Nick Kronov


Chapter 1. Evolution of a picture of the world.

The change in notions about the form and sizes of the universe during centuries until today is described in the initial chapters of many popular scientific books on cosmology. Until an epoch of the Great geographical openings by Columbus, Magellan and others the majority of people considered that the Earth is «a circle» (this is written in Bible: Isaiah 40:22), up to its edges it is possible to reach and to glance from its edge «downwards» - in «the chasm». On edge of the circle of the Earth the firmament («the expance»), like a tent, bases on the Earth. Over the firmament the Sun and Moon are moving. And stars are the heads of silver nails knocked in the solid dome of heaven (a Russian word «zvesdy» is «gvezdy» - nails).

However already the philosophers of Ancient Greece understood that «a circle» of the Earth is not a plane but convex and is bent in all the points and along all the directions equally. Now it is named «the Cosmology Principle» - that the universe - if to compare its large parts - is homogeneous and isotropic. They tried to calculate a radius of this sphere by measuring the change of height above horizon of a Polar star and distances during long trips along the river Nile flowing from the south to north. They had calculated precisely. But had not believed themselves: can a radius of Globe be really so great? What then the distances up to the Sun and Moon will be! And what then their sizes: it seems they are such small and not so distant. The public, certainly, will not have understood it. And they had lessened the figure in the report. Soon Ptholemey had drawn a bold picture of the universe. It turns out we live not on a plane but on a sphere - on a surface of a ball! At that time this was very difficult to imagine. It turns out that if you will go at one direction - you will go back to the same place from the opposite direction! The picture of the curved and closed space of the universe, depicted by Einstein, causes today exactly the same bewilderment to the public. 

Around of the spherical Earth, according to Ptholemey’s model, it are placed one into another several skies - the rotating transparent crystal spheres, to which are attached: a flat lantern the Moon - to the sky, nearest from the Earth, to the next sky - Mercury, further Venus, then the Sun, to the next - Mars, Jupiter, Saturn, and to the latter - whether seventh, whether ninth sky - the familiar to us silver nails - stars. Though it was unintelligible how the inhabitants of the opposite party of the Earth can live there head over heels and to be kept from fall «downwards», in «a chasm» but it should be trusted because the Ptholemey’s model was based on the elementary measurements and calculations made on the Nile. An insignificant mote - the mortal man can measure the universe! Then it very much delighted.

Therefore it looked like an indecent escapade in attitude to the respectable authoritative «experts» the statement of some Nicholas Kopernik when he after the lapse of more than a thousand years suddenly has paid attention to some absurdities in the Ptholemey’s model and has offered the own model - with the Sun at the center of the world. The modern skeptics and postmodernists could yawn and tell to Kopernik: well, there may be many models, everyone is free to draw the own one, «a text should not cause any consequences». But at that time it was not accepted simply to classify and to collect the different points of view, people were listening to arguments, considering that the truth may be only one, and were accepting close to heart everything that had been told. And then Galilei, who discovered the forces of inertia, also «impertinently» had declared: if the awfully removed seventh sky with stars makes one revolution for a day, it will have disintegrated to pieces because of such speed of rotation, - not the sky but the Earth rotates! And at last Jordano Bruno had summed up: «It means, there is no solid sky with stars-nails, the stars are the same suns like the our one. It means also, the universe has not any center». It was impossible to endure this!

But also it was impossible to refute it convincingly. The new ideas were being caught up and were being developed. On the basis of the laws of Galilei’s dynamics and of universal gravitation law of Newton the distances from the Sun up to planets rotating around it and also their sizes and weights had been calculated. And then by the same method, that the travellers along the Nile have calculated the size of Globe, now, by «travelling» on Globe around the Sun and, by measuring from the opposite points of already measured orbit a corner between the Sun and stars, the distances up to nearest of them had been calculated. For the majority of stars the changes of the corner (named as parallax) were so small, that its could not be measured - so these stars had turned out far.

Thus the Newton’s model had appeared. It prevailed till the 20-s of XX century. According it, the universe is infinite in space and in time, that is to say is eternal. The stars rotate around of the center of the galaxy. The groups of galaxies rotate around of the center of the group. The congestions of groups of galaxies form in turn the congestions of the higher level etc. Recently it have been found out that the congestions of galaxies form in the space of the universe the cellular structure like a honeycomb. But also it does not change that fact that along all directions on distance up to 12 billions light years from us up to which the modern telescopes have reached the same is everywhere. Also there are no reasons to think that outside of visibility there is something other. On borders of visibility have found the huge luminous congestions of matter named quazars, that are not present near of us. It is possible to explain it that we see the border areas such what these - and, obviously, all our universe - were 10-12 billions years ago. The changeability of the universe in time undermines an idea of its eternity, also the Newton’s model as whole.

