Infinite space or infinite time? Does infinity exist?

So, by accepting the reality of time, we can explain the non-trivial structure of the Universe. But how long can it remain complex and structured? Can a nonequilibrium state persist indefinitely? Maybe we live in a “complexity bubble” in an equilibrium Universe?

This brings us to the most slippery topics in modern cosmology: the infinity of space and time. There is no concept more romantic than infinity, but in science the concept leads to confusion. Imagine that the Universe is infinite in space, and the same laws apply everywhere, but the initial conditions are chosen randomly. This is the Boltzmann Universe. Almost everything in the infinite Universe is in thermodynamic equilibrium. If something interesting happens, it is only due to fluctuations. These fluctuations occur somewhere in the Universe, and if there are infinitely many “somewheres”, then every fluctuation, no matter how unlikely, occurs demon final number once.

Therefore, our part of the observable Universe may simply be a statistical fluctuation. If the Universe is infinite, and the extent of our observable Universe is about 93 billion light years, then such a part will repeat itself endlessly in infinite space. So if the universe is a Boltzmann model, we exist an infinite number of times.

This violates Leibniz's principle: there are not and cannot be two identical places in the Universe. But not only him. Imagine that today could have been completely different. I might not have been born. You would marry your first girlfriend. Someone, not heeding the advice of friends, got behind the wheel drunk and killed a child. Your cousin was born into another, dysfunctional family and ended up committing mass murder. Intelligent dinosaurs evolved and solved the problem climate change, and mammals did not take over reptiles. All this could happen and change the current configuration of the Universe. Each such configuration is a possible configuration of atoms. Therefore, in infinite space, each of them appears an infinite number of times.

A terrifying prospect! The question arises, for example: why should I care about the consequences of my decisions if all other decisions have already been made by other instances of me in other areas of the infinite Universe? I can raise my child in this world, but should I care about other children suffering because of others?

In addition to these ethical issues, there are also those concerning the usefulness of science. If everything that can happen happens, the areas requiring explanation are greatly reduced. The principle of sufficient reason requires that rational reason in each case when one scenario is realized in the Universe and not another. But if all the scenarios in the Universe have already been realized, there is no need to explain anything. Of course science can give us insight local conditions, but this is also futile, because the true law will say: everything that can happen happens an infinite number of times, right now. It's kind of reductio ad absurdum The Newtonian paradigm applied to cosmology is another example of a cosmological fallacy. I call it Boltzmann's endless tragedy.

One of its reasons is that predictive power physics is significantly reduced: the meaning of the concept of probability is no longer what you think. Suppose you conduct an experiment for which quantum mechanics predicts outcome A 99% of the time, B 1% of the time. The experiment has been performed a thousand times. In about 990 cases you can expect result A. If you bet on A, you would feel confident because A will come up in about 99 out of 100 cases, and B in 1 out of 100 cases. A good chance to confirm the predictions of quantum mechanics! But in an infinite Universe there is an infinite number of copies of the experiment. An infinite number of times you observe outcome A, an infinite number of times you observe outcome B. So the prediction of quantum mechanics that one outcome of an experiment is observed 99 times more often than another cannot be verified in an infinite Universe.

IN quantum cosmology it is called measurement problem. After reading and asking experts, I came to the conclusion that the problem is unsolvable. I prefer to accept it as a fact: quantum mechanics proves that we live in a finite universe containing only one instance of me.

We can avoid the tragedy of an infinite universe by denying that the universe is infinite in space. Taking into account, of course, that we can observe it up to a certain distance, we can safely hypothesize that the Universe is finite, but unlimited, as Einstein believed. Consequently, the Universe has a topologically closed surface in the form of a sphere or torus.

This does not contradict observations. Which topology is true depends on the average curvature of space. If the curvature is positive (the case of a sphere), there is only one possibility - a three-dimensional analogue of the surface of a sphere in two-dimensional topology. If the average curvature of space is zero (the case of a plane), then for a finite Universe there is only one possibility: a three-dimensional analogue of the surface of a torus (doughnut) in two-dimensional topology. If the curvature is negative (the saddle case), there is an infinite number of possibilities for its topology. (They are too complex to describe here.) Their cataloging is a triumph of late 20th century mathematics.

Einstein's proposal is a hypothesis that must be confirmed. If the Universe is closed and small enough, light should circle around it several times, and we should see multiple images of the same distant galaxies. This has not yet been discovered. There are, however, good reasons think that cosmological theory is modeled in space-time, the space of which is closed. If the Universe is not closed, it is infinite. This is counterintuitive and means there is a boundary in space. It is infinitely far away, but, nevertheless, information cannot overcome it. Therefore, spatially infinite universe cannot be considered a self-sufficient system and must be considered part larger system, which includes any information coming from the boundary.

