The Generalized Copernican Cosmological Principle

The Generalized Copernican Cosmological Principle

Konrad Rudnicki

(Chapter 4)

4.01. Simple generalization

The generalization of the Genuine Copernican Cosmological Principle was an obvious step to take. It required only two small changes in its formulation to turn into another, more general principle. Copernicus was of the opinion that the Universe observed from every planet looks roughly the same. It is enough to replace the word planet with point and to add in every direction to get the Generalized Copernican Cosmological Principle, which is also called the Ordinary, Narrow, or Weak Cosmological Principle: The Universe observed from every point and in every direction looks roughly the same.

Another, more exact, formulation is: the Universe is roughly homogeneous and isotropic. Still another, not quite equivalent formulation is: the Universe possesses spatial homogeneity. Due to that last formulation, the Copernican Cosmological Principle is sometimes called Spatial Homogeneity Principle.

4.02. Other forms of the Generalized Copernican Principle.

Principles of Neyman-Scoff, Mandelbrot and Einstein

There are many other formulations of the Copernican Principle. For example, Jerzy Neyman and Elisabeth Scott (1959) give the following probabilistic definition: "the... Universe is a single realization of a stochastic process which is stationary with respect to displacement in space but, probably, not stationary in time" but they restrict its validity to the "observable Universe" and thus arouse some doubts whether it can actually be regarded as a cosmological principle. Furthermore, this formulation can be accepted only when one shares Neyman's probabilistic world view; then this formulation becomes essentially identical to the versions given above, and even explains in what sense the Universe is only roughly homogeneous - namely in the probabilistic sense. So it is the matter of additional assumptions and personal predi1ections whether to consider it as another version of the Generalized Copernican Principle, or maybe as a separate Probabilistic Cosmological Principle or Cosmological Principle of Neyman-Scott. Yet another probabilistic version of the Copernican Principle (the matter distribution in the Universe satisfies the same statistical laws, irrespective of the reference system from which the Universe is observed) was proposed by Mandelbrot (1977). But this form of the principle is not consistent with most of its other formulations, as it was shown by Zabierowski (1988a). It can be, however, considered as a separate Fractal Cosmological Principle.

When Einstein used the Copernican Principle for the first time, he postulated that no average property of the cosmic medium defines a preferred place or preferred direction in space. He assumed, in fact, that all the observers connected with the typical partic1es of the Universe (fundamental particles in the sense of 4.14; see below) are equivalent to each other. This formulation of the principle was sometimes called the Cosmological Principle of Einstein. Milne (1935b) was of the opinion that this principle is more general than the theory of General Relativity, which is just one of a number of its possible realizations. However, in fact, it proved not to be so. The theory of General Relativity can be applied to cosmology without acceptance of this principle (cf. 4.07). The form of the Copernican Principle used by Einstein can also be helpful today for understanding some properties of the Universe. But the name the Cosmological Principle of Einstein is rather seldom used.

4.03. Genuine and Generalized Copernican Principles

In early considerations of cosmological principles the genuine and the generalized Copernican principles were usually considered as the same principle. The difference was seen only in the fact that Copernicus, Tycho Brahe, Kepler and their contemporaries constructed their universe models using planets, whereas today we construct them using extragalactic objects. It was Edmund Skariynski (1970a, 1970b) who showed that the difference between those two formulations is a much deeper one. I am not going to relate Skarzynski's papers here. It is enough to say that models of the Universe having definite, favored centers are allowed in the genuine formulation (the genuine model of Copernicus has also its center - the Sun) but not allowed in the generalized one. Thus, the two cosmological principles have to be investigated separately, even though they have a number of common attributes.

The Generalized Principle is a special case of the Genuine Copernican Principle, and not the other way around. If a model is homogeneous and isotropic, if it looks roughly the same from every point, then, of course, it looks roughly the same when seen from each planet. But it can happen that it looks the same from every planet but not from every point in general. For example, the Tychonian model fulfills the conditions of Genuine Copernican Principle but not those of Generalized Copernican Principle, since it has a distinguished, central point. The kind of generalization applied in the Generalized Copernican Principle yields not a wider but a narrower c1ass of models.

The question is whether the generalization of the Genuine Copernican Principle could be made in just a one way. Is it not possible that a different generalization could have been made, perhaps even still can be made today? If we consider this principle as a geometrical rule, from which the spatial, mechanical models can be constructed, then the generalization involved in the Generalized Copernican Principle is the most straightforward one. But when we consider that in the statements of Copernicus the idea that planets are bodies similar to our Earth is also inherent, then a generalization tending toward accepting physical life on other planets and, in consequence, in the entire Universe, would also be possible. Some philosophical considerations of that kind after Copernicus are actually known. However, they were of no significance for cosmology. Only recently, the Anthropic Principle refers to this latter meaning of the Copernican Principle (See Chapter 6). Still another direction of generalization is possibly realized in the Uniformity Principle (cf. 7.5).

When in the literature, the name Copernican Cosmological Principle or just Copernican Principle is used, without the term genuine or generalized, usually the Generalized Principle is understood. Also in this book, I will use this shorter designation for Generalized Principle wherever there is no danger of misunderstanding.

4.04. Precursors of Generalization

The Generalized Copernican Cosmological Principle was formulated in its strict form only in the 20th century, but existed much earlier as a vague idea concerning the structure of the Universe. Some scholars trace it back to the beliefs of Anaximenes of Miletus (ea. 585-525 B.C.), who maintained that there were earthlike bodies that circle "with stars." However, according to the same Anaximenes, the stars were like nails in the celestial sphere. Thus it seems that he considered those earthlike bodies to move together "with stars" around the Earth rather than to move "around stars." Whether elevating "earthlike bodies" (material bodies?) under the celestial sphere was a step towards Copernicanism or just a reverberation of ancient Indian views that there were many "earths" (not necessarily material ones) in the Cosmos, remains quite unknown, since we know Anaximenes only from scarce quotations by later Greek writers.

Demokritus (ca. 460-370 B.C.) is regarded sometimes as an early adherent of the opinion that the Universe is roughly homogeneous. He maintained that the Milky Way consisted of "many small" stars. However, since it is not known what his views on the nature of stars were, the question cannot be answered whether his cosmic outlook can be considered as a step towards Copernicanism.

