Paradoxes of the Universe: mass continues to surprise physicists. The Axis of the Worlds or the Paradoxes of the Universe The Paradox of the Flying Arrow

Paradoxes can be found everywhere, from ecology to geometry and from logic to chemistry. Even the computer on which you are reading the article is full of paradoxes. Here are ten explanations of curious paradoxes. Some of them are so strange that it is difficult to immediately understand what the point is...

1. Banach-Tarski paradox

Imagine that you are holding a ball in your hands. Now imagine that you start tearing this ball into pieces, and the pieces can be any shape you like. Then put the pieces together so that you get two balls instead of one. How big will these balls be compared to the original ball?
According to set theory, the two resulting balls will be the same size and shape as the original ball. In addition, if we take into account that the balls have different volumes, then any of the balls can be transformed in accordance with the other. This suggests that a pea can be divided into balls the size of the Sun.
The trick to the paradox is that you can break the balls into pieces of any shape. In practice, this is impossible to do - the structure of the material and, ultimately, the size of the atoms impose some restrictions.
In order for it to be truly possible to break the ball the way you like, it must contain an infinite number of available zero-dimensional points. Then the ball of such points will be infinitely dense, and when you break it, the shapes of the pieces can turn out to be so complex that they will not have a certain volume. And you can assemble these pieces, each containing an infinite number of points, into a new ball of any size. The new ball will still be made of infinite points, and both balls will be equally infinitely dense.
If you try to put the idea into practice, nothing will work. But everything works out wonderfully when working with mathematical spheres - infinitely divisible numerical sets in three-dimensional space. The solved paradox is called the Banach-Tarski theorem and plays a huge role in mathematical set theory.

2. Peto's paradox

Obviously, whales are much larger than us, which means they have many more cells in their bodies. And every cell in the body can theoretically become malignant. Therefore, whales are much more likely to get cancer than humans, right?
Not this way. Peto's Paradox, named after Oxford professor Richard Peto, states that there is no correlation between animal size and cancer. Humans and whales have about the same chance of getting cancer, but some breeds of tiny mice have a much higher chance.
Some biologists believe that the lack of correlation in Peto's paradox can be explained by the fact that larger animals are better able to resist tumors: a mechanism that works to prevent cells from mutating during the process of division.

3. The problem of the present time

For something to physically exist, it must be present in our world for some time. There cannot be an object without length, width and height, and there cannot be an object without “duration” - an “instant” object, that is, one that does not exist for at least some amount of time, does not exist at all.
According to universal nihilism, the past and future do not occupy time in the present. Moreover, it is impossible to quantify the duration that we call "present time": any amount of time that you call "present time" can be divided into parts - past, present and future.
If the present lasts, say, a second, then this second can be divided into three parts: the first part will be the past, the second - the present, the third - the future. The third of a second that we now call the present can also be divided into three parts. Surely you already understand the idea - you can continue like this endlessly.
Thus, the present does not really exist because it does not continue through time. Universal nihilism uses this argument to prove that nothing exists at all.

4. Moravec's paradox

What is the chance that a random number will start with the number "1"? Or from the number "3"? Or with "7"? If you know a little about probability theory, you can guess that the probability is one in nine, or about 11%.
If you look at the actual numbers, you will notice that “9” occurs much less frequently than in 11% of cases. Also, far fewer numbers than expected start with “8,” but a whopping 30% of numbers start with “1.” This paradoxical pattern plays out in all sorts of real-life cases, from population size to stock prices to the length of rivers.
Physicist Frank Benford first noted this phenomenon in 1938. He found that the frequency of a digit appearing first fell as the digit increased from one to nine. That is, "1" appears as the first digit about 30.1% of the time, "2" appears about 17.6% of the time, "3" appears about 12.5% of the time, and so on until "9" appears in as the first digit in only 4.6% of cases.
To understand this, imagine that you are numbering lottery tickets sequentially. When you number your tickets from one to nine, there is an 11.1% chance of any number being number one. When you add ticket number 10, the chance of a random number starting with "1" increases to 18.2%. You add tickets #11 through #19, and the chance of a ticket number starting with "1" continues to increase, reaching a maximum of 58%. Now you add ticket number 20 and continue numbering the tickets. The chance of a number starting with a "2" increases, and the chance of a number starting with a "1" slowly decreases.
Benford's law does not apply to all cases of number distribution. For example, sets of numbers whose range is limited (human height or weight) are not covered by the law. It also doesn't work with sets that only have one or two orders.
However, the law applies to many types of data. As a result, authorities can use the law to detect fraud: when the information provided does not follow Benford's Law, authorities can conclude that someone has fabricated the data.