If the borders were a finite distance from us, you could imagine that outside visible universe there is still space. Information about a boundary can be conveyed through what comes from the world beyond that boundary.

The infinitely distant border makes it impossible to imagine the world beyond it. We simply have to indicate which information comes and which comes from us, but the choice is arbitrary. There can be no further explanation. Consequently, nothing can be explained within the framework of any model of the Universe with infinitely distant boundaries. The principle of closed explanations is violated, and with it the principle of sufficient reason is violated.

There are technical subtleties here. But this argument is decisive, although, as far as I can tell, it is ignored by cosmologists who believe that the Universe is spatially infinite. I see no other way out other than this: any model of the Universe must be spatially closed and without boundaries. There is nothing infinitely distant, just as there is no infinite space.

Now let's talk about the infinity of time.

The literature on cosmology is full of thoughts about the future. If the Universe is more like the model of Leibniz than of Boltzmann, then perhaps its lifetime is finite? Perhaps in the long run, not only will we die, but also the Universe? The assumption that it is finite in space frees us from the paradoxes inherent in the Boltzmann Universe. However, not from everyone. A spatially finite and closed Universe can live indefinitely, and if it does not contract, it will expand forever. There is infinite time to achieve thermal equilibrium. If so, it doesn't matter how long it takes. There will still be time for fluctuations to appear and incredible structures to be created. Thus, we can say that everything that can happen will happen an infinite number of times. This again leads to the Boltzmann brain paradox. If the principles of sufficient reason and the identity of indiscernibles are to hold, the universe must avoid such a paradoxical end.

IN scientific literature Attempts have been made to speculate about the distant future of the Universe. But to think about the distant future, you have to make some significant assumptions. One of them is that the laws of nature should not change, because if they did, we would be unable to predict anything. And there should be no undiscovered phenomena that can change the course of the history of the Universe. For example, there may be forces so weak that we have not yet discovered them, but nevertheless they come into play on long distances and large time intervals exceeding the current age of the Universe. It's possible. But such a scenario invalidates any prediction made based on existing knowledge. There should be no surprises like space “bubbles” coming towards us at the speed of light from beyond the horizon.

So we can reliably deduce the following.

Galaxies will stop producing stars. Galaxies are giant machines for turning hydrogen into stars. And not very efficient: a typical spiral galaxy produces only about one star each year. Now the Universe (almost 14 billion years old) consists mainly of primordial hydrogen and helium. Even though there is a lot of hydrogen, a finite number of stars will come out of it. Even if all the hydrogen turns into stars, there will always be the last star. And that's the upper limit. Most likely, the nonequilibrium processes involved in star formation will cease long before the hydrogen runs out.

The last stars will burn out. Stars have a limited lifespan. Massive stars live for several million years and die when they go supernova. Most stars live for billions of years and end up as white dwarfs. The time will come when the last star will go out. And then what?

The universe will be filled with matter and dark matter, radiation and dark energy. What happens in the Universe in the long term depends largely on dark energy, which we know the least about. It is associated with empty space. According to the latest data, it makes up about 73% of the total mass-energy of the Universe. Its nature is not yet known, but we are observing its influence on the movement of distant galaxies. In particular, dark energy needed to explain the recently discovered acceleration of universal expansion. Other than that, we don't know anything about dark energy. It may simply be a cosmological constant or an exotic form of energy with a constant density. Although the density of dark energy is roughly the same, we don't know if that's really the case—or if it's changing more slowly than we detect.

The future of the universe varies greatly depending on the density of dark energy. Let's first consider a scenario in which the dark energy density is conserved as the Universe expands. If the density is constant, then it behaves like Einstein's cosmological constant. It does not decrease, despite the fact that the Universe continues to expand. The density of the rest - all matter and all radiation - decreases as the Universe expands and the energy density of these sources steadily decreases. After a few tens of billions of years, everything will become insignificant except the energy density associated with the cosmological constant.

Clusters of galaxies due to exponential expansion will disband so quickly that they will soon be able to see each other. Photons, leaving one cluster and spreading at the speed of light, do not move fast enough to catch up with other clusters. Observers in each cluster are surrounded by a horizon that hides its neighbors. Each cluster will turn into closed system. Each horizon is like a box, the walls of which separate the subsystem from the Universe. Therefore, the methods of physics “in a box” are applicable to such a subsystem, and we can apply the methods of thermodynamics to them.

