Do black holes have charges? Solutions of field equations describing black holes

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 what exactly such holes are is not known to everyone.

From the history of black holes

1783 The first hypothesis of 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 of Newton's creations - optics and mechanics. Michell's idea was this: if light is a stream of tiny particles, then, like all other bodies, the 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 colleague) 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 the curvature of spacetime caused by matter, and a few months later, German astronomer and theoretical physicist Karl Schwarzschild used it to solve a specific astronomical problem. He explored the structure of 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 it with the term “Black Hole”.

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

2013 The latest research into 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 will most likely be absorbed by it, observing a unique process provides enormous opportunities for new discoveries of the features of black holes.

(The 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 multi-country collaboration Event Horizon Telescope, connecting eight telescopes from different points on the Earth’s continents, observed a black hole, which is a supermassive object located in the M87 galaxy, constellation Virgo. The mass of the object is 6.5 billion (!) solar masses, gigantic times larger than the massive object Sagittarius A*, for comparison, with a diameter slightly less than the distance from the Sun to Pluto.

Observations were carried out in several stages, starting in the spring of 2017 and throughout the periods of 2018. The volume of information amounted to petabytes, which then had to be decrypted and a genuine image of an ultra-distant object obtained. Therefore, it took another two whole years to thoroughly process all the data and combine them into one whole.

2019 The data was successfully decrypted and displayed, 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)

The 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 ultra-long baseline interferometric observations. These are 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 the same direction.

What black holes actually are

A laconic explanation of the phenomenon goes 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.

The black hole was once a massive star. As long as thermonuclear reactions maintain high pressure in its depths, everything remains normal. But over time, the energy supply 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. A black hole ejects a jet at high speed


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


• 3. Black hole


• 4. Detailed diagram of the black hole region


• 5. Size of new observations found

The most common theory is that similar phenomena exist in every galaxy, including the center of our Milky Way. The hole's enormous gravitational force 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 turned into a black hole, and can be thousands of light years.

Schwarzschild radius

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

• 1. Moving matter at half the speed of light


• 2. Photon ring


• 3. Inner photon ring


• 4. Event horizon in a black hole

Based on Einstein's General Theory of Relativity, if a body approaches a critical distance to the center of the hole, it will no longer be able to return. This distance is called the Schwarzschild radius. What exactly happens inside 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 infinitesimal point, and at its center there is an object with infinite density, which scientists call a singular perturbation.

How does falling into a black hole happen?

(In the picture, the black hole 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. This glow may be due to friction in the gas and dust caused by the Sagittarius A* black hole, which orbits it as an accretion disk. Thus, we become observers of the amazing phenomenon of absorption of a gas cloud by a supermassive black hole.

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

• 1. When first appearing 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 from the black hole. It falls into it at a speed of 2500 km/s.


• 3. The cloud is expected to pass by the black hole, but tidal forces caused by the difference in gravity acting on the leading and trailing edges of the cloud will cause it to take on an increasingly elongated shape.


• 4. After the cloud is torn apart, most of it will most likely flow into the accretion disk around Sagittarius A*, generating shock waves in it. The temperature will jump 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 next, but it is expected that as it falls it will emit powerful streams of X-rays and will never be seen again.

Video: black hole swallows a gas cloud

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

What's inside a black hole

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

According to another theory, which has existed for half a century, everything that falls into a black hole passes 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 an event horizon? It is impossible to look inside a black hole even with a super-powerful telescope, since even light, entering the giant cosmic funnel, has no chance of emerging back. Everything that can be at least somehow considered is located in its immediate vicinity.

The event horizon is a conventional surface line 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.

What is the electric charge of a black hole? For “normal” black holes of astronomical scale this question is stupid and meaningless, but for miniature black holes it is very 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 ends up inside dense matter?

First, 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 tiriampampation that led to its appearance, and begin to sum up their electrical charges: proton - +1, electron - -1. Let's 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 not much - such a charge passes through the filament of a 20-watt light bulb in a second. But for a static charge, the value is not bad (a bunch of protons with such a total charge weighs 0.121 nanograms), and for the static charge of an object the size of an elementary particle, the value is simply awesome.

Let's see what happens when a charged black hole gets inside relatively dense matter. First, let's consider the simplest case - gaseous diatomic hydrogen. We will consider the pressure 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. Let's take the distance to which electrons need to be pulled away from the atoms to be 10 -10 m, which seems to be enough. Thus, the force acting on a pair of electrons in a hydrogen molecule during the ionization process 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 very quickly be absorbed by the black hole. It would be interesting to calculate how much energy particles of opposite charge will have time to emit into the environment, but calculating integrals breaks :-(and I don’t know how to do this without integrals :-(Offhand, visual effects will vary from very small ball lightning to quite decent ball lightning.

A black hole does roughly the same thing with other dielectrics. For oxygen, the ionization radius is 2.55 cm, for nitrogen - 2.32 cm, neon - 2.21 cm, helium - 2.07 cm. In liquids, the dielectric constant of the medium is noticeably greater than unity, and in water, the ionization radius by a 14-kiloton black hole is only 2.23 mm. Crystals have different dielectric constants 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 dielectric constant) will be 8.39 mm. Surely he lied about little things almost everywhere, but the order of magnitude should be like this.

