How can a closed universe form? Continuation of the previous article

In the last article “Closed Universe - how it turns out, how to explain it more simply, and why it is impossible to get out of it” I tried to explain, within the general physics course, the properties of a closed Universe. The fact that our Universe is with a very high probability is closed, it follows from the data on the measurement of the Hubble constant, the average density of matter, the age of stars and the equations of the general theory of relativity (hereinafter GR). However, while the excess of the measured density over the critical, at which the Universe would be infinite, is small - only 2% . Therefore, it is better to postpone discussions on this issue until there is new data. Nevertheless, if for the formation of a closed world of finite volume (and at the very beginning - a microscopic one), several possible mechanisms can be proposed, it is rather difficult to come up with an explanation for the appearance of an open universe of infinite dimensions. Meanwhile, numerous experimental confirmations of the Big Bang prove that the universe had a beginning. Now we will consider the question of how exactly a closed universe (not necessarily ours), described in the previous article, can arise. The options here can be different, for example, there was a model of the emergence of a closed world as a result of quantum fluctuations. However, after Hawking discovered the evaporation of a black hole in 1974, a less exotic mechanism appeared. In 1976, Ya.B. Zeldovich suggested that a closed universe arises as a result of evaporation.

As well as in the previous article, a certain minimum of knowledge in physics is supposed, by all means GR, at least in a small volume. Here the material is much more complicated, so much needed is needed. A school course in physics will not be enough. Reading popular books on cosmology will not replace a high school physics course, for some reason not learned at the institute.

Black hole evaporation

Consider a black hole (BH), which evaporates in accordance with the theory of Hawking. The time of complete evaporation according to the clock of a distant observer is very long, so we assume for simplicity that it is located in an open Universe, whose lifetime is infinite. What happens to the substance inside the black hole in the process of evaporation, with particles falling below the horizon? Without thinking, many will say that they evaporate. The answer is obviously wrong, since no particle, no radiation, can leave the BH. The horizon can be crossed only in one direction - inward. Then the question arises, where does the radiation that comes to the remote observer come from? In the previous article, I showed with simple, though weak, considerations how a gravitational field can reduce the mass or energy of rest of a substance down to zero, without changing its number, that is, the number of particles. The evaporation of a black hole is the transfer of the rest energy of the particles inside the black hole to the outside. Moreover, in the process of evaporation of a black hole, the amount of matter in it should increase, it should just be recalled again that, as explained in the previous article, an increase in the amount of matter can lead to a decrease in mass / energy and size!

Consider a distant observer slowly approaching the horizon of a black hole. If at a long distance the observer determines the temperature T of the horizon by radiation according to Hawking's formula T = where Is Planck's constant, is the speed of light, k is the Boltzmann constant, r _g is the gravitational radius of a black hole, r is the radial coordinate of the observer in the Schwarzschild metric, then when approaching the horizon the observed frequency of photons will increase in (1 - r _g / r ) ^-1/2 times [1], §102. The observed temperature of the horizon will also increase. As a result, an observer near the horizon will see a very hot surface. The radiation that occurs near the horizon does not all go to infinity — part of it falls inside a black hole, because it is directed towards the horizon or at a large angle to the radius. But this is only part of the increase inside the black hole. Since its external observable mass falls as a result of evaporation, then, from the point of view of a distant observer at rest, a paradox arises - no mass / energy can leave the horizon sphere, and the black hole mass / energy falls. The paradox was noticed, oddly enough, rather late. The only possible mechanism for decreasing the mass / energy of a black hole is the absorption of particles with negative energy.
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Let's start with an explanation of the most basic - how a gravitational field can give rise to matter. For the first time this idea was put forward by Ya.B. Zeldovich. For illustration, there is the well-known Schwinger effect — the production of electron-positron pairs by a strong constant electric field [2]. A pair of virtual particles "electron - positron" can become real, having received an energy of about mc ² from an external dc electric field, m is the electron mass. To do this, the electron must tunnel through the potential barrier, passing a distance in the accelerating electric field of the order of , E is the field strength, e is the electron charge. Of course, this is a very simplified picture, unrelated to the method developed by Schwinger.

Also, as in the Schwinger effect, the virtual particles near the horizon are subject to the force of the force, which can give them additional energy sufficient to become real. There is an analogue of the Schwinger effect - the Unruh effect: an observer flying with constant acceleration in vacuum (vacuum from the point of view of an observer at rest), it is not vacuum that is observed, but thermal radiation corresponding to temperature T , [3]. Since the constant acceleration is equivalent to a uniform gravitational field, the similarity of the mechanisms is obvious.