6. C-paradox

Genes contain all the information necessary for the creation and survival of an organism. It goes without saying that complex organisms should have the most complex genomes, but this is not true.
Single-celled amoebas have genomes 100 times larger than those of humans; in fact, they have perhaps the largest genomes known. And in species that are very similar to each other, the genome can differ radically. This oddity is known as the C-paradox.
An interesting conclusion from the C-paradox is that the genome may be larger than necessary. If all the genomes in human DNA were used, the number of mutations per generation would be incredibly high.
The genomes of many complex animals like humans and primates include DNA that codes for nothing. This huge amount of unused DNA, varying greatly from creature to creature, seems to depend on nothing, which is what creates the C-paradox.

7. Immortal ant on a rope

Imagine an ant crawling along a rubber rope one meter long at a speed of one centimeter per second. Also imagine that the rope stretches one kilometer every second. Will the ant ever reach the end?
It seems logical that a normal ant is not capable of this, because the speed of its movement is much lower than the speed at which the rope stretches. However, the ant will eventually reach the opposite end.
When the ant has not even begun to move, 100% of the rope lies in front of it. After a second, the rope became much larger, but the ant also walked some distance, and if you count it as a percentage, the distance it must travel has decreased - it is already less than 100%, albeit not by much.
Although the rope constantly stretches, the small distance traveled by the ant also becomes greater. And, although overall the rope lengthens at a constant rate, the ant's path becomes a little shorter every second. The ant also continues to move forward at a constant speed all the time. Thus, with every second the distance he has already covered increases, and the distance he must travel decreases. As a percentage, of course.
There is one condition for the problem to have a solution: the ant must be immortal. So, the ant will reach the end in 2.8×1043.429 seconds, which is slightly longer than the existence of the Universe.

In cosmology, the question of the finitude or infinity of the Universe is of great importance:

if the Universe is finite, then, as Friedman showed, it cannot be in a stationary state and must either expand or contract;
if the Universe is infinite, then any assumptions about its compression or expansion lose any meaning.

It is known that the so-called cosmological paradoxes were put forward as objections to the possibility of the existence of an infinite Universe, infinite in the sense that neither its size, nor the time of existence, nor the mass of the matter contained in it can be expressed by any, no matter how large, numbers. Let's see how justified these objections turn out to be.

Cosmological paradoxes - essence and research

It is known that the main objections to the possibility of the existence of a Universe infinite in time and space are as follows.

1. “In 1744, the Swiss astronomer J.F. Shezo was the first to doubt the correctness of the idea of an infinite Universe: if the number of stars in the Universe is infinite, then why doesn’t the whole sky sparkle like the surface of a single star? Why is the sky dark? Why are the stars separated by dark spaces? . It is believed that the same objection to the model of an infinite Universe was put forward by the German philosopher G. Olbers in 1823. “Albers’s counter-argument was that the light coming to us from distant stars should be weakened due to absorption in the matter in its path. But in this case, this substance itself should heat up and glow brightly, like stars.” . However, this is how it really is! According to modern ideas, vacuum is not “nothing”, but is “something” that has very real physical properties. Then why not assume that light interacts with this “something” in such a way that each photon of light, when moving in this “something,” loses energy in proportion to the distance it travels, as a result of which the photon’s radiation shifts to the red part of the spectrum. Naturally, the absorption of photon energy by vacuum is accompanied by an increase in the temperature of the vacuum, as a result of which the vacuum becomes a source of secondary radiation, which can be called background radiation. When the distance from the Earth to a emitting object - a star, a galaxy - reaches a certain limiting value, the radiation from this object receives such a large red shift that it merges with the background vacuum radiation. Therefore, although the number of stars in the infinite Universe is infinite, the number of stars observed from the Earth, and in general from any point in the Universe, is finite - at any point in space the observer sees himself as if in the center of the Universe, from which a certain limited number of stars (galaxies) are observed. At the same time, at the frequency of background radiation, the entire sky sparkles like the surface of a single star, which is actually observed.