So, a black hole, once in a dielectric, quickly loses its electrical charge, without producing any special effects other than converting a small volume of the 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 the black hole in the bowels of the 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. We are blatantly ignoring helium for now. The thermal speed of protons under these conditions is 498 km/s, but electrons fly at almost relativistic speeds - 21,300 km/s. It is almost impossible to catch such a fast electron by gravity, so the black hole will quickly gain a positive electrical charge until an equilibrium is reached between the absorption of protons and the absorption of electrons. Let's see what kind of balance this will be.

The proton is subject to gravitational force from the black hole.

F p = (GMm p - kQq)/R 2

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

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

v p1 = sqrt((GMm p - kQq)/mR)

The second "electrocosmic" speed of the proton is

v p2 = sqrt(2)v 1 = sqrt(2(GMm p - kQq)/(m p 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 equal to

R e = 2(GMm e + kQq)/(m e v e 2)

In order 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

It’s already surprising that nothing depends on the temperature of the plasma. We decide:

Q = GM(m p - m e)/(kq)

We substitute the numbers and are surprised to get Q = 5.42*10 -22 C - less than the charge of the electron.

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 a 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 be not three orders of magnitude less than the elementary one, but one, so what?

So, the general conclusion: 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 look at the dynamics of a miniature black hole - how it rushes through the bowels of a planet or star and devours matter on its way.

Black holes

Starting in the middle of the 19th century. development of the theory of electromagnetism, James Clerk Maxwell had large amounts of information about electric and magnetic fields. In particular, what was surprising was the fact that electric and magnetic forces decrease with distance in exactly the same way as 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 the nuclei of atoms together - the forces of strong and weak interactions - have a short range of action. Nuclear forces make themselves felt only in a very small area surrounding nuclear particles. The large range of electromagnetic forces means that, far from the black hole, experiments can be made to find out whether the 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 south magnetic pole), then a distant observer can use sensitive instruments to detect the existence of these charges. In the late 1960s and early 1970s, astrophysicists -theorists have been working hard on the problem: which properties of black holes are preserved and which 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 geometry of space-time 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 except mass, charge and angular momentum. Thus, black holes are very simple objects; they are much simpler than the stars from which black holes arise. G. Reisner and G. Nordström discovered a solution to Einstein's gravitational field equations, which completely describes a “charged” black hole. Such a black hole may have an electric charge (positive or negative) and/or a magnetic charge (corresponding to the north or south magnetic pole). If electrically charged bodies are common, then magnetically charged ones 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 during one of their experiments they had discovered a magnetic monopole. If these results are confirmed, it turns 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 possibility of a black hole having a monopole magnetic field. Regardless of how the black hole acquired its charge, all the properties of that charge in the Reisner-Nordström solution are combined into one characteristic - the number Q. This feature is analogous to the fact that the Schwarzschild solution does not depend on how the black hole acquired its mass. Moreover, the geometry of space-time in the Reisner-Nordström solution does not depend on the nature of the charge. It can be positive, negative, correspond to the north magnetic pole or the south - 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, on 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 is not very significant, because electromagnetic forces are much stronger than gravitational forces. For example, the electric field of an electron or proton is trillions of trillions of times stronger than its gravitational field. This means that if a black hole had a large enough charge, then enormous forces of electromagnetic origin would quickly scatter gas and atoms “floating” in space in all directions. In a very short time, particles with the same charge sign as the black hole would experience powerful repulsion, and particles with the opposite charge sign would experience an equally powerful attraction towards it. By attracting particles with opposite charges, 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| should be much less than M. In fact, from calculations it follows that black holes that could actually exist in space should have a mass M at least a billion billion times greater than the value |Q|.

An analysis of the evolution of stars has led astronomers to the conclusion that black holes can exist both in our Galaxy and in the Universe in general. In the two previous chapters, we examined a number of properties of the simplest black holes, which are described by the solution to the gravitational field equation that Schwarzschild found. A Schwarzschild black hole is characterized only by mass; it has no electric charge. It also lacks a magnetic field and rotation. All properties of a Schwarzschild black hole are uniquely determined by the task mass alone that star that, dying, turns into a black hole during gravitational collapse.

There is no doubt that Schwarzschild's solution is an overly simple case. Real the black hole must at least be spinning. However, how complex can a black hole really be? What additional details should be taken into account and which ones can be neglected in a complete description of the black hole that can be detected when observing the sky?

Let's imagine a massive star that has just run out of all its nuclear energy resources and is about to enter a phase of catastrophic gravitational collapse. One might think that such a star has a very complex structure and that a comprehensive description of it would have to take into account many characteristics. In principle, an astrophysicist is able to calculate the chemical composition of all layers of such a star, the change in temperature from its center to the surface, and obtain all the data on the state of matter in the interior of the star (for example, its density and pressure) at all possible depths. Such calculations are complex, and their results depend significantly on the entire history of the star’s development. The internal structure of stars formed from different clouds of gas and at different times must obviously be different.

However, despite all these complicating circumstances, there is one indisputable fact. If the mass of a dying star exceeds approximately three solar masses, that star certainly will turn into a black hole at the end of its life cycle. There are no physical forces that could prevent the collapse of such a massive star.