But it is clear that when Giordano Bruno claimed that the fixed stars seen in all directions are remote Suns with their own planetary systems, he had in mind the idea that everywhere in the Universe much the same can be encountered (cf. Michel 1973). Also the famous cosmological paradoxes, photometric (the sky should shine in every point with the mean stellar surface brightness), gravitational (the force of gravity from all directions of the sky should be infinitely large), had among their premises that the Universe in all directions is roughly homogeneous.

This kind of thinking about the Universe is strictly related to the materialism developed by and after Copernicus. When the physical, material world is to be taken as a self-dependent entity, then it has to possess some stateliness formerly attributed to God; its laws should be universal and grandiose [1]. In the beginning of the 20th century this opinion began to be considered as a self-evident and often even unmotivated one. Only a few scientists considered it as just one of many possible cases.

4.05. Einstein's excuse

In any case, when Einstein (1917) used the homogeneity of the Universe as one of the assumptions in the mathematical construction of his first relativistic model of the Universe, he justified it in a rather primitive way. He said that as long as we are concerned with large scale structures, we can imagine that matter is distributed over immense regions of space. The density of matter varies, but that variability is very slow, so his simplifying procedure is similar to accepting the ellipsoidal shape of the Earth instead of the very complicated (in small scale), real shape of the Earth. Another fact important for Einstein was that the velocities of stars (sic!) are very low in comparison with the velocity of light. Thus he based his cosmological investigations on the following approximating assumptions: 1: there is a frame of reference in which matter can be considered as remaining at rest. 2: the scalar of the mean density of matter can be accepted as constant. Of course, a priori this scalar may be (sic!) a function of coordinates, but if we accept that the Universe is finite, then one can incline to the hypothesis that it does not depend on its location in space.

This kind of explanation shows most clearly that Einstein did not realize how serious an impact that "modest" assumption could have on the results of calculations. Besides, he had in mind rather a small, closed Universe, where an observer could see "around the world", filled with individual stars. He did not admit the possibility that celestial bodies (e.g. galaxies) can have large velocities (cosmological or peculiar); he had no idea about gross irregularities in distribution of stars throughout the Universe. Initially he regarded both as small and preserved the same assumptions even when he already knew of galaxies and their large redshifts. He made only one exception in his later works, assuming that matter remains in rest only in the local frame of reference. But he retained the assumption of constant matter density (homogeneity and isotropy) in its initial form. In fact, the assumption of homogeneity and isotropy puts strong constraints on the possibility of motion in space. Only radial movements with velocity proportional to the relative distance of two points in space are permitted. When the impact of that "simple" assumption was later discovered, it was designated a Cosmological Principle, for the first time in the history of cosmology. After admitting the possibility of other cosmological principles, it was called Ordinary Cosmological Principle, Narrow Cosmological Principle or Weak Cosmological Principle. The last name shows that, for a long time, it was not realized how strong the assumption in fact is, how much it constrains the models. Bondi (1948) was the first one to grasp the connection of this cosmological principle with the Copernican ideas and to use the name of Copernican Cosmological Principle. After Skarzynski's paper, the name Generalized Copernican Cosmological Principle won wide acceptance.

4.06. The Copernican Principle and Hubble's Law

The Copernican Cosmological principle is closely connected with the so-called Hubble's law. There are many false opinions and deep-rooted superstitions in this matter.

The positive correlation between the distance of a galaxy and its redshift was actually known before Hubble, i.e. before the scale of extragalactic distances was established. Stellar magnitudes of what were then called extragalactic nebulae, or their angular sizes, were used as indicators of relative distances. Edwin Hubble, in his very early papers on redshifts, did not discuss any deeper regularities at all. And later, in his first paper devoted to wider problems of redshifts of galaxies (Hubble 1929), the relation between distance and redshift was presented as a linear dependence; in this form it became known as Hubble's Law. Here its history begins. Only a few people actually witnessed the "prehistory" of the law's formulation.

In fact, like his predecessors, Hubble first tried to find a polynomial form fitting a regression curve of redshifts on the distance axis. Only after acquainting himself with the first relativistic models of the Universe did he drop the terms involving higher powers of distance and adopted the linear form (Gates 1962, cf. also: Rudnicki 1991). Of course, any correlation and any empirical function can be represented in a linear form as a first approximation. In this case, accepting a linear form was quite natural. For nearby galaxies, the numerical value of the coefficient by the linear term in the polynomial form considered was much larger than by the higher terms.

It may be of interest to know that not only Einstein's method of modeling the Universe influenced Hubble's investigations, but Hubble's method also affected Einstein's theory. Einstein felt at first quite unhappy with a Universe which evolved by either expansion or contraction. He introduced the famous cosmological constant to the General Relativity equations. The aim of this constant was to secure the stabilization of the static Universe. Only after getting acquainted with Hubble's correlation between redshifts and distances did he accept the expanding model of the Universe, dropping the cosmological constant. It is said that R.C. Tolman, who collaborated first with Einstein and then with Hubble, contributed much to this interaction between the famous theorist and the famous observer.

4.07. Hubble's Law and the expansion of the Universe

In its early years, Hubble's Law was viewed by many as an observational corroboration of the theory of General Relativity. Today, second-rate popular treatments often make this claim. Even in a very interesting book written by a known scientist in 1986, I found a statement: the Big Bang cosmological model is a prediction of Einstein s general theory of relativity, which of course is not altogether true. It became clear that the linear expansion in most relativistic models (e.g. models of Alexander Alexandrowich Friedman (1922, 1924)) results not from General Relativity but from the mathematical assumption of homogeneity and isotropy (i.e. from the Copernican Cosmological Principle). The principle can be reconciled with radial motions only, the observed velocity of this movement being proportional to the distance from the observed body.