2. In 1850, the German physicist R. Clausius “... came to the conclusion that in nature heat passes from a warm body to a cold one... the state of the Universe should increasingly change in a certain direction... These ideas were developed by the English physicist William Thomson, according to which all physical processes in the Universe are accompanied by the conversion of light energy into heat." Consequently, the Universe faces “thermal death”, so the endless existence of the Universe in time is impossible. In reality, this is not the case. According to modern concepts, matter is converted into “light energy” and “heat” as a result of thermonuclear processes occurring in stars. “Thermal death” will occur as soon as all the matter in the Universe “burns up” in thermonuclear reactions. Obviously, in an infinite Universe, the reserves of matter are also infinite, therefore, all the matter of the Universe will “burn out” in an infinitely long time. “Heat death” threatens rather the finite Universe, since the reserves of matter in it are limited. However, even in the case of a finite Universe, its “heat death” is not obligatory. Newton also said something like this: “Nature loves transformations. Why shouldn’t there be some in a series of different transformations in which matter turns into light, and light into matter?” Currently, such transformations are well known: on the one hand, matter turns into light as a result of thermonuclear reactions, on the other hand, photons, i.e. Light, under certain conditions, turns into two completely material particles - an electron and a positron. Thus, in nature there is a circulation of matter and energy, which excludes the “heat death” of the Universe.

3. In 1895, the German astronomer H. Seeliger “... came to the conclusion that the idea of an infinite space filled with matter at a finite density is incompatible with Newton’s law of gravitation... If in an infinite space the density of matter is not infinitesimal, and every two particles, according to Newton’s law, are mutually attracted, then the force of gravity acting on any body would be infinitely large, and under its influence the bodies would receive an infinitely large acceleration.”

As explained, for example, by I.D. Novikov, the essence of the gravitational paradox is as follows. “Let the Universe be, on average, uniformly filled with celestial bodies, so that the average density of matter in very large volumes of space is the same. Let's try to calculate, in accordance with Newton's law, what gravitational force caused by all the infinite matter of the Universe acts on a body (for example, a galaxy) placed at an arbitrary point in space. Let's first assume that the Universe is empty. Let us place a test body at an arbitrary point in space A. Let us surround this body with a substance of density that fills a ball of radius R to the body A was in the center of the ball. It is clear without any calculations that, due to symmetry, the gravitation of all particles of matter of the ball in its center balances each other, and the resulting force is zero, i.e. on the body A no force is applied. We will now add more and more spherical layers of matter of the same density to the ball... spherical layers of matter do not create gravitational forces in the internal cavity and adding these layers does not change anything, i.e. still the resultant gravitational force for A equal to zero. Continuing the process of adding layers, we ultimately arrive at an infinite Universe, uniformly filled with matter, in which the resulting gravitational force acting on A, is equal to zero.

However, the reasoning can be carried out differently. Let us again take a uniform ball of radius R in an empty universe. Let's place our body not in the center of this ball with the same density of matter as before, but on its edge. Now the force of gravity that acts on the body A, will be equal according to Newton's law

F = GMm/R 2 ,

Where M– mass of the ball; m– mass of the test body A.

We will now add spherical layers of matter to the ball. Once a spherical shell is added to this ball, it will not add any gravitational forces within itself. Therefore, the gravitational force acting on the body A, will not change and is still equal F.

Let's continue the process of adding spherical shells of matter of the same density. Force F remains unchanged. In the limit, we again get a Universe filled with homogeneous matter with the same density. However, now on the body A force acts F. Obviously, depending on the choice of the initial ball, one can obtain the force F after the transition to a Universe uniformly filled with matter. This ambiguity is called the gravitational paradox... Newton’s theory does not make it possible to unambiguously calculate gravitational forces in an infinite Universe without additional assumptions. Only Einstein’s theory allows us to calculate these forces without any contradictions.”

