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Authors: Michael Heller

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BOOK: Ultimate Explanations of the Universe
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2.6  
The Crisis of Einstein’s Philosophy

From the theoretical point of view the situation was paradoxical. Einstein’s model had a non-zero density of matter but did not predict the moving away of the galaxies (spiral nebulae). De Sitter’s model predicted the moving away of the galaxies but had a zero density of matter. Nonetheless the argument that the mean density of matter in the real world was smaller than the best vacuum we could obtain in laboratories on Earth, in other words that we could treat de Sitter’s model as a close approximation to reality, was a dodge. And scientists knew it. After all, theoretical zero density is not the same as a very small density.

But the paradox was soon resolved. The Russian mathematician and meteorologist Aleksandr Aleksandrovich Friedman published two papers presenting his discovery of a whole class of spatially homogeneous and isotropic solutions to Einstein’s equations of which Einstein’s and de Sitter’s solutions were special cases.
11
In this class there was only one static model (Einstein’s); all the others were either expanding or shrinking. He also explained the apparently paradoxical status of de Sitter’s solution: all the models expanding out to infinity (monotonically) tended to de Sitter’s empty model as time tended to plus infinity. Thus de Sitter’s state was effectively an asymptotic state for the expanding models, in which the density of matter tended to zero in outcome of the expansion.

Gradually the situation was starting to clear up. Einstein’s proposition that there was only one unique, uniquely possible cosmological model concordant with all the philosophical expectations turned out not to hold. In cosmology, as in all the other branches of physics, many models can be constructed and only observation will tell which of them corresponds best to the reality in the world.

Cosmology would not become a fully experimental science until the last decades of the twentieth century, but it started to mature already by the 1920s. In 1929 Edwin Hubble collated about 40 results for the red shift measurements in the spectra of galaxies and published his famous law: the velocity at which a galaxy is moving away is directly proportional to its distance from us.
12
These results were already in circulation among scientists. In 1927 Georges Lemaître compared the results of measurements of the red shift with predictions for one of the solutions discovered by Friedman, which he had found independently of Friedman, and confirmed that there was no discrepancy between the theory and observations.
13

In the 1930s the paradigm of an expanding universe became firmly established. Even Einstein, who for a long time would not accept it, finally had to concede in the face of facts. The reason for his opposition had been that an expanding universe suggested the idea that it must have had a beginning. Knowing the distance to a few galaxies and the velocities at which they were receding, on the basis of Hubble’s law it is easy to estimate how long ago all the galaxies were situated “at one point.” For Einstein this was a difficult conclusion to accept. A universe which was supposed to be self-explanatory should not have a beginning.

Michael Heller,
Ultimate Explanations of the Universe
, DOI: 10.1007/978-3-642-02103-9_3, © Springer-Verlag Berlin Heidelberg 2009
3. A Cyclical Universe

Michael Heller

(1) 
ul. Powstańców Warszawy 13/94, 33-110 Tarnów, Poland
Michael 
Heller
Email:
[email protected]
Abstract
By about the mid-1920s it was evident that an eternal universe could not be kept on in relativistic cosmology “at a low cost.” Not only does Einstein’s static model contradict observations and experimentally measured values for the red shift in galactic spectra, but – as Eddington showed – it is also unstable: it cannot persist in a state of “static equilibrium,” and the occurrence of even a slight perturbation, which is what gravitation does by its very nature, would give rise either to its collapse or expansion. Thus all the indications are that the universe is not static but dynamic. But a dynamic universe implies the question of a beginning. The first measurements of the red shift suggested, and Hubble’s subsequent work confirmed, that the universe is expanding; and if it is expanding, then by extrapolating back in time we reach a conclusion that the process must have started from a state in which all the matter and energy in the universe now were in a state of gigantic compression. The name “Big Bang” had not come into use yet, but the idea itself was crystallising out and becoming well-established.
3.1  
The Problem of the Beginning

By about the mid-1920s it was evident that an eternal universe could not be kept on in relativistic cosmology “at a low cost.” Not only does Einstein’s static model contradict observations and experimentally measured values for the red shift in galactic spectra, but – as Eddington showed – it is also unstable: it cannot persist in a state of “static equilibrium,” and the occurrence of even a slight perturbation, which is what gravitation does by its very nature, would give rise either to its collapse or expansion. Thus all the indications are that the universe is not static but dynamic. But a dynamic universe implies the question of a beginning. The first measurements of the red shift suggested, and Hubble’s subsequent work confirmed, that the universe is expanding; and if it is expanding, then by extrapolating back in time we reach a conclusion that the process must have started from a state in which all the matter and energy in the universe now were in a state of gigantic compression. The name “Big Bang” had not come into use yet, but the idea itself was crystallising out and becoming well-established.