This is where it appears new effect quantum mechanics, due to which, inside each horizon, the space is filled with a gas of photons in thermal equilibrium: a kind of fog formed in the same way as the radiation of a Hawking black hole is formed. Temperature and Density radiation horizon extremely low, but remain unchanged as the Universe expands. Meanwhile, everything else, including matter and cosmic microwave background radiation, becomes less and less dense, and after enough big time the only thing that will fill the Universe is the radiation of the horizon. The universe must come into balance forever. There will, of course, be fluctuations and their relapses, and from time to time one or another configuration of the Universe will be exactly repeated (including the Boltzmann brain paradox, which I described in Chapter 16 as reductio ad absurdum Newtonian paradigm). According to this scenario, the apparent complexity of our Universe is just a short blip before the transition to eternal equilibrium.

We can almost say with certainty that we are not Boltzmann brains, because then we probably wouldn’t see a large, orderly Universe. This means that the scenario for the future of the Universe does not correspond to reality. The principle of sufficient reason, acting through the principle of the identity of indiscernibles, also rejects it.

The easiest way to avoid the death of the Universe is to stop its expansion. This is possible if the density of the matter is sufficient to cause compression. Matter gravitationally attracts matter, and this slows down expansion, so that if there is enough matter, the universe will collapse to a singularity. Or perhaps quantum effects will stop the collapse, turning compression into expansion, and lead to the emergence new universe. But there will probably not be enough matter to slow down the expansion.

Next the simplest way avoiding “thermal death” is realized in a scenario in which the cosmological constant is not constant. While there is evidence that dark energy (which for our purposes is identified with the cosmological constant) has not changed over the lifetime of our Universe, there is no evidence that it will not change in the long term. This change may be a consequence of deeper laws that operate so slowly that their effects are visible only on long time scales, or the change may simply be a consequence general trend changes in the laws themselves. Indeed, the principle of mutual influence states that the cosmological constant must be influenced by the Universe, which it itself decisively influences.

The cosmological constant can decrease to zero. If so, then the expansion of the Universe will slow down, but most likely will not turn it into compression. The universe can exist forever, but be static. At least this will help avoid the Boltzmann brain paradox.

Whether a Universe without a cosmological constant will expand or collapse forever depends on initial conditions. If the expansion energy is sufficient to overcome mutual gravitational attraction of all matter in the Universe, the latter will not be compressed. But even if the Universe is eternal, there is ample opportunity for rebirth, since every black hole can give rise to the embryo of the Universe. As noted in Chapter 11, there are strong theoretical indications that this should happen. If so, our Universe, which is far from dying, has already produced a billion billion descendants. Each of the new Universes will produce offspring, and the fact that it may die after that is no longer significant.

There is potential for a renaissance that involves not just black holes, but the entire universe. This hypothesis was studied in a class of cosmological models called cyclic models. This task solves one of Paul Steinhardt's cyclic models from Princeton University and Neil Turok from the Perimeter Institute. It is assumed that the cosmological constant decreases to zero and then continues to decrease to significant negative values. This leads to the collapse of the Universe. However, Steinhardt and Turok argue that collapse is accompanied by expansion. This may be due to the effects quantum gravity, or the final singularity may not be reached due to extreme values ​​of dark energy.

Theoretical indications that the cosmological singularity will not be achieved due to quantum effects, leading to a new expansion of the Universe, stronger than in the case of a singularity associated with a black hole. In the theory of loop quantum gravity, several models of quantum effects near cosmological singularity. It turned out that such a rebound is a universal phenomenon. It should be noted, however, that these are only models and are based on significant assumptions. The key assumption is that the Universe is spatially homogeneous. We know for sure that homogeneous regions - without gravitational waves and black holes - cannot give rise to new Universes.

In the worst case, highly heterogeneous areas will not experience rebound. They will simply collapse into a singularity where time stops. However, it does give us a principle to determine in which parts of the Universe the rebound will occur and the Universe will reproduce itself. If the rebound can only occur in more homogeneous regions, at the birth of new Universes, immediately after the rebound, these Universes will also be highly homogeneous. This predicts that the early Universe immediately after the rebound in highest degree homogeneous and there are no black or white holes in it, there are no gravitational waves (the case of our Universe).

But for a cyclical scenario to be scientific, there needs to be at least one testable prediction against which hypotheses can be tested. There are at least two scenarios associated with the spectrum of fluctuations of the IFI. Cyclical scenarios offer explanations for those fluctuations that do not require a short period of extremely rapid inflation (this is often accepted as the main cause of fluctuations). The observed spectrum of fluctuations is successfully reproduced, but there are two differences between the predictions of the cyclical and inflationary models, and these predictions can be experimentally tested now or in the near future. First, will gravitational waves be observed in the IFI spectrum? The inflation model says yes, but the cyclical models deny it. The latter predict that the spectrum cosmic microwave background radiation is not entirely random, that is, that the shape of such a spectrum will deviate from the shape of the Gaussian distribution.