The contradictions, however, immediately disappear if we remember that an infinite Universe is not the same as a very large one:

in an infinite Universe, no matter how many layers of matter we add to the ball, an infinitely large amount of matter remains outside it;
in the infinite Universe, a ball of any, no matter how large, radius with a test body on its surface can always be surrounded by a sphere of an even larger radius in such a way that both the ball and the test body on its surface will be inside this new sphere filled with matter of the same density, as inside the ball; in this case, the magnitude of the gravitational forces acting on the test body from the side of the ball will be equal to zero.

Thus, no matter how much we increase the radius of the ball and no matter how many layers of matter we add, in an infinite Universe uniformly filled with matter, the magnitude of the gravitational forces acting on the test body will always be equal to zero. In other words, the magnitude of the gravitational forces created by all matter in the Universe is zero at any point. However, if there is no substance outside the sphere on the surface of which the test body lies, i.e. if all the matter of the Universe is concentrated inside this ball, then a gravitational force proportional to the mass of the matter contained in the ball acts on a test body lying on the surface of this body. Under the influence of this force, the test body, and in general all the outer layers of the ball’s matter, will be attracted to its center - a ball of finite dimensions, uniformly filled with matter, will inevitably compress under the influence of gravitational forces. This conclusion follows both from Newton’s law of universal gravitation and from Einstein’s general theory of relativity: A Universe of finite dimensions cannot exist, since under the influence of gravitational forces its matter must continuously contract towards the center of the Universe.

“Newton understood that, according to his theory of gravity, stars should be attracted to each other and therefore, it would seem... should fall on each other, approaching at some point... Newton said that So(hereinafter it is emphasized by me - V.P.) really there should have been if we only had final number of stars in ultimate areas of space. But... if the number of stars endlessly and they are more or less evenly distributed across infinite space, then this never will not happen, since there is no central point where they need to fall. These arguments are an example of how easy it is to get into trouble when talking about infinity. In an infinite Universe, any point can be considered the center, since on both sides of it the number of stars is infinite. (Then you can - V.P.) ... take a finite system in which all the stars fall on each other, tending to the center, and see what changes will happen if you add more and more stars, distributed approximately evenly outside the region under consideration. No matter how many stars we add, they will always tend to the center." Thus, in order not to “get into trouble,” we must select a certain finite region from the infinite Universe, make sure that in such a finite region the stars will fall towards the center of this region, and then extend this conclusion to the infinite Universe and declare, that the existence of such a Universe is impossible. Here is an example of how “... to the universe as a whole...” is transferred “... as something absolute, such a state... to which... only part of matter can be subject” (F. Engels. Anti-Dühring), for example, a single star or a cluster of stars. In fact, since “in an infinite Universe any point can be considered a center,” the number of such points is infinite. In the direction of which of this infinite number of points will the stars move? And one more thing: even if such a point is suddenly discovered, then an infinite number of stars will move in the direction of this point for an infinite time and the compression of the entire infinite Universe at this point will also occur in an infinite time, i.e. never. It's a different matter if the Universe is finite. In such a Universe, there is a single point, which is the center of the Universe - this is the point from which the expansion of the Universe began and in which all the matter of the Universe will again concentrate when its expansion is replaced by compression. Thus, it is the finite Universe, i.e. The Universe, the dimensions of which at each moment of time and the amount of matter concentrated in it can be expressed by some finite numbers, is doomed to contraction. Being in a state of compression, the Universe will never be able to exit this state without some kind of external influence. Since, however, there is no matter, no space, no time outside the Universe, the only reason for the expansion of the Universe can be the action expressed in the words “Let there be light!” As F. Engels once wrote, “We can twist and turn as we please, but... .. we return again every time... to the finger of God” (F. Engels. Anti-Dühring). However, the finger of God cannot be the subject of scientific study.

Conclusion

Analysis of the so-called cosmological paradoxes allows us to conclude the following.

