HOW THE LIMIT VALUES WORK

Yuri L. Bolotina, Vladimir V. Yanovskyb,c,* National Science Center "Kharkov Institute of Physics and Technology", 1, Akademicheskaya str., 61108, Kharkiv, Ukraine Institute for Single Crystals, NAS Ukraine, 60 Nauky Ave., Kharkiv, 61001, Ukraine V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine *Corresponding Author: yanovsky@isc.kharkov.ua, tel. +380-97-614-28-13 Received November 4, 2020; accepted December 15, 2020

The observed Universe represents a hierarchy of structures with a gigantic spread of characteristic parameters (sizes, energies) from Planck to cosmological. The search for the mysterious connection between large and small scales that describe the world around us is an exciting game of chance. "Random" coincidences caused a storm of enthusiasm and the illusion of unraveling the innermost secrets of nature. The roots of the game go back to antiquity. The first "number theory" -Kabbalah denied the possibility of random coincidences, considering numbers to be a symbol of the spiritual nature of things.
A new era in "physical numerology" began in the first half of the twentieth century, when the characteristic sizes of macro (Universe) and micro (atom, electron) objects became known. The greats entered the game: Weyl [1], Edington [2], Dirac [3].
Already in 1923, in an appendix to his book, "Space, Time, Matter", G. Weyl estimated the ratio of the radius of the Universe to the classical radius of an electron as 40 10 . As he writes, this suggests that the enormous numerical value of the constant is related to the difference in the sizes of the Universe and the electron. P. Dirac belongs to the observation that in the hydrogen atom the ratio of electric to gravitational forces is close to the ratio of the size of the Universe to the classical radius of the electron.
No physical explanation of this result has yet been found, although it is obvious that any numerological coincidence is of interest only if it can be explained (obtained) in the framework of fundamental physical theory. The purpose of this article is to introduce a new approach [4] to explain the relationship between different scales. We show that the so-called limit values are an effective tool for achieving this goal. Due to their universality, the limiting values make it possible to establish connections between, at first glance, distant fields of physics.

PHYSICAL MOTIVATIONS
The traditional formulation of the ultimate goal of physics (in the narrower sense of physical axiomatics) represents the derivation of physical laws from first principles. However, the search for first principles is not inferior in complexity to the original problem. An alternative way to understanding the world around us is to interpret the fundamental limit values as a factor determining the structure of physical laws. Limit values play a central role in both phenomenology and axiomatic theories. It is well known that quantum mechanics can be built on the basis of the existence of a minimum quantum of action  , and the special theory of relativity -maximum speed c . From relatively recently, it became clear that a similar approach can be implemented in GR, which can be constructed by postulating the existence of a new limit value -the minimum length [5,6]

EEJP. 1 (2021)
Yuri L. Bolotin, Vladimir V. Yanovsky Limit values play a key role in the analysis of the measurement process in quantum mechanics. In particular, these quantities are included as parameters in the uncertainty relations, determining the permissible accuracy of measuring physical quantities.
Over the nearly century-long history of its existence, the uncertainty principle (the Heisenberg uncertainty principle (HUP)) has undergone a complex evolution related to the inclusion of gravitational interaction (generalized uncertainty principle (GUP)) and the transition to macroscopic spatial scales (extended uncertainty principle (EUP)) [7-11] 2 2 Here * l is a new fundamental length scale, the value of which should be determined. The need to modify the HUP is dictated by the fact that the HUP fails both at short and long distances. The common cause of failure is gravity: a "weak" gravitational interaction can transform into strong either at very small distances (Planck scales), or for very large masses (cosmological scales) Probably the most important consequence of the transition from the HUP to the GUP is the appearance of a minimum spatial length and a minimum time scale GUP imposes fundamental restrictions on the accuracy   l t   of measuring length (time)   l t regardless of the measurement method used [12][13][14]     The discreteness of space-time generated by the existence of a minimum length and a minimum time is inextricably linked with the concept of an event horizon. The appearance of singularities in the theory is considered to be the first signal that the theory has gone beyond its applicability and needs to be modernized. A natural variant of such modernization is to take into account effects that have fallen from consideration, which allows us to hope (although hopes are not always justified) to make the theory free of divergences. There is also an alternative point of view, formulated by R. Penrose [15] in the form of the principle of cosmic censorship: "Nature has an aversion to naked singularity." According to this principle, the singularities of space-time appear in places that, like the inner regions of black holes, are hidden from the observer. The natural question: is it possible to generalize the principle of cosmic censorship to the level of the "principle of physical censorship", making it a universal physical principle? The existence of a minimum length allows an affirmative answer to this question, and the formation of an event horizon serves as a mechanism for generating a minimum length. In the simplest case, the Schwarzschild solution, the event horizon is a hypothetical sphere around the point of gravitational singularity, dividing the space into two causally unrelated areas. Gravitational radius -the radius of this sphere for a body of mass m Now consider the GUP structure The right-hand side of this relation represents the sum of two length scales: the wavelength of the observer particle (photon, electron) p   and its effective gravitational radius An important consequence of GUP is the appearance on the physical scene of a whole set of limit values: minimum length, minimum time, maximum force, maximum power. Consider the nature of the last two quantities. This will help to better understand the role of the event horizon as the boundary of the observed physical reality.
The assertion of the existence of a maximum (limit) force in the framework of GR as a principle was first formulated by G. Gibbons [16]. This force is equal to 4 43 max 3.25 10 4 The limit does not depend on the nature of the forces and is satisfied for gravitational, electromagnetic, nuclear and any other forces. It is interesting to note that, up to a factor of 2π, the reciprocal of the ultimate force coincides with the coefficient in front of the pulse energy tensor in the field equations of GR.
The statement about the existence of maximum power is absolutely equivalent to this.

