The Perfect Cosmological Principle

By Konrad Rudnicki

5.01. Time horizon

The Generalized Copernican Cosmological Principle produces the kinematic Hubble Law. The simplest interpretation of redshifts of extragalactic objects is that of the Doppler effect. If we combine them both, we come to the conclusion that the Universe is in a state of expansion. Again, the simplest interpretation of that expansion is the diminishing of mean density. A decrease of density with time (toward the future) is equivalent to an increase against time (in direction toward the past). The models based on General Relativity claim that this increase will go to infinity, so going back into the past far enough we reach a stage of infinitely high density. The time when the density was infinite is called the initial singularity and, at first, was considered to be singular in a mathematical sense only. Mathematics is, after all, only an approximation of reality. Georges E. Lemaitre (1927, 1946) came early to the conclusion that the initial singularity can be interpreted as a violent origin of the Universe. The first not only to consider that singularity to be of physical significance, but also to realize how to draw physical conclusions from it, was a student of Alexander Friedman, George Gamoff, who, in the middle of 20th century, showed how important conclusions can be made about the stages of evolution close to the singularity involved in Friedman-type models. This line of research was very fruitful and was continued by followers of Gamoff (cf. Melchiorri and Ruffini 1986).

Earlier cosmological theories were concerned only with mean mass density in cosmic space. Other features of matter -like chemical composition, types and dimensions of aggregates of matter in consecutive epochs of cosmic evolution, or forces other than the gravitational one - were irrelevant for them. Only after Gamoff did cosmology come into contact with almost all areas of physics and chemistry. This line of research is still alive, and some of its results became classic long ago (cf. Weinberg 1977). In this way one could explain the mean abundance of chemical elements in the Universe. It is possible even to propose some scenarios of formation of galaxies, and - what is most striking - the origin of physical constants or even the origin of physical laws as such (cf.: e.g. Reeves 1986). There are, in fact, immense possibilities yet to be explored. Nevertheless, the fact remains that present day physics cannot grasp states of matter with densities of arbitrary high value. For very high densities beyond a certain limit (today this limit is considered to be about 10100 g.m-3), physics is completely incapable of producing even a single physical formula since any physical interactions no longer have any meaning and relationship to space and time. As long as the explanations of physical phenomena consist of deriving physical equations, it is very unlikely that this limit of density (called Planck density) could be overcome by physical consideration. And no other style of physics can be proposed, or, at least, has been proposed yet. This situation created another limitation for our knowledge: time horizon. The limiting density 10100 g.m-3) was achieved, according to contemporary theories, in 10-4 of a second after the singularity.

Whether or not there was the Planck density long ago or even the singularity itself, most contemporary cosmologists are convinced that the history of our Universe is, beyond some point in its earlier past, impenetrable for scientific investigation. Thus, the problem of how to approach the question of the time horizon has been considered to be one of the most fundamental.

5.02. Overcoming horizons with a suitable cosmological principle

The difficulties with the spatial cosmological horizon can be removed using the Generalized Copernican Principle. This principle, once adopted, automatically overcomes all the difficulties which the spatial horizon itself created. The horizon is still there, but it no longer delimits our knowledge about the Universe; it vanishes, as it were, at least when considering the most general aspects of the Universe. But what to do with the cosmological time horizon? In order to remove it two cosmologists, Herman Bondi and Thomas Gold (1948), proposed a new cosmological principle called the Perfect Cosmological Principle or Strong Cosmological Principle. It states:

The Universe observed from every point, in every direction, and at every time looks roughly the same.

Or using other words:

The universe is (roughly) homogeneous in space and time and isotropic in space.

As is easy to see, this principle removes the problems of both cosmological horizons. It retains the assumptions belonging to the Copernican Principle and adds another one: homogeneity in time. Of course the Hubble Law must still be fulfilled: it does follow from spatial homogeneity and isotropy. It could be said that the Perfect Principle is merely one particular case of the Generalized Copernican Principle, just as the Generalized one is merely one particular case of the Genuine Copernican Principle. The Perfect Principle puts more constraints on the models of the Universe.

5.03. Spatial infinity of the Universe

To the assumptions of Generalized Copernican Principle, the Perfect Principle adds only one more assumption, that of the invariability in time. But, just as the Hubble Law follows from the assumptions of the Copernican Principle, so, too, from this additional assumption follows infinity not only of time but, in most cases, also of space. Since the Perfect Principle is one special case of the Copernican Principle, one could deduce this spatial infinity of the Universe from the General Relativity models. Since invariability in time also means infinite time duration, an everlasting expansion or contraction with constant velocity needs unlimited space; otherwise, the Universe would, in time, be either larger or smaller. Only infinity has the property that its dimensions remain the same even when multiplied or divided by some finite factor.