Cyclic models are good examples of how postulating the fundamental nature of the concept of time (in the sense that time does not begin with big bang, and existed before) leads to a cosmology capable of making reliable predictions. MFI fluctuations are also described within the framework of theories suggesting that in early universe the speed of light was higher than today. These variable speed of light theories choose to emphasize the concept of time in such a way that it violates the principle of relativity. They are not as popular, but they also offer an explanation for IFI fluctuations without invoking inflation.

Roger Penrose proposed another scenario: the Universe gives rise to a new Universe. Penrose accepts the scenario of an eternal Boltzmann universe with a fixed cosmological constant and asks what will happen infinite time later. (Only Roger could ask such a question!) What if, after everything elementary particles, with mass (including protons, quarks and electrons) will decay, leaving only photons with other massless particles? If so, then the transition to eternity cannot be detected, since photons propagating at the speed of light do not need time. For a photon, the eternity of the late Universe is indistinguishable from the early Universe - the only difference is the temperature. True, this difference is huge. Penrose thinks it doesn't matter. Within the framework of the relational description of photon gas, only the relationships between objects that exist at this time matter, since there is no sensitivity to a common scale. The late Universe, filled with a gas of cold photons and other massless particles, is indistinguishable from the early Universe, filled with a hot gas of the same particles. According to the principle of the identity of indiscernibles, the late Universe is the same as the newly born one.

The Penrose scenario is implemented only after an infinite period of time and does not solve the Boltzmann brain paradox. However, he predicts that traces of the former Universe are present in the remnants of the Big Bang. Although most of information will be destroyed during an infinite time spent in a state of thermal equilibrium, one information carrier will not disappear anywhere - gravitational radiation. The information carried by gravitational waves does not disappear in cyclic models. It is preserved at the moment of rebound and is transferred to the new Universe.

The most powerful signal transmitted by gravitational waves is the imprint of the collision of large black holes that were at the center of long-extinct galaxies. These signals, like ripples on water, spread throughout the new Universe. Therefore, Penrose believes, circles should be visible in the MFI, the structure of which was fixed on early stage evolution of our Universe. These are shadows of events in the former Universe.

In addition, Penrose suggests the presence of many concentric circles originating from galaxy clusters in which more than one pair of galactic black holes have collided. This striking prediction is quite different from those made based on most cosmological scenarios for the IFI.

Now there is a debate about whether it is possible or not to observe concentric Penrose circles in MFIs. However, as we see, cosmological scenarios in which our Universe evolved from a pre-Big Bang universe are capable of predictions that can be confirmed or refuted. On the contrary, in scenarios in which the Universe is one of many at the same time existing worlds, there are not and most likely there will be no testable predictions.

In Chapter 10 I stated: rational explanation why the specific laws and initial conditions implemented in our Universe require that choices be made several times. Otherwise, we could know why exactly such a choice was made, because there is no reason for choosing the same initial conditions and the same laws of nature, made many times in a row. I considered two scenarios with multiple Big Bangs - simultaneous and sequential. Only in the latter case can we build a cosmological model that would answer why these particular laws were chosen, and at the same time remain scientific in the sense of the ability to make experimentally verifiable predictions. In this chapter I returned to this issue, and we saw: only in the case of sequential rebirth of the Universes can predictions be obtained that can be verified in experiment.

Thus, when we work with time as fundamental concept, cosmological model becomes scientific, and ideas are testable. Those who are burdened with metaphysical assumptions that the purpose of science is to discover eternal truths may think that by eliminating time and making the universe like a mathematical object, they will arrive at a scientific cosmology. But it turns out that the opposite is true. Charles S. Pierce more than a century ago I realized: we can explain the laws of nature if they evolve.

Brief summary of the work

Space without infinity

And, indeed, if the Universe is not infinite...

Could this be?

It turns out it can.

And not even in the sense that it occupies part of the space. The universe may occupy all of space, but this space does not have places in mathematics designated by the sign ∞ (infinity).

To understand this, we only have three steps to take.

First, let's depict such a space in general outlines, and then begin to draw all the details.

So, step one.

One-dimensional space.

In everyday understanding, it appears to us as something like a number line.

On the straight line, mark the beginning of the countdown - point O and from it in one direction with a plus sign (+), in the other with a minus sign (-), at equal intervals, called a unit of measurement, we make markings +1, +2, +3, ... ,+ ∞ and, accordingly, -1, -2, -3, …, - ∞. That is, on both sides there are ∞ signs - this is a one-dimensional infinite space.