1. World space is not empty, but is filled with some medium, whether we call this medium ether or physical vacuum. When moving in this medium, photons lose energy in proportion to the distance they travel and the distance they travel, as a result of which the photon emission shifts to the red part of the spectrum. As a result of interaction with photons, the temperature of the vacuum or ether rises several degrees above absolute zero, as a result of which the vacuum becomes a source of secondary radiation corresponding to its absolute temperature, which is actually observed. At the frequency of this radiation, which is indeed the background radiation of the vacuum, the entire sky turns out to be equally bright, as J.F. assumed. Shezo.

2. Contrary to the assumption of R. Clausius, “heat death” does not threaten the infinite Universe, which includes an infinite amount of matter that can turn into heat in an infinitely long time, i.e. never. “Heat death” threatens a finite universe containing a finite amount of matter that can be converted into heat in a finite time. That is why the existence of a finite Universe turns out to be impossible.

3. In an infinite Universe, the dimensions of which cannot be expressed by any, no matter how large a number, uniformly filled with matter at a non-zero density, the magnitude of the gravitational forces acting at any point in the Universe is equal to zero - this is the true gravitational paradox of the infinite Universe. The equality of gravitational forces to zero at any point in an infinite Universe, uniformly filled with matter, means that the space in such a Universe is Euclidean everywhere.

In the finite Universe, i.e. in the Universe, the dimensions of which can be expressed by some, albeit very large numbers, a test body located “at the edge” of the Universe is subject to an attractive force proportional to the mass of the matter contained in it, as a result of which this body will tend to the center of the Universe - the finite Universe, the matter of which is uniformly distributed throughout its limited volume, is doomed to compression, which will never give way to expansion without some external influence.

Thus, all objections or paradoxes that are believed to be directed against the possibility of the existence of a Universe infinite in time and space are actually directed against the possibility of the existence of a finite Universe. In reality, the Universe is infinite in both space and time; infinite in the sense that neither the size of the Universe, nor the amount of matter contained in it, nor its lifetime can be expressed by any, no matter how large numbers - infinity, it is infinity. The Infinite Universe never arose either as the result of a sudden and inexplicable expansion and further development of some “pre-material” object, nor as a result of Divine creation.

It must be assumed, however, that the above arguments will seem completely unconvincing to supporters of the Big Bang theory. According to the famous scientist H. Alfven, “The less scientific evidence there is, the more fanatical the belief in this myth becomes. It seems that in the current intellectual climate the great advantage of Big Bang cosmology is that it is an affront to common sense: credo, quia absurdum (I believe because it is absurd)” (quoted in ). Unfortunately, for some time now, “fanatical faith” in this or that theory has been a tradition: the more evidence of the scientific inconsistency of such theories appears, the more fanatical the belief in their absolute infallibility becomes.

At one time, polemicizing with the famous church reformer Luther, Erasmus of Rotterdam wrote: “Here, I know, some, holding their ears, will certainly shout: “Erasmus dared to fight with Luther!” That is, a fly with an elephant. If anyone wants to attribute this to my feeble-mindedness or ignorance, then I will not argue with him, only even if the weak-minded - even for the sake of learning - are allowed to argue with those whom God has gifted richer... Maybe my opinion is deceiving me ; therefore I want to be an interlocutor, not a judge, an explorer, not a founder; I am ready to learn from everyone who offers something more correct and reliable... If the reader sees that the equipment of my essay is equal to that of the opposite side, then he himself will weigh and judge what is more important: the judgment of all enlightened people..., all universities..., or the private opinion of this or that person... I know that in life it often happens that the larger part defeats the best. I know that when investigating the truth it is never a bad idea to add your diligence to what has been done before.”

With these words we will conclude our brief study.

Information sources:

Klimishin I.A. Relativistic astronomy. M.: Nauka, 1983.
Hawking S. From the big bang to black holes. M.: Mir, 1990.
Novikov I.D. Evolution of the Universe. M.: Nauka, 1983.
Ginzburg V.L. About physics and astrophysics. Articles and speeches. M.: Nauka, 1985.