The factor of proportionality may be positive (general expansion), negative (contraction) or zero value (a static Universe, with no systematic motions). In fact, these three possibilities can be all considered as expansion when considering contraction as negative expansion and the static state as the intermediate (zero) case. Every model of the Universe based on this assumption, independently of accepted physical theories, obeys Hubble's law of proportionality. That it is necessarily so was proved by Bondi (1961). On the other hand, relativistic models constructed without assumption of this cosmological principle do not necessarily fulfill Hubble's Law, as for example Kurt Goedel's model (1949). This shows that Hubble's Law is not related to General Relativity or to any other physical theory but to a cosmological principle, a mathematical assumption based on philosophical conviction.

The implication here goes in one direction only. The Generalized Copernican Principle results in Hubble's Law, but Hubble's Law can also be valid when this principle is not fulfilled.

The actual strength of the Copernican Cosmological Principle was not recognized for a long time. This fact is also 'reflected in one of its alternate names: Weak Cosmological Principle. Contrary to that unassuming name, it requires a very specific property of its models; it allows only relative systematic radial motions with velocity proportional to distance, as seen by any (actual or imagined) observer located in any part of the Universe. The proportionality constant, called Hubble's Constant, may take, in theory, any possible finite value, i.e. it may also be zero. This value, which in general is a function of time, depended on the initial conditions and the course of evolution (by fixed initial conditions - on the age) of the modeled Universe. If Hubble�s Law is accepted, the numeral value of the constant can be established from observation.

It must be said to Hubble and his collaborators' credit that even as they formally adjusted the redshift-distance relation to a relativistic, expanding model, they were aware that the observational data did not necessarily have to be regarded as confirmation of the expansion of the Universe. They merely regarded expansion as the simplest of many possible hypotheses.

In one of the first papers devoted to this problem, Milton L. Humason (1931) writes:

It is not at all certain that the large red-shifts observed in the spectra are to be interpreted as a Doppler effect, but for convenience they are expressed in terms of velocity and referred to as apparent velocities.

Edwin Hubble and Richard C. Tolman (1935) wrote the following about the redshift-distance relation:

The most obvious explanation of this finding is to regard it as directly correlated with a recessional motion of�� the nebulae, and this assumption has been commonly adopted in the extensive treatments of nebular motion that have been made with the help of the relativistic theory of gravitation and also in the more purely kinematic treatment proposed by Milne. Nevertheless, the possibility that the redshift may be due to some other cause, connected with the long time or distance involved in the passage of light from nebula to observer, should not be prematurely neglected, and several investigators have indeed suggested such other cases, although without as yet giving an entirely satisfactory detailed account of their mechanism. Until further evidence is available, both the present writers wish to express an open mind with respect to the ultimately most satisfactory explanation of the nebular red-shift and, in presentation of purely observational findings, to continue to use the phrase �apparent� velocity of recession. They both incline to the opinion however, that if the red-shift is not due to recessional motion, its explanation will probably involve some quite new physical principles.

In the above statement, not only the nature of redshifts is considered to be uncertain, but, even assuming the Doppler interpretation to be correct, the authors do not see any need to connect it with General Relativity. They refer to a very general kinematic theory put forward by E.A. Milne (1935a). Moreover, in the same paper they propose tests of the nature of galactic redshifts that might be performed by future investigators. These tests are today considered too primitive, but no better tests have been proposed till now.

The last sentence of the paper contains the following statement:

It...seemed desirable to express an open-minded position as to the true cause of the nebular redshift....

It is appropriate to add here that Milne's theory and its formulae were developed from purely kinematic considerations, without recourse to the assumption of the existence of "laws of nature" or appealing to any specific theory of gravitation (Milne 1935a). Milne was one of the first scientists (Milne 1932) who was courageous enough to raise doubts as to the validity of relativistic cosmology; he set forth his reasons in detail a few years later (Milne 1935b).

Both Hubble himself and Humason, who was still active in the 1960's, up to their deaths hesitated to accept the "simplest interpretation" of the Hubble Law. Hubble was even urged by some physicists to accept such an interpretation, but he never gave in (Arp 1991). There is no need to quote all the scientists who entertained doubts about this interpretation of the correlation of redshift and distance. I want, however, to mention Fritz Zwicky, who several decades later, up to his death, continued to use the symbol Vs (i.e. symbolic velocity expressed in km.s^-1) instead of Vr (radial velocity expressed in the same units).

These examples show that from the time the Hubble Law was enunciated its interpretation as a confirmation of Friedman-type models of the Universe was readily accepted by people less familiar with astronomy but certainly not by some of the more reputable scientists.

4.08. Hubble's Law, Doppler Effect, relativistic models

There are two different aspects of Hubble's Law that have to be distinguished: the observational, linear correlation between redshift and distance, and the law of linear expansion or contraction of the Universe. In fact, they are two different statements called by the same name. To avoid misunderstanding, I propose to call them the spectral Hubble Law and the kinematic Hubble Law, respectively.

Before the spectral Hubble Law can be accepted as an observational confirmation of any relativistic model of the Universe, or more generally, as the overall expansion implied by the Copernican Cosmological Principle (homogeneity and isotropy), the following independent evidence is required:

l. Proof that the redshift-distance relation is (in a fixed moment of time!) linear not only in the first approximation but also in its intrinsic sense. This does not mean that the observed redshifts are strictly linear with distance. The light from various objects takes different times to reach the observer, who contemplates those objects as they were in various epochs of the evolution of the Universe. This departure of observed dependence of redshifts on distance due to the variability of the Hubble constant with time (it is supposed to be constant only in spatial coordinates) has to be allowed. Of course, some dispersion due, for example, to the peculiar motions of galaxies, to certain accidental error dispersion, or to some systematic deviations, also has to be taken into account. This poses no serious problems for practical calculations. The redshift of extragalactic objects can be measured to great accuracy today, in sharp contrast to the imprecision of contemporary methods in establishing distances. Distances to only a few nearby galaxies were established with good reliability, i.e. using several different, independent methods applied to the same objects. These few objects are not enough to deduce a strict functional dependence between redshifts and distance. Some observations originated by Halton Arp (1987) seem to show that objects obviously located at the same distance may have quite different redshifts. Also, the so-called Rubin-Ford effect alone (cf. Rubin 1986) causes some doubts about the fundamental linear character of this law. This effect consists in small but real dependence of the Hubble constant on direction and distance. Some not fully explored phenomena like the so-called Great Attractor make the situation even more complicated. Undoubtedly, there is a correlation between distance and redshift. But how strict this correlation is and how precise it can be approximated with a linear function is still a matter of further investigations. It may also be mentioned here that, according to a space-time theory originated by A.A. Robb (1936) and developed by Le. Segal (1972), the redshifts should be proportional to the square of distance and only in small distance intervals can be approximated by linear functions. It is known to every student of mathematics that every three points can be connected with a straight line, provided the line is thick enough.