To better understand the meaning of this statement, remember that a black hole is such a curved region of space-time that nothing can escape from it, not even light! In other words, no information can be obtained from a black hole. Once an event horizon has emerged around a dying massive star, it becomes impossible to figure out any details of what is happening below that horizon. Our Universe forever loses access to information about events below the event horizon. That's why a black hole is sometimes called grave for information.

Although a huge amount of information is lost when a star collapses with the appearance of a black hole, some information from the outside remains. For example, the extreme curvature of space-time around a black hole indicates that a star has died there. The mass of a dead star is directly related to specific properties of the hole, such as the diameter of the photon sphere or event horizon (see Figs. 8.4 and 8.5). Although the hole itself is literally black, the astronaut will detect its existence from afar by the gravitational field of the hole. By measuring how much his spacecraft's trajectory deviates from a straight line, an astronaut can accurately calculate the total mass of the black hole. Thus, the mass of a black hole is one element of information that is not lost during collapse.

To support this statement, consider the example of two identical stars that form black holes when they collapse. Let's place a ton of stones on one star, and an elephant weighing one ton on the other. After the formation of black holes, we will measure the strength of the gravitational field at large distances from them, say, by observing the orbits of their satellites or planets. It turns out that the strengths of both fields are the same. At very large distances from black holes, Newtonian mechanics and Kepler's laws can be used to calculate the total mass of each of them. Since the total sums of the masses of the constituent parts entering each of the black holes are identical, the results will also be identical. But what is even more significant is the impossibility of indicating which of these holes swallowed the elephant and which the stones. This information is gone forever. No matter what you throw at a black hole, the result will always be the same. You will be able to determine how much of the substance the hole swallowed, but information about what shape, what color, what chemical composition this substance was is lost forever.

The total mass of a black hole can always be measured because the hole's gravitational field affects the geometry of space and time at vast distances from it. A physicist located far from the black hole can conduct experiments to measure this gravitational field, for example by launching artificial satellites and observing their orbits. This is an important source of information that allows a physicist to say with confidence that it is a black hole Not absorbed. In particular, everything that this hypothetical researcher can measure far from the black hole is did not have completely absorbed.

Starting in the middle of the 19th century. development of the theory of electromagnetism, James Clerk Maxwell had large amounts of information about electric and magnetic fields. In particular, what was surprising was the fact that electric and magnetic forces decrease with distance in exactly the same way as gravity. Both gravitational and electromagnetic forces are forces long range. 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 short range. Nuclear forces make themselves felt only in a very small area surrounding nuclear particles.

The large range of electromagnetic forces means that a physicist, far from a black hole, can undertake experiments to find out charged this hole or not. If a black hole has an electric charge (positive or negative) or a magnetic charge (corresponding to the north or south magnetic pole), then a physicist located in the distance can detect the existence of these charges using sensitive instruments. Thus, in addition to information about mass, information about charge black hole.

There is a third (and final) important effect that a remote physicist can measure. As will be seen in the next chapter, any rotating object tends to involve the surrounding space-time in rotation. This phenomenon is called or the drag effect of inertial systems. When our Earth rotates, it also carries space and time with it, but to a very small extent. But for rapidly rotating massive objects this effect becomes more noticeable, and if the black hole was formed from rotating star, then the drag of space-time near it will be quite noticeable. A physicist in a spacecraft far from this black hole will notice that he is gradually drawn into rotating around the hole in the same direction in which it itself is rotating. And the closer our physicist gets to the rotating black hole, the stronger this involvement will be.

When considering any rotating body, physicists often talk about it Momentum momentum; this is a quantity determined by both the mass of the body and the speed of its rotation. The faster a body rotates, the greater its angular momentum. In addition to mass and charge, the angular momentum of a black hole is one of its characteristics about which information is not lost.

In the late 1960s and early 1970s, theoretical astrophysicists worked hard on the problem: which properties of black holes are preserved and which are lost in them? The fruit of their efforts was the famous theorem that “a black hole has no hair,” first formulated by John Wheeler of Princeton University (USA). We have already seen that 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 geometry of space-time near it. The work of Stephen Hawking, Werner Israel, Brandon Carter, David Robinson and other researchers has shown that only these characteristics are preserved during the formation of black holes. 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 except mass, charge and angular momentum. Thus, black holes are very simple objects; they are much simpler than the stars from which black holes arise. To fully describe a star requires knowledge of a large number of characteristics, such as chemical composition, pressure, density and temperature at different depths. A black hole has nothing like this (Fig. 10.1). Really, a black hole has no hair at all!

Since black holes are completely described by three parameters (mass, charge and angular momentum), there should be only a few solutions to Einstein’s gravitational field equations, each describing its own “respectable” type of black hole. For example, in the previous two chapters we looked at the simplest type of black hole; this hole has only mass, and its geometry is determined by the Schwarzschild solution. Schwarzschild's solution was found in 1916, and although many other solutions have been obtained since then for mass-only black holes, All they turned out to be equivalent to it.

It is impossible to imagine how black holes could form without matter. Therefore, any black hole must have mass. But in addition to mass, the hole could have an electrical charge or rotation, or both. Between 1916 and 1918 G. Reisner and G. Nordström found a solution to the field equations that describes a black hole with mass and charge. The next step along this path was delayed until 1963, when Roy P. Kerr found a solution for a black hole with mass and angular momentum. Finally, in 1965, Newman, Koch, Chinnapared, Exton, Prakash and Torrance published a solution for the most complex type of black hole, namely one with mass, charge and angular momentum. Each of these solutions is unique - there are no other possible solutions. A black hole is characterized, at most, three parameters- mass (denoted by M) charge (electric or magnetic, denoted by Q) and angular momentum (denoted by A). All these possible solutions are summarized in table. 10.1.

