernicus taught us that the Earth does not occupy a special place in space. Why should it have a special place in time? Cosmology is based on the assumption that the same laws of physics are valid throughout the entire universe. If, in accordance with relativistic cosmology, we assume that at the beginning of its evolution the universe experienced a superdense phase, then we can hardly expect the same laws of physics that we have today to apply in densities of the order of 10
⁹³
g/cm
³
. But “if the universe presents the same aspect to every fundamental observer, wherever he is and at all times, then none of these difficulties and doubts arises.”
⁶

For the steady-state model the observations of the red shift in galactic spectra merely confirm that the theory’s deductive reasoning is right.

But the agreement of the perfect cosmological principle with the observed expansion of the universe can only be upheld at the cost of the assumption that matter is continually being created in space, so as to maintain a constant mean density throughout the universe, despite its expansion. Bondi stresses: “It should be clearly understood that the creation here discussed is the formation of matter not out of radiation but out of nothing.”
⁸

Of course, the creation of matter understood in this sense is in breach of the principle of the conservation of energy. Bondi and Gold are well aware of the fundamental role this principle plays in physics, but they emphasise that what is really important in physics is agreement with what is observed, “there is, however, no observational evidence whatever contradicting continual creation at the rate demanded by the perfect cosmological principle,”
⁹
which requires that a mass equivalent to an atom of hydrogen be created in every litre of volume at a mean rate of once in 5.10
¹¹
years – and there is no experiment sensitive enough to detect such an amount.

The perfect cosmological principle turned out to be a powerful enough assumption to allow for a determination of the geometry of the universe without recourse to gravitational field equations, which Bondi and Gold could not use. The steady state postulate almost immediately leads to a conclusion that the curvature of space should be zero,
¹⁰
and that the expansion must proceed at an exponential rate,
¹¹
which effectively gives de Sitter’s space-time. In the relativistic cosmology de Sitter’s model is empty, but this is an outcome of the field equations, which imply that the density of matter must be zero. In Bondi and Gold’s version of the steady-state cosmology there are no field equations, so there is no need for this conclusion to hold.

As we see, in Bondi and Gold’s model everything is an elegant outcome of the initial assumptions. But even the most elegant outcome would not have impressed anyone if the model could not claim to make any empirical predictions. The fact that it could earned it a considerable degree of authority, despite initial reluctance. Its predicted observations were also an outcome of the perfect cosmological principle. New galaxies were appearing to replace those which were moving away, at a rate sufficient to keep mean galactic density constant. Moreover, the model predicted that there would be a uniform mean distribution of young and old galaxies in space; somewhat later the statistical distribution of young and old galaxies was determined for the steady-state model. This prediction differed essentially from the predictions of relativistic cosmology, according to which young galaxies were expected to be systematically further away than old galaxies, since in making observations at increasing distances we would be looking at a universe younger than what it was now, and there could be no old galaxies in a young universe. In the years immediately following the publication of the steady-state model the verification of the predictions made by the two theories was beyond the reach of astronomical observation, and the debate continued on the basis of theoretical arguments and each side pointing out their rival’s weak points.

Without doubt, one of the weak points of relativistic cosmology was the problem of the age of the universe. On the basis of Hubble’s law and the available data for the red shift the age of the universe was estimated at about 2 billion years, while estimates for some of the rocks on Earth, meteorites and stellar systems gave values of up to 5 billion years. The creators of steady-state cosmology did not fail to turn this argument to their advantage. A steady-state universe has an infinite age, of course, and there is no clash with any other estimates on a time scale.

Hoyle was more conservative in his revolutionary ideas than Bondi and Gold. He wanted to preserve the conceptual framework of relativistic cosmology as far as possible, departing from it only at the point where the introduction of the hypothesis of the continual creation of matter called for it. In compliance with his philosophy he thought that in this way he would maintain all the advantages of a cosmology based on the general theory of relativity while avoiding the conceptual and observational problems associated with it.

As we know, there is a local principle of conservation of energy built into the general theory of relativity. To infringe it Hoyle introduced into the field equations a new tensor term, which he called the creation tensor. The creation tensor was inserted in the place of the term with the cosmological constant, which does not appear in Hoyle’s equations. The creation tensor’s mathematical properties are similar to those of the cosmological constant component, except that it does not obey the principle of conservation of energy. Thanks to this Hoyle’s version of de Sitter’s solution, which is “empty” in relativistic cosmology (viz. the density of matter has a zero value), is filled up with matter which is continually being created. Hoyle also showed that this solution was stable. Hoyle’s equations also have other solutions apart from de Sitter’s solution, but it was chiefly de Sitter’s solution that was the focus of Hoyle’s attention and of the discussion that ensued. It is precisely in this solution that Hoyle’s vision of the world is identical with Bondi and Gold’s vision; despite the differences, or even controversies, that later emerged between these scientists, the two theories were later treated as just two variants of the same cosmology. But discussions with Hoyle’s version were easier: with its more elaborate mathematical apparatus it could be more readily amended and improved, but it was also more liable to criticism in the form of specific objections. And it was Hoyle’s version that found itself in the centre of the debate that soon emerged.