2. Proof that the observed redshifts are to be accounted for predominantly by the Doppler Effect, the other causes remaining insignificant for large distances. Such a proof is very difficult because nobody can be sure that we already know all the possible causes underlying the shifting of spectral lines. Especially in last decades the discovered phenomenon of periodicity (or quantization) of redshifts (I do not refer to any specific paper because the problem belongs to the forefront of astronomy, and by the time this book becomes available to readers, any paper of today will be out of date) shows that the Doppler effect cannot explain at least some features of extragalactic redshifts.

3. Furthermore, a separate proof is needed that the Universe conforms to an expanding model based on the assumptions of homogeneity and isotropy, for example to a model of the Friedman type. Such proof is possible, if at all, only in an observational way - only for the observable part of the Universe.

Accomplishing the first of these proofs would be of great importance in and of itself; it would qualify redshift as a simple, secure indicator of extragalactic distances and not just in a statistical sense. This would have an enormous practical significance. A large part of the modem debate about Hubble's Law is devoted to this practical problem. But the proof would also be very important for the physical understanding of the Universe. If it turns out that the effect is strictly proportional to the distance independent of the nature of the objects, we can conclude that the phenomenon is due to the structure of space-time alone. Or, if we do not like to accept the notion of space-time, we can conclude it is due either to space or to time by itself. If, for example, we decide to attribute the redshift effect to some particular kind of intergalactic matter, we should conclude that this matter is distributed completely homogeneously over the entire Universe accessible to our observations. (It is worth noting that this implication is not valid in reverse. The redshift may be caused by space or time alone, even if it is not strictly proportional to distance, provided space-time is not homogeneous.) The problem whether some component of redshift may depend on the nature of the object observed is now under vivid discussion (Arp 1987), so let us wait until the discussion ends.

It may be very difficult to carry out a positive Proof No. 2, provided Proof No.1 turns out negative, but such a possibility cannot be rejected a priori. It is possible that redshifts are Dopplerian in origin but related to the nature and/or history of bodies, not space or time. For example, there could be some (unknown) effects which make bodies moving toward us invisible. Then even out of completely accidental movements, without any privileged directions, only shifts toward the red end of the spectrum could be observed. Another possibility would be a real but irregular expansion of the Universe. It may be mentioned here that Milne (1935a) showed that a phenomenon of general expansion observed in terms of radial velocities is not equivalent to a purely physical expansion of a system of celestial bodies. Out of formulae derived by him it became clear that most kinds of systematic motions of a system of points, observed from one of these points, also cause observed systematic radial motions. Milne's formulae linking observed radial velocities with spatial motions of celestial bodies are quite complicated even with the assumption that there are no peculiar motions within the system. The observed general expansion (or contraction) can be due to quite complicated spatial motion conditions. In this case, however, one should bear in mind that no actual Universe showing systematic velocities with distance but revealing no linear proportionality could ever be reconciled with the Copernican Cosmological Principle.

Sometimes in the definition of redshifts the Doppler Effect is involved. So, for example, Hawking (1988) provides the following popular scientific definition: Redshift - the reddening of light from a star that is moving away from us, due to the Doppler effect. If this definition is accepted, then the proof must be given whether the shifts of galaxy and quasar spectra towards the red end is the "redshift" in sense of the above definition.

Proof No. 3 can be positive without No.1 and No.2 in only one case, namely, when we suppose a static relativistic model of the Universe. Under these circumstances, a Universe model may be constructed according to General Relativity even if the redshifts are neither Dopplerian in origin nor strictly proportional to distance. Of course, other physical theories must be involved in explaining the nature of redshifts in this case.

Summing up, the spectral Hubble Law, with its simplest Dopplerian explanation as a confirmation of a Friedman-type relativistic model or any other model based on the Copernican Cosmological Principle, is not a monolithic statement. Its three main components - the phenomenological part, the Doppler explanation, and its application as an argument confirming the homogeneity and isotropy of the Universe - are logicalIy independent, and may be verified or falsified, accepted or rejected, independently of one another.

4.09. Can the Copernican Principle be proved observationally?

The Copernican Cosmological Principle can be proved at most in the observable parts of the Universe. Even if such a proof could be provided positively in the future, we cannot exclude the possibility that only our part of the Universe has this particular property of homogeneity and isotropy, whereas the Universe at large does not reveal this property. On the other hand, even if it turned out that our parts of the Universe are altogether neither homogeneous nor isotropic, the possibility is not excluded that this feature is local, typical only for this particular region of space accessible to our observation. The Universe considered as a whole (infinitely large or only much larger) may still be roughly homogeneous and isotropic - in principle.

If there was a good reason to accept the Copernican Principle for the entire Universe, any observational fact concerning the observable part of it could not be accepted as an argument against the Copernican Principle. Again, if there was a good reason to reject the principle, no observational arguments could force us to keep it. This, along with the fact that the nature of the redshift of extragalactic objects remains unknown, makes it a not insignificant minority of astronomers unconvinced that the Universe actually expands. Before they accept the expansion of the Universe, they want first to scrutinize all the other possibilities (e.g. Vigier, Festschrift, Keys el al. 1991). I will not repeat this again when, in later chapters, the possibility of a negative Hubble Constant etc. is mentioned. It does not mean that such a group believes that the Universe does not expand. Most of them are just waiting for more c1ear evidence for or against.

4.10. Radial versus circular motions

Whatever we think about radial movements as the only systematic movements allowed in the Universe, it may be of interest to notice that this view is the opposite of the conviction of the ancient Greeks that only circular motions are allowed in the Universe.