Some other absurdities in the Newton’s model of the world were found out earlier, and the main was a conclusion from the equations of electrodynamics about a constancy of speed of light that is always equal 300,000 km/s. And at the beginning of XX century Einstein in his «special theory of relativity» - considering only the uniform motion - has managed to bring to the Newton’s mechanics the changes connected to a constancy of speed of light as the top speed of any motion in general. But the consequences of it were very important. From the Einstein’s «General Theory of relativity», that considered already both the accelerated motion and the forces of gravitation, it followed, that the three-dimensional space of the universe is not infinite - as are infinite, for example, an one-dimensional direct line and a two-dimensional plane - but is finite in volume and is closed on itself, as are finite and are closed an one-dimensional line - a circumference also a two-dimensional surface of a ball - a sphere.

But the one-dimensional line - a circumference can be curved and is closed only because the plane, on which it is placed, has two dimensions. Also the two-dimensional surface - a sphere can be closed only because the space, where it is placed, has three dimensions. But the three-dimensional space of the universe can be bent and closed because our world actually is four-dimensional, and its fourth dimension is time. Time appeared as the fourth dimension already in the earlier - «special» theory of relativity.

Soon A. A. Fridman had found out one more consequence of the Einstein’s theory: that the closed three-dimensional space of the universe cannot be stationary but must being expanded, being inflated - as the closed two-dimensional surface of a toy-balloon is being stretched when the balloon is being inflated. Whether our universe actually expands and why expands - to prove and to explain it Fridman has offered to the others. He spoke that it is his affair to decide the equations but to ascertain the physical sense of the decisions other experts - physicists, astronomers - should. And soon Edwin Habble showed that the unintelligible displacement to the red party in spectra of all remote stars and galaxies is possible to explain only that all of them leave from us and that the speed of this universal «scattering» of galaxies is proportional to their remoteness from us.

Thus the today’s generally accepted «Theory of the Hot Universe» was born. It is also named as the «Theory of Big Bang». It is describing in detail the conditions and evolution of matter towards its complication at different periods of existence of the universe from the shares of the first second of its expansion from a point.

However the theory has not explained why the space of universe expands. I hope that this my article will have given the explanation. The important consequences - amendments to the generally accepted model of the world follow from my explanation. 

Chapter 2. Lacks of the generally accepted theory.

              In England one student - an astronomer had made complex calculations and had shown that it is possible completely to explain unusual behavior of Uranium by influence on it of a planet more distant from the Sun than Uranium. Unfortunately, due attention had not been paid to results of calculations of this young man because he was only a student.
                (W. Kaufmann «The Space Boundaries of the Theory of Relativity», Moscow, 1981, p.70)
I assert: the universe expands because in ours four-dimensional world three dimensions of space and one dimension - time geometrically are connected thus that out of the flowing of time the expansion of space inevitably follows. In particular, it means that the expansion of the universe could have stopped only if the flowing of time will stopped. Consequently the statement of the generally accepted theory is incorrect that it depends on amount of matter in the universe - whether the expansion of the universe will stop sometime by being replaced with compression, or the expansion will be continuing always. 

I. A. Klimishin in the book «The Relativism Astronomy» («Science» M. 1981, p. 247) writes: «The theory of «Big Bang» that is to say the theory of the Hot Universe cannot give an answer to a question «why the universe expands». As the academician J. B. Zeldovich had noted: «Into this theory the expansion is incorporated initially. As the theorists express themselves, is incorporated «by hands», by arbitrary setting of the initial conditions. To the questions, why the universe expands, why the galaxies scatter nowadays, the answer is, that already at the first second (but can be also earlier) there was appropriate to the scattering the initial distribution of speeds».

And here I undertake to answer a question, why the universe expands. I shall do without this mystical, «incorporated by hands», «the initial distribution of speeds» for everything of the unimaginable huge number of nucleuses formed per the first second of existence of the universe and constituting now the universe - the initial distribution of speeds, conformed for some reason as if just to their «scattering» (what is this?!), instead of, for example, their «concentrating» or, say, «to flying past».