Table 10.1

Solutions of field equations describing black holes.

Types of black hole Description of a black hole Solution name Year received Mass only (parameter M) The "simplest" black hole. It only has mass. Spherically symmetrical. Schwarzschild solution Mass and charge (options M And Q) Charged black hole. It has mass and charge (electric or magnetic). Spherically symmetrical Reisner-Nordström solution Mass and angular momentum (parameters M And a) Rotating black hole. It has mass and angular momentum. Axisymmetric Kerr's solution Mass, charge and angular momentum (options M, Q And a) A spinning charged black hole, the most complex of all. Axisymmetric Kerr-Newman solution

The geometry of a black hole depends crucially on the introduction of each additional parameter (charge, spin, or both). The Reisner-Nordström and Kerr solutions are very different from each other and from the Schwarzschild solution. Of course, in the limit when the charge and angular momentum vanish (Q -> 0 and A-> 0), all three more complex solutions reduce to the Schwarzschild solution. Yet black holes that have charge and/or angular momentum have a number of remarkable properties.

During the First World War, G. Reisner and G. Nordström discovered a solution to Einstein's gravitational field equations, which completely describes a “charged” black hole. Such a black hole may have an electric charge (positive or negative) and/or a magnetic charge (corresponding to the north or south magnetic pole). If electrically charged bodies are common, then magnetically charged ones are not at all. Bodies that have a magnetic field (for example, an ordinary magnet, a compass needle, the Earth) have both north and south poles. immediately.љљ 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 in the course of one of their experiments they had discovered . If these results are confirmed, it turns 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 possibility of a black hole having a monopole magnetic field. 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 analogous to the fact that the Schwarzschild solution does not depend on how the black hole acquired its mass. It could be composed of elephants, stones or stars - the end result will always be the same. Moreover, the geometry of space-time in the Reisner-Nordström solution does not depend on the nature of the charge. It can be positive, negative, correspond to the north magnetic pole or the south - only its full value is important, which can be written as | Q|. So, the properties of a black hole depend on only two parameters - the total mass of the hole M and its full charge | Q|љљ (in other words, from its absolute value). By thinking about real black holes that could actually exist in our Universe, physicists have come to the conclusion that the Reisner-Nordström solution turns out to be Not good significant, because electromagnetic forces are much greater than gravitational forces. For example, the electric field of an electron or proton is trillions of trillions of times stronger than its gravitational field. This means that if a black hole had a large enough charge, then enormous forces of electromagnetic origin would quickly scatter gas and atoms “floating” in space in all directions. In a very short time, particles with the same charge sign as the black hole would experience powerful repulsion, and particles with the opposite charge sign would experience an equally powerful attraction towards it. By attracting particles with opposite charges, 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 | Q| should be much less than M. In fact, from calculations it follows that black holes that could actually exist in space should have a mass M at least a billion billion times greater than the value | Q|. Mathematically this is expressed by the inequality

Despite these unfortunately unfortunate limitations imposed by the laws of physics, it is instructive to conduct a detailed analysis of the Reisner-Nordström solution. This analysis will prepare us for a more thorough discussion of Kerr's decision in the next chapter.

To make it easier to understand the features of the Reisner-Nordström solution, let's 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 in the center of the hole (at r=0), and the event horizon is at a distance of 1 Schwarzschild radius (exactly at 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 geometry of spacetime. The Reisner-Nordström solution contains two event horizon. Namely, from the point of view of a remote observer, there are two positions at different distances from the singularity, where time stops its run. At the most insignificant charge, the event horizon, which was previously at the “height” of 1 Schwarzschild radius, shifts slightly lower towards the singularity. But even more surprising is that immediately near the singularity a second event horizon appears. Thus the singularity in a charged black hole is surrounded by two event horizons - external and internal. Structures of an uncharged (Schwarzschild) black hole and a charged Reisner-Nordström black hole (at M>>|Q|) are compared in Fig. 10.2.

If we increase the charge of the black hole, the outer event horizon will begin to shrink, and the inner one will expand. Finally, when the charge of the black hole reaches a value at which the equality M=|Q|, both horizons merge with each other. If you increase the charge even more, the event horizon will completely disappear, and all that remains is "naked" singularity. At M<|Q| event horizons missing, so the singularity opens directly into the outer Universe. This picture violates the famous “rule of space ethics” proposed by Roger Penrose. This rule (“you cannot expose the singularity!”) will be discussed in more detail below. The sequence of circuits in Fig. Figure 10.3 illustrates the location of event horizons for black holes that have the same mass but different charge values.

Rice. 10.3 illustrates the position of event horizons relative to the singularity of black holes in space, but it is even more useful to analyze the space-time diagrams for charged black holes. To construct such diagrams—graphs of time versus distance—we will start with the “straight-line” approach used at the beginning of the previous chapter (see Figure 9.3). The distance measured outward from the singularity is plotted horizontally, and time, as usual, is plotted vertically. In such a diagram, the left side of the graph is always limited by a singularity, described by a line running vertically from the distant past to the distant future. World lines of event horizons are also verticals and separate the outer Universe from the inner regions of the black hole.