“The abyss full of stars has opened;
The stars have no number, the bottom of the abyss.”
M. V. Lomonosov, “Evening reflection on God’s Majesty...”

This stanza from the ode of the brilliant Lomonosov has become the most famous of the entire creative heritage of the great scientist, poet and philosopher. The writing of this philosophical work dates back to 1747. Let us only note that since then, scientific thought has not decided on the interpretation of the infinity of the universe, leaving Lomonosov’s hypothesis about the infinity of the universe unproven.

Since that time, a whole branch of fundamental science, cosmology, has emerged. But she is not yet able to answer this eternal question; moreover, the more information we have, the more insoluble paradoxes arise before us. The scientific concept of the infinity of the universe is expressed in the sense that its dimensions, lifetime, mass of matter contained in universal volumes cannot be expressed in finite numerical values of any size. The logic of such an understanding of the infinity of space puts forward the following two incompatible logical conclusions:

According to Friedman's theory, the finite universe cannot be stationary and must either expand or contract;

The concept of expansion or contraction of the universe in the case of its infinity does not make sense, since the expanding infinity is equal to the contracting one, and the emergence of the universe at the moment of the Big Bang from a volume of one neutron with a finite, albeit arbitrarily large mass is refuted by its infinity in terms of mass, size and time of existence.

It is known that the process of expansion of the universe has been proven experimentally based on experiments in measuring distances to the nearest star systems using the radar method. It is also proven that there was a moment when space came into existence as a result of the Big Bang. It is definitely known that time is a vector concept and has no reverse direction. Consequently, until this moment, time did not exist, and as follows from Einstein’s theory, space and time cannot exist without each other. This means that there was a moment when there was no space. For most scientists - cosmologists, this paradox is the basis for asserting the absence of God or other supreme power that was the impetus for the emergence of the universe. And yet, let us leave ourselves with hope of resolving this paradox, since our understanding of cause-and-effect relationships, and therefore all philosophy, suffers from bias. The genius of Lomonosov lies in the fact that in his ode he connected science, philosophy and the divine principle, creating a precedent for foreseeing not even the future itself, but a scientific idea of the universe in this future.

Cosmological paradoxes of the Universe

Cosmological paradoxes— difficulties (contradictions) that arise when extending the laws of physics to the Universe as a whole or to sufficiently large areas of it. The classical picture of the world of the 19th century turned out to be quite vulnerable in the field of cosmology of the Universe, due to the need to explain 3 paradoxes: photometric, thermodynamic and gravitational. You are invited to explain these paradoxes from the point of view of modern science.

Photometric paradox (J. Chezo, 1744; G. Olbers, 1823) boiled down to an explanation of the question “Why is it dark at night?”
If the Universe is infinite, then there are countless stars in it. With a relatively uniform distribution of stars in space, the number of stars located at a given distance increases in proportion to the square of the distance to them. Since the brilliance of a star decreases in proportion to the square of the distance to it, the weakening of the general light of the stars due to their distance should be exactly compensated by the increase in the number of stars, and the entire celestial sphere should glow uniformly and brightly. This contradiction with what is observed in reality is called the photometric paradox.
This paradox was first formulated in its entirety by the Swiss astronomer Jean-Philippe Louis de Chaizeau (1718-1751) in 1744, although similar thoughts were expressed earlier by other scientists, in particular Johannes Kepler, Otto von Guericke and Edmund Halley. The photometric paradox is sometimes called Olbers' paradox, after the astronomer who brought it to attention in the 19th century.
The correct explanation of the photometric paradox was proposed by the famous American writer Edgar Allan Poe in the cosmological poem “Eureka” (1848); a detailed mathematical treatment of this solution was given by William Thomson (Lord Kelvin) in 1901. It is based on the finite age of the Universe. Since (according to modern data) more than 13 billion years ago there were no galaxies and quasars in the Universe, the most distant stars that we can observe are located at distances of 13 billion light years. years. This eliminates the main premise of the photometric paradox - that the stars are located at any, no matter how large, distances from us. The Universe, observed at great distances, is so young that stars have not yet formed in it. Note that this does not in any way contradict the cosmological principle, from which the boundlessness of the Universe follows: it is not the Universe that is limited, but only that part of it where the first stars managed to be born during the arrival of light to us.
The red shift of galaxies also makes some (significantly smaller) contribution to the decrease in the brightness of the night sky. Indeed, distant galaxies have (1+ z) longer radiation wavelength than galaxies at close distances. But the wavelength is related to the energy of light according to the formula ε= hc/λ. Therefore, the energy of photons received by us from distant galaxies is (1+ z) times less. Further, if from a galaxy with redshift z two photons are emitted with a time interval δ t, then the interval between the reception of these two photons on Earth will be another (1+ z) times greater, therefore, the intensity of the received light is the same number of times less. As a result, we get that the total energy coming to us from distant galaxies is (1+ z)² times less than if this galaxy had not moved away from us due to cosmological expansion.