4.11. Actual but unobservable regions of the Universe

The Generalized Copernican Principle in its most common form - the kinematic Hubble Law and the expansion of the Universe - has produced the notion of a cosmological horizon: a surface surrounding every observer and situated at such a distance from him that the velocity of recession is equal to the velocity of the fastest physical signal, to the upper limit of physical velocities. (The velocity of light is considered today to be just that limiting value). No physical signal can reach the observer from the regions of the Universe located beyond the cosmological horizon. In the ancient models of the Universe, only the non-physical, purely spiritual regions were not accessible to sensual observations. Now, for the first time, some parts of the physical Universe have become unobservable. But the same Copernican Principle underlying the Hubble Law also gives one the means of forming a judgment about those unobservable regions, of overcoming the cosmological horizon mentally. According to this principle, beyond the cosmological horizon there is, in every direction, much the same as what we observe here in the neighborhood of our native Earth.

4.12. Models based on the Generalized Copernican Principle

There are many models consistent with the Copernican Principle. Most of them are relativistic ones. But there are also models of the Robertson-Walker type (H.P. Robertson 1935, A.G. Walker 1936) where the Cosmological Principle alone, without invoking any physical theory, is sufficient for producing a model. The prevailing number of models based on the Copernican Principle begins in time from a kind of primordial explosion called the Big Bang. Some of them end in time with a general squeeze called the Big Crunch. In this case, the Universe exists only during a finite interval of time.

Some Universe models are infinite in space, some are finite (finite but not limited space, due to its curvature). The infinity in space always corresponds to that of time. Reviews of such models were given in many books, beginning from the third decade of this century (e.g. Rindler 1924, 1967).

On the other hand, all contemporary models (even non-relativistic ones) of the Universe which have a beginning, or a beginning and an end, are based on the Copernican Principle. It is remarkable that there is no known model which has no beginning but possesses an end in time. The Copernican Principle allows such models. Thus it is rather a matter of philosophical convictions that models of this kind are not constructed. Are these models too pessimistic, or is it simply that those cosmologists who do not accept redshifts as an argument for expansion of the (entire) Universe do not accept the Copernican Principle either?

4.13. The Copernican Cosmological Principle for space

Einstein was sure that his General Relativity theory fulfilled a principle called, by Einstein himself, Mach's Principle. This principle, based on the philosophical considerations of Ernst Mach (1838-1916), claims that local physical conditions are unambiguously determined by the entire Universe. For cosmology this meant that local physical properties, revealed in the curvature of space, are univocally determined by the distribution of matter in the entire Universe and vice versa. Derek J. Raine and Michael Heller (1981) proved that this is not true. For example, the relativistic model of the Universe constructed by Wilhelm de Sitter (1917a,b,c,d) yields zero density of matter (i.e. empty) and has the same curvature of space as Einstein's model which consists of a finite amount of matter. Similarly, in those models by Friedmann which expand to infinity, the mean density of matter drops, with time, to zero, but the expansion rate remains positive; it means that the expansion of space is (with time tending to infinity) independent of matter.

All this showed that when discussing relativistic models of the Universe or any models based on theories involving curvature (or any other structure) of space, the distribution of matter and the features of space have to be treated separately. It follows logically from this that there can be a Universe model fulfilling a cosmological principle in respect to distribution of matter but not in respect to structure (e.g. curvature) of space, or vice versa. In principle, this can be stated about every cosmological principle in connection with every physical theory including a developed theory of space. In practice, this revealed itself as a problem only in discussing the Copernican Principle in connection with General Relativity.

Since properties of space and those of distribution of matter are, as we saw, more of less independent, Heller maintains that two kind of cosmological principles have to be taken into consideration, one cosmological principle for space and another for matter. It is relatively easy to formulate a cosmological principle for space, because a mathematical formulation is sufficient here. For the Generalized Copernican Cosmological Principle, it is enough to require that space have a constant and isotropic curvature (constant in spatial coordinates, not necessarily in time).

4.14. The Copernican Cosmological Principle for matter

More complicated is the issue of a cosmological principle for matter. The Ancient Indian Principle had produced no definite model, at least up to the present. For the construction of cosmological models based on the other two historical cosmological principles (Ancient Greek and Genuine Copernican), individual celestial bodies and their trajectories were used. This was possible because these principles proclaimed only some privileged positions (the Earth by Ancient Greek) or equipollence (planets by Copernicus) of certain celestial bodies.

It is not so with the Generalized Copernican Principle. It states some property (self similarity in every point and in every direction) of the Universe as such, not of any particular kind of objects. Any celestial objects can be considered here. Whichever "constructing material" one chooses can be useful, provided it fulfills the accepted cosmological principle. Einstein, who was a physicist rather than an astronomer, applied stars. It was already known in Einstein's time that the spatial distribution of stars is not homogeneous. Since Hubble, it has been clear that stars which are aggregated into stellar systems (islands, as Immanuel Kant (1724-1804) called them, galaxies, as we call them now) are of no use for the purpose. The spatial distribution of stars cannot be considered as obeying, even in a very gross approximation, the law of homogeneity. In the course of time galaxies also turned out not to be good enough because Fritz Zwicky (1938) showed that all the galaxies participated in clustering. Clusters of galaxies also revealed themselves as not good "bricks" for constructing a homogeneous Universe. But there is a quite common belief among cosmologists that some celestial bodies (or agglomerations of celestial bodies) must exist whose spatial distribution obeys the Copernican Principle. Whatever their nature, they can be called fundamental bodies.

With the development of extragalactic astronomy, irregularities of ever higher order were discovered. Of course, statements of homogeneity in the observable part cannot be taken as an argument for universal homogeneity over the entire Universe, but even proving this for the regions accessible to physical perception would be of great importance. Suggestions of such proofs are sometimes given (Stoeger, Ellis and Helaby 1987), but have not been conclusively accomplished to date. Most contemporary observations contradict homogeneity over any scale of dimensions. Some adherents of easy cosmology say that this is of no importance because the contrast between density of matter in large galaxy clustering structures and in intervening voids is very low. This could be valid by assuming that the density in voids is still considerable. If we assume that it is close to zero, then it produces an enormous contrast in proportion to any finite density. It has to be noted that all the considerations about spatial distribution of extragalactic objects rely on the validity of the spectral Hubble Law, which is still disputable.