Here we ask our question: “Can there be a one-dimensional space that does not contain ∞?”

It turns out it can.

In the initial sketch we will give only those examples that will be necessary and sufficient for us to understand the essence and further logical description of the next steps. At the same time, we will try to avoid introducing any new definitions.

Let's draw a circle.

This is also a one-dimensional space.

But no matter how you mark out such a space, if we take a certain finite value as a unit of measurement, then the sign ∞ cannot be placed anywhere in such a space.

This circle is a local example of a one-dimensional space that does not contain the ∞ sign.

Step two.

Two-dimensional space.

Let us draw two mutually perpendicular lines on the plane. Let's mark them in exactly the same way as the straight line in the first step, taking the intersection point as the starting point for each. Thus we define two-dimensional infinite space.

Here again we ask our question: “Can there exist a two-dimensional space that does not contain ∞?”

It turns out that it can too.

Pick up the globe.

No matter how you mark its surface, you won’t be able to place the ∞ sign anywhere.

This sphere is a local example of a two-dimensional space that does not contain ∞.

Let's move on to the third step.

Through the point of intersection of two mutually perpendicular lines we draw a third line perpendicular to the first two. Let's mark it in exactly the same way as in the first two steps. We get a three-dimensional infinite space, or more precisely, a way to display it - a Cartesian coordinate system.

We ask the initial question: “Can there exist a space that does not contain the sign ∞?”

It turns out it can.

A local example similar to the examples in the first two steps cannot be given here.

These local examples were given only in order to obtain a way to display such a space in Cartesian system coordinates, which will allow us to determine the method of calculating an ideally defined space - a space that does not contain the ∞ sign, in the global sense.

Let's move on to the method of displaying an ideally defined space in a Cartesian coordinate system.

Let's return to one-dimensional space.

How can you display a circle on a line?

Let's mark any point on the circle and take it as the origin, denoting it exactly the same as on the straight line - O (with a zero value). From point O we measure half a circle in any direction and designate this mark as point M (that is, OM - half a circle in any direction). From point O in one direction with a sign (+), in the other with a minus sign (-), with exactly the same identical in......

The theory of relativity views space and time as unified education, the so-called “space-time”, in which time coordinates play as significant a role as spatial ones. Therefore, in the very general case we, from the point of view of the theory of relativity, can only talk about the finitude or infinity of this particular united “space - time”. But then we enter the so-called four-dimensional world, which has completely special geometric properties that differ most significantly from geometric properties of the three-dimensional world in which we live.

And the infinity or finitude of four-dimensional “space-time” still says nothing or almost nothing about the spatial infinity of the Universe that interests us.

On the other hand, the four-dimensional “space-time” theory of relativity is not just a convenient mathematical apparatus. It fully reflects certain properties, dependencies and patterns of the real Universe. And therefore, when solving the problem of the infinity of space from the point of view of the theory of relativity, we are forced to take into account the properties of “space-time”. Back in the twenties of the current century, A. Friedman showed that within the framework of the theory of relativity, a separate formulation of the question of the spatial and temporal infinity of the Universe is not always possible, but only under certain conditions. These conditions are: homogeneity, that is, the uniform distribution of matter in the Universe, and isotropy, that is, the same properties in any direction. Only in the case of homogeneity and isotropy is the unified “space-time” split into “ homogeneous space" and universal "world time".

But, as we have already noted, the real Universe is much more complex than homogeneous and isotropic models. This means that the four-dimensional ball of the theory of relativity, corresponding to the real world in which we live, in the general case does not split into “space” and “time”. Therefore, even if with increasing observational accuracy we can calculate average density(and therefore local curvature) for our Galaxy, for a cluster of galaxies, for the observable region of the Universe - this will not yet be a solution to the question of the spatial extent of the Universe as a whole.

It is interesting, by the way, to note that some regions of space may indeed turn out to be finite in the sense of closure. And not only the space of the Metagalaxy, but also any region in which there are sufficiently powerful masses that cause strong curvature, for example, the space of quasars. But, we repeat, this still does not say anything about the finitude or infinity of the Universe as a whole. In addition, the finiteness or infinity of space depends not only on its curvature, but also on some other properties.

Thus, when current state general theory of relativity and astronomical observations, we cannot obtain a sufficiently complete answer to the question of the spatial infinity of the Universe.

They say that the famous composer and pianist F. Liszt provided one of his piano works with the following instructions for the performer: “fast”, “even faster”, “fast as possible”, “even faster”...

This story involuntarily comes to mind in connection with the study of the question of the infinity of the Universe. Already from what was said above, it is quite obvious that this problem is extremely complex.

And yet it is still immeasurably more complex...