COSMOLOGICAL APPLICATIONS
The surfaces on which the maximum force (maximum momentum flux) or maximum power (maximum energy flux) is realized are event horizons. Any attempt to exceed the force limit creates a horizon that prevents a further increase in force.
It is fundamentally important that on any event horizon, regardless of its scale (i.e., its gravitational radius), maximum force 4 / c G is realized. Therefore, for the Planck and current cosmological horizons Those, this ratio is satisfied throughout the evolution of the Universe. Limit values can be used as an alternative way of finding a solution to the problem of the cosmological constant. For this, the condition for limiting the concentration of energy can be used, in other words, the condition for preventing the formation of an event horizon: the total energy contained in a region with a linear size L should not exceed the mass of a black hole of the same size. If the volume is filled with energy in the form of a cosmological constant, then this condition can be written in the form [18] Here   is the energy density in volume 3 L . If this inequality is violated, a black hole is formed with an event horizon that prevents a further increase in the energy density.
We apply inequality (15) to the Universe as a whole. In this case, it is natural to identify the IR scale with the Hubble radius 10 GeV This value is close (the difference is only two orders of magnitude and not 120) with the observed density of dark energy. Let us now try at a more fundamental level to understand what our hopes are for solving the problem of the cosmological constant along this path. If we treat the cosmological constant as the energy of zero-point oscillations of a vacuum, then Within the framework of this model, the only way to bring this gigantic value of the density of dark energy closer to the observed one is to reduce the value of ultraviolet cutoff max k or, what is the same, to degrade the spatial resolution 1 max k  . Within the framework of the proposed approach, it seems natural to switch from cutoff max~Pl k m to cutoff associated with the existence of a minimum length.
In any effective quantum field theory, defined in a spatial region with a characteristic size l and using ultraviolet cutoff max k , entropy is The new units perfectly reproduce the mass, size and lifetime of the observed Universe, but solving the problem of the cosmological constant, they give rise to the problem of Planck constant, 122 88 7.35 10 7.75 10 J s All the same 120 orders! Just the difficulty is "buried" in another place, and, apparently, to solve the problem of the cosmological constant, one should look for fundamentally new approaches.

CONCLUDING REMARKS
Historical experience teaches that the most interesting discoveries in physics occur during the transition to new characteristic scales of quantities that describe the problem under study. The boundary of the region in which a certain paradigm operates is determined by fundamental constants. For example, the transition from classical mechanics to relativistic is controlled by the speed of light, and the transition from classical mechanics to quantum is controlled by the Planck constant. The transition to the Planck scale is much more complicated, both quantitatively (this area is separated from the parameters currently achieved by tens of orders of magnitude) and qualitatively. (Is the concept of continuous space-time compatible with quantum mechanics). These difficulties make us look for workarounds to solve the problem. One of these areas may be the physics of limit quantities. Some specific examples of the use of limit values were considered in this paper. It is fundamentally important that on any event horizon, regardless of its scale (i.e., its gravitational radius), the universal value of limit force 4 / 4 c Gis realized. This allows you to relate the characteristics of the Universe related to various stages of its evolution.