Black holes with electric charge. Black hole - the most mysterious object in the universe Charged black hole

Black holes

Starting in the middle of the XIX century. development of the theory of electromagnetism, James Clerk Maxwell had a large amount of information about the electric and magnetic fields. In particular, it was surprising that the electric and magnetic forces decrease with distance in exactly the same way as the force of gravity. Both gravitational and electromagnetic forces are long-range forces. They can be felt at a very great distance from their sources. On the contrary, the forces that bind together the nuclei of atoms - the forces of strong and weak interactions - have a short radius of action. Nuclear forces make themselves felt only in a very small area surrounding nuclear particles. The large range of electromagnetic forces means that, being far from a black hole, experiments can be undertaken to find out whether this hole is charged or not. If a black hole has an electric charge (positive or negative) or a magnetic charge (corresponding to the north or young magnetic pole), then an observer located in the distance is able to detect the existence of these charges using sensitive instruments. In the late 1960s and early 1970s, astrophysicists -theorists have worked hard on the problem: what properties of black holes are stored and what properties are lost in them? The characteristics of a black hole that can be measured by a distant observer are its mass, its charge, and its angular momentum. These three main characteristics are preserved during the formation of a black hole and determine the space-time geometry near it. In other words, if you set the mass, charge and angular momentum of a black hole, then everything about it will already be known - black holes have no other properties than mass, charge and angular momentum. So black holes are very simple objects; they are much simpler than the stars from which black holes emerge. G. Reisner and G. Nordström discovered the solution of Einstein's equations of the gravitational field, which completely describes a "charged" black hole. Such a black hole may have an electrical charge (positive or negative) and/or a magnetic charge (corresponding to the north or south magnetic pole). If electrically charged bodies are commonplace, then magnetically charged bodies are not at all. Bodies that have a magnetic field (for example, an ordinary magnet, a compass needle, the Earth) necessarily have both north and south poles at once. Until very recently, most physicists believed that magnetic poles always occur only in pairs. However, in 1975 a group of scientists from Berkeley and Houston announced that they had discovered a magnetic monopole in one of their experiments. If these results are confirmed, then it will turn out that separate magnetic charges can exist, i.e. that the north magnetic pole can exist separately from the south, and vice versa. The Reisner-Nordström solution allows for the existence of a monopole magnetic field in a black hole. Regardless of how the black hole acquired its charge, all the properties of this charge in the Reisner-Nordström solution are combined into one characteristic - the number Q. This feature is similar to the fact that the Schwarzschild solution does not depend on how the black hole acquired its mass. In this case, the space-time geometry in the Reisner-Nordström solution does not depend on the nature of the charge. It can be positive, negative, correspond to the north or south magnetic pole - only its full value is important, which can be written as |Q|. So, the properties of a Reisner-Nordström black hole depend only on two parameters - the total mass of the hole M and its total charge|Q| (in other words, from its absolute value). Thinking about real black holes that could actually exist in our Universe, physicists came to the conclusion that the Reisner-Nordström solution turns out to be not very significant, because the electromagnetic forces are much greater than the forces of gravity. For example, the electric field of an electron or a proton is trillions of trillions of times stronger than their gravitational field. This means that if the black hole had a sufficiently large charge, then the huge forces of electromagnetic origin would quickly scatter in all directions the gas and atoms "floating" in space. In the shortest possible time, particles with the same charge sign as the black hole would experience a powerful repulsion, and particles with the opposite charge sign would experience an equally powerful attraction to it. By attracting particles with a charge of the opposite sign, the black hole would soon become electrically neutral. Therefore, we can assume that real black holes have only a small charge. For real black holes, the value of |Q| must be much smaller than M. Indeed, it follows from the calculations that black holes that could actually exist in space must have a mass M at least a billion billion times greater than |Q|.

Researchers from the universities of Valencia and Lisbon decided to look beyond general relativity to solve the main problem with black holes - the strange phenomena at their center.

electrically charged black holes

The black hole they consider is a special case that does not exist in nature, since it is electrically charged and does not revolve around itself. This strange object is not a point of infinite density, but a wormhole - a kind of bridge to another place in time and space.

To come to this conclusion, scientists equated a black hole with graphene or a crystal. Their geometry can be used to reproduce space and time.

Space-Temporal Anomaly

Just as crystals are imperfect in their microstructure, the central region of a black hole can be interpreted as an anomaly in space and time, and this requires new geometric elements to describe it more accurately. Scientists explored all possible options, taking into account the facts that they observed in nature.

Describing the features of black holes is still an incredibly difficult task. To provide it, it is necessary to combine the theory of relativity and quantum mechanics, and they work rather poorly together.

The scientists' theory naturally solves several problems in interpreting electrically charged black holes. First of all, they solved the singularity problem, since there is a "door" in the center of a black hole - a wormhole through which time and space can continue.

The role of the wormhole

In the interpretation of scientists, the place in the center of a black hole is replaced by a wormhole, the size of which is directly proportional to its electric charge. The larger the charge, the larger the wormhole. Theoretically, some brave explorer could jump into this black hole, where he would be sucked in by intense tidal forces (a process called spaghettification), pass through the wormhole, and be able to return back to the universe.

This discovery is rather curious. Although wormholes are usually predicted in general relativity, they require some exotic matter to remain stable. Instead, they manifest themselves in ordinary matter and energy.

Electrically charged black holes are not supposed to form in nature, especially if they lead to idiosyncratic results, such as the formation of a stable wormhole. But in the end, even real black holes were once considered just a fancy theoretical idea.

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Introduction

1.1 The concept of a black hole

Conclusion

References

Application

Introduction

A black hole is a region in space-time, the gravitational attraction of which is so great that even objects moving at the speed of light, including quanta of light itself, cannot leave it. The boundary of this region is called the event horizon, and its characteristic size is called the gravitational radius.

Theoretically, the possibility of the existence of such regions of space-time follows from some exact solutions of Einstein's equations, the first of which was obtained by Karl Schwarzschild in 1915. The exact inventor of the term is unknown, but the designation itself was popularized by John Archibald Wheeler and first used publicly in the popular lecture "Our Universe: Known and Unknown" on December 29, 1967. Previously, such astrophysical objects were called "collapsed stars" or "collapsars" (from the English collapsed stars), as well as "frozen stars" (English frozen stars).

Relevance: In the literature devoted to the physics of black holes, the description of black holes by Reissner-Nordström is strictly formalized and is mainly of a theoretical nature. In addition, an astronomer observing celestial bodies will never see the structure of a charged black hole. Insufficient illumination of this issue and the impossibility of physical observation of charged black holes became the basis for the study of the work.

Purpose of the work: to build a model of a black hole according to the Reissner-Nordström solution for visualizing events.