If in the frame of reference of a distant observer there is a stream of particles of positive energy FROM a black hole, then its mass / energy decreases. Then there must be a stream of particles with negative total energy (from the point of view of the distant observer, in its frame of reference) falling inside the BH. In no other way can the mass of a black hole decrease - the movement of any body, any carrier of energy is possible only inside the horizon. These can be virtual particles tunneling into the horizon, where they, after receiving energy from the gravitational field, become real. Such a picture, of course, is overly simplified for purposes of understanding. More precisely, under the horizon there should appear a stream of particles that could not exist as real, outside the BH, since they would have negative energy. Only in this way it is possible to ensure the flow of energy out of the BH, recorded by a remote observer. I do not know so well the literature on this issue to indicate the article where such a problem was solved, but there is a mention in the literature about this effect.

Thus, the BH evaporation increases the number of particles inside the horizon, reducing its mass. There is no contradiction here, as explained in the previous article. But this is not the only mechanism for the birth of matter by a gravitational field. When even the ideal sphere collapses, the gravitational field inside matter changes rapidly. Such a variable field should give rise to particles, which for the most part remain inside the BH. Further, during the collapse of a real non-spherical mass near the singularity, when matter shrinks to a point, the Mixmaster Universe mode [4] opened by V.A. Belinsky, E.M.Liffshits and I.M. Khalatnikov. At the same time, inhomogeneous chaotic compression and tension occur, accompanied by the birth of particles. True, this does not change the mass of the black hole observed from the outside.

As a result, by the end of the BH evaporation there will be much more substance in it than at the beginning, although the mass and size observed from outside vanish, similarly to the example disassembled in the previous article. Gravity pumped all of its own energy out of the substance, bypassing the ban on leaving the horizon. As explained in the previous article, this is the closed universe. It should be remembered that the mass / energy of matter inside the closed world is zero ONLY for an external observer. The internal observer will register the gravitational field of the surrounding substance, its mass, etc., the surrounding substance is absolutely real for him, just like for us. This is one of the most amazing effects of GR.

AFTER EVAPORATION

As proved mathematically, for example, by Penrose, within the framework of the classical GR, collapse leads to compression of a substance to a point. Further continuation of the solution for singularity from the point of view of mathematics is impossible. However, it is clear that on a very small scale the classical (non-quantum) theory is not applicable, if only because it is impossible to measure distances smaller than the Planck length. 1.6x10 ^-35 m, G is the gravitational constant, since on such scales the quantum fluctuations of the gravitational field lead to the fact that the metric tensor has no definite value like the electron coordinate in the atom. From general considerations, it can be assumed that compression will stop at similar sizes or earlier. The fact is that, strictly speaking, the GR equations written by Einstein contain only the first, linear in the curvature tensor terms. There were good reasons for this. First, it would be very difficult to investigate more complex equations. Secondly, the Newtonian theory predicted the motion of the planets quite accurately, therefore the next approximation should have been enough, and it turned out that way for macroscopic scales. Thirdly, the curvature of our space, resulting from GR, is very small, the corresponding radius of curvature is billions of light years. Therefore, a linear approximation is valid until the curvature of space becomes too large. Then the following terms, quadratic in the curvature tensor, should be taken into account in the equations; however, the corresponding coefficients can no longer be determined from comparison with Newton's theory.

It is necessary to make a meaningful assumption about the nature of gravity. For the first time, such corrections in general form were written and estimated by ADD. Sakharov in 1966 [5], based on his hypothesis of gravity, as a result of the action of quantum fluctuations of all fields. As expected, based on the proposed nature of the amendments, they became significant when the radius of curvature is on the order of the Planck length. Unfortunately, this wonderful work did not continue, and was forgotten. After 13 years, these ideas were again put forward and brought to a complete theory in the works of Starobinsky, Mukhanov [6], and others ... It was shown that such corrections in cosmology work against attraction. Therefore, it is reasonable to assume that they can stop the collapse for very large values ​​of the curvature of space, for very small sizes. Then, in accordance with the GR equations, the expansion should begin — the equations simply do not have any other reasonable solutions. This is the beginning of the evolution of a closed universe. If expansion did not take place in a closed world, but under a horizon of non-zero size, then the corresponding solution would be a “white hole” - as if a black hole in reverse. There is such a region in the complete solution of the equations of general relativity on the spherical compression of matter [1], §103, where matter moves only from the center, and it is impossible to move to the center under the horizon, only outwards. Apparently, the white hole is impossible, because it looks quite absurd - it does not attract, but repels. Therefore, it should be assumed that the expansion occurs already in a closed world, a breakaway from the external universe. That is, if for an external observer, the process of evaporation of BH and the departure of matter into a closed world takes an insanely long time: 10 ⁷⁴ seconds for the mass of the Sun, then for an observer falling in BH on the border of collapsing matter, this process takes as much as falling to the center - very quickly, in a time order of r _g / c .