A reader may say: «But can be, really, something had been blown up there - then the matter of the universe had begun to scatter». However an academician Zeldovich rejects such a simple explanation. He writes:

«Is it possible to speak that the high pressure is a reason of the expansion of the universe and that a powerfully compressed substance expands for the same reason that gases of high pressure scatter when its have been formed as a result of detonation of an explosive substance? No, such a point of view is completely wrong. The qualitative distinction is that the explosive substance is surrounded with air at the atmospheric pressure. The expansion is caused by a difference between enormous pressure of gases (of the products of explosion) and rather weak pressure of the surrounding air. But when we consider pressure in the homogeneous universe, it is supposed that the pressure is distributed strictly homogeneously! Hence, between various particles at the same moment there is no difference of pressure, hence, there is no also a force, that could influence the expansion and furthermore to be the reason of the expansion. The fact of expansion in the existing theory is a result of initial distribution of speeds. The reason of this initial distribution of speeds is unknown today». (I. A. Klimishin, page 191)

But it is ridiculous even to speak about the reason of something, if this something is thought up, «is incorporated by hands», «is an arbitrary setting» as Zeldovich himself recognizes. Moreover, the «incorporated in the theory by hands» turn out inconceivable even in the theory. Because what is the unimaginable initial direction of speeds of the particles, that result its scattering under its own momentum in the homogeneous universe, that is to say in the such universe, where matter in regular intervals is distributed along all its volume, where is present neither center, nor periphery? In other words, where the speeds of particles should be directed in order its just would scatter? There is no such direction! In my opinion, it should be completely clear, that the initial distribution of speeds of particles in the homogeneous universe can be only chaotic, and the seeming effect of «the scattering» can arise, only if distance between particles will be increasing because of expansion of space. Therefore the seeming, visible today's scattering of galaxies can be explained only by appropriate model of space-time, say, likening it to a being inflated air-ball, the being stretched membrane of which represents the two-dimensional model of the three-dimensional space of universe - but not that «galaxies now are moving under its own momentum, and their speed is being braked by gravitation», as, for example, I. D. Novikov, the author of the book «Evolution of the universe» («Science», M. 1990, p.23) asserts. If the scattering of galaxies is seeming (because of expansion of space), it cannot also be braked by any force. Besides, the resulting force of all gravitation forces in the homogeneous universe equal zero, hence also there is nothing that can brake. Thus the today's theory is unsatisfactory at any rate.

But if the forces of inertia do not accept participation in expansion of the universe, and force of gravitation - in braking this expansion, why does the universe expand and how? How can the toy-balloon model explain it?

It is impossible visually to imagine space of the universe as three-dimensional, not having borders, and at the same time - as finite, closed upon itself like a ring. But let's reject one of three dimensions of space and, as model of the universe, imagine the two-dimensional space with the same characteristics - for example, a surface of a toy-balloon. Let's draw many points, distributed more or less in regular intervals on all the surface of the balloon. The points represent particles of substance per the first seconds of existence of the homogeneous universe. Once again we shall ask the supporters of the generally accepted theory, who too compare the space of the universe with a surface of a toy-balloon: WHAT should be «the initial distribution of speeds» so as the points would being scattered? WHERE should be directed the speeds of motion of these points in space of the model, that is to say on a surface of the balloon? The answer from the theorists is absent. But the answer is clear: there is not such a direction. But it means, other statement of the generally accepted theory is incorrect too: that the scattering - already not of particles but galaxies formed of them - the scattering continuing ostensibly by its own momentum is being braked by force of gravitation. But the braking is required by the law of preservation of energy! Because potential energy of galaxies, that gravitate each to other, is being increased when its are moving off each from other - its kinetic energy and the speeds of its scattering should being decreased! But it can't be helped: we can see it visually that in two-dimensional space, that is to say on a surface of a balloon, there is no especial point, whence the particles originally have taken off and where the braking force would attract back today's galaxies. Because this «original point» - that is to say the initial universe - has been expanded up to the sizes of the present universe. The absence in present space of such an especial point, such a «center of Big Bang» follows also from the Cosmological Principle, that is to say from the large-scale uniformity and isotropy of the universe. And now - as Einstein has neglected a universality of the Newton's law about addition of speeds for the sake of a constancy of speed of light, as it is necessary to neglect a universality of the law of preservation of energy for the sake of a constancy of speed of expansion of the universe that, as I shall show below, is connected to a constancy of speed of light and flowing of time.

The psychologist Max Vertheimer in the book «Productive thinking» wrote: «Physicists of the past was trying to construct a perpetual mobile. After many unsuccessful attempts suddenly a question had arisen: how will physics look, if the fundamental laws of nature would made impossible an existence of the perpetual mobile? By becoming central, this question had resulted in huge changes. Also Einstein had a question: how will physics look, if on nature of things under all conditions the speed of light will be constant?».

Now I offer a question too: how will physics and the universe look, if the truth is that a «radius» of the universe grows with constant speed and this speed is equal to speed of light?