As he once admitted in a discussion with Lemaître,
1
Einstein did not like the concept of an expanding universe because it conjured up a conclusion “too reminiscent of the idea that the universe had been created.” Admittedly, in the early years of the development of relativistic cosmology notions derived from philosophy and researchers’ worldviews played a substantial part. But it is also true that in scientific cosmology the idea of a beginning is unwelcome, not the least for purely methodological considerations. In classical physics, to which the theory of relativity and cosmology certainly belong, whenever we have any kind of evolutionary process we explain it by means of dynamic equations. The following rule applies to such equations: if we know the state the system is in and its rate of change at a given time (or its state at two different moments in time), we are able to calculate its state at any arbitrary moment. This is known as the principle of determinism. In accordance with this principle, the dynamic equations for a given system and its rate of change at any given instant in time (or its state at two instants in time) give a full explanation of the particular evolutionary process. The equations Friedman used in his 1922 and 1924 papers to describe the evolution of cosmological models are dynamical systems in the sense used in classical physics, but in spite of this they fail to give a classical (deterministic) explanation of the evolution of the universe: knowing the state of the universe and its rate of change at any arbitrary instant in time (or its state at any two instants in time), it is not possible to calculate the states of the universe “prior to the beginning” using these equations. The classical principle of determinism breaks down at the beginning (and also at “the end,” for models which envisage an “end”).

In Friedman’s models the “beginning” and, by analogy, the “end,” may be described as follows: if time tends to the “beginning” (or “end”), then the mean density of matter tends to infinity. In more detailed models the same holds for certain other physical parameters, e.g. temperature and pressure. But in physics states in which some values tend to infinity are considered “non-physical,” since “you cannot do anything” with such values. In particular they defy all attempts to measure them, even estimates carried out on a purely theoretical basis. Not only does “the beginning” violate the classical deterministic explanation, but it also brings non-physical components into the model. For this reason the term
singularity
(
initial
or
fina
l) was introduced for the “beginning” and “end” in cosmological models. As we shall see in due course, the occurrence of singularities in models of the universe became a serious problem in cosmology.

3.2  
An Oscillating Universe

In view of the fiasco of Einstein’s static model, which was to represent a universe that had always been in existence, impelled by scientific and/or philosophical considerations, cosmologists and other thinkers turned their attention to oscillating relativistic models. The idea of a pulsating world passing through an endless series of “beginnings” and “ends,” had been present in the history of human thought for a long time. It was also an embodiment, albeit in a different manner, of the concept of an eternal world, in other words a self-explanatory world (again in a certain sense).

The solutions Friedman discovered in his 1922 and 1924 publications comprise an infinite number of solutions, among them also ones representing oscillating worlds. In the class of models with a constant, positive curvature of space (closed models) oscillating solutions exist for values of the cosmological constant Λ less than Λ
E,
the cosmological constant for Einstein’s static model. In particular, there exists an oscillating solution for Λ = 0. Oscillating solutions also exist for negative values of the cosmological constant.
2
In the class of models with a zero or a negative curvature of space (open models) oscillating solutions exist for negative values of the cosmological constant.
3
In both classes, for open and closed models, the smaller the value of the cosmological constant, the shorter the period of oscillation.

To describe a cosmological model’s evolution let us introduce a function
R(t)
called the
scale factor
. We may envisage it as the mean intergalactic distance. The scale factor is a function of time
t
. For the static model it is a constant function,
R(t
) = const. If the model is expanding,
R(t)
is a function which increases with time; if the model is shrinking,
R(t)
is a function decreasing with time. The typical path of evolution for an oscillating model is as follows. The cycle begins with the initial singularity, for which
R(t)
= 0. Then comes the expansion phase, during which all the galaxies recede from each other until
R(t
) reaches its maximum; from that moment
R(t)
begins to decrease and the expansion passes into contraction, the galaxies move closer and closer to one another, until finally, when
R(t)
again goes to the zero value they collapse to the final singularity.

Strictly speaking, at both the initial and final singularity the solution to the dynamic equations breaks down. There is no known method of extending the solution beyond the singularities. Essentially what we have is not an infinite number of cycles, but just one: not an oscillating or pulsating model, but just one pulse. The solution gives us no information on what was there before the initial singularity, or what will happen after the final singularity. Since we have no knowledge on this subject, we may imagine whatever we like. On these grounds many cosmologists have imagined an infinite number of oscillating cycles, making the reservation that although we do not know the mechanisms for “rebound,” viz. the passage from one cycle to the next, it would be reasonable to assume that before an expansion there was a contraction phase. Often a remark would be added that at the turning point the scale factor
R(t)
was not quite zero, but had a very small value, not actually equal to zero. In this way there would be no singularity at the beginning of each cycle, only a state “of very high density.” Such assumptions would be supported merely by the expectation that the future development of the theory would confirm them.

3.3  
The Recurrence Theorem

The question of “eternal returns” appeared in science independently of cosmology thanks to Poincaré’s famous theorem, known as the recurrence theorem, which this scientist proved in 1890 in his paper on the three-body problem.
4
As formulated by Poincaré, the theorem says that in a system of material points subject to forces which depend only on position in space, the state of motion, determined by configuration and velocity, after a certain time will return, with an arbitrary approximation, to its initial state once more or even an infinite number of times, providing the coordinates and velocities do not increase to infinity.