CONCLUSION. AND A LITTLE EXPLANATION

BH evaporation provides the simplest mechanism for the formation of a closed universe. Other hypotheses like quantum fluctuations look much more exotic. This mechanism allows us to explain the emergence of a universe like ours, containing a huge amount of matter. It turned out that an astronomically large amount of matter in BH is not required. In the initial stage after the Big Bang, there is an intensive birth of matter by a gravitational field - this idea was first put forward by Ya.B. Zeldovich, the theory was built in the works of A.A. Starobinsky, V.F. Mukhanova, G.V. Chibisov and others. With reference to our Universe, it turns out that the best agreement with the data on the relic radiation is obtained if we assume that there was no or almost no matter at the initial moment. In [6] V.F. Mukhanov considered the option when all matter is born of a gravitational field.
I do not cite any references to the original works, because here I tried to give just a summary of the available ideas that I learned from the reviews. According to the link [3], the reader will find the most simple and accessible presentation of the output of Hawking radiation and the Unruh effect. Click [7] for a good review of the literature and a simplified presentation of a number of modern theories. There are enough books and articles on the Internet, and a reader educated in physics will easily find a more detailed account of any issue raised in this article. My task was to give an idea of ​​the overall picture. I will add, finally, a very accessible review article by Ya.B. Zeldovich [9], where he, in particular, set forth his ideas in favor of the closedness of the Universe. I highly recommend reading.

Along the way, I will take the opportunity to clarify terms that often appear in popular articles and books on cosmology. These are “dark energy” and “dark matter”, about which most readers have a dark idea. “Dark energy” is a scientific name for a long-known thing, the cosmological constant [1], §111. It was first introduced by Einstein, considering a possible form of the equations of general relativity in the form

Where Are the curvature, energy-momentum and metric tensors, R is the trace of the curvature tensor, and - cosmological constant. Einstein initially believed non-zero, in order to get a static solution for the Universe, but later refused this term as having no clear physical meaning and not confirmed by observations. Nevertheless, such a term does not contradict any physical principles. In the work of Sakharov [5] is the energy density of zero oscillations of all fields in a flat space (up to a constant factor). This interpretation has been adopted by many today, perhaps slightly tampered with, although it is not the only possible one. With a positive there are serious changes in the solutions of the equations of general relativity for the universe. In this case, the closed Universe does not necessarily stop expanding with subsequent compression - it can go to the accelerated expansion mode, called de Sitter, by the name of de Sitter, who received this solution. The radius of the universe grows exponentially with time. It is precisely to such results that astronomers who studied supernovae came in 1998, the observations .

Sometimes they write about the negative pressure of "dark energy". This hazy statement has a simple physical meaning. If you have a closed volume, outside of which the field is absent, then the energy of zero field oscillations increases with increasing volume - new energy levels appear [8]. If one of the walls is a movable piston, then to increase the volume you will have to do work on the piston - to increase the energy of zero oscillations, that is, the piston tends to move inwards. For the electromagnetic field, this is called the Casimir effect and was confirmed experimentally. The gas tends to push the walls, it has a positive pressure, moving the piston out. In this sense, zero pressure has negative pressure, but there is no antigravity here.

Dark matter has nothing to do with "dark energy". This is a real substance with a mass, which for unknown reasons is not visible to astronomers directly. Its presence and its share in the total mass in the Universe are proved by the gravitational influence on the trajectories of stars and light rays. At first they thought it was a cold gas, but even it could not be so transparent to light and radio waves. Apparently, these particles do not interact with the electromagnetic field. Despite the fact that its share in the total mass of the observed cosmos is estimated at over 70%, there is practically no dark matter in the immediate vicinity of the Solar System, so it is impossible to register its particles.