The supporters of the generally accepted theory diligently reveal all the consequences which appears from its algebraic formulas but for some reason they do not wish to see the consequences which appears from their own geometrical model - of the toy balloon model. You see, could they ask themselves some questions: what is the radius of the balloon, that is to say an additional dimension, that is perpendicular to the dimensions of space, may it be time? What is the speed that radius of the balloon is growing and why it is growing, that is to say why space of the universe is expanding? Where on the ball there is a horizon of a visible part of the universe and what is located behind the horizon? What is the speed of the galaxies, that are stationed behind horizon and are moving off from us? Is the speed greater than speed of light? 

Unfortunately, the supporters of the generally accepted theory prefer in answer to criticize my model instead of at first to answer in essence to all criticism stated here about them. To escape the answer, they are even ready to refuse from their own model of a toy balloon and from a visual method in general, by hinting that the absence of a visual model is attribute of depth and complexity of their theory.

As a matter of fact, they refuse in general any model of the universe. As A. Tursunov wrote («Philosophy and the Modern Cosmology», M. 1977, p.179): «By judging by the present literature, the majority of authors prefer the most natural and at the same time the easiest - namely an interpretation of the universe as a part of the world, and accordingly, a restriction of a subject of the cosmology researching by limits of the methagalaxy, deliberately finite both in space and in time1. Formally such approach seems twice radical: because in this case it is not simply at once is split the Gordiy's knot of conceptual difficulties of the modern cosmology but at the same time all the previous philosophical discussions about «the beginning of the world» are being depreciated too. However, actually, the methodological radicalism result in narrowing the problem».

FOOTNOTE 1: Actually all the volume of space of the universe should be named as «the universe» - otherwise it turns out a mess. Especially because this volume is finite, as it follows from the theory of «Big Bang». But «our metagalaxy» is just the part of the universe, the part that we can see. And a word «world» is not at all a name for space but for an absolutely other essence - for the four-dimensional space-time. (N. K.) 

By the way, D. I. Mendeleyev spoke: «It is better to hold by such hypothesis, that can turn out to be incorrect, than to hold by nothing».

Chapter 3. Why a radius of «the balloon» grows just with speed of light.

My hypothesis, my model of the universe has not the above contradictions of the generally accepted theory. In my model the «scattering» of particles and galaxies is seeming. It means, the scattering cannot be braked by force of gravitation. The seeming scattering appears because space of the universe - a surface of the balloon - is stretching when the balloon is being inflated. Space can being stretched only because our world, except three dimensions of space, has fourth dimension - time flowing constantly and evenly. We can visually see it on a model of the universe - on a toy-balloon. Except two dimensions of the surface, that is to say two dimensions of space, the balloon has the third dimension - a line passing through each point of the surface and the center of the ball, from there the inflating was begun. In the theory of relativity this line is known as «the Axis of Time» or «the World Line» of the given point of space2. Each point of a surface of a balloon, that is to say each point of two-dimensional space is moving along its «World Line» AT THE SPEED OF LIGHT when the balloon is being inflated. Because according to the theory of relativity, factor of proportionality between change of coordinate of a point on its «world line» and time is speed of light. But in the theory of relativity it is supposed that the world lines of all points of space are parallel, however according my model, all of them come from one center, that now is (on model) already very far - on my calculation, at a distance of 14 billions light years. And this is the radius of the universe model - of our balloon. It corresponds to age of the universe of 14 billions years. 

FOOTNOTE 2: In the Theory of relativity the line in the fourth dimension, that (the line) is perpendicular to the three dimensions of space and drawn through any its point, is named as «the Axis of Time» or «the World Line» of this point. 

Thus: 1. From the Special Theory of relativity (STR) the principle (model) of the «world line» is known. Simultaneously it is a stationary model of a flat world (space-time): 



Fig.2 depicts the same that Fig.1, but on Fig.2 the plane representing the universe, is visible «from an edge». On these figures a point «Ŕ», in that we are, is being moved AT A SPEED OF LIGHT from the past to the future along its world line (vertical line). Similarly all other points of the plane (represented on Fig.2 as an horizontal line) are being moved together with all - on a model two-dimensional - planets, stars and galaxies, that are taking place in this infinite plane representing our three-dimensional space and extending unlimitedly along all the directions. Still two planes represented on Fig.2 as two trait horizontal lines are the same space of the universe but not at the present moment but at any moments of the past and of the future. At this Newton’s model the time has not the beginning in the past also the end in the future as it is visible on the figures.

2. This already known model of a «world line» (afore-cited Fig.2 is taken from the section «The Theory of relativity» of the textbook «General Physics» for universities) I connect with also known - from the General Theory of relativity (GTR) - principle of curvature and circularity of space of the universe. I as though am curving a horizontal line on Fig.2, closing up its ends and I receive a model of already non-stationary universe that permanently is expanding. The next Fig.3 is not present in the textbooks, it contains my opening, that has the important consequences.


