Do black holes have charges? A black hole is the most mysterious object in the universe. Reissner-Nordström solution

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.

When man began to study space, he encountered a mysterious phenomenon. It is called "black hole". It turns out that in space-time there is a certain region with a high gravitational attraction. As a result, even objects moving at the speed of light cannot get out of it.

We are talking about the quanta of the light itself. These areas are really black, absorbing everything around and never letting go. We can only guess about their nature and capabilities, and the lack of information about this phenomenon gives rise to some myths.

Myths about black holes

Albert Einstein was the first to announce the existence of black holes. It would seem, who, if not this great scientist, theoretician of time and space, should declare the existence of black holes? In fact, he was not the first to make such an assumption, but John Mitchell. This happened back in 1783, while Einstein created his theory in 1916. However, in those days, the theory turned out to be unclaimed, the English priest Mitchell simply did not find application for it. He himself began to think about black holes, accepting Newton's theory of the nature of light. In those days, it was believed that it consists of the smallest material particles, photons. Thinking about their movement, Mitchell realized that it completely depends on the gravitational field of the star from where the particles start their journey. The scientist wondered what would happen to photons if the gravitational field was so large that it didn't release light at all. Interestingly, it is Mitchell who is considered the founder of seismology as we know it. The English priest was the first to suggest. That earthquakes spread over the surface like waves.

Black stars do not absorb space. Space can be thought of as a sheet of rubber. Then the planets will be some kind of balls that put pressure on him. As a result, deformation occurs, and straight lines disappear. This is how gravity appears, which explains the movement of planets around stars. With increasing mass, the deformation only increases. Additional perturbations of the field appear, which determine the force of attraction. Orbital velocities are increasing, which implies more and more rapid movement of bodies around the object. For example, the planet Mercury moves around the Sun at a speed of 48 km / s, and stars move in space near black holes 100 times faster! In the case of a strong gravitational force, a collision between the satellite and larger objects is possible. And all this mass tends to the center - to the black hole.

All black holes are the same. It seems to many of us that this term belongs to objects that are essentially the same. However, astronomers have come to the conclusion that black holes have several varieties. There are holes that rotate, some have an electric charge, and there are those who have both features. Typically, such objects appear by absorbing matter, while a rotating black hole appears when two ordinary ones merge. Such formations, due to the increased perturbation of space, begin to consume much more energy. A charged black hole turns into one huge particle accelerator. A classic example of an object of this class is GRS 1915+105. This black hole spins at a speed of 950 revolutions per second, and it is located at a distance of 35,000 light years from our planet.

The density of black holes is low. These objects, for their size, need to be very heavy in order to generate an attractive force to hold light within them. So, if the mass of the Earth is compressed to the density of a black hole, you get a ball with a diameter of 9 millimeters. A dark object, 4 million times the mass of the Sun, could fit between Mercury and our star. Those black holes that are in the center of galaxies can weigh 10-30 million times more than the Sun. Such a grandiose mass in a relatively small volume means that black holes have a huge density and the processes occurring inside are very strong.

Black holes are very quiet. It is hard to imagine that a huge dark object, sucking everything around into itself, was also noisy. In fact, everything that falls into this abyss moves with constant acceleration. As a result, at the edge of space-time, which we can still feel due to the finiteness of the speed of light, the particles are accelerated almost to the speed of light. When matter begins to move to its limiting speeds, a gurgling sound appears. It is a consequence of the transformation of the energy of movement into sound waves. As a result, the black hole turns out to be a very noisy object. In 2003, astronomers working at the space-based Chandra X-ray Observatory were able to detect sound waves emanating from a massive black hole. But it is located at a distance of 250 million light years from us, which once again indicates the noisiness of such objects.

Nothing can escape the attraction of black holes. This statement is correct. After all, when some large or small objects are near a black hole, they will certainly be captured by its gravitational field. At the same time, it can be both a small particle and a planet, a star or even a galaxy. However, if this object is affected by a force greater than the attraction of the black hole, then it will be able to avoid death captivity. It could be, for example, a rocket. But this is possible before the object reaches the event horizon, when light can still escape from captivity. After this border, it will be impossible to escape from the embrace of the all-consuming space monster. After all, in order to escape beyond the horizon, it is necessary to develop a speed greater than the speed of light. And this is impossible even theoretically. So black holes are truly black - since light can never get out, we can't look inside this mysterious object. Scientists believe that even a small black hole will tear an unwitting observer into particles before reaching the event horizon. The force of attraction grows not only as we approach the center of the planet and star, but also towards the black hole. If you fly feet first towards it, then the force of gravity in the feet will be much higher than in the head, and will lead to an instant rupture of the body.