To achieve the goal set in the work, the following tasks should be solved:

· Perform a theoretical review of the literature on the physics of black holes and their structure.

· Describe the Reissner-Nordström information model of a black hole.

· Build a computer model of a Reissner-Nordström black hole.

Research Hypothesis: A charged black hole exists if the mass of the black hole is greater than its charge.

Research method: computer simulation.

The object of research are black holes.

The subject is the structure of a black hole according to the Reissner-Nordstrom solution.

The educational-methodical, periodical and printed literature of Russian and foreign researchers, physicists and astrophysicists of black holes served as an information base. The bibliographic list is presented at the end of the work.

The structure of the work is determined by the tasks set in the study and consists of two chapters. The first chapter is devoted to a theoretical review of the physics of black holes. The second chapter discusses the stages of Reissner-Nordström black hole modeling and the result of the computer model.

Scientific novelty: the model allows you to observe the structure of the Reissner-Nordström black hole, study its structure, investigate its parameters and visually present the simulation results.

Practical significance of the work: presented in the form of a developed Reissner-Nordstrom model of a charged black hole, which will allow demonstrating the result of the model in the educational process.

Chapter 1. Theoretical review of ideas about black holes

1.1 The concept of a black hole

At present, a black hole is commonly understood as a region in space, the gravitational attraction of which is so strong that even objects moving at the speed of light cannot leave it. The boundary of this region is called the event horizon, and its radius (if it is spherically symmetrical) is called the gravitational radius.

The question of the real existence of black holes is closely related to how correct the theory of gravity, from which their existence follows. In modern physics, the standard theory of gravity, best confirmed experimentally, is the general theory of relativity (GR), which confidently predicts the possibility of the formation of black holes. Therefore, observational data are analyzed and interpreted primarily in the context of general relativity, although, strictly speaking, this theory is not experimentally confirmed for conditions corresponding to the space-time region in the immediate vicinity of stellar-mass black holes (however, it is well confirmed under conditions corresponding to supermassive black holes). Therefore, statements about direct evidence of the existence of black holes, strictly speaking, should be understood in the sense of confirming the existence of astronomical objects that are so dense and massive, as well as having some other observable properties, that they can be interpreted as black holes of general relativity.

In addition, black holes are often called objects that do not strictly correspond to the definition given above, but only approach in their properties to such a black hole - for example, these can be collapsing stars in the late stages of collapse. In modern astrophysics, this difference is not given much importance, since the observational manifestations of an “almost collapsed” (“frozen”) star and a “real” (“eternal”) black hole are almost the same. This is because the differences in the physical fields around the collapsar from those for the "eternal" black hole decrease according to power laws with a characteristic time of the order of the gravitational radius divided by the speed of light .

A very massive star can continue to shrink (collapse) beyond the pulsar stage before becoming a mysterious object called a black hole.

If the black holes predicted by the theory really exist, then they are so dense that a mass equal to the sun is compressed into a ball with a diameter of less than 2.5 km. The gravitational force of such a star is so great that, according to Einstein's theory of relativity, it sucks everything that comes close to it, even light. A black hole cannot be seen, because neither light, nor matter, nor any other signal can overcome its gravity.

X-ray source Cygnus X-1, located at a distance of 8000 sv. years (2500 pc) in the constellation Cygnus, a possible candidate for a black hole. Cygnus X-1 is an invisible eclipsing binary star (period 5-6 days). Its observable component is a blue supergiant whose spectrum changes from night to night. X-rays recorded by astronomers may be emitted when Cygnus X-1, with its gravitational field, sucks matter from the surface of a nearby star into a rotating disk that forms around a black hole.

Rice. 1.1. Black hole NGC 300 X-1 as seen by an artist.

What happens to a spacecraft that makes an unfortunate approach in space to a black hole?

The strong gravitational pull of the black hole will pull the spacecraft inward, creating a destructive force that will intensify as the ship falls and eventually tear it apart.

1.2 Analysis of ideas about black holes

In the history of ideas about black holes, three periods can be conditionally distinguished:

The second period is associated with the development of the general theory of relativity, the stationary solution of the equations of which was obtained by Karl Schwarzschild in 1915.

The publication in 1975 of Stephen Hawking's work, in which he proposed the idea of radiation from black holes, begins the third period. The boundary between the second and third periods is rather arbitrary, since all the consequences of Hawking's discovery did not immediately become clear, the study of which continues to this day.

Newton's theory of gravity (on which the original theory of black holes was based) is not Lorentz invariant, so it cannot be applied to bodies moving at near-light and light speeds. Deprived of this shortcoming, the relativistic theory of gravity was created mainly by Einstein (who finally formulated it by the end of 1915) and was called the general theory of relativity (GR), . It is on it that the modern theory of astrophysical black holes is based.

General relativity suggests that the gravitational field is a manifestation of the curvature of spacetime (which thus turns out to be pseudo-Riemannian, and not pseudo-Euclidean, as in special relativity). The connection between the curvature of space-time and the nature of the distribution and movement of the masses contained in it is given by the basic equations of the theory - the Einstein equations.

Since black holes are local and relatively compact formations, when constructing their theory, the presence of the cosmological constant is usually neglected, since its effects for such characteristic dimensions of the problem are immeasurably small. Then the stationary solutions for black holes in the framework of general relativity, supplemented by known material fields, are characterized by only three parameters: mass (M), angular momentum (L) and electric charge (Q), which are the sum of the corresponding characteristics of those entering the black hole during the collapse and falling into it later than bodies and radiations.

Solutions of the Einstein equations for black holes with the corresponding characteristics (see Table 1.1):

Table 1.1 Solutions of the Einstein equations for black holes

Schwarzschild's solution (1916, Karl Schwarzschild) is a static solution for a spherically symmetric black hole with no rotation and no electric charge.

The Reissner-Nordström solution (1916, Hans Reissner (1918, Gunnar Nordstrom) is a static solution of a spherically symmetric black hole with a charge but no rotation.

Kerr's solution (1963, Roy Kerr) is a stationary, axisymmetric solution for a spinning black hole, but without charge.

The Kerr-Newman solution (1965, E.T. Newman, E. Couch, K. Chinnapared, E. Exton, E. Prakash and R. Torrens) is the most complete solution at the moment: stationary and axisymmetric, depends on all three parameters.

According to modern concepts, there are four scenarios for the formation of a black hole:

1. Gravitational collapse of a fairly massive star (more than 3.6 solar masses) at the final stage of its evolution.

2. Collapse of the central part of the galaxy or pragalactic gas. Modern concepts place a huge black hole at the center of many, if not all, spiral and elliptical galaxies.