To explain means to reduce to what is known. A similar technique is used in almost every scientific research. And when we try to solve the question of the geometric properties of the Universe, we also strive to reduce these properties to familiar concepts.

The properties of the Universe are, as it were, “measured” to those existing in this moment abstract mathematical representations about infinity. But are these ideas sufficient to describe the Universe as a whole? The trouble is that they were developed largely independently, and sometimes completely independently of the problems of studying the Universe, and in any case on the basis of research limited area space.

Thus, the solution to the question of the real infinity of the Universe turns into a kind of lottery, in which the probability of winning, i.e. random coincidence at least enough large number properties of the real Universe with one of the formally derived standards of infinity is very insignificant.

The basis of modern physical ideas about the Universe is the so-called special theory relativity. According to this theory, the spatial and temporal relationships between the various environments around us real objects are not absolute. Their character depends entirely on the state of motion of a given system. Thus, in a moving system, the pace of time slows down, and all length scales, i.e. the sizes of extended objects are reduced. And this reduction is stronger, the higher the speed of movement. As we approach the speed of light, which is the maximum possible speed in nature, all linear scales decrease without limit.

But if at least some geometric properties of space depend on the nature of the movement of the reference system, that is, they are relative, we have the right to pose the question: aren’t the concepts of finitude and infinity also relative? After all, they are most closely related to geometry.

IN last years The famous Soviet cosmologist A.L. Zelmapov studied this curious problem. He managed to discover a fact that, at first glance, was absolutely amazing. It turned out that space, which is finite in a fixed reference frame, at the same time can be infinite relative to a moving coordinate system.

Perhaps this conclusion will not seem so surprising if we remember about the reduction of scales in moving systems.

Popular presentation complex issues modern theoretical physics is made very difficult by the fact that in most cases they do not allow for visual explanations and analogies. Nevertheless, we will now try to give one analogy, but while using it, we will try not to forget that it is very approximate.

Imagine that a spaceship is rushing past the Earth at a speed equal to, say, two-thirds the speed of light - 200,000 km/sec. Then, according to the formulas of the theory of relativity, a reduction in all scales should be observed by half. This means that from the point of view of the astronauts on the ship, all segments on Earth will become half as long.

Now imagine that we have, although a very long, but still finite straight line, and we measure it using some unit of length scale, for example, a meter. For an observer located in spaceship, rushing at a speed approaching the speed of light, our reference meter will shrink to a point. And since there are countless points even on a finite straight line, then for an observer in a ship our straight line will become infinitely long. Approximately the same thing will happen with regard to the scale of areas and volumes. Consequently, finite regions of space can become infinite in a moving frame of reference.

We repeat once again - this is by no means a proof, but only a rather rough and far from complete analogy. But it gives some idea physical entity phenomenon of interest.

Let us now remember that in moving systems not only do scales decrease, but the flow of time also slows down. It follows from this that the duration of existence of some object, finite in relation to a fixed (static) coordinate system, may turn out to be infinitely long in a moving reference system.

Thus, from Zelmanov’s works it follows that the properties of “finitude” and “infinity” of space and time are relative.

Of course, all these at first glance rather “extravagant” results cannot be considered as the establishment of some universal geometric properties of the real Universe.

But thanks to them you can do extremely important conclusion. Even from the point of view of the theory of relativity, the concept of the infinity of the Universe is much more complex than it was previously imagined.

Now there is every reason to expect that if a theory more general than the theory of relativity is ever created, then within the framework of this theory the question of the infinity of the Universe will turn out to be even more complex.

One of the main provisions modern physics, its cornerstone is the requirement of the so-called invariance of physical statements with respect to transformations of the reference system.

Invariant—means “not changing.” To better imagine what this means, let's take some geometric invariants as an example. Thus, circles with centers at the origin of the rectangular coordinate system are rotation invariants. At any turn coordinate axes relative to the origin, such circles transform into themselves. Straight lines perpendicular to the “OY” axis are invariants of transformations of the coordinate system transfer along the “OX” axis.

But in our case we are talking about invariance in more in a broad sense words: any statement only then has physical meaning, when it does not depend on the choice of reference system. In this case, the reference system should be understood not only as a coordinate system, but also as a method of description. No matter how the method of description changes, the physical content of the phenomena being studied must remain unchanged and invariant.

It is easy to see that this condition is not only purely physical, but also fundamental, philosophical meaning. It reflects the desire of science to clarify the real, true course of phenomena, and to exclude all distortions that can be introduced into this course by the process of scientific research itself.