Black holes don't change time. Light goes around the event horizon, but eventually it penetrates inside and disappears into non-existence. So what happens to the clock if it falls into a black hole and continues to work there? As they approach the event horizon, they will begin to slow down until they finally stop. Such a stop of time is associated with its gravitational slowdown, which explains Einstein's theory of relativity. A black hole has such a great gravitational force that it can slow down time. From the point of view of the watch, nothing will change, but it will disappear from view, and the light from it will be stretched under the influence of a heavy object. Light will begin to move into the red spectrum, its wavelength will increase. As a result, he will finally become invisible.

A black hole does not produce any energy. It is known that these objects draw in the entire surrounding mass. Scientists suggest that everything inside is compressed so much that even the space between atoms is reduced. As a result, subatomic particles are born that can fly out. In this they are helped by magnetic field lines that cross the event horizon. As a result, the release of such particles generates energy, and the method itself is quite effective. The conversion of mass into energy in this case gives a 50 times greater return than in the course of nuclear fusion. The black hole itself appears as a huge reactor.

There is no relationship between stars and the number of black holes. Once Carl Sagan, a famous astrophysicist, said that there are more stars in the universe than grains of sand on the beaches of the whole world. Scientists believe that this number is still finite and is 10 to the power of 22. What does it have to do with black holes? It is their number that determines the number of stars. It turns out that the particle streams released by black objects expand into bubbles of some kind that can spread through star formation sites. These areas are located in gas clouds, which, when cooled, give rise to luminaries. And the particle streams heat the gas clouds and prevent new stars from appearing. As a result, there is a constant balance between the activity of black holes and the number of stars in the universe. After all, if there are too many stars in the galaxy, then it will turn out to be too hot and explosive, it will be difficult for life to arise there. And, on the contrary, a small number of stars also will not help life to arise.

Black holes are made of different material than we are. A number of scientists believe that black holes help with the birth of new elements. And this can be understood, given the splitting of matter into the smallest subatomic particles. They then participate in the formation of stars, which eventually leads to the appearance of elements heavier than helium. We are talking about carbon and iron, necessary for the appearance of solid planets. As a result, these elements are part of everything that has mass, that is, the person himself. It is likely that the true builder of our body is some distant black hole.

The concept of a black hole is known to everyone - from schoolchildren to the elderly, it is used in science and fiction literature, in the yellow media and at scientific conferences. But not everyone knows what exactly these holes are.

From the history of black holes

1783 The first hypothesis for the existence of such a phenomenon as a black hole was put forward in 1783 by the English scientist John Michell. In his theory, he combined two creations of Newton - optics and mechanics. Michell's idea was this: if light is a stream of tiny particles, then, like all other bodies, particles should experience the attraction of a gravitational field. It turns out that the more massive the star, the more difficult it is for light to resist its attraction. 13 years after Michell, the French astronomer and mathematician Laplace put forward (most likely independently of his British counterpart) a similar theory.

1915 However, all their works remained unclaimed until the beginning of the 20th century. In 1915, Albert Einstein published the General Theory of Relativity and showed that gravity is a curvature of space-time caused by matter, and a few months later, the German astronomer and theoretical physicist Karl Schwarzschild used it to solve a specific astronomical problem. He explored the structure of the curved space-time around the Sun and rediscovered the phenomenon of black holes.

(John Wheeler coined the term "black holes")

1967 American physicist John Wheeler outlined a space that can be crumpled, like a piece of paper, into an infinitesimal point and designated the term "Black Hole".

1974 British physicist Stephen Hawking proved that black holes, although they swallow matter without a return, can emit radiation and eventually evaporate. This phenomenon is called "Hawking radiation".

2013 The latest research on pulsars and quasars, as well as the discovery of cosmic microwave background radiation, has finally made it possible to describe the very concept of black holes. In 2013, the gas cloud G2 came very close to the black hole and is likely to be absorbed by it, observing the unique process provides great opportunities for new discoveries of the features of black holes.

(Massive object Sagittarius A *, its mass is 4 million times greater than the Sun, which implies a cluster of stars and the formation of a black hole)

2017. A group of scientists from the Event Horizon Telescope collaboration of several countries, linking eight telescopes from different points of the Earth's continents, carried out observations of a black hole, which is a supermassive object and is located in the M87 galaxy, the constellation Virgo. The mass of the object is 6.5 billion (!) solar masses, gigantic times larger than the massive object Sagittarius A *, for comparison, the diameter is slightly less than the distance from the Sun to Pluto.

The observations were carried out in several stages, starting from the spring of 2017 and during the periods of 2018. The amount of information was calculated in petabytes, which then had to be deciphered and a genuine image of an ultra-distant object obtained. Therefore, it took another two whole years to pre-scan all the data and combine them into one whole.