3. Formation of black holes at the moment of the Big Bang as a result of fluctuations of the gravitational field and/or matter. Such black holes are called primordial.

4. Emergence of black holes in nuclear reactions of high energies - quantum black holes.

Stellar-mass black holes form as the final stage in the life of some stars. After the complete burnout of thermonuclear fuel and the termination of the reaction, the star should theoretically begin to cool, which will lead to a decrease in internal pressure and compression of the star under the influence of gravity. Compression can stop at a certain stage, or it can turn into a rapid gravitational collapse. Depending on the mass of the star and the rotational moment, it can turn into a black hole.

The conditions (mainly mass) under which the final state of stellar evolution is a black hole have not been studied well enough, since for this it is necessary to know the behavior and states of matter at extremely high densities that are inaccessible to experimental study. Various models give a lower estimate of the mass of a black hole resulting from gravitational collapse, from 2.5 to 5.6 solar masses. The radius of the black hole is very small - a few tens of kilometers.

Supermassive black holes. Expanded very massive black holes, according to modern concepts, form the core of most galaxies. These include the massive black hole at the core of our galaxy.

Primordial black holes currently have the status of a hypothesis. If at the initial moments of the life of the Universe there were sufficient deviations from the homogeneity of the gravitational field and the density of matter, then black holes could form from them by means of collapse. At the same time, their mass is not limited from below, as in stellar collapse - their mass could probably be quite small. The detection of primordial black holes is of particular interest in connection with the possibility of studying the phenomenon of black hole evaporation.

Quantum black holes. It is assumed that stable microscopic black holes, the so-called quantum black holes, can appear as a result of nuclear reactions. A mathematical description of such objects requires a quantum theory of gravity, which has not yet been created. However, from general considerations, it is very likely that the mass spectrum of black holes is discrete and there is a minimal black hole - the Planck black hole. Its mass is about 10 -5 g, radius - 10 -35 m. The Compton wavelength of a Planck black hole is equal in order of magnitude to its gravitational radius.

Even if quantum holes exist, their lifetime is extremely short, making their direct detection very problematic. Recently, experiments have been proposed to find evidence of the appearance of black holes in nuclear reactions. However, for the direct synthesis of a black hole in an accelerator, an energy of 10 26 eV, unattainable today, is required. Apparently, virtual intermediate black holes can appear in superhigh-energy reactions. However, according to string theory, much less energy is required and synthesis can be carried out.

1.3 Black holes with electric charge Reissner-Nordström

During the First World War, G. Reisner and G. Nordström discovered a solution to Einstein's equations of the gravitational field, which completely describes a "charged" black hole. Such a black hole may have an electrical charge (positive or negative) or a magnetic charge (corresponding to the north or south magnetic pole). If electrically charged bodies are commonplace, then magnetically charged bodies are not at all. Bodies that have a magnetic field (for example, an ordinary magnet, a compass needle, the Earth) necessarily have both north and south poles at once. Until very recently, most physicists believed that magnetic poles always occur only in pairs. However, in 1975 a group of scientists from Berkeley and Houston announced that they had discovered a magnetic monopole in one of their experiments. If these results are confirmed, then it will turn out that separate magnetic charges can exist, i.e. that the north magnetic pole can exist separately from the south, and vice versa. The Reisner-Nordström solution allows for the existence of a monopole magnetic field in a black hole. Regardless of how the black hole acquired its charge, all the properties of this charge in the Reisner-Nordström solution are combined into one characteristic - the number Q. This feature is similar to the fact that the Schwarzschild solution does not depend on how the black hole acquired its mass. It could be made up of elephants, stones or stars - the end result will always be the same. In this case, the space-time geometry in the Reisner-Nordström solution does not depend on the nature of the charge. It can be positive, negative, correspond to the north or south magnetic pole - only its full value is important, which can be written as |Q|. So, the properties of a Reisner-Nordström black hole depend only on two parameters - the total mass of the hole M and its total charge |Q| (in other words, from its absolute value). Thinking about real black holes that could actually exist in our Universe, physicists came to the conclusion that the Reisner-Nordström solution turns out to be not very significant, because the electromagnetic forces are much greater than the forces of gravity. For example, the electric field of an electron or a proton is trillions of trillions of times stronger than their gravitational field. This means that if the black hole had a sufficiently large charge, then the huge forces of electromagnetic origin would quickly scatter in all directions the gas and atoms "floating" in space. In the shortest possible time, particles with the same charge sign as the black hole would experience a powerful repulsion, and particles with the opposite charge sign would experience an equally powerful attraction to it. By attracting particles with a charge of the opposite sign, the black hole would soon become electrically neutral. Therefore, we can assume that real black holes have only a small charge. For real black holes, the value of |Q| must be much less than M. Indeed, it follows from the calculations that black holes that could actually exist in space must have a mass M at least a billion billion times greater than |Q|. Mathematically, this is expressed by the inequality

Despite these, alas, unfortunate limitations imposed by the laws of physics, it is very instructive to conduct a detailed analysis of the Reisner-Nordström solution.

To make it easier to understand the features of the Reisner-Nordström solution, consider an ordinary black hole without a charge. As follows from Schwarzschild's solution, such a hole consists of a singularity surrounded by an event horizon. The singularity is located at the center of the hole (for r = 0), and the event horizon is at a distance of 1 Schwarzschild radius (namely, for r = 2M). Now imagine that we gave this black hole a small electrical charge. Once the hole has a charge, we must turn to the Reisner-Nordström solution for the space-time geometry. There are two event horizons in the Reisner-Nordström solution. Namely, from the point of view of a distant observer, there are two positions at different distances from the singularity, where time stops running. With the smallest charge, the event horizon, which was previously at the "height" of 1 Schwarzschild radius, shifts a little lower to the singularity. But even more surprising is the fact that immediately near the singularity, a second event horizon appears. Thus, a singularity in a charged black hole is surrounded by two event horizons - outer and inner. The structures of an uncharged (Schwarzschild) black hole and a charged Reisner-Nordström black hole (for M>>|Q|) are compared in Fig. 1.2.

If we increase the charge of the black hole, then the outer event horizon will shrink, and the inner one will expand. Finally, when the charge of the black hole reaches a value at which the equality M=|Q| is satisfied, both horizons merge with each other. If the charge is increased even more, then the event horizon will completely disappear, and a "bare" singularity remains. At M<|Q| горизонты событий отсутствуют, так что сингулярность открывается прямо во внешнюю Вселенную. Такая картина нарушает знаменитое "правило космической этики", предложенное Роджером Пенроузом. Это правило ("нельзя обнажать сингулярность!") будет подробнее обсуждаться ниже. Последовательность схем на рис. 1.3 иллюстрирует расположение горизонтов событий у черных дыр, имеющих одну и ту же массу, но разные значения заряда.