As we have seen, from the works of A.L. Zelmanov it follows that neither infinity in space nor infinity in time satisfy the requirement of invariance. This means that the concepts of temporal and spatial infinity that we currently use do not fully reflect the real properties of the world around us. Therefore, apparently, the very formulation of the question of the infinity of the Universe as a whole (in space and time) with modern understanding infinity has no physical meaning.

We have received yet another convincing evidence that the “theoretical” concepts of infinity, which the science of the Universe has used so far, are very, very limited in nature. Generally speaking, this could have been guessed before, since the real world is always much more complex than any “model” and we can only talk about a more or less accurate approximation to reality. But in this case, it was especially difficult to gauge, so to speak, by eye, how significant the approach achieved was.

Now at least the path to follow is emerging. Apparently, the task is, first of all, to develop the very concept of infinity (mathematical and physical) based on the study real properties Universe. In other words: “to try on” not the Universe to theoretical ideas about infinity, but, on the contrary, these theoretical ideas to the real world. Only this research method can lead science to significant advances in this area. No abstract logical reasoning or theoretical conclusions can replace facts obtained from observations.

It is probably necessary, first of all, to develop an invariant concept of infinity based on the study of the real properties of the Universe.

And, in general, apparently, there is no such universal mathematical or physical standard of infinity that could reflect all the properties of the real Universe. As knowledge develops, the number of types of infinity known to us will itself grow indefinitely. Therefore, most likely, the question of whether the Universe is infinite will never be given a simple “yes” or “no” answer.

At first glance, it may seem that in connection with this, studying the problem of the infinity of the Universe generally loses any meaning. However, firstly, this problem in one form or another confronts science at certain stages and has to be solved, and secondly, attempts to solve it lead to a number of fruitful discoveries along the way.

Finally, it must be emphasized that the problem of the infinity of the Universe is much broader than just the question of its spatial extent. First of all, we can talk not only about infinity “in breadth”, but, so to speak, also “in depth”. In other words, it is necessary to get an answer to the question of whether space is infinitely divisible, continuous, or whether there are some minimal elements in it.

Currently, this problem has already faced physicists. The question of the possibility of the so-called quantization of space (as well as time), i.e., the selection of certain “elementary” cells in it that are extremely small, is being seriously discussed.

We must also not forget about the infinite variety of properties of the Universe. After all, the Universe is, first of all, a process. characteristic features of which are continuous movement and incessant transitions of matter from one state to another. Therefore, the infinity of the Universe is also an infinite variety of forms of movement, types of matter, physical processes, relationships and interactions, and even properties of specific objects.

Does infinity exist?

In connection with the problem of the infinity of the Universe, it appears at first glance unexpected question. Does the very concept of infinity have real meaning? Isn't it just conditional? mathematical construction, to which nothing corresponds at all in the real world? This point of view was held by some researchers in the past, and it still has supporters today.

But scientific data indicate that when studying the properties real world we are in any case faced with what can be called physical, or practical, infinity. For example, we encounter quantities so large (or so small) that, from a certain point of view, they are no different from infinity. These quantities lie beyond the quantitative limit beyond which any further changes no longer have any noticeable effect on the essence of the process under consideration.

Thus, infinity undoubtedly exists objectively. Moreover, both in physics and in mathematics we are faced with the concept of infinity at almost every step. This is not an accident. Both of these sciences, especially physics, despite the apparent abstractness of many provisions, ultimately always start from reality. This means that nature, the Universe, actually has some properties that are reflected in the concept of infinity.

The totality of these properties can be called the real infinity of the Universe.

Dissimilar infinities

Infinity is one of those mathematical images, which is difficult to imagine not only for non-specialists, but also for scientists. One famous mathematician, who taught geometry at the physics department of Moscow University, confidentially admitted to students that when he tries to imagine infinity, he feels his mind begin to cloud.

Nevertheless, mathematicians, physicists, and astrophysicists in their research have to deal with infinities, with infinitely large quantities and operate with them. Moreover, it turns out that infinities can be different, and they can even be compared with each other.

The simplest, most “elementary” infinity and at the same time the “smallest” is the infinity of numbers in the natural series. It can be obtained by adding one unit after another to one over and over again.

Since such an operation is not limited by anything and can be repeated for as long as desired, the result is an infinite set of integers - a “countable” set, as mathematicians call it. This infinity, convenient in many respects, plays the role of a kind of “measuring ruler”, a kind of standard for measuring other infinities. To do this, you need to try to simply number their elements. And see what comes of it...

Just? Why not? We know how to count from one and so on. But here a completely unexpected surprise awaits us. One of those that we encounter almost at every step when we deal with infinities. For example, let us “apply” our standard to the infinite set of all even numbers. Let us number two, the smallest even number, as one, four as two, six as three, and so on, and so on... And we will be surprised to discover that there are not only enough numbers to designate all even numbers - this was to be expected, – but there are still available rooms.