In Fig.3 a point «Ŕ» (where we are) again is moving AT SPEED OF LIGHT from the past to the future ALONG ITS WORLD LINE, hence the sphere is being stretched - its radius grows AT SPEED OF LIGHT. Thus two spherical surfaces represented by trait lines are also the same space of our universe but - what it was some billions years ago and what it will some billions years after the present moment. Radius ÎŔ of the model today equal, according my calculations, 14 billions light years.

At this model the space is unrestricted (that is to say has not borders) but finite in its sizes. Therefore the amount of galaxies in space is finite too. Time has no end in the future, but has the beginning in the past - the point «O». We, as well as in case of the previous, Newton's model, are in the point «A» and with all our space and with all galaxies we are being relocated along a dimension of time from the past to the future. Thus the space in this model is being expanded - a membrane of a balloon (as the balloon is being inflated) is being stretched, and the galaxies are scattering. George Lemetre had offered the similar model at 1927 but he had not identified the growing radius of curvature of space with the time - with the fourth dimension of space-time.


was meeting such an objection: the universe will not expand always, the radius of the universe (radius of the balloon) is a projection of radius of one more circumference. Therefore radius of the universe will being grown up to a certain size, then will begin to be decreased up to zero, then again will begin to grow, such the oscillating universe exists eternally. But if it would be truth, this new mysterious circumference should be in some fifth dimension. All this is just fantasy. What is the physical sense of this fifth dimension? There is not such!

Chapter 4. What follows from my model.

Let's consider more in detail the offered by me spherical model of the universe expanding at a speed of light - not under its own momentum, as it is accepted to think today but owing to the time flowing:


Fig.4 shows the spherical surface (at a vertical section), that is a two-dimensional model of our three-dimensional space. If the space of the universe would not being expanded, at radius of the sphere equal in my model nowadays 14 billions light years, a ray of light has gone round the universe and has returned to the initial point from the opposite direction in 2 ő 3,14 ő 14 = 88 billions years. At similar stationary model of Einstein - who didn’t recognize at first the expansion of the universe - this figure was equal 70 billions years. 

R - is a radius of curvature of space (two-dimensional - on a model or three-dimensional - in nature), the same radius is the third coordinate - on a model, but in a nature it is the fourth coordinate of the four-dimensional world of Minkovsky, R = ő4 = ict. There i is imaginary one, i is a square root of minus one; c is the speed of light, c=300.000 km/s.; t is time in any point of the universe, that is to say the time having flowed since the moment «zero» - the moment of the beginning of the universe expansion, the moment of so-called «Big Bang». The sphere is being stretched, the radius is being lengthened at a speed of light. We are in a point «Ŕ». A dashed line, that is passing through a point «Ŕ» from a point «Î», is the fourth dimension corresponding to time, ő4=ict, named also by «an Axis of Time» or by «a World Line» of the point «Ŕ». At each moment of existence of any point of the universe a coordinate of this point on its «axis of time» ő4 equal a radius of the sphere, ő4=R. A distance Rbn on a surface of the sphere from a point «Ŕ» up to points «Â» and «Ń» (that move off from a point «A» at a speed of light) is a radius of the space visible by us, Rbn=R. From galaxies located from us farther than points «Â» and «Ń», light up to us never will reach, as these galaxies move off from us faster than the speed of light, move off owing to expansion of space but not because of galaxies motion under its own momentum. But even the galaxies, that are taking place to us little bit closer of points «Â» and «Ń», we see by not such what they are now - similar to our galaxy but we see them what they were many billions years ago - as so-called «quasars».

Let's calculate distance from us up to points «Â» and «Ń», using the Habble's law, that says that the speed of removal of galaxies from us (V) is proportional to the distance (r) from these galaxies up to us, V = Hr, Í is a factor of proportionality named «a Habble's constant». It is known, for example, that one of galaxies in a constellation «the Eagle» leaves from us at a speed V = 2650 km/s., also it is on a distance r = 125 millions light years from us. By making a proportion, we have got: if some other galaxy (taking place in a point «B» or «C») leaves from us with a speed of light 300.000 km/s., hence it is in a distance of 14 billions light years from us. This distance is the radius Rbn of the space, that we can see. Also the time of existence of the universe comes out 14 billions years.