In most cases when an assumption turns out not to correspond to reality, as here, there are two ways out. The first is to reject the perplexing assumption. The second is to assume that the assumption is fulfilled in some other realm, e.g. in a different dimension range. Since this chapter is devoted to the Generalized Copernican Principle, I wil1 consider only the second case, which does not require rejecting it.

Instead of real celestial bodies and their agglomerations, some abstract substratum can be used which ex definitione satisfies the needed criterion. One can consider the substratum to consist of such large agglomerations of matter that, in the corresponding scale, fulfill the conditions of homogeneity and isotropy. Or, one can say that the substratum is an abstract notion of homogeneously distributed matter. This matter has its density equal to the mean density of the matter actually existing within the Universe. One can also say, in a picturesque way, that the notion of substratum is that of the real matter of the Universe ground finely and then dispersed in a strictly homogeneous way throughout the Universe.

This notion is a highly abstract one, and to make a bridge between it and the actual celestial bodies requires some mental endeavor. Michael Heller (Heller 1975, Heller et al. 1974) proposes for this purpose a procedure which can be described in a slightly simplified form as follows.

I. description of theoretical concepts.

Definition 1: A substratum is a set of material points obeying the Cosmological Principle.

Definition 2: Material points which are the elements of the substratum are called fundamental particles.

II. description of empirical reality.

Definition 3: A fundamental body is the matter included in a fundamental region.

Definition 4: A fundamental region is the part of the momentary (t = constant) space resulting from the following procedure:

A) We divide the (three-dimensional) space in as many ways as possible into relatively compact parts, i.e., parts bounded gravitationally so that the ratio of inner to outer gravity forces is rather high). Furthermore, these parts ought to have equal volumes, limited diameters, and limited differences of contained mass.

B) We choose from all possible partitions described under A the one which shows the smallest differences of contained masses and highest ratio of inner to outer gravity forces.

III. Postulate connecting the theoretical and empirical part: The fundamental particles defined in 2 can be identified with the centers of fundamental bodies defined in 3.

Heller said about the last postulate that at the present level of our cosmological knowledge (it) remains wishful thinking.

All this looks to be complicated indeed, and nobody has performed this procedure to date. But any other attempts at identifying the theoretical substratum with any individual fundamental bodies have also failed. Up to now, no scale is known in which the distribution of matter could be regarded as homogeneous and isotropic; but the Copernican Cosmological Principle is still preferred among other cosmological principles, and most contemporary cosmological considerations rely on it. One can yet hope that a scale will be discovered in which the principle will appear to correspond strictly to reality.

4.15. Hierarchical Universe

We can examine a case where no scale homogeneity is obtained but the aggregates of matter (clusters of higher order) still form an infinite sequence. Such a model of the Universe was proposed first by Johann H. Lambert (1728-1777) and was made more popular in the beginning of our century by C.V.L. Charlier (1908a, 1908b). There are two possibilities for a hierarchical Universe: the first, that the mean density tends to zero as scale increases, and the second, that the mean density tends to a constant non zero value but the distribution of matter tends to total homogeneity as the scale increases (the density contrast between consecutive agglomerations tends to zero; cf. Maciejewski 1991).

The hierarchical models of the Universe are infinite in their spatial extensions. In the latter case (the mean density contrast tending to zero) they fulfill the Copernican Principle not in a finite scale but only as the limit in infinity. If our Universe is similar to those models, then the Copernican Principle can be legitimately accepted, but the observable part of the Universe, as well as any finite region of the Universe, must be considered as filled with matter in a non-homogeneous way. Of course, in the case of a hierarchical Universe, the procedure proposed by Heller cannot work.

4.16. Frame of absolute rest. Neoether

The Generalized Copernican Cosmological Principle was introduced into cosmology during the elaboration of relativistic models of the Universe. As stated above, it is not related to General Relativity in any logical way, but its history is, nonetheless, strictly connected with that of General Relativity. Even more astonishingly, any model based on the Copernican Principle and thus accepting the kinematic Hubble Law is, in a certain sense, antirelativistic or, to put it in a less extreme way, involves the conception of an absolute frame of reference. Such concepts are quite opposite to the concept of relativity of all motions.

There exists in astronomy the phenomenon of astronomical aberration (see any book on general astronomy). Its essence consists in superposing the velocity of the observer and the velocity of light. The strict theory of aberration is very complicated because every mixture of the velocity of light with another velocity needs a very exact treatment; simple vector addition is not adequate here. Thus, even in some scientific books the similitude of falling rain is given instead of exact formulae ^[2]. The phenomenon causes the line of sight of every observer to any celestial body to change its direction, inclining itself towards the direction of the observer's velocity. In effect, due to the aberration, more objects are seen on the hemisphere towards which the observer moves.

This phenomenon is observationally proved in connection with the annual movement of the Earth, with the orbital movement of the Sun within our Galaxy, and probably (the interpretation is not certain!) with the velocity of our Galaxy or systems of galaxies. It reveals itself in the positions of celestial bodies. However, the most distinct effect can be noticed by observing the 3-Kelvin background radiation. Out of observations of aberrational effects, an absolute frame of reference, an absolute rest system can be derived; it is the only one in which the Universe is isotropic and homogeneous, manifesting the same mean density of objects in all directions and in all its points. The theory of General Relativity claims that there are no absolute, no preferred frames of reference, but when combined with the Copernican Principle, it does provide such an absolute frame of reference locally. Heller (Heller et al. 1974) calls it Neoether. The existence of some absolute frame of reference does not logically contradict the theory of General Relativity. It can be said that the relativistic theory provides the possibility of describing motion and gravity phenomena in any coordinate systems, and thus deals with relative motions only and that the Copernican Principle selects out of many possibilities the coordinates connected with the neoether as a preferred frame of reference. The Copernican Principle destroys the idea of equipollence of all frames of reference. Thus it is in opposition not only to General Relativity, not only to Einstein's Special Relativity, but even to the basic relativity principle of Galileo. Galileo promoted the idea that a motion with constant velocity is strictly equivalent to being at rest. The frame of reference connected with the neoether distinguishes not only between motions with a constant and non-constant velocity but also distinguishes absolute rest from all other constant velocities. In this way, the Copernican Principle brings us to the old view of Aristotle that the natural state for a body is (absolute) rest. The Copernican cosmology does not state that it is the natural state, but only that it is a favored one.