Rice. 1.2. Charged and neutral black holes. The addition of even a negligible amount of charge leads to the appearance of a second (inner) event horizon directly above the singularity.

We know that Fig. Figure 1.3 illustrates the position of event horizons relative to the singularity of black holes in space, but it is even more useful to analyze space-time diagrams for charged black holes. In order to build such charts - plots of time versus distance, we will start with a "straight line" approach.

Rice. 1.3. Image of charged black holes in space. As charge is added to the black hole, the outer event horizon gradually shrinks, while the inner one expands. When the total charge of the hole reaches the value |Q|= M, both horizons merge into one. At even larger values of the charge, the event horizon disappears altogether and an open, or "bare" singularity remains.

The distance measured outward from the singularity is plotted horizontally, while time, as usual, is plotted vertically. In such a diagram, the left side of the graph is always limited to a singularity, described by a line running vertically from the distant past to the distant future. World lines of event horizons also represent verticals and separate the outer Universe from the inner regions of the black hole.

On fig. Figure 1.4 shows space-time diagrams for several black holes with the same masses but different charges. Above for comparison is a diagram for a Schwarzschild black hole (recall that the Schwarzschild solution is the same as the Reisner-Nordström solution for |Q|=0). If a very small charge is added to this hole, then the second (inner) horizon will be located directly near the singularity. For a black hole with a moderate charge (M > |Q|), the inner horizon is located farther from the singularity, and the outer horizon has decreased its height above the singularity. For a very large charge (M=|Q|; in this case, one speaks of the Reisner-Nordström limit solution), both event horizons merge into one. Finally, when the charge is exceptionally large (M< |Q|), горизонты событий просто исчезают.

Rice. 1.4. Space-time diagrams for charged black holes. This sequence of diagrams illustrates the appearance of space-time for black holes that have the same mass but different charges. Above for comparison is a diagram for a Schwarzschild black hole (|Q|=0).

Rice. 1.5. "Naked" singularity. A black hole whose charge is monstrous (M<|Q|), вообще не окружает горизонт событий. Вопреки "закону космической этики" сингулярность красуется на виду у всей внешней Вселенной.

As can be seen from fig. 1.5, in the absence of horizons, the singularity opens directly into the outer universe. A distant observer can see this singularity, and an astronaut can fly straight into a region of arbitrarily curved space-time without crossing any event horizons. A detailed calculation shows that, immediately next to the singularity, gravity begins to act as a repulsion. Although the black hole attracts the astronaut to itself, as long as he is far enough away from it, but as soon as he approaches the singularity at a very small distance, he will be repelled. The complete opposite of the case of the Schwarzschild solution is the region of space immediately near the Reisner-Nordström singularity - this is the realm of antigravity.

The surprises of the Reisner-Nordström solution are not limited to two event horizons and gravitational repulsion near the singularity. Recalling the above detailed analysis of the Schwarzschild solution, we can think that diagrams like those shown in Figs. 1.4 does not describe all sides of the picture. So, in the Schwarzschild geometry, we encountered great difficulties caused by the superposition of different regions of space-time in a simplified diagram (see Fig. 1.9). The same difficulties await us in diagrams like Fig. 1.4, so it's time to move on to identifying and overcoming them.

It is easier to understand the global structure of space-time by applying the following elementary rules. The diagram, called the Penrose diagram, is shown in fig. 1.6a.

Rice. 1.6a. Penrose diagram for a Schwarzschild black hole. Here you can see the most remote outskirts of the two Universes (I - , I 0 , and I + for each of them).

black hole charged reissner

It can also be called the Penrose diagram for the special case of a Reisner-Nordström black hole when there is no charge (|Q|=0). Moreover, if we deprive the Reisner-Nordström hole of charge (i.e., go to the limit |Q|->0), then our diagram (whatever it is) will necessarily reduce in the limit to the Penrose diagram for the Schwarzschild solution. From this follows our first rule: there must be another Universe, opposite to ours, the achievement of which is possible only along forbidden space-like lines.

When constructing a Penrose diagram for a charged black hole, there are grounds to expect the existence of many universes. Each of them must have five types of infinity (, and).

This I is a timelike infinity in the past. It is the "place" from which all material objects originated (Borya, Vasya, Masha, the Earth, galaxies and everything else). All such objects move along timelike world lines and must go to I + - timelike infinity of the future, somewhere in billions of years after "now". In addition, there is I 0 - spacelike infinity, and since nothing can move faster than light, nothing can ever fall into I 0 . If no objects known to physics move faster than light, then photons move exactly at the speed of light along world lines inclined by 45 degrees on the space-time diagram. This makes it possible to introduce - the light infinity of the past, from where all light rays come. There is, finally, and - the light infinity of the future (where all the "light rays" go).

In addition, each of these outer universes must be drawn as a triangle, since the Penrose conformal mapping method works in this case like a brigade of small bulldozers, "raking" all of space-time into one compact triangle. Therefore, our second rule will be the following: any outer universe must be represented as a triangle with five types of infinities. Such an external Universe can be oriented either to the right (as in Fig. 1.6b) or to the left.

Rice. 1.6b. Outer Universe. In the Penrose diagram for any black hole, the outer universe is always represented by a triangle with five infinities (I", S~, I 0 ,S + , I +). Such an outer universe can be oriented with an angle to the right (as shown in the figure) or to the left.

To arrive at the third rule, recall that in the Penrose diagram (see Fig. 1.6a), the event horizon of a Schwarzschild black hole had a slope of 45 degrees. So, the third rule: any event horizon must be light-like, and therefore always has a slope of 45 degrees.

To derive the fourth (and last) rule, recall that when passing through the event horizon, space and time changed roles in the case of a Schwarzschild black hole. From a detailed analysis of spacelike and timelike directions for a charged black hole, it follows that the same picture will be obtained here as well. Hence the fourth rule: space and time reverse roles whenever the event horizon crosses.

On fig. 1.7, the fourth rule just formulated is illustrated for the case of a black hole with a small or moderate charge (M>|Q|). Away from such a charged black hole, the spacelike direction is parallel to the space axis, and the timelike direction is parallel to the time axis. Passing under the outer event horizon, we will find the roles of these two directions reversed - the spacelike direction is now parallel to the time axis, and the timelike direction is parallel to the spatial axis. However, as we continue to move towards the center and descend under the inner event horizon, we are witnessing a second role reversal. Near the singularity, the orientation of the spacelike and timelike directions becomes the same as it was away from the black hole.

Rice. 1.7. Change of roles of space and time (for М>|Q|). Every time you cross the event horizon, space and time reverse roles. This means that in a charged black hole, due to the presence of two event horizons, the complete reversal of the roles of space and time occurs twice.