2019 The data was successfully decoded and brought into view, producing the first ever image of a black hole.

(The first ever image of a black hole in the M87 galaxy in the constellation Virgo)

Image resolution allows you to see the shadow of the point of no return in the center of the object. The image was obtained as a result of interferometric observations with an extra long baseline. These are the so-called synchronous observations of one object from several radio telescopes, interconnected by a network and located in different parts of the globe, directed in one direction.

What are black holes really?

A laconic explanation of the phenomenon sounds like this.

A black hole is a space-time region whose gravitational attraction is so strong that no object, including light quanta, can leave it.

A black hole was once a massive star. As long as thermonuclear reactions maintain high pressure in its bowels, everything remains normal. But over time, the supply of energy is depleted and the celestial body, under the influence of its own gravity, begins to shrink. The final stage of this process is the collapse of the stellar core and the formation of a black hole.

• 1. Ejection of a black hole jet at high speed


• 2. A disk of matter grows into a black hole


• 3. Black hole


• 4. Detailed scheme of the black hole region


• 5. Size of found new observations

The most common theory says that there are similar phenomena in every galaxy, including in the center of our Milky Way. The huge gravity of the hole is capable of holding several galaxies around it, preventing them from moving away from each other. The "coverage area" can be different, it all depends on the mass of the star that has turned into a black hole, and can be thousands of light years.

Schwarzschild radius

The main property of a black hole is that any matter that gets into it can never return. The same applies to light. At their core, holes are bodies that completely absorb all the light that falls on them and do not emit their own. Such objects can visually appear as clots of absolute darkness.

• 1. Moving matter at half the speed of light


• 2. Photon ring


• 3. Inner photon ring


• 4. The event horizon in a black hole

Based on Einstein's General Theory of Relativity, if a body approaches a critical distance from the center of the hole, it can no longer return. This distance is called the Schwarzschild radius. What exactly happens within this radius is not known for certain, but there is the most common theory. It is believed that all the matter of a black hole is concentrated in an infinitely small point, and in its center there is an object with infinite density, which scientists call a singular perturbation.

How does it fall into a black hole

(In the picture, the black hole of Sagittarius A * looks like an extremely bright cluster of light)

Not so long ago, in 2011, scientists discovered a gas cloud, giving it the simple name G2, which emits unusual light. Such a glow can give friction in gas and dust, caused by the action of the black hole Sagittarius A * and which rotate around it in the form of an accretion disk. Thus, we become observers of the amazing phenomenon of the absorption of a gas cloud by a supermassive black hole.

According to recent studies, the closest approach to a black hole will occur in March 2014. We can recreate a picture of how this exciting spectacle will play out.

• 1. When it first appears in the data, a gas cloud resembles a huge ball of gas and dust.


• 2. Now, as of June 2013, the cloud is tens of billions of kilometers away from the black hole. It falls into it at a speed of 2500 km / s.


• 3. The cloud is expected to pass the black hole, but the tidal forces caused by the difference in attraction acting on the leading and trailing edges of the cloud will cause it to become more and more elongated.


• 4. After the cloud is broken, most of it will most likely join the accretion disk around Sagittarius A*, generating shock waves in it. The temperature will rise to several million degrees.


• 5. Part of the cloud will fall directly into the black hole. No one knows exactly what will happen to this substance, but it is expected that in the process of falling it will emit powerful streams of X-rays, and no one else will see it.

Video: black hole swallows a gas cloud

(Computer simulation of how much of the G2 gas cloud will be destroyed and consumed by the black hole Sagittarius A*)

What's inside a black hole

There is a theory that claims that a black hole inside is practically empty, and all its mass is concentrated in an incredibly small point located in its very center - a singularity.

According to another theory that has existed for half a century, everything that falls into a black hole goes into another universe located in the black hole itself. Now this theory is not the main one.

And there is a third, most modern and tenacious theory, according to which everything that falls into a black hole dissolves in the vibrations of strings on its surface, which is designated as the event horizon.

So what is the event horizon? It is impossible to look inside a black hole even with a super-powerful telescope, since even light, getting inside a giant cosmic funnel, has no chance to emerge back. Everything that can be somehow considered is in its immediate vicinity.

The event horizon is a conditional line of the surface from under which nothing (neither gas, nor dust, nor stars, nor light) can escape. And this is the very mysterious point of no return in the black holes of the Universe.

We now turn to the story of how a black hole can work as an electrical machine (electric motor, dynamo, etc.).

First of all, we must get acquainted with the amazing properties of the boundary of a black hole, which, with

Rice. 5. Lines of force of the electric field of a charge near a black hole. Pluses and minuses denote fictitious surface charges at the edge of a black hole

from the point of view of an external observer, manifests itself as a "membrane", endowed with certain electrical properties.