4.17. Absolute time

Apart from the absolute rest frame, all the Copernican Principle models, except for the models with a Hubble Constant equal to zero, also produce an absolute time called cosmic time. Due to the kinematic Hubble Law (the Hubble constant not equal to zero), the density of matter in the Universe changes constantly. Thus, the momentary density of the Universe univocally determines time, a time which is the same for all points in the Universe. The same fact may be formulated in a more strict way as follows. In the four-dimensional space-time of these models, a family of three-dimensional surfaces can be distinguished in a unique way; the averaged density of matter is constant. The vector perpendicular to these three-dimensional surfaces shows in every spatial-temporal point the direction of absolute time.

If the substratum is to possess the basic properties of homogeneity and isotropy, it has to be considered in the coordinate system connected with the neoether. In other words, the substratum complying with the Copernican Cosmological Principle implies the neoether frame of reference, which is a preferred one.

According to General Relativity, each observer has his own time which depends on the local value of his gravity field as well as his velocity in respect to other bodies. This time can be used in all scientific considerations with the same validity as the individual time of any other observer. But such an observer can see the Universe in a very complicated way, in various stages of overall evolution, with various densities of matter, in various directions and space points. However, he can transform his picture of the Universe to a reference frame in which the Universe agrees with the Copernican Cosmological Principle, and then he sees the Universe as possessing (roughly) equal overall density. Thus, he now considers everything in the neoether frame of reference, and his time becomes cosmic time. Such an observer, connected with a substratum and thus with the absolute frame of reference, is called a basic observer.

4.18. Copernicanism versus General Relatively

From these considerations, it is clear that General Relativity is not responsible for producing Hubble's Law (and thus the Big bang hypothesis); on the contrary, it is rather difficult to reconcile the basic "world view" of General Relativity with the conclusions of Hubble's Law which result from the Generalized Copernican Cosmological Principle alone. In fact, General Relativity and the Copernican Principle tend to quite opposite directions, relativity towards making all more and more diversified, more "individual", and "Copernican" towards making everything as homogeneous as possible.

This peculiar misalliance of the highly intellectual, sophisticated and "democratically oriented" (every coordinate system has equal rights) General Relativity with the "common, rough", so to say "Communistic" (there is only one pertinent system), Copernican Principle conceived in 1917 by the very father of the former, Albert Einstein, created a tightly married couple remaining together in the common cosmological life for more than 75 years in spite of difficulties and inner controversies which it causes in trying to understand the Universe. Was this actually another great invention of Einstein, greater than either he, his contemporaries, or even we, now realize? Or was it just another great mistake he made, leading cosmology altogether astray? The problem of combining these two ideas was never fully analyzed from a methodological, philosophic, or even historical point of view. Of course, the question - what if Einstein did apply some other cosmological principle in his first relativistic model of the Universe is not a scientific one either from a cosmological or from any other standpoint. However, it will be of interest to see, when the future development of mathematics permits, what kind of Universe models could be obtained by combining General Relativity with, for example, the Ancient Indian Cosmological principle.

4.19. Zero case of the Hubble Constant

Most of the above conclusions were based on the assumption that the Hubble Constant does not equal zero. The zero case can be analyzed as a particular case in the frame of the Generalized Copernican Principle. But it may also be considered as a special case of the Perfect Cosmological Principle, and so it will be discussed in the next chapter.

4.20. The Copernican Principle and Kaluza-Klein type models

Relativistic models of the Universe are constructed not only in 3+1 space-time but also in spaces with a higher number of spatial dimensions. This involves theories of the Kaluza-Klein type where the non-gravitational interactions are combined with additional space dimensions. Most known models of this type operate with a 10+1 dimensional space-time (ten space dimensions and time). Those models usually accept the Copernican Cosmological Principle as the initial condition (for the first stage after the Big Bang) for all the space dimensions. In the course of the evolution of the Universe, only three dimensions (the "normal" space dimensions that we perceive with our senses) keep fulfilling the Copernican Principle while other dimensions contract (the so called spontaneous reduction of dimensions). But among these others dimensions, some symmetries are preserved as well.

It remains problematic whether the cosmological principle assumed here for many dimensions in the beginning of evolution is the same Generalized Copernican Cosmological Principle formulated for 3 space dimensions or whether it is a modification. Another problem is how to comprehend symmetries remaining in the further course of evolution. Such symmetries are connected with general simplifying assumptions involved in the model. These assumptions are, in fact, strictly equivalent to the cosmological principles of the "ordinary" 3-dimensional models.

At present, no terminology for such multidimensional cosmological principles is created yet. Sometimes one wonders if the authors of Kaluza-Klein type models are aware that their work involves cosmological principles of a new kind, or, at least, new versions of the old ones.

4.21. 3- Kelvin background radiation

The presence of the 3-Kelvin background radiation is sometimes considered an important argument for the reality of the Big Bang. Specifically, after improving observational methods to such sensitivity that local differences of temperature became measurable, many cosmologists claimed that this is a direct confirmation of the hypothesis. I do not want to enter into discussion about this problem here. But it should be noticed that no matter to what extent the temperature and the differences in intensity and spectral characteristics in various directions are or are not in agreement with this or that Big Bang model, some other hypotheses can explain all those properties (e.g. Davies 1972, Skarzy�ski 1975, Rana 1979,� 980a, 1980b).

When we want to stay within our main area of interest, namely within the issue of cosmological principles, not of particular cosmological models, the most important conclusions concerning the existence and characteristics of background radiation are the following. It is impossible to test any cosmological principle in a general way without any additional assumptions. And we can add the supplementary assumption that the observable part of the Universe is so large that any given cosmological principle can be applicable for what can be observed. So, when the Generalized Copernican principle is under consideration, we can expect to observe something that is homogeneous. Cosmologists, from the very birth of cosmology, have looked for celestial bodies or phenomena which are distributed in this way. Einstein supposed that such homogeneity is found in the distribution of stars. Hubble presumed that this condition is fulfilled by galaxies. In the middle of 20th century, the hope was to find homogeneity in the distribution of clusters of galaxies. All those hopes proved to be futile.