In Fig. Figure 10.4 shows space-time diagrams for several black holes that have the same masses but different charges. Above, for comparison, is a diagram for a Schwarzschild black hole (remember that the Schwarzschild solution is the same as the Reisner-Nordström solution for | Q| =0). If you add a very small charge to this hole, then the second

The (inner) horizon will be located directly near the singularity. For a black hole with a moderate charge ( M>|Q|) the inner horizon is located further from the singularity, and the outer horizon has decreased its height above the singularity. With a very large charge ( M=|Q|; in this case we talk about limit solution of Reisner-Nordström) both event horizons merge into one. Finally, when the charge is exceptionally large ( M<|Q|), the event horizons simply disappear. As can be seen from Fig. 10.5, 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 directly 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 repulsion. Although the black hole attracts the astronaut to itself as long as he is far enough away from it, if he approaches the singularity at a very short distance, he will be repulsed. The exact opposite of the case of the Schwarzschild solution is the region of space immediately around the Reisner-Nordström singularity - this is the realm of antigravity.

The surprises of the Reisner-Nordström solution go beyond two event horizons and gravitational repulsion near the singularity. Recalling the detailed analysis of the Schwarzschild solution made above, one can think that diagrams like those shown in Fig. 10.4 describe far Not all sides of the picture. Thus, in Schwarzschild geometry we encountered great difficulties caused by the overlap in the simplified diagram different regions of space-time (see Fig. 9.9). The same difficulties await us in diagrams like Fig. 10.4, so it’s time to move on to identifying and overcoming them.

Easier to understand global structure space-time, applying the following elementary rules. Above we figured out what the global structure of the Schwarzschild black hole is. The corresponding picture, called , shown in Fig. 9.18. 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 may be) will necessarily be reduced in the limit to the Penrose diagram for the Schwarzschild solution. Hence our first rule follows: there must be another Universe, opposite to ours, the achievement of which is possible only along forbidden space-like lines. and ), discussed in the previous chapter. In addition, each of these outer Universes must be depicted as a triangle, since the Penrose conformal mapping method works in this case like a team of small bulldozers (see Fig. 9.14 or 9.17), “raking” all space-time into one compact triangle. Therefore, our second rule will be the following: any external Universe must be represented as a triangle, having five types of infinities. Such an outer Universe can be oriented either to the right (as in Fig. 10.6) or to the left.

To arrive at the third rule, recall that in the Penrose diagram (see Fig. 9.18), the event horizon of the Schwarzschild black hole had a slope of 45°. So, the third rule: any event horizon must be light-like, and therefore always have a slope of 45º.

To derive the fourth (and last) rule, remember 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 the spacelike and timelike directions for a charged black hole, it follows that the same picture will be obtained here. Hence the fourth rule: space and time change roles every time, when the event horizon is crossed.

In Fig. 10.7 illustrates the fourth rule just formulated for the case of a black hole with a small or moderate charge ( M>|Q| ). Far 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. Having passed under the outer event horizon, we will find a change in the roles of these two directions - the space-like direction has now become parallel to the time axis, and the time-like direction has now become parallel to the spatial axis. However, continuing the Movement towards the center and descending below the internal horizon of events, we become witnesses of a second change of roles. Near the singularity, the orientation of the spacelike and timelike directions becomes the same as it was far from the black hole.

The double reversal of the roles of the spacelike and timelike directions is crucial 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 switch roles just once. Within a single event horizon, lines of constant distance are directed in a spacelike (horizontal) direction. This means that the line depicting the location of the singularity ( r= 0), must be horizontal, i.e. directed spatially. However, when there are two event horizon, 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 must be oriented in a time-like manner. Therefore, we arrive at a conclusion of paramount importance: the singularity of a charged black hole must be timelike!

Now you can use the above rules to construct a Penrose diagram for the Reisner-Nordström solution. Let's start by imagining an astronaut located in our Universe (let's 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. 10.8, our Universe looks like a triangle with five infinities on the Penrose diagram. Any permissible path of an astronaut must always be oriented on the diagram at an angle of less than 45° to the vertical, since he cannot fly at superluminal speed.

In Fig. 10.8 such admissible world lines are depicted by dotted lines. As the astronaut approaches the charged black hole, he descends below the outer event horizon (which should have a slope of exactly 45º). Having passed this horizon, the astronaut will never be able to return to our The Universe. However, it can sink further below the inner event horizon, which also has a slope of 45°. Beneath this inner horizon, an astronaut might foolishly encounter a singularity where he would be subject to gravitational repulsion and where spacetime would be infinitely curved. Let us note, however, that the tragic outcome of the flight is by no means not 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 directing his spacecraft away from the singularity along the allowed timelike path, as shown in Fig. 10.8. The rescue trajectory takes him away from the singularity, and he again crosses the inner event horizon, which also has a slope of 45º. Continuing the flight, the astronaut goes beyond the outer event horizon (and it has an inclination of 45°) and enters the outer Universe. Since such a journey obviously takes time, the sequence of events along the world line must go from the past to the future. Therefore the astronaut can not

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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 whose gravitational attraction is so strong 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 publicly used in the popular lecture "Our Universe: Known and Unknown" on December 29, 1967. Previously, such astrophysical objects were called “collapsed stars” or “collapsed stars”, as well as “frozen stars”.