Thus the space of the universe «at the given moment», that is to say all its points is (on the model, fig.4) the SURFACE of the balloon - a sphere, though it is deformed by holes and funnels (the points «D» and «E»), about these will be told below. All points of space are named as «the events» because these are not only the points of space but the points of space-time too. Also all the points of a VOLUME inside the ball are «all events of the past». The unlimited volume OUTSIDE OF the balloon is «the events of the future». Light reaches us from the galaxies, distances up to which (along a surface of a ball) are no more than Rbn = 14 billions light years. This «horizon», that we see (the points «B» and «C»), is moving off from us at a speed of light because the balloon is being inflated. But also the galaxies located on this horizon is moving off from us at a speed of light too. Therefore we can always see only the same, limited by this horizon, area of the universe. Besides, the removed objects we see with delay, that equal to time of passing of light from them to us. The rest, approximately 5 times larger part of the universe (behind the points «B» and «C») always remain unobservable for us. But according to the Cosmological Principle - the principle of large-scale uniformity and isotropy of the universe - in inaccessible to our supervision the 5/6 of the universe everything is, in general, the same as in the part visible by us.

Thus the circumference of the «balloon», that is to say a circumference of the universe comes out equal 2ďR = 88 billions light years. Interestingly - and it is not present in the textbooks, it follows from my model - that galaxies which are taking place in the part of universe inaccessible to our supervision, that is to say on distance from 14 up to 44 billions light years from us, is moving from us at speeds larger than speed of light. It is possible, as these speeds of scattering are seeming and are not the speeds of motion of matter in space. Actually all galaxies remain on the same places, and its «scattering» turns out because the space of the universe is being expanded, is being inflated as a membrane of a being inflated balloon is stretching. The distances in the universe between the material objects gravitating each to other are now increasing for one year just by 1/14,000,000,000. Hence the amount of gravitation energy (potential energy) in the universe grows too. But any «braking of speed» of seeming scattering of galaxies and of reduction its kinetic energy cannot be. It means, the energy is being born continuously! Really only about a quarter (1-3.14/4) of the specified amount of potential energy is being born, but 3.14 / 4 of this amount is arising at the expense of reduction of kinetic energy of rotation of all galaxies, stars, planets etc. around the center of mass of the appropriate systems with expansion of their orbits owing to expansion of space. At an initial period of existence of the universe the energy and matter were being born in huge amounts. But also now the energy arising thus inside the Sun3 equal, according to my calculations, not less than 1/1200 of its radiation. Thus, according to my model, the fundamental Law of Concervation of energy is not absolute: the expansion of space increases the amount of energy in the universe.

FOOTNOE 3: The arisen thus additional potential energy of gravitation between atoms inside the stars, planets and us with you at once is being converted into thermal energy. Therefore the bodies are not expanding.

As according to the Cosmological Principle, in the homogeneous and isotropic universe all the points of space have the equal standing (on the model this space is a two-dimensional surface of a sphere), the point «Î», from which space began to expand, should be nowhere in the present space but should be equally removed from all the points of present space. Certainly, such removal can be only in the fourth dimension. But not a flat space and not a saddle-looking space corresponds to this condition - but just and only a spherical model of space of the universe - a model of a being stretched surface of a toy-balloon.

At such a model, as well as in the Newton's model, space is boundless also it is almost flat, that is to say its curvature k=1/R is very small because the radius of curvature R is huge, R = 14 billions light years. And at the same time space in this model is closed, is finite in its volume - this corresponds to the Einstein's model.

My model also settles «the photometric paradox», that is to say it answers a question, why, with finality and circularity of space, the sky is black, instead of to be filled with continuous light of stars: in my model the light cannot have gone round the universe because the radius R of a spherical space of the universe is being increased at speed of light, but length of the circumference 2ďR of space of the universe, hence, is being increased at the speed that equal 6,28 speed of light.

Now let's look how the picture of spherical space (Fig.4) is being deformed under influence of concentrated mass of substance. In a point «D» the space had been sagged under influence of a concentrated mass (for example, of the mass of a star) because the radius RD is reduced because of delay of time (tD < t). More visually the situation in a point «D» is shown on Fig.5 and Fig.6


A light ball is rolling down («is being gravitated») to a heavy one.

The planet is being rotated around a star owing to curvature of space.
The following citations can be an explanation to Fig.5 and Fig.6: 
«According to the General Theory of relativity the speed of flowing of time depends from the force of gravitation. Now it is checked up experimentally»

                                                                             ( Academician J. Zeldovich)

      «According to the gravitation theory by Einstein, the true gravitational field is not anything else 
      but the evincing of curvature (of difference the geometry from the Euclid's one) of the 
      four-dimensional space-time».