Only background radiation revealed a high degree of isotropy, and if we assume that this isotropy is preserved in all other points of the observed part of the Universe, then we can conclude that the sources which emitted (or, in some hypotheses, still emit) this radiation were (are) distributed homogeneously. Thus, to date, background radiation can and does serve as the best observational support for the Generalized Copernican Principle.

4.22. The Softened Copernican Principle

The Copernican Principle consists of two independent assumptions: homogeneity and isotropy. Andrzej Zieba (1975) discussed a remarkable kind of relativistic model where only the former condition is satisfied. His models, called the models of the Zone-Universe, consist of concentric spherical shells of equal thickness possessing alternately finite and zero density and filling in this way the entire (finite or infinite) space of the Universe. The mean density in regions sufficiently thicker than individual shells is obviously constant here. But in every region, a certain direction (perpendicular to the shell surface) is distinguished. Zieba himself claimed that his models fulfilled the Copernican Principle globally but not locally. He had in mind here that the favored directions are various in various regions and thus, overall, there is no preferred direction. However, the mean density is attained here by a very simple procedure, whereas, in order to obtain an average mixture of directions, the fields must be selected in a rather particular way. Thus, it seems more accurate to say that only a Softened Copernican Principle, or a softened version of the Generalized Copernican Principle, can be applied locally as well as globally, because, in fact, only the approximate homogeneity of distribution of matter is preserved here.

The Zone-Universe models have a number of remarkable features, some of them elaborated by Zbigniew Dulewicz (1971). The Softened Cosmological Principle allows that a model might be infinite in space but finite in time.

The cosmological model of Kurt Goedel (1949) also fulfills that softened version of the Copernican Principle. Here the assumption of homogeneity is fulfilled, but all over the Universe there is a preferred direction which can be interpreted as a rotation axis. Goedel's Universe has no center but is not isotropic. This model contains world lines which form loops; it means that the same point of space-time may appear several times in the history of a particle, thus violating the causal order of events.

Nobody knows how many other remarkable features the models based on that softened version of the Copernican Principle may have. This version of the principle and the cosmological consequences of its models seem to be of interest at least methodologically. But this version could be particularly worth remembering if any large-scale anisotropies are subsequently discovered in the observed region of the Universe.

Similar anisotropies can be expected if the topology of the Universe is not too simple. In thinking about the Universe, which is finite but not limited, one has usually in mind a three-dimensional closed space immersed in a four-dimensional Euclidean space. When we think of the space of the Universe, we often think of a sphere or a Moebius Band as an adequate model. In fact, a finite, unlimited three dimensional space can possess a much more complicated structure, involving not only different geometries but also different topologies. Such topologies can allow quite complicated local inhomogeneities while preserving mean (!) constant density of matter over sufficiently large regions. These possibilities, based on the so called Clifford-Klein spatial forms, were elaborated in the last decades by George F.R. Ellis (1971). Some cases go beyond any Copernican way of thinking, but some could be considered as further examples of models satisfying the Softened Copernican Principle. Models of the Universe based on that (as yet almost unexplored) principle could give some interesting results and shed new light on some methodological issues. It is also possible that some of the results could somehow correspond to our actual Universe.

4.23. Isotropy without homogeneity

Another way of reducing the Copernican Principle, that is, accepting isotropy but rejecting homogeneity (Cosmological Principle of Isotropy), leads to highly sophisticated geometrical and topological models if isotropy has to be fulfilled all over the Universe. The Schur's theorem states that if isotropy is preserved in every point of some space in which the concept of parallel lines has any sense, then this space must be a homogeneous one. To construct a Universe model where isotropy is preserved but homogeneity not, we have to turn to some highly sophisticated space constructions where the usual concept of parallel lines can no longer be applied. Thus the notion of 'direction' must be defined in an unusual way. Universe models obtained using such procedures are rather far from reality, or at least from the kind of reality that we can imagine today.

If, however, we admit that this isotropy has to be kept only in a specific, distinguished point of the Universe, then we, accepting our Earth as this point, arrive at the Ancient Greek Cosmological Principle or to its generalized form if we accept other points (other centers of the Universe, cf.2.13).

From the last discussion we can see that when we restrict ourselves to mathematically trivial topologies and geometries of space (e.g. to manifolds only), the assumption of isotropy observed from every point in the Universe produces by itself the homogeneity of the Universe. With such a restriction, one can shorten the definition of the Generalized Copernican Principle to the following formulation: The Universe is isotropic when seen from every point. For a practical use in constructing cosmological models this form seems to be sufficient. Nevertheless in most cosmological papers and books, homogeneity is considered an independent assumption; this is completely correct if we have in mind not just practical purposes but all the possible exotic topologies of the Cosmos.

The assumption of isotropy in every point is stronger than the assumption of homogeneity. As can be seen from the Zone-Model, even on all types of manifolds the latter assumption can stand alone. It does not imply the former one.

One could question whether each kind of existing symmetries (e.g. the 9 Bianchi types) should be considered to be another version of some cosmological principle or an independent cosmological principle.

�[1] I leave it to the reader to think over why as long as the Universe was considered to be the body of a supreme spiritual being (ancient India), it was considered as infinitely heterogeneous, but when considered to be a supreme being in itself it is considered as infinitely homogeneous.

[2] When drops of rain fall freely towards the earth in the absence of any wind, a standing person sees, on average, the same number of rain drops in all directions. However, if she (or he) starts to run, she encounters more drops from the direction in which she moves. If she has an umbrella, she has to tilt it forward when running.

Bibliography
© Konrad Rudnicki

Konrad Rudnicki is a professor at Jagiellonian University, Cracow, Poland. He is a member of the Free European Academy of Science (Holland), member of the Commission of Galaxies of the International Astronomical Union and member of the Mathematical-Astronomical Section at the Goetheanum, (Switzerland). Prof. Rudnicki has been Senior Research Fellow at the California Institute of Technology (1965-67), visiting professor at Rice University, USA (1988-89). His areas of interest are: extragalactic astronomy, cosmology, philosophy of science and methodology of science.

Continued in the next issue of SCR.