Relevance: In the literature devoted to the physics of black holes, the description of Reissner-Nordström black holes 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. The insufficient coverage of this issue and the impossibility of physically observing charged black holes became the basis for the study of the work.

Purpose of the work: to build a black hole model according to the Reissner-Nordström solution to visualize 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 black hole information model.

· Construct a computer model of the 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 modeling.

The object of study is black holes.

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

The educational and methodological, periodical and printed literature of Russian and foreign researchers, physicists and astrophysicists of black holes served as the information base. A bibliography is presented at the end of the work.

The structure of the work is determined by the objectives set in the study and consists of two chapters. The first chapter is devoted to a theoretical overview of the physics of black holes. The second chapter discusses the stages of modeling the Reissner-Nordström black hole 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, explore its parameters and visually present the simulation results.

Practical significance of the work: presented in the form of a developed model of a charged Reissner-Nordström black hole, which will make it possible to demonstrate the result of the model in the educational process.

Chapter 1. Theoretical overview of ideas about black holes

1.1 The concept of a black hole

Currently, a black hole is usually 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 symmetric) is called the gravitational radius.

The question of the real existence of black holes is closely related to how correct the theory of gravity is, 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, first of all, in the context of general relativity, although, strictly speaking, this theory is not experimentally confirmed for conditions corresponding to the region of space-time in the immediate vicinity of black holes of stellar masses (however, it is well confirmed in 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 confirmation of the existence of astronomical objects so dense and massive, as well as having certain other observable properties, that they can be interpreted as black holes of the general theory of relativity.

In addition, black holes are often called objects that do not strictly correspond to the definition given above, but only approach such a black hole in their properties - 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 happens because the differences between the physical fields around the collapsar and 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 contract (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 less than 2.5 km across. The gravitational force of such a star is so strong that, according to Einstein's theory of relativity, it sucks in everything that comes close to it, even light. A black hole cannot be seen because no light, no matter, no 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 double star (period 5-6 days). Its observable component is a blue supergiant whose spectrum changes from night to night. The X-rays detected by astronomers may be emitted when Cygnus X-1, with its gravitational field, sucks material from the surface of a nearby star onto a rotating disk that forms around the black hole.

Rice. 1.1. An artist's impression of the black hole NGC 300 X-1.

What happens to a spaceship that unsuccessfully approaches a black hole in space?

The black hole's strong gravitational pull will pull the spacecraft in, creating a destructive force that will increase 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 roughly 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 of Stephen Hawking's work in 1975, 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 is still ongoing.

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. The relativistic theory of gravity, devoid of this drawback, was created mainly by Einstein (who finally formulated it by the end of 1915) and was called the general theory of relativity (GTR). It is on this that the modern theory of astrophysical black holes is based.

General relativity assumes that the gravitational field is a manifestation of the curvature of spacetime (which thus turns out to be pseudo-Riemannian, rather than 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 - Einstein's equations.

Since black holes are local and relatively compact formations, when constructing their theory, the presence of a cosmological constant is usually neglected, since its effects for such characteristic dimensions of the problem are immeasurably small. Then stationary solutions for black holes within 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 who entered the black hole during collapse and those who fell into it later than bodies and radiations.

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

Table 1.1 Solutions of Einstein's equations for black holes

Schwarzschild solution (1916, Karl Schwarzschild) is a static solution for a spherically symmetric black hole without rotation and without electric charge.

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

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

The Kerr-Newman solution (1965, E. T. Newman, E. Couch, K. Chinnapared, E. Exton, E. Prakash and R. Torrance) 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 progalactic gas. Current ideas 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. The emergence of black holes in high-energy nuclear reactions - quantum black holes.

Stellar-mass black holes form as the final stage in the life of some stars. After complete burnout of thermonuclear fuel and the cessation 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. The compression can stop at a certain stage, or it can turn into rapid gravitational collapse. Depending on the mass of the star and its rotational moment, it may 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 this requires knowledge of 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 the black hole resulting from gravitational collapse from 2.5 to 5.6 solar masses. The radius of the black hole is very small - several tens of kilometers.

Supermassive black holes. Overgrown very massive black holes, according to modern ideas, form the cores 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 uniformity of the gravitational field and matter density, then black holes could form from them through collapse. Moreover, their mass is not limited from below, as in a stellar collapse - their mass could probably be quite small. The discovery of primordial black holes is of particular interest due to the possibility of studying the phenomenon of black hole evaporation.

Quantum black holes. It is assumed that stable microscopic black holes, so-called quantum black holes, can arise 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 that there exists a minimal black hole - a 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, which makes their direct detection very problematic. Recently, experiments have been proposed to detect evidence 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, in reactions of ultra-high energies, virtual intermediate black holes can arise. However, according to string theory, much less energy is required and synthesis can be achieved.