                                                           ( I. D. Novikov «Evolution of the Universe», page 74)

      «The Principle of proportionality of inert and heavy masses logically results in the statement that 
      the fall of bodies in a gravitational field can be considered as their free movement along geodetic (extreme) line in the curved four-dimensional space-time. Let's consider such an example. Let's 
      imagine a horizontally stretched rubber film (membrane), on which the Cartesian grid of 
      coordinates is marked. Here an light ball, having initial speed V, will move evenly and 
      rectilinearly (the geodetic line is here a direct one). If to put a heavy body over the membrane 
      (see Fig.5), under action of its weight the membrane will sag. In this curved two-dimensional 
      world the geometry any more will not be Euclid's one. The ball, moving with initial speed V
      along a membrane, will be rolling down («will being gravitated») to a heavy body». 
      «Here pertinently to recollect an aphorism of American physicist A. Weeler that in the General 
      Theory of relativity «substance tells to space how to be bent, also the space tells to the 
      substance how to move».

                                                                                             (I. A. Klimishin, page 115)

Thus in my model (Fig.4, a point «D») we can see visually that space-time is bent just because of time delay created by gravitation (RD< R, as RD= ictD, R = ict, but tD< t).

Now from a point «D» we pass to a point «E» (Fig.4). In a point «E» at a certain moment t1 (several billions years back - judging by the scale of Fig.4) in small volume so large mass had concentrated that a «black hole» had arisen. Time here in the moment t1 had stopped in general, from the point of view of the external observer. But such «the external observer», as a matter of fact, it is all the universe. After us in the universe will have elapsed some more 10 billions years, the universe will have been even more expanded (a radius of the sphere R will been increased 1,5 times), stars will go out etc., but in the black hole (in a point «E») during several billions already gone years and for subsequent 10 billions years nothing has occurred and will not have occurred, and in particular, radius RE will remain constant, what it was in at a moment t1 and what it is now. But after the moment t1 as the sphere radius R (see Fig.4) is increasing at a speed of light, a funnel of curved space in a point «E» all time is going deeper, that is to say the black hole actually is leaving from the External Observer at a speed of light! It visually shows why light cannot go out from a black hole, why fall into a black hole is never being finished and why, with approach(approximation) to a black hole, the linear size in a direction to it is being stretched (from a point of view of the external observer).

I emphasize that the shown above visual picture of black holes would not turned out, if a radius of the «balloon» was being increased at speed not equal the speed of light, as it is imagined by the supporters of the generally accepted theory.

As it is visible from Fig.4 and Fig.7,


the space near to a black hole is not flat both not spherical any more and not slightly curved, as in a point «D» but hyperbolic - a funnel. Time near to a point «E» is slowed-up, and it is more slowed-up, than closer to the black hole (because, the closer to a black hole, the stronger a field of its gravitation is). The vector of time (radius RE) in a point «E» is constant (time has been stopped), and in vicinities of a black hole, for example, in a point «F», the vector RF though grows, but slower, than radius of sphere R (time is slowed-up).

At Fig.4 the funnel of a black hole is, certainly, exaggerated, too wide.If to draw in scale, the funnel would be turned out many millions times thinner than a human hair. Therefore, though there are a lot of black holes but on a huge body of the universe these are just rare most thin punctures. Can be, the sole large black hole, nearest to us - that is located at the center of our Galaxy.

Here some citations are for the explanation the situation in a point «E» and near it:

«According to the General Theory of relativity the speed of flowing of time depends from the force of gravitation. Now it is checked up experimentally. At the collapsing super-dense stars («black holes») the effect of delay of time reaches 100 percents! In result these stars for the external observer «freeze» at the certain compression stage, that is not balanced, - it is alike a jumper on a photo as though he has hung in air. For us the star has been fixed in that condition, in which at its compression it was caught by the stopping of time».

                                         (Academician J. Zeldovich, «Science and Life» No 4, 1973)

      «From the point of view Of the External Observer, the process of fall of the Traveller into a 
      black hole is being prolonged up to infinity. The Schwarzschild's sphere (a border of a black 
      hole) is named also as horizon of events. A traveller, who has crossed a Schwarzschild's 
      sphere, already by any way can not establish contacts with the far Observer. The truth, it 
      will seem to him (if to consider he stayed alive in a monstrous field of gravitation, where 
      even the protons and neutrons cease to exist as such - N. K.), that the signals, sent by 
      him, are spreading normally. But thus R-area (all universe being outside a black hole) will 
      be moving off from him at speed of light, hence, the light signals of the Traveller never will 
      achieve the border T- and R-area».

                                                    (I. A. Klimishin «The Relativist Astronomy», page 153, 157)

      «The Schwarzschild's sphere (a border of a black hole) has a number of interesting 
      properties - it brightly demonstrates a relativity of concept of time, any material object, 
      including light, cannot escape its surface to outside. A Kerre's black hole (being rotated), 
      besides Schwarzschild's sphere has a so-called limit of static character, that is placed 
      over this sphere. Physically the presence of a limit of static character means that to remain 
      in immobility, by reaching a limit of static character, it is possible just by having speed of light». 