The double reversal of the roles of the spacelike and timelike directions is of decisive importance for the nature of the singularity of a charged black hole. In the case of a Schwarzschild black hole, which has no charge, space and time change roles only once. Within a single event horizon, lines of constant distance point in a spacelike (horizontal) direction. This means that the line representing the location of the singularity (r = 0) must be horizontal, i.e. directed spatially. However, when there are two event horizons, lines of constant distance near the singularity have a timelike (vertical) direction. Therefore, the line describing the position of the singularity of a charged hole (r = 0) must be vertical, and it must be oriented in a timelike manner. So we come to a conclusion of paramount importance: the singularity of a charged black hole must be timelike!

Now, using the above rules, we can construct a Penrose diagram for the Reisner-Nordström solution. Let's start by imagining an astronaut in our Universe (say, just on Earth). He gets into his spaceship, turns on the engines and heads towards the charged black hole. As can be seen from fig. 1.8, our Universe looks like a triangle with five infinities on the Penrose diagram. Any admissible path of an astronaut must always be oriented on the diagram at an angle of less than 45 degrees to the vertical, since he cannot fly at a superluminal speed.

Rice. 1.8. Section of the Penrose diagram. Part of the Penrose diagram for the Reisner-Nordström solution can be constructed by considering the possible world lines of an astronaut heading out of our universe into a charged black hole.

On fig. 1.8 such admissible world lines are shown by a dotted line. As an astronaut approaches a charged black hole, he descends below the outer event horizon (which should be tilted exactly 45 degrees). Having passed this horizon, the astronaut will never be able to return to our Universe. However, it could drop further below the inner event horizon, which also has a 45-degree slope. Beneath this inner horizon, an astronaut could foolishly encounter a singularity where he would be subject to gravitational repulsion and where space-time is infinitely curved. Note, however, that the tragic outcome of the flight is by no means inevitable! Since the singularity of a charged black hole is timelike, it should be represented by a vertical line on the Penrose diagram. An astronaut can avoid death by simply steering his spacecraft away from the singularity along a permitted time-like path, as depicted in Fig. 1.8. The rescue trajectory takes him away from the singularity, and he again crosses the inner event horizon, which also has a 45-degree slope. Continuing the flight, the astronaut goes beyond the outer event horizon (and it has an inclination of 45 degrees) and enters the outer Universe. Since such a journey obviously takes time, the sequence of events along the world line must proceed from the past to the future. Therefore, an astronaut cannot return to our Universe again, but will fall into another Universe, the Universe of the future. As one would expect, this future Universe should look like a triangle with the usual five infinities on the Penrose diagram.

It should be emphasized that in constructing these Penrose diagrams we again encounter both black and white holes. The astronaut can jump out through the event horizons and find himself in the outer universe of the future. Most physicists are convinced that in principle there can be no white holes in nature. But we will still continue the theoretical analysis of the global structure of space-time, including the existence of black and white holes side by side with each other.

The described flight episodes and diagrams in fig. 1.8 should be no more than a fragment of a whole. The Penrose diagram for a charged black hole needs to be supplemented with at least one instance of another Universe opposite to ours, which is reachable only along (forbidden) spacelike world lines. This conclusion is based on our rule 1: if we remove its charge from the black hole, then the Penrose diagram should be reduced to the image of the Schwarzschild solution. And although no one from our Universe will ever be able to penetrate this "other" Universe due to the impossibility of moving faster than light, we can still imagine an astronaut from that other Universe traveling to the same charged black hole. Its possible world lines are shown in Fig. 1.9.

Rice. 1.9. Another section of the Penrose diagram. This new section of the Penrose diagram for the Reisner-Nordström solution can be constructed by considering the possible world lines of an astronaut from an alien universe.

Such a journey of an alien astronaut from another Universe looks exactly the same as the journey of an astronaut who has taken off from our Universe, from the Earth. The Alien Universe is also depicted on the Penrose diagram as a familiar triangle. On the way to the charged black hole, the alien astronaut crosses the outer event horizon, which should have a 45-degree slope. Later, it also descends under the inner event horizon, also with a 45-degree tilt. The alien is now faced with a choice: either crash into a time-like singularity (it is vertical on the Penrose diagram), or collapse and cross the inner event horizon again. To avoid an unfortunate end, the alien decides to leave the black hole and exit through the inner event horizon, which, as usual, has a 45-degree slope. Then it flies through the outer event horizon (tilted on the Penrose diagram by 45 degrees) into the new Universe of the future.

Each of these two hypothetical journeys covers only two parts of the complete Penrose diagram. The complete picture is obtained by simply combining these parts with each other, as shown in Fig. 1.10.

Rice. 1.10. The complete Penrose diagram for a Reisner-Nordström black hole (M > > |Q|). A complete Penrose diagram for a black hole with a small or moderate charge (M > |Q|) can be constructed by connecting the sections shown in Fig. 1.8 and 1.9. This diagram repeats ad infinitum both in the future and in the past.

Such a diagram must be repeated an infinite number of times in the future and in the past, since each of the two astronauts considered could decide to leave the Universe in which he surfaced again and again go into a charged black hole. Thus, astronauts can penetrate into other universes, even more distant in the future. Similarly, we can imagine other astronauts from Universes in the distant past arriving in our Universe. Therefore, the full Penrose diagram repeats in both directions in time, like a long tape with a repeating stencil pattern. In general, the global geometry of a charged black hole combines an infinite number of universes in the past and in the future with our own universe. This is as amazing as the fact that, using a charged black hole, an astronaut can fly from one universe to another. This incredible picture is closely related to the idea of a white hole, which will be discussed in one of the following chapters.

The just described approach to the elucidation of the global structure of space-time concerned the case of black holes with a small or small charge (M>|Q|). However, in the case of the ultimate Reisner-Nordström black hole (when M=|Q|), the charge turns out to be so large that the inner and outer horizons merge with each other. This combination of two event horizons leads to a number of interesting consequences.

Recall that far from a charged black hole (outside the outer event horizon), the spacelike direction is parallel to the space axis, and the timelike direction is parallel to the time axis. Recall also that near the singularity (below the inner event horizon - after space and time have switched roles twice) the spacelike direction is again parallel to the space axis, and the timelike direction is again parallel to the time axis. As the charge of the Reisner-Nordström black hole increases more and more, the region between the two event horizons gets smaller and smaller. When, finally, the charge increases so much that M = |Q|, this intermediate region will shrink to zero. Consequently, when passing through an externally-internally united event horizon, space and time do not change roles. Of course, one can just as well speak of a double reversal of roles in space and time, occurring simultaneously on the only event horizon of the ultimate Reisner-Nordström black hole. As shown in fig. 1.11, the timelike direction in it is everywhere parallel to the time axis, and the spacelike direction is everywhere parallel to the spatial axis.