1.3 Black holes with Reissner-Nordström electric charge

During the First World War, G. Reisner and G. Nordström discovered a solution to Einstein's gravitational field equations, which completely describes a “charged” black hole. Such a black hole may have an electric charge (positive or negative) or a magnetic charge (corresponding to the north or south magnetic pole). If electrically charged bodies are common, then magnetically charged ones 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 during one of their experiments they had discovered a magnetic monopole. If these results are confirmed, it turns 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 possibility of a black hole having a monopole magnetic field. Regardless of how the black hole acquired its charge, all the properties of that charge in the Reisner-Nordström solution are combined into one characteristic - the number Q. This feature is analogous 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. Moreover, the geometry of space-time in the Reisner-Nordström solution does not depend on the nature of the charge. It can be positive, negative, correspond to the north magnetic pole or the south - 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, on 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 is not very significant, because electromagnetic forces are much stronger than gravitational forces. For example, the electric field of an electron or proton is trillions of trillions of times stronger than its gravitational field. This means that if a black hole had a large enough charge, then enormous forces of electromagnetic origin would quickly scatter gas and atoms “floating” in space in all directions. In a very short time, particles with the same charge sign as the black hole would experience powerful repulsion, and particles with the opposite charge sign would experience an equally powerful attraction towards it. By attracting particles with opposite charges, 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| should be much less than M. In fact, from calculations it follows that black holes that could actually exist in space should have a mass M at least a billion billion times greater than the value |Q|. Mathematically this is expressed by the inequality

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

To make it easier to understand the features of the Reisner-Nordström solution, let's 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 (at r = 0), and the event horizon is at a distance of 1 Schwarzschild radius (precisely at 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 geometry of spacetime. There are two event horizons in the Reisner-Nordström solution. Namely, from the point of view of a remote observer, there are two positions at different distances from the singularity, where time stops its run. At the most insignificant charge, the event horizon, which was previously at the “height” of 1 Schwarzschild radius, shifts slightly lower towards the singularity. But even more surprising is that immediately near the singularity a second event horizon appears. Thus, the singularity in a charged black hole is surrounded by two event horizons - external and internal. 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, the outer event horizon will begin to shrink, and the inner one will expand. Finally, when the charge of the black hole reaches a value at which the equality M=|Q| holds, both horizons merge with each other. If you increase the charge even more, the event horizon will completely disappear, and what remains is a “bare” singularity. 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. Adding even an insignificant charge leads to the appearance of a second (internal) 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. To construct such diagrams - graphs of time versus distance, we will start with the "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 contracts and the inner one expands. When the total charge of the hole reaches the value |Q|= M, both horizons merge into one. At even higher values ​​of charge, the event horizon disappears altogether and an open, or “naked” singularity remains.

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

In Fig. Figure 1.4 shows space-time diagrams for several black holes that have the same masses but different charges. Above, for comparison, is a diagram for a Schwarzschild black hole (remember 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 further from the singularity, and the outer horizon has decreased its height above the singularity. At a very large charge (M=|Q|; in this case we speak 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 spacetime 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, the charge of which 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 directly 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 repulsion. Although the black hole attracts the astronaut to itself as long as he is far enough away from it, if he approaches the singularity at a very short distance, he will be repulsed. The exact opposite of the case of the Schwarzschild solution is the region of space immediately around the Reisner-Nordström singularity - this is the realm of antigravity.

The surprises of the Reisner-Nordström solution go beyond two event horizons and gravitational repulsion near the singularity. Recalling the detailed analysis of the Schwarzschild solution made above, one can think that diagrams like those shown in Fig. 1.4 does not describe all aspects of the picture. Thus, in Schwarzschild geometry we encountered great difficulties caused by the overlapping 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. A diagram called the Penrose diagram is shown in Fig. 1.6, a.

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

black hole charged reissner

It can also be called a 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 may be) will necessarily reduce in the limit to the Penrose diagram for the Schwarzschild solution. Hence our first rule follows: 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 is reason to expect the existence of many Universes. Each of them must have five types of infinities (, and).

This is I - time-like infinity in the past. It is the “place” from which all material objects (Borya, Vasya, Masha, Earth, galaxies and everything else) originated. All such objects move along timelike world lines and must go to I + - the timelike infinity of the future, somewhere billions of years after “now”. In addition, there is I 0 - spacelike infinity, and since nothing can move faster than light, nothing can ever get into I 0. If no object known to physics moves faster than light, then photons move exactly at the speed of light along world lines tilted 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. Finally, there is the light infinity of the future (where all the “light rays” go).

In addition, each of these outer Universes must be depicted as a triangle, since the Penrose conformal mapping method works in this case like a team of small bulldozers, “raking” all of space-time into one compact triangle. Therefore, our second rule will be the following: any external Universe must be represented as a triangle, having 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. 1.6, b. Outer Universe. In a Penrose diagram for any black hole, the outer Universe is always depicted as a triangle with five infinities (I", S~, I 0 ,S + , I +). Such an outer Universe can be oriented at an angle to the right (as shown in the figure) or to the left.

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

To derive the fourth (and last) rule, remember 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 the spacelike and timelike directions for a charged black hole, it follows that the same picture will be obtained here. Hence the fourth rule: space and time switch roles whenever the event horizon intersects.

In 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|). Far 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. Having passed under the outer event horizon, we will find a change in the roles of these two directions - the space-like direction has now become parallel to the time axis, and the time-like direction has now become parallel to the spatial axis. However, as we continue to move toward the center and descend below the inner event horizon, we become witnesses to a second role reversal. Near the singularity, the orientation of the spacelike and timelike directions becomes the same as it was far from the black hole.