                                        (V. Lebedev, Ph.D. «Knowledge is Power» monthly, No 10, 1983)

For a conclusion there are just two remarks about the situation probably causing bewilderment - the situation near to a point of the Beginning, that is to say at time, almost equal to zero.

There can be a question: why at the initial period of existence of the universe, when huge density of matter, the gravitation had not compressed all the matter into a black hole but on the contrary, the matter evenly had been expanded forming the homogeneous universe?

Because, according to the existing theory of «the Hot Universe» (theory of «Big Bang»), the particles of heavy weight (protons) were formed of rigid radiation only by the end of the first second of existence of the universe. But because of extremely high temperature the protons were distributed in space completely evenly. . The local condensations of matter could be formed much later, in process of general expansion and cooling of the matter. These condensations subsequently had turned into the galaxies. At the centers of galaxies the presence of black holes is supposed.

The variety of possible intermediate steps and variants of a condensation of matter generates all the Mendeleyev's table of elements and further - at the molecular and corporal level - all variety of forms of substance, including alive forms, though the final form of the condensation is, obviously, a black hole. Thus the flowing of time that had caused expansion of space - as though thus «had wound up a spring», appears as the initial engine of all processes in the universe. Space continues to expand - it means «the spring is being wound up further», that is to say the potential energy of the substance particles, that are moving off each from other but are gravitating each to other, is increasing. As according to the Einstein's formula E = mc2 the energy is equivalent to a certain mass, it is turned out that there is the creation not only of energy but of matter too. From my model follows, that the distance between the gravitating particles of substance for one year is being increased now only by 1/14,000,000,000. It would seem, the gain of energy (or matter) is slight. However a simple calculation gives such the figures: at age of the universe of 10 billions years the gain of gravitation energy per one year equal almost 1/40,000,000,000, at age of 1 billion years it equal 1/4,000,000,000 per one year, that is to say 10 time more. At age of 100 millions years it equal even 10 time more etc. And at the first instants of existence of the universe the energy - also, hence, matter - were born in huge amounts. Whether by such the way had the matter arisen in general? Because it is more reasonable to assume that at «zero» moment the matter (energy) only beginnings to arise and now continues to arise - than to agree with the generally accepted theory, that though space and time have been begun from zero, but all the matter of the universe ostensibly existed from the very beginning, and then just was extending and was cooling down.

I will notice also that the delay of time, according to the Theory of relativity, is conceivable only about of «the External Observer». But if the universe is considered as a whole, in relation to it there is no «the External Observer». Therefore it is impossible to speak about delay of time wholly in all universe, - for example, when it means the initial moment, moment of huge density of matter in the universe.

I want also to emphasize a difference of my model from the theory of an astronomer N. A. Kozyrev, who considered that «the rotation of interacting bodies results in occurrence of additional forces, and consequently, also of additional energy». I assert something another: that the additional potential energy is being generated by expansion of space of the universe, and the expansion is a consequence of flowing of time.

Why my model of the universe had not been offered earlier by somebody else? A physicist D. I. Kazakov wrote: «Owing to power of mathematics the physicists look where the imagination does not penetrate but after this on a place of mathematical abstraction the human thought generates a visual image again». But the point is that the supporters of the generally accepted theory of «Big Bang» who were charmed by the algebraic equations4 (FOOTNOTE 4: About so-called «Piphagor’s syndrome» see an article by R. A. Aronov «The Theater of Absurdity: is it necessary to the modern physics?» («Questions of Philosophy Monthly» No 12, 1997), have neglected the more visual geometry and even have been distracted from the physical reality. Because the four-dimensional world of Minkovsky with which they operate is a mathematical design, its four dimensions have equal standing but it is incorrect physically. In the real physical world the fourth dimension - time has the nature, that is distinct from of nature of three dimensions of space. Though it is possible at a pinch to speak about «moving» of the particles and galaxies in four-dimensional space-time but the vectors of their speeds, accelerations and forces, braking or accelerating, can be located just and only in the three-dimensional space. Unequal standing of the dimensions of space and the dimension of time becomes apparent even according to the generally accepted theory: at the large-scale geometry of the physical four-dimensional world each of three dimensions of space is closed on itself as a ring, and the ring expands from zero at flowing of time but the dimension of time is not closed and has a point of the beginning - the point of so-called «Big Bang», that is to say the point of a beginning of existence not only of space but also of time, and of matter.