Rice. 1.11. Space-time diagram for the limiting Reisner-Nordström black hole (M=|Q|). When the black hole's charge becomes so large that M=|Q|, the inner and outer event horizons merge. This means that when passing through the resulting (double) horizon, there is no change of roles for space and time.

Although the ultimate Reisner-Nordström black hole has only one event horizon, the situation here is completely different than in the case of the Schwarzschild black hole, which also has only one event horizon. With a single event horizon, there is always a change in the roles of space- and time-like directions, as can be seen in Fig. 1.12. However, the event horizon of the ultimate Reisner-Nordström black hole can be interpreted as a "double", i.e. as superimposed inner and outer horizons. That is why there is no change in the roles of space and time.

Rice. 1.12. Space-time diagram for a Schwarzschild black hole (|Q|=0). Although a Schwarzschild black hole (which has no charge) has only one event horizon, as it moves from one side to the other, space and time change roles. (Compare with Fig. 1.11.)

The fact that the outer and inner event horizons merge at the ultimate Reisner-Nordström black hole means that a new Penrose diagram is required. As before, it can be constructed by considering the world line of a hypothetical astronaut. At the same time, the list of rules remains the same, with the significant exception that when crossing the event horizon, space and time do not change roles. Imagine an astronaut taking off from the Earth and falling into the ultimate Reisner-Nordström black hole. Our universe, as usual, is depicted as a triangle on a Penrose diagram. After diving under the event horizon, the astronaut is free to make a choice: he can either crash into the singularity, which is timelike and therefore should be depicted vertically on the Penrose diagram, or (Fig. 1.13) take his spacecraft away from the singularity along the allowed timelike world line.

Rice. 1.13. Penrose diagram for the ultimate Reisner-Nordström black hole (M=|Q|). A diagram of the global structure of space-time can be constructed if we consider the possible world lines of an astronaut diving into the ultimate Reisner-Nordström black hole and emerging from it.

If he chose the second path, then later he would cross the event horizon again, entering another universe. He will again have an alternative - to stay in this future Universe and fly to some planets, or turn back and go into a black hole again. If the astronaut turns back, he will continue his way up the Penrose diagram, visiting any number of universes of the future. The full picture is shown in Fig. 1.13. As before, the diagram repeats an infinite number of times in the past and in the future, like a ribbon with a repeating stencil pattern.

From the point of view of mathematics, a black hole with a huge charge M<|Q|; правда, она не имеет смысла с точки зрения физики. В этом случае горизонты событий попросту исчезают, остается лишь "голая" сингулярность. Ввиду отсутствия горизонтов событий не может быть и речи о каком-то обмене ролями между пространством и временем. Сингулярность просто находится у всех на виду. "Голая" сингулярность - это не закрытая никакими горизонтами область бесконечно сильно искривленного пространства-времени.

If an astronaut, having taken off from the Earth, rushes to a "naked" singularity, he does not have to descend under the event horizon. It remains all the time in our universe. Near the singularity, powerful repulsive gravitational forces act on it. With sufficiently powerful engines, the astronaut under certain conditions could crash into the singularity, although this is pure madness on his part.

Rice. 1.14. "Naked" singularity. At the "naked" singularity (M<|Q|) горизонтов событий нет. Черная дыра этого типа не связывает нашу Вселенную с какой-либо другой Вселенной.

A simple fall on a singularity - a "bare" singularity does not connect our Universe with any other Universe. As in the case of any other charged black holes, here the singularity is also timelike and therefore should be depicted as a vertical line on the Penrose diagram. Since, apart from our Universe, there are no other Universes now, the Penrose diagram for the "bare" singularity looks quite simple. From fig. 1.14 it can be seen that our Universe, as usual, is depicted as a triangle with five infinities, bounded on the left by a singularity. Whatever is to the left of the singularity is cut off from us completely. No one and nothing can pass through the singularity.

Since real black holes can only have very weak charges (if they have any at all), much of what has been described above is only of academic interest. However, as a result, we have established trouble-free rules for constructing complex Penrose diagrams.

Chapter 2

2.1 Mathematical description of the model

The Reissner-Nordström metric is given by:

where the metric coefficient B(r) is defined as follows:

This is an expression in geometric units, where the speed of light and Newton's constant of gravity are both equal to one, C = G = 1. In conventional units, .

The horizons converge when the metric coefficient B(r) is equal to zero, which occurs on the outer and inner horizons r + and r-:

In terms of horizon location r ± , the metric coefficient B(r) is defined as follows:

Figure 2.1 shows a diagram of the Reissner-Nordström space. This is a space diagram of the Reissner-Nordström geometry. The horizontal axis represents radial distance and the vertical axis represents time.

The two vertical red lines are the inner and outer horizons, at the radial positions r+ and r-. The yellow and ocher lines are world lines of light rays moving radially in and out respectively. Each point at radius r in the space-time diagram is a 3-dimensional space sphere of a circle, as measured by observers at rest in Reissner-Nordström geometry. The dark purple lines are the Reissner-Nordström constant time lines, while the vertical blue lines are the constant circle lines of radius r. The bright blue line marks the zero radius, r = 0.

Rice. 2.1. Diagram of the Reissner-Nordström space

Like the Schwarzschild geometries, the Reissner-Nordström geometry exhibits poor behavior at its horizons with light rays tending to the asymptotes at the horizons without passing through them. Again, pathology is a sign of a static coordinate system. Incident light rays actually pass through horizons, and have no features on either horizon.

As in Schwarzschild geometry, there are systems that behave better on horizons, and which show more clearly the physics of the Reissner-Nordström geometry. One of these coordinate systems is the Finkelstein coordinate system.

Rice. 2.2. Finkelstein space scheme for Reissner-Nordström geometry

As usual, Finkelstein's radial coordinate r is the radius of a circle, defined so that the corresponding circle of the ball at radius r is 2pr , while Finkelstein's time coordinate is defined so that the radially incident light rays (yellow lines) move at an angle of 45o on space-time diagram.

The Finkelstein time t F is related to the Reissner-Nordström time t by the following expression:

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Gravitational g(r) at radial position r is the internal acceleration

g(r) =

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dt ff

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Coloring of the lines, as in the case of the Schwarzschild black hole: the red horizon line, the blue line is the line at zero radius, the yellow and ocher lines are respectively the world lines for radially incident and outgoing light rays, while the dark magenta and blue lines are respectively lines of constant Schwarzschild time and constant circle radius.