To understand what is at stake here, consider the electric field of a charge located near a non-rotating uncharged black hole. As we have already said, the three-dimensional space in the vicinity of a black hole is curved, and therefore the field lines of this field look very unusual, as shown in Fig. 5. This drawing, of course, is schematic, since it is impossible to depict the configuration of lines in a curved space on a flat piece of paper. We see that part of the field lines, bending, goes into space away from the black hole. Other field lines rest against the black hole.

If the matter were limited to this, then this would mean that the black hole is charged. Indeed, we know that Gauss's law states that the number of lines of force crossing a closed surface determines the total charge inside it. But our black hole as a whole is not charged; this means that if there are lines of force entering the black hole, then there must be lines coming out of it. Indeed, we see in the figure that the lines of force of the electric field come out of the black hole from the side opposite to the charge and go away from the black hole. Such a complex configuration of the field is associated with a strong curvature of space.

The lines of force in fig. 5 look as if the surface of the black hole is an electrically conductive sphere and approaching it from outside the charge causes polarization of free charges in the electrically conductive sphere. Charges that have opposite

Rice. 6. Fictitious surface current at the boundary of a black hole. Black hole oblate due to rotation

sign compared to the approached one, are attracted by it and are collected on one side of the sphere. Charges of the same sign as the approaching one are repelled and collected from the opposite side (see Fig. 5). Such an analogy allows us to conditionally assume that there are (fictitious) charges on the surface of a black hole, on which the lines of force of the external electric field end.

Let us consider in more detail the process of approaching an electric charge to a black hole. In the course of approaching the charge, the distribution of the fictitious surface charge of the black hole will change - charges of the opposite sign are drawn to a point located directly under the approaching charge. So, we can assume that a (fictitious) current flows on the surface of a black hole! Further, we can relate the strength of this current to the strength of the electric field that acts along the surface of the black hole when the charge approaches, as seen by a distant observer:

This relationship has the form of the well-known Ohm's law. Here we have denoted by the (fictitious) surface resistance of the black hole. A detailed examination shows that or in ordinary units it is equal to 377 ohms.

So, already consideration of the simplest electrodynamic problems shows that the surface of a black hole behaves like a membrane endowed with certain

electrical properties. Consideration of more complex problems confirms this point of view. For example, let two streams of charges of the opposite sign fall into different parts of the black hole surface (Fig. 6), so that the total charge of the black hole does not change. Then we can assume that from the place where positive charges A fall to the place where negative charges B fall, a surface electric current flows, as shown in Fig. 6.

We must once again remind the reader that in reality there are no surface charges and currents (as well as the material surface itself) for a black hole. If some observer falls into a black hole, then he does not encounter any material surface, no charges, no currents when crossing the horizon. The introduction of these fictitious quantities is simply a visual method of representing the behavior of the field lines of an electric (and, as we shall see, also magnetic) field near the boundary of a black hole, from the point of view of an observer located “far from the black hole. Such a representation is very convenient, visual, and allows our intuition, accustomed to the analysis of laboratory experiments with conducting spheres, to work. This allows us, without resorting to complicated ideas and calculations concerning the curved four-dimensional space-time that general relativity deals with, to imagine the behavior of a black hole in certain conditions in a relatively simple way.

In the future, we will use the described representation, without specifying each time the fictitiousness of the concepts of surface charges and currents for a black hole.

Let us now turn to the consideration of how a black hole can play the role of various elements of an electrical circuit and electrical machines. This line of research is now being actively developed by the American physicist Kip Thorne and his colleagues. Of course, we will not dwell on the technical details of the structures, but will present only general schemes.

The existing ideas about black holes are based on theorems proved by means of the differential geometry of manifolds. The presentation of the results of the theory is available in books and we will not repeat them here. Referring the reader for details to monographs and collections, as well as original papers and reviews, we confine ourselves to a brief enumeration of the main provisions underlying modern ideas about black holes.

The most general family of vacuum solutions of the Einstein equations, describing stationary asymptotically flat space-times with a nonsingular event horizon and regular everywhere outside the horizon, has axial symmetry and coincides with the two-parameter Kerr family. Two independent parameters and a define the mass and angular momentum of the black hole. Theorems supporting this statement were formulated in works for a non-rotating black hole and generalized to the Kerr metric in . The solutions of Einstein's non-vacuum equations describing black holes can be characterized by a large number of parameters. So, in the case of the Einstein-Maxwell system of equations, the family of Kerr-Newman solutions possesses the listed properties, which has four parameters where electric, magnetic charges, the uniqueness of this family was proved in . There are solutions to the Einstein-Yang-Mills system of equations describing black holes carrying gauge (color) charges, as well as the Einstein-Yang-Mills-Higgs system with spontaneously broken symmetry, describing point gravitating monopoles and dyons hidden under the event horizon. In extended supergravity, solutions have been found that describe extremely charged black holes with a fermionic structure. It is essential that all the listed solutions are known for fields of zero mass, which cannot have massive external fields of a black hole.

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