In the so-called Steady State model based on the Perfect Principle, the infinity of space is derived from relativistic equations. But the problem is a more general one. Not only relativistic models of the Universe can be constructed. The Copernican Principle produces the Hubble Principle. Regardless of whether the Hubble Constant is positive or negative, with the new assumption, this parameter must also be constant in time (no time changes are allowed!). If the Universe expands or contracts with constant speed and is nonetheless everlasting, its spatial dimensions must be infinite, whatever physical theories we accept.

There remains, however, the zero case of the Hubble Constant. There is no mathematical necessity to accept the spatial infiniteness of the Universe. One could imagine a finite but not limited cosmic space lasting from eternity to eternity. However, the Universe consists not only of space which can be considered mathematically, but it also has to contain some matter. Sir Isaac Newton, who was an adherent of the Static Universe model, was of the opinion that a finite Universe would shrink to one great lump of matter due to gravity alone. He had in mind a finite material Universe immersed in infinite Euclidean space. The concept of the curvature of space was not yet known at the time. Besides, he was not aware that an infinite but static Universe would be inherently unstable as long as we applied classic (or relativistic) mechanics to it; that instability would lead to systematic changes. Anyway, it is easier to imagine an infinite than a finite static Universe, and all the adherents of zero Hubble Constant are of the opinion that in a finite Universe some overall variability would be unavoidable. All the matter would contract to one point or all the celestial bodies would evolve in one direction (thus there would be a global evolution of the Universe). The problem of the stabilization of an infinite static Universe remains still under discussion. In any case, if we accept as a reality a Universe model with the Hubble Constant equal to zero, we also have to consider this Universe as infinite in space. The opposite case, an invariable Universe infinite in time but finite in space, remains still as a logical but completely abstract possibility without any theoretical elaboration.

As we showed above, in the case of a finite, positive, or negative value of Hubble Constant and within the Perfect Cosmological Principle, spatial infinity of the Universe can be mathematically proved. The zero case is the limiting case from both sides. When the mathematical functions are not too exotic (and most of the macrocosmic physical functions are not) we should have the same result for both limiting cases. Is this pseudo-mathematical proof more convincing, or the fact that all adherents of zero case claim so? Whichever is the case, all considerations known to me which are based on the Perfect Principle always involve infinite space.

5.04. Metagalaxy

(This Principle allows for the evolution of particular celestial bodies, their systems, and their supersystems. Any global evolution of the Universe is, however, excluded. According to Jaakkola (1989), systems do evolve and have centers. But the Universe is neither a system nor a supersystem. It is totality, it is infinity and as such without a center and in all its history ever self-similar. In order to emphasize the fact that all we can observe is just an inconspicuous speck in comparison to the infinite Universe, the term Metagalaxy was introduced for the observable part of the Universe. Of course knowledge of the Metagalaxy belongs to astronomy. Cosmology can only make use of astronomical facts concerning the Metagalaxy. If we accept the Perfect Principle, no observational evidence can convince us of the evolution of the entire Universe. Thus, there exists no possibility to convince (by logical argumentation) the adherents of the Perfect Principle that they are not right.

The notion of the Metagalaxy proved very useful during the Stalinist period as well as during the first years of the post-Stalinist period in the Soviet Union and other countries under Soviet domination. At that time, it was considered an ideological crime to support the hypothesis of an expanding Universe. But it was permitted to speak and even lo publish papers on the expansion of the Metagalaxy. Thus, Metagalaxy became for many cosmologists from the Soviet block the cryptonym for Universe. The censors, happily enough, took no notice. In many Russian, Ukrainian, Estonian, Czech, Slovak and other papers from that epoch, the secret name 'Metagalaxy' should be replaced with the proper one by a present-day reader.

Of course the term 'Metagalaxy' is meaningful in cosmology not only as a cryptonym. It reminds us constantly of the necessity of distinguishing between astronomy and cosmology.

5.05. The only possibility of knowing everything

The Perfect Principle, when accepted, gives the feeling of knowing all. I can look in any direction of space; I can think about regions of the Universe located as far beyond the cosmological horizon as I wish; I can imagine epochs as faraway as I wish and know exactly how it was then. Literally everything, everywhere, and in any time is such as it is here and now. In a scientific sense, it means that I do not need to investigate any exotic stages of matter or strange geometries of space. The best thing I can do is to become acquainted in ever greater detail with those parts of the Universe which are easily accessible to my sensual perception and to the reach of my instruments.