Consider the waterfall model of the Reissner-Nordström space. The waterfall model works well for a charged black hole in Reissner-Nordström geometry. However, while in Schwarzschild geometry the waterfall falls at an ever increasing speed all the way to the central singularity, in Reissner-Nordström geometry the waterfall slows down due to the gravitational repulsion produced by the tension or negative pressure of the electric field.

The Reissner-Nordström waterfall is described by exactly the same Gulstrand-Pineleave metric as for the Schwarzschild metric, but the mass M for space velocity is replaced by the mass M(r) of the inner radius r:

Fig 2.3. Reissner-Nordström waterfall.

The internal mass M(r) is equal to the mass M as seen at infinity, minus the mass-energy Q 2 / (2r) in the electric field

The electromagnetic mass Q 2 / (2r) is the mass outside r, associated with the energy density E 2 / (8p) of the electric field E = Q / r 2 surrounding the charge Q.

The speed of the incoming space v exceeds the speed of light c on the outer horizon r + = M + (M 2 - Q 2) 1 / 2, but slows down to a lower speed than the speed of light on the inner horizon r - = M - (M 2 - Q 2 ) 12 . The speed slows down until the zero point r 0 = Q 2 /(2M) inside the inner horizon. At this point, space turns around and accelerates back, reaching the speed of light once again at the inner horizon r - . Space is now entering the white hole, where space is moving outward faster than light. Rice. Figure 2.3 shows a white hole in the same place as a black hole, but in fact, as seen from the Penrose diagram, a white hole and a black hole are different regions of space-time. As space falls outward in the white hole, the gravitational repulsion produced by the negative pressure of the electric field weakens relative to the gravitational attraction of the mass. Outgoing space slows down to the speed of light at the outer horizon r + of the white hole. This space opens into a new region of space-time, possibly a new universe.

2.2 Results of Reissner-Nordstrom charged black hole simulation in Delphi programming environment

Modeling was carried out according to the block method. The program works in five modes in which it is possible to view the space of a black hole from different points of view.

1. Viewing the structure of a black hole. Allows you to simulate the change in the position of the inner and outer horizons depending on the charge of the black hole. At the minimum charge Q = 0, only one outer horizon is observed, as shown in Fig. 2.4.

Rice. 2.4. The outer horizon of a black hole at zero charge.

As the charge increases, an inner horizon appears. In this case, the outer horizon shrinks as the inner horizon increases. You can increase the charge by dragging the slider marker to the desired position (see Fig. 2.5).

Rice. 2.5. The outer and inner horizons of a black hole in the presence of a charge.

As the charge increases to a value equal to the mass of the black hole, the inner and outer horizons merge into one, as shown in Fig. 2.6.

Rice. 2.6. The outer and inner horizons merge into one at a charge equal to the mass of the black hole.

When the value of the black hole's mass charge is exceeded, the horizons disappear and a bare singularity opens up.

2. Modeling the space diagram in Reissner-Nordström. This mode allows you to see the change in the direction of the incoming and outgoing light rays, represented in the Reissner-Nordström geometry. As the charge changes, the picture changes. The change in the rays of light can be traced in Fig. 2.7, 2.8 and 2.9.

Rice. 2.7. Space diagram of the Reissner-Nordström geometry at zero charge.

The two vertical red lines are the inner and outer horizons. Yellow lines are world lines of light rays moving radially inward from bottom to top, ocher lines are world lines of light rays moving radially outward also from bottom to top.

The change in direction (top-down) of the yellow incoming rays between the two horizons demonstrates the change in space and time on the outer and inner horizons, which occurs twice.

The incoming yellow light rays have asymptotes at the horizons, which does not reflect the real picture due to the peculiarities of the Reissner-Nordström geometry. In fact, they pass through the horizons, and have no asymptotes on them.

Rice. 2.8. Space diagram of the Reissner-Nordström geometry in the presence of a charge.

Kerr-Newman field

Postponing the discussion of solutions with magnetic and gauge charges until § 18, let us consider in more detail the Kerr-Newman solution describing a rotating electrically charged

black hole. In the Boyer-Lindqvist coordinates, the square of the space-time interval has the form

where the standard notation is introduced

4-potential (-form) of the electromagnetic field, defined by the relation

for does not differ from the potential of a point charge in the Minkowski space. An additional term proportional to a coincides at spatial infinity with the potential of the magnetic dipole. The nonzero components of the contravariant metric tensor are

For the Kerr-Newman metric, there are thirty non-zero Christoffel symbols, of which twenty-two are pairwise equal

where indicated

The Christoffel symbols are even difference functions and do not vanish in the equatorial plane of the Kerr metric. The rest of the connectivity components are odd with respect to reflection in the plane, where they take on zero values. It is useful to keep this in mind when solving the equations of particle motion.

The non-zero components of the electromagnetic field tensor are equal to

which corresponds to the superposition of the Coulomb field and the magnetic dipole field.

The line element (1) does not depend on the coordinates, so the vectors

are Killing vectors generating shifts in time and rotations around the axis of symmetry. Killing vectors and are not orthogonal to each other

The symmetry of the electromagnetic field with respect to the transformations given by the Killing vectors is expressed in the equality to zero of the Lie derivatives of the 4-potential (3) along the vector fields (8),

The vector of time is similar in the region bounded by the inequality

and becomes isotropic on the surface of the ergosphere

which is an ellipsoid of revolution. Inside the ergosphere, the vector is spacelike, but there is a linear combination of the Killing vectors

which is a timelike Killing vector inside the ergosphere if the inequality

The surface on which they merge is the event horizon, its position is determined by the large root of the equation

where do we find where

The value plays the role of the angular velocity of the horizon rotation; in accordance with the general theorem, it does not depend on the angle

The event horizon is an isotropic hypersurface whose spatial section has the topology of a sphere. The area of the two-dimensional surface of the horizon is calculated by the formula

which leads to the result

According to Hawking's theorem, the surface area of the event horizon of a black hole immersed in a material medium whose energy-momentum tensor satisfies the energy dominance conditions cannot decrease. The mass and moment of rotation of the hole can individually decrease, while, having completely lost the rotational moment, the black hole will turn out to have a mass of at least

which has been called the "irreducible" mass of a black hole. The law of non-decreasing of the area of the event horizon has a common nature with the law of increasing entropy, it can be associated with the loss of information about the state of matter that is under the event horizon. If a black hole did not have some

entropy, then the absorption of, say, a heated gas in the outer space would lead to a decrease in entropy. Invoking quantum considerations eliminates the danger of a contradiction with the second law of thermodynamics, because it turns out that in quantum gravity the entropy of a black hole is indeed proportional to the surface area of the event horizon (21) in units of the square of the Planck length

This also corresponds to earlier calculations of the effect of particle production in black holes in the framework of the semiclassical theory. The total entropy of the black hole and the absorbed matter does not decrease in this case, since the mass (and, possibly, the rotational moment) of the black hole increases during absorption, as a result of which the surface area of the event horizon increases. It should be noted that the denominator in (23) is extremely small; therefore, with a macroscopic change in the horizon area, the black hole entropy changes by a very large value.