The natural application of Poincaré’s theorem was classical mechanics. The theorem holds for a finite mechanical system
5
in the phase space on which there is defined a finite measure needed to determine the evolution of the system.
6
Poincaré was aware of the possibility of exceptions occurring in which the system will return to its initial state only a finite number of times or even no times at all. This issue was clarified by Constantin Carathéodory, who showed that the measure zero should be assigned for the set of exceptions.
7

The conclusions to be drawn from Poincaré’s theorem were too much of a temptation not to be applied to speculations on the history of the universe. However, the idea of a universe that every so often returned to its “starting point” appeared to contradict the second law of thermodynamics which, if extrapolated to the cosmic scale, suggested a one-way cosmic history, running from a state of minimum entropy to a state of thermodynamic equilibrium characterised by maximum entropy and referred to as heat death. Already William Thomson had reached such conclusions on the grounds of the second law of thermodynamics, writing of “some finite epoch [with] a state of matter derivable from no antecedent by natural laws.”
8
The hypothesis of heat death had appeared as such in the work of Hermann Helmholtz.
9
Later speculations of this kind gave rise to a long series of discussions and debates
10
in the course of which Ernst Zermelo observed that there was a certain inconsistency between Poincaré’s theorem and the heat death hypothesis: a cyclic history of the universe and a one-way process towards heat death cannot both be true simultaneously.

The problem was clarified by Ludwig Boltzmann, who showed that the laws of thermodynamics were statistical in character, what is more only over a long time scale. Even if the universe were to reach a state of thermodynamic death, statistical fluctuations could displace it from that state. Furthermore, according to Boltzmann, we may conceive of the universe as having reached a state of heat death long since, with only “our world” as a “small fluctuation” in it with a rising entropy.
11

Although originally Poincaré’s theorem was applied to classical mechanics, after the emergence of relativistic cosmology it seemed natural to apply the theorem to spatially closed models of the universe. The spatial closedness should be regarded as a counterpart of the concept of an isolated system in thermodynamics, and Friedman’s oscillating models provide the relativistic version of the cyclic nature of the universe. We shall return to this issue in due course.

3.4  
Tolman’s Universes

Oscillating models of the universe and thermodynamic problems in the context of the general theory of relativity pretty soon attracted the attention of the American physicist Richard Tolman. Already in 1934 he published a monograph entitled
Relativity, Thermodynamics and Cosmology
,
12
which is sometimes still referred to even today, not only for historical reasons. In this book he conducted a detailed analysis of the laws of thermodynamics from the point of view of their agreement with the special and general theory of relativity.

Tolman realised that in relativistic cosmology it was an overstatement to speak of oscillating models of the universe. Every phase of contraction would have to end in a singularity at which the solution to Einstein’s equations broke down and, strictly speaking, it was no longer possible to predict what would happen afterwards. However, he thought that this was a weakness of the contemporary theory rather than a fundamental obstacle. Thus he made the working assumption that when the universe reached its “minimum volume” some sort of hitherto unknown physical mechanisms would emerge to “make the universe rebound” and initiate the expansion phase. On the basis of this assumption Tolman proceeded to examine sequences of cosmic oscillations unlimited by time.

We should distinguish between reversible and irreversible oscillations in these sequences. In the cosmological models studied hitherto it had been assumed (usually tacitly) that entropy remained constant, viz. there were no processes taking place in them involving the dissipation of energy. But for the analysis of the thermodynamics of oscillation we should introduce energy dissipation. In spatially isotropic models, viz. ones in which the expansion is uniform in all directions, neighbouring layers of cosmic matter “do not rub against each other” and therefore there is no dissipation of energy. However, Tolman assumed that energy could be dissipated at the expense of the potential energy stored in the gravitational field, although he did not name any specific physical mechanisms which might carry out this dissipation. It was not until much later that researchers realised that these mechanisms were associated with the so-called bulk viscosity, which causes the dissipation of energy in outcome of a rapid change of volume.
13
On the basis of his assumption, Tolman showed that if irreversible processes (associated with the dissipation of energy) were taken into consideration, then the amplitude of successive cycles in the oscillating model would increase. This was something of a surprise, since in classical mechanics if energy dissipation is envisaged in an oscillating system (an oscillator), the oscillations diminish and die down. But according to Tolman’s calculations the reverse should happen in the general theory of relativity – the amplitude of the oscillations should increase. Tolman was right – this happened because the system could draw an unlimited amount of energy from the gravitational field (viz. the curvature of space-time). Years later, when terms responsible for bulk viscosity were brought into the cosmological equations, in accordance with the strict rules of the game, it turned out that Tolman had missed another effect: not only does amplitude increase for successive cycles, but also the cycles become asymmetric with time; contraction takes place faster than expansion. Time asymmetry means that the processes associated with energy dissipation determine the arrow of time. It also turned out that Tolman’s type of solutions included ones for which the oscillations could not be extrapolated back in time to infinity (quite apart from the singularity problem): the earlier an oscillation, the shorter its period, until it was eventually “reduced to zero.” Here, too, loomed a vision of a beginning.
14

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