In the times when the Perfect Principle used to be fashionable, it was said that this principle is like a lonely street lamp on an otherwise dark street. If a man walking home at night loses his house key, he has to search for it where the lamp sheds some light. If the key dropped from his pocket into the light, then he may be lucky and find it. If, however, the key was lost somewhere else, then he has no chance. Similarly if the Perfect Principle is not true, then we have no other possibility of knowing what is beyond the time horizon. Due to progress in physics, that time horizon may be shifted still further, but some initial stages of the Universe will always be altogether inaccessible to us. The horizons would be impenetrable for us. And thus, humanity would have to abandon its pretensions of knowing everything. The Universe would always remain only partially known. So we had better hope that the Perfect Principle is true...

5.06. Creation of matter

But when we accept the Perfect Principle, which is just a narrowed version (involving an additional assumption) of the Copernican Principle, and, as a consequence of this fact, we accept the Hubble Law, then, due to this law alone, the density of matter must change with time, which would contradict homogeneity over time. Only three possibilities remain to fulfill simultaneously the old requirements and the new demand of homogeneity in time: to compensate these changes in the density of matter with the creation of matter by a positive value of the Hubble constant, to compensate through the vanishing of matter in the case of a negative value, or to ascribe a zero value for the constant.

The first possibility is realized in the Steady State model, postulated in 1948 by Herman Bondi and Thomas Gold as well by Sir Fred Hoyle (1948, 1949). This model accepts the expansion of the Universe as a fact and considers the rate of that expansion (the Hubble constant) to be constant not only in space but, in accordance with the additional assumption, also in time. Thus, the creation of matter has to be assumed in order to make the mean density of the Universe invariable. Such a creation may be regarded as some fundamental law of nature. As a matter of fact, to keep the density constant, the creation of one hydrogen atom per year in a volume of about one cubic kilometer proved to be sufficient if a Hubble constant of 100 km.s-1.Mpc-1 is accepted. Such a tiny process cannot be discovered with today's measuring instruments. The creation would have had to be roughly homogeneous (i.e. the newly created elementary partic1es or atoms should be distributed homogeneously over very large areas of space), but some places can be locally distinguished. For example, matter can be created just in maximal distances from the existing galaxies and aggregate into protogalaxies. Or, it can be created just in the nuc1ei of galaxies and then be sent off as protogalaxies. This can make the possibility of observational or laboratory confirmation of the creation process even more difficult.

The Perfect Cosmological Principle originated from an extremely materialistic world-view. The attitude of its adherents can be described in a simplified way as follows: if one accepts that all knowledge must be attained through physical means only, if one accepts that the human mind is the highest intelligence throughout the Universe, and if one accepts that the truth about all the Universe should be attainable for humanity, then all the physically construed cosmological horizons have to be overcome. The Perfect Principle is considered as a method of overcoming those horizons. But when we adopt the Steady-State model, the creation of matter must be also acknowledged. Could such a creation be reconciled with the materialistic world view? Some people (cf. Rudnicki 1982) are of the opinion that the materialistic world view is self-destructive.

For the benefit of that model, it can be said that it does remove the photometric cosmological paradox x in the most straightforward way by not involving any additional assumptions.

5.07. Static Universe

Another possibility of materialistic thinking is the homogeneously populated everlasting Universe with no expansion (the Hubble constant equals zero). It is usually called the static or quasistatic model of the Universe. The prefix 'quasi' means that over detached, even immensely vast areas of such a Universe non-static processes are going on, but the Universe at large remains ever the same. This model had many advocates in the epoch of classic materialism, especially in the 19th century when the Perfect Principle and the Hubble Law had not yet been formulated. Also, the official Soviet cosmology in Stalin's time, supported by the Communist Party, proclaimed that model as the only one corresponding to the actual Universe. Propagation of other models was prohibited by Soviet law [1]. At present it has little appeal; notwithstanding that a modern version of it was presented by Toivo Jaakkola (1989).

There are two very difficult questions to be addressed. The first one is: how to limit the action of gravitational forces in such a way that they do not cause ever increasing condensations of matter in the Universe. And the second: how to reconcile the evolution of celestial bodies with the assumed globally constant composition of chemical elements in the Universe. Jaakkola (1989) proposes some ways of doing this using some remarkable ad hoc assumptions.

The models based on the Perfect Principle differ from those based on the Copernican Principle in the necessity of securing their (rough) invariability in time; one must, so to speak, neutralize the Hubble Law. But in the zero case, the Hubble Law turns into a stationary state by itself. This zero case for the Perfect Principle is thus identical with that of the Generalized Copernican Principle. I discuss it here because it is more closely related to the Perfect Principle in philosophy.

Sometimes the Copernican or the Perfect Principle with the Hubble constant equal to zero is considered as a separate cosmological principle and goes by the name of the Lucretian Principle (cf. 7.2).