At the event horizon, a linear combination of the components of the 4-potential is constant, which has the meaning of the electrostatic potential of the horizon for an observer rotating with the horizon

Also constant is the quantity called the “surface gravity” of a black hole, which is equal to the acceleration (in units of coordinate time) of a particle held at rest on the horizon, in an invariant form

where the vector is determined by formula (14). at (i.e., is an isotropic vector lying on the hypersurface

Another isotropic vector normalized by the condition For the Kerr-Newman metric, the surface gravity of the horizon is

What is the electric charge of a black hole? For "normal" black holes of astronomical scales this question is silly and meaningless, but for miniature black holes it is quite relevant. Let's say a miniature black hole ate a little more electrons than protons and acquired a negative electrical charge. What happens when a charged miniature black hole is inside dense matter?

To begin with, let's roughly estimate the electric charge of a black hole. Let's number the charged particles falling into the black hole starting from the very beginning of the tiryampampation that led to its appearance, and start summing up their electric charges: proton - +1, electron - -1. Consider this as a random process. The probability of getting +1 at each step is 0.5, so we have a classic example of a random walk, i.e. the average electric charge of a black hole, expressed in elementary charges, will be equal to

Q = sqrt(2N/π)

where N is the number of charged particles absorbed by the black hole.

Let's take our favorite 14-kiloton black hole and calculate how many charged particles it ate.

N = M/m proton = 1.4*10 7 /(1.67*10 -27) = 8.39*10 33
Hence q = 7.31*10 16 elementary charges = 0.0117 C. It would seem a little - such a charge passes in a second through the filament of a 20-watt light bulb. But for a static charge, the value is not sickly (a bunch of protons with such a total charge weighs 0.121 nanograms), and for a static charge of an object the size of an elementary particle, the value is simply fucking.

Let's see what happens when a charged black hole gets inside relatively dense matter. To begin with, consider the simplest case - gaseous diatomic hydrogen. The pressure will be assumed to be atmospheric, and the temperature to be room temperature.

The ionization energy of a hydrogen atom is 1310 kJ/mol or 2.18*10 -18 per atom. The covalent bond energy in a hydrogen molecule is 432 kJ/mol or 7.18*10 -19 J per molecule. The distance to which the electrons need to be dragged away from the atoms, we will take as 10 -10 m, it seems to be enough. Thus, the force acting on a pair of electrons in a hydrogen molecule during ionization should be equal to 5.10 * 10 -8 N. For one electron - 2.55 * 10 -8 N.

According to Coulomb's law

R = sqrt(kQq/F)

For a 14 kiloton black hole we have R = sqrt (8.99*10 9 *0.0117*1.6*10 -19 /2.55*10 -8) = 2.57 cm.

Electrons torn from atoms receive a starting acceleration of at least 1.40 * 10 32 m/s 2 (hydrogen), ions - at least 9.68 * 10 14 m / s 2 (oxygen). There is no doubt that all particles of the required charge will be absorbed by the black hole very quickly. It would be interesting to calculate how much energy particles of the opposite charge will have time to throw into the environment, but counting integrals breaks :-(I don’t know how to do this without integrals :-(Offhand, visual effects will vary from very small ball lightning to completely decent fireball.

With other dielectrics, a black hole does about the same thing. For oxygen the ionization radius is 2.55 cm, for nitrogen it is 2.32 cm, for neon it is 2.21 cm, and for helium it is 2.07 cm. For crystals, the permittivity is different in different directions, and the ionization zone will have a complex shape. For diamond, the average ionization radius (based on the table value of the permittivity constant) will be 8.39 mm. I'm sure I lied about little things almost everywhere, but the order of magnitude should be like this.

So, a black hole, having got into a dielectric, quickly loses its electric charge, without producing any special effects, except for the transformation of a small volume of dielectric into plasma.

If it hits a metal or plasma, a stationary charged black hole neutralizes its charge almost instantly.

Now let's see how the electric charge of a black hole affects what happens to a black hole in the bowels of a star. In the first part of the treatise, the characteristics of the plasma in the center of the Sun were already given - 150 tons per cubic meter of ionized hydrogen at a temperature of 15,000,000 K. For now, we brazenly ignore helium. The thermal speed of protons under these conditions is 498 km/s, while electrons fly at almost relativistic speeds – 21,300 km/s. Capturing such a fast electron by gravity is almost impossible, so the black hole will quickly gain a positive electric charge until an equilibrium is reached between the absorption of protons and the absorption of electrons. Let's see what kind of balance it will be.

The force of gravity acting on the proton from the side of the black hole

F p \u003d (GMm p - kQq) / R 2

The first "electrospace" :-) speed for such a force is obtained from the equation

mv 1 2 /R = (GMm p - kQq)/R 2

v n1 = sqrt((GMm n - kQq)/mR)

The second "electrocosmic" speed of the proton is

v n2 = sqrt(2)v 1 = sqrt(2(GMm n - kQq)/(m n R))

Hence, the proton absorption radius is equal to

R p = 2(GMm p - kQq)/(m p v p 2)

Similarly, the electron absorption radius is

R e \u003d 2 (GMm e + kQq) / (m e v e 2)

For protons and electrons to be absorbed with equal intensity, these radii must be equal, i.e.

2(GMm p - kQq)/(m p v p 2) = 2(GMm e + kQq)/(m e v e 2)

Note that the denominators are equal, and reduce the equation.

GMm p - kQq = GMm e + kQq

Surprisingly, nothing depends on the plasma temperature. We decide:

Q \u003d GM (m p - m e) / (kq)

We substitute the numbers and with surprise we get Q \u003d 5.42 * 10 -22 C - less than the electron charge.

We substitute this Q into R p = R e and with even greater surprise we get R = 7.80 * 10 -31 - less than the radius of the event horizon for our black hole.

PREVED MEDVED

The conclusion is equilibrium at zero. Each proton swallowed by the black hole immediately leads to the swallowing of an electron and the charge of the black hole again becomes zero. Replacing a proton with a heavier ion does not fundamentally change anything - the equilibrium charge will not be three orders of magnitude less than the elementary one, but one, so what?

So, the general conclusion is that the electric charge of a black hole does not significantly affect anything. And it looked so tempting...

In the next part, if neither the author nor the readers get bored, we will consider a miniature black hole in dynamics - how it rushes through the bowels of a planet or star and devours matter on its way.