Thermodynamic paradox (Clausius, 1850), is associated with the contradiction of the second law of thermodynamics and the concept of the eternity of the Universe. According to the irreversibility of thermal processes, all bodies in the Universe tend to thermal equilibrium. If the Universe exists for an infinitely long time, then why has thermal equilibrium in nature not yet arrived, and why are thermal processes still continuing?

Gravitational paradox

Mentally select a sphere of radius R 0 so that the cells of inhomogeneity in the distribution of matter inside the sphere are insignificant and the average density is equal to the average density of the Universe r. Let there be a body of mass on the surface of the sphere m, for example, Galaxy. According to Gauss's theorem on a centrally symmetric field, the gravitational force from a substance of mass M, enclosed inside the sphere, will act on the body as if all the matter were concentrated at one point located in the center of the sphere. At the same time, the rest of the matter of the Universe does not make any contribution to this force.

Let's express the mass through the average density r: . Let Then - the acceleration of free fall of a body to the center of the sphere depends only on the radius of the sphere R 0 . Since the radius of the sphere and the position of the center of the sphere are chosen arbitrarily, uncertainty arises in the action of the force on the test mass m and the direction of its movement.

(Neumann-Seliger paradox, named after the German scientists K. Neumann and H. Zeliger, 1895) is based on the provisions of infinity, homogeneity and isotropy of the Universe, has a less obvious character and consists in the fact that Newton’s law of universal gravitation does not give any a reasonable answer to the question about the gravitational field created by an infinite system of masses (unless we make very special assumptions about the nature of the spatial distribution of these masses). For cosmological scales, the answer is given by A. Einstein’s theory, in which the law of universal gravitation is refined for the case of very strong gravitational fields.

In cosmology, the question of the finitude or infinity of the Universe is of great importance:

if the Universe is finite, then, as Friedman showed, it cannot be in a stationary state and must either expand or contract;
if the Universe is infinite, then any assumptions about its compression or expansion lose any meaning.

Cosmological paradoxes - essence and research

It is known that the main objections to the possibility of the existence of a Universe infinite in time and space are as follows.

F = GMm/R 2 ,

Where M– mass of the ball; m– mass of the test body A.

The contradictions, however, immediately disappear if we remember that an infinite Universe is not the same as a very large one:

in an infinite Universe, no matter how many layers of matter we add to the ball, an infinitely large amount of matter remains outside it;
in the infinite Universe, a ball of any, no matter how large, radius with a test body on its surface can always be surrounded by a sphere of an even larger radius in such a way that both the ball and the test body on its surface will be inside this new sphere filled with matter of the same density, as inside the ball; in this case, the magnitude of the gravitational forces acting on the test body from the side of the ball will be equal to zero.

Conclusion

Analysis of the so-called cosmological paradoxes allows us to conclude the following.

With these words we will conclude our brief study.

Information sources:

Klimishin I.A. Relativistic astronomy. M.: Nauka, 1983.
Hawking S. From the big bang to black holes. M.: Mir, 1990.
Novikov I.D. Evolution of the Universe. M.: Nauka, 1983.
Ginzburg V.L. About physics and astrophysics. Articles and speeches. M.: Nauka, 1985.

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