5.08. Vanishing of matter

The case of a Universe fulfilling the Perfect Principle with a negative value of the Hubble constant is for cosmologists a matter of coffee-break talks rather than of scientific publications. The advocates of the Perfect Principle usually accept the Dopplerian interpretation of redshifts and thus the positive value of Hubble's constant. No model

with vanishing matter is known. But constructing one could be instructive, just as a methodological exercise. Some ideas as to where the excess mass vanishes (within all existing galaxies? or in only some of them?) have to be developed in order to secure the overall invariability of the Universe. Certainly it will not be merely a Steady-State model with reversed time.

5.09. The Perfect Principle and absolute frame of reference

In all models based on the Perfect Principle, the evolution of individual celestial bodies is possible and even inevitable, but any evolution of the Universe at large is excluded. In the models based on the Copernican Principle, the overall state of the Universe is a kind of clock measuring cosmic time. All the models based on the Perfect Principle involve no cosmic time in this sense. However, they do possess an absolute frame of reference. In order to be able to see the Universe as (roughly) the same in any direction, only one relative velocity must be chosen (different from point to point in space) and, in fact, this velocity may even be absolute rest as well. Thus, the neoether is admissible also in this class of models. Due to the aberration effect, the Universe will be seen as having various densities in various directions during any movement relative to this state of rest.

When there is a preferred absolute frame of reference, there is also a preferred time direction in space-time, and it may be designated cosmic time. In four-dimensional space-time it is a direction perpendicular to the three-dimensional surfaces connected with neoether. It differs from the case based on the Copernican Principle only in that there are no global zero points in time. The different fundamental observers have to synchronize their docks in an arbitrary way, but their times run para11el. The duration of a time unit and simultaneity can be established in the same way for all observers.

5.10. Possible generalizations of the Perfect Principle

The premises of the Generalized Copernican Principle were deliberately determined and discussed for the needs of a relativistic outlook on the Universe. In fact, this cosmological principle is not based on relativistic concepts. It proclaims only some properties of space, not of time, whereas relativity employs in its considerations a unified concept - space-time. Thus, any assumption which does not concern all four dimensions of space-time spoils the elegant generality and symmetry of the relativistic picture of the world.

In this sense, the Perfect Principle is much more relativistic because it puts similar conditions on space and on time. According to this principle, the Universe should be homogeneous in respect to space and time. However, some asymmetry still remained. The Perfect Cosmological Principle requires the homogeneity of both, but it limits the requirement of isotropy to space only; it does not require such from time.

At first glance, the requirement of isotropy in time seems to be impossible of realization in the actual Cosmos. Even in the stationary Universe of Jaakkola, the isotropy in time can be construed only in a metaphorical sense. To be sure, the overall view of the universe is the same when we move in time to the positive or to the negative direction, but the local phenomena are not reversible. Gravitation, the pulling forces and the explosive, dispersing forces, all act in opposite directions in time and produce, in Jaakkola's Universe as well, phenomena by no means the same but merely directed the opposite way in time. The same is the case with electromagnetic radiation. The time arrow does still exist.

Nevertheless, one cannot exclude the possibility that, in the course of the further search for similarities and identities in various physical interactions, theories will emerge involving phenomena perfectly symmetrical in time. Then a model of a completely static universe, fulfilling what could be called the Generalized Perfect Cosmological Principle would be possible. Of course, a model does not necessarily correspond to reality. One can think over how it would be to live in a Universe with no arrow of time or, rather, with equal arrows in two opposite directions.

But even such a generalized principle is not fully equitable in respect to the space-time concept. It sets the same requirements for space as for time but still does so separately for space and for time. The Fully Perfect Cosmological Principle should set one unitary requirement for space-time as such. At first glance such a total isotropy is not possible, at least as long as we retain the ordinary notion of relativistic space-time, because the metric signature of space-time itself distinguishes the time with the sign opposite to the signs of spatial dimensions. But there is a mathematical trick of using imaginary time. Then the metrics of space-time becomes fully symmetrical in respect to all four dimensions, and the Fu11y Perfect Principle can be introduced. This possibility is explored in the Hartle-Hawking model of the Universe. Besides, there are other space-time theories (e.g. the theory of Segal 1972). Thus, the Fully Perfect Principle can be used for constructing models in many ways.

I hope the reader has already noticed that the last versions of cosmological principles make no claims to be fulfilled in reality. Rather, they are formulated here to draw our attention to the fact that the more elegant, symmetrical, and simple a principle, the narrower and the further from reality it is.

This is the property of all theoretical considerations. The simpler, the more elegant a theory, the less is it concerned with reality. Without any idealization and simplification, reality would be utterly incomprehensible for us, but if the idealization and simplification proceed too far, then, though the theory can be readily grasped, it hardly fits reality.

Keeping a balance between idealization and complexity is the task of the theoretician and not just in cosmology.