It turns out that both infinities - countable and the infinity of all even numbers - are the same? How so? After all, of every two consecutive numbers in the natural series, only one is even. This means that there should be half as many such numbers as all integers! In other words, the set of all even numbers is only part of the set of all integers. And the corresponding infinities are the same, have, as mathematicians say, the same power.

But that doesn’t happen, it can’t be! The set of any object cannot be equal to its own part! Yes, indeed, it cannot, as long as we are dealing with finite formations. But infinities have their own laws - bizarre, of course, from an ordinary point of view - but nevertheless quite strict. By the way, Galileo drew attention to the fact that infinite sets can be equal to their own subsets... Much to his surprise!

However, any discovery, as we already know, inevitably entails new questions. The one in question is no exception. For example, the following question arises: are there infinite sets more “powerful” than countable ones? Here is a straight line segment. How many dots can fit on it? It is clear that there are countless of them. But how much exactly?

Let us once again resort to the help of our standard - a countable set. And in the end we will find that this time there are too few numbers in the natural series to number all the points of the segment we have chosen. In mathematics, a strict theorem is proven in this regard: no matter how many points of a segment we number, there will always be points for which there are not enough numbers in the natural series. Thus we have discovered infinity more high order, than a countable set - infinity, called continuum. But the continuum is not the limit. In principle, one can construct infinities of arbitrarily high rank.

Let's return to the question of the geometric properties of the Universe. You may have noticed that when discussing this problem, either the possible infinity of world space or its unlimitedness is mentioned. In the “ordinary” world, for which Euclidean geometry is valid, the same geometry that we study in school, these concepts are essentially equivalent, meaning the same thing. Although there are still some differences. Strictly speaking, infinity is a quantitative, “metric” property: infinity of length, area, volume. What about unlimited?..

“What do we want to express when we say that our space is infinite? – wrote Einstein, who had the fortunate ability to express the most abstract ideas with the help of visual images. - Nothing other than the fact that we can apply one thing to another equal bodies, say, cubes in any number, and at the same time we will never fill the space. This kind of construction will never end. There will always be room to add one more cube..."

This is what infinite space is. As for unlimitedness, this property is structural, as mathematicians say, topological. This circumstance was especially emphasized at one time outstanding mathematician Bernhard Riemann.

"By revising spatial constructions in the direction of the infinitely large,” he noted, “one should distinguish between the properties of unlimitedness and infinity: the first of them is the property of extension, the second is the metric property.”

In Euclidean space, any line that can be extended indefinitely is infinite. But we live in a curved world... In such a world, infinity and unlimitedness differ in an even more significant way. To the point - another unexpected paradox - that unlimited space can be both infinite, that is, without a boundary, an “edge,” or finite!

To somewhat soften this latest blow to common sense, let’s use an analogy. Analogies in science are not strict evidence, but they allow us to better understand the essence of certain complex phenomena.

Imagine an ordinary ball of finite radius. A spherical surface is a two-dimensional formation curved in three-dimensional space. Imagine some fantastic flat creature living on this surface and not even suspecting that there is some kind of third dimension. Traveling through its curved world in any direction, this creature will never come across any border. And in this sense, the surface of the ball is unlimited space. But since the radius of our ball is finite, its surface area is also finite. Thus, the unlimited and at the same time finite world appeared before us in all its reality. It turned out to be possible what at first glance seemed absolutely impossible.

The next step will require more from us greater strength imagination. We will talk about a three-dimensional ball that is located in four-dimensional space... Unfortunately, it is no less difficult for us, creatures of the three-dimensional world, to visualize such a situation than for an imaginary inhabitant of a spherical surface to imagine a two-dimensional sphere curved in three-dimensional space.

But in the theory of relativity our world looks exactly like this: it is curved in four-dimensional space, where, however, the role fourth dimension time plays out. According to Einstein, we live in four-dimensional “space-time.” Wherein great physicist believed that our curved world has a finite volume, it is, as it were, closed in on itself.

The history of studying the geometric properties of the Universe has taken another sharp turn. The classical Newtonian ideas of infinite and boundless space had to be abandoned. They played their role, but the world turned out to be more complicated.

Thus, another extremely important step was taken in understanding the hidden properties of our world. However, the mathematical, or more precisely, geometric, model of our Universe, built general theory relativity, in itself could not yet be considered proof of the finitude of real space. But Einstein himself considered this option the most reasonable.

However, this was not the end of the road. It was still very, very far away. The new level to which the study of the geometric properties of our world has reached has given rise to a whole series of questions to which answers have not yet been found.