Rice. 1.7. Change of roles of space and time (for M>|Q|). Whenever the event horizon is crossed, space and time change roles. This means that in a charged black hole, due to the presence of two event horizons, a complete change of roles for space and time occurs twice.

The double reversal of the roles of the spacelike and timelike directions is crucial 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 switch roles only once. Within a single event horizon, lines of constant distance are directed in a spacelike (horizontal) direction. This means that the line depicting 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 charged hole singularity (r = 0) must be vertical, and it must be oriented in a time-like manner. Therefore, we come to a conclusion of utmost importance: the singularity of a charged black hole must be timelike!

Now you can use the above rules to construct a Penrose diagram for the Reisner-Nordström solution. Let's start by imagining an astronaut located in our Universe (let's 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 permissible 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 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 traveling from our Universe into a charged black hole.

In Fig. 1.8 such admissible world lines are depicted by dotted lines. As the astronaut approaches the charged black hole, he descends below the outer event horizon (which must be tilted exactly 45 degrees). Having passed this horizon, the astronaut will never be able to return to our Universe. However, it can sink further below the inner event horizon, which also has a 45-degree slope. Beneath this inner horizon, an astronaut might foolishly encounter a singularity where he would be subject to gravitational repulsion and where spacetime would be infinitely curved. Let us 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 directing his spacecraft away from the singularity along the allowed timelike path, as shown in Fig. 1.8. The escape trajectory takes him away from the singularity, and he again crosses the inner event horizon, which also has a slope of 45 degrees. 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 go from the past to the future. Therefore, the astronaut cannot return again to our Universe, but will end up in another Universe, the Universe of the future. As you would expect, this future Universe should look like a triangle with the usual five infinities on a Penrose diagram.

It should be emphasized that when constructing these Penrose diagrams we again encounter both black and white holes. An 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 cannot be white holes in nature. But we will still continue our theoretical analysis of the global structure of space-time, which includes the existence of black and white holes side by side with each other.

The flight episodes and diagrams shown in Fig. 1.8 should be nothing 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 ours, which is only reachable along (forbidden) spacelike world lines. This conclusion is based on our rule 1: if you remove its charge from a black hole, then the Penrose diagram should be reduced to an 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 traveling 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 flew out of our Universe, from Earth. The Alien Universe is also depicted on the Penrose diagram by the usual triangle. On the way to the charged black hole, the alien astronaut crosses the outer event horizon, which should have an inclination of 45 degrees. Later it descends below the inner event horizon, also with an inclination of 45 degrees. The alien now faces a choice: either crash into the timelike singularity (which is vertical on the Penrose diagram), or roll up and cross the inner event horizon again. To avoid an unfortunate end, the alien decides to leave the black hole and exits through the inner event horizon, which, as usual, has a slope of 45 degrees. It then flies through the outer event horizon (tilted by 45 degrees on the Penrose diagram) into the new Future Universe.

Each of these two hypothetical journeys covers only two parts of the full Penrose diagram. The full picture is obtained if you simply combine these parts with each other, as shown in Fig. 1.10.

Rice. 1.10. Complete Penrose diagram for the 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 into the future and into the past.

Such a diagram must be repeated an infinite number of times into the future and into the past, since each of the two astronauts considered could decide again to leave the Universe in which he emerged and again go into a charged black hole. Thus, astronauts can penetrate into other Universes, even further into the future. In the same way, we can imagine other astronauts from Universes in the distant past arriving in our Universe. Therefore, a complete Penrose diagram repeats in both directions in time, like a long ribbon with a repeating stencil pattern. Overall, the global geometry of a charged black hole unites an infinite number of past and future Universes 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 concept of a white hole, which will be discussed in a later chapter.

The approach to elucidating the global structure of spacetime just described concerned the case of black holes with 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. Let us also remember that near the singularity (under the internal 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 parallel to the time axis. As the Reisner-Nordström black hole's charge 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 the united externally-internal event horizon, space and time do not change roles. Of course, we can just as well talk about a double change of roles for space and time, occurring simultaneously on the single event horizon of the ultimate Reisner-Nordström black hole. As shown in Fig. 1.11, the time-like direction in it is everywhere parallel to the time axis, and the space-like direction is everywhere parallel to the spatial axis.

Rice. 1.11. Space-time diagram for the ultimate 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, the roles of space and time do not change.

Although the ultimate Reisner-Nordström black hole has only one event horizon, the situation here is completely different from the case of a 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 “double”, i.e. as internal and external horizons superimposed on each other. 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, when moving from one side to the other, space and time switch 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. In this case, the list of rules remains the same, with the significant exception that when crossing the event horizon, space and time do not change roles. Let's imagine an astronaut leaving the Earth and falling into the ultimate Reisner-Nordström black hole. Our Universe, as usual, is depicted as a triangle on the Penrose diagram. After diving below the event horizon, the astronaut is free to make a choice: he can either crash into a singularity, which is timelike and therefore must be depicted vertically on a Penrose diagram, or (Fig. 1.13) take his spacecraft away from the singularity along a permitted 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 by considering the possible world lines of an astronaut diving into and emerging from the ultimate Reisner-Nordström black hole.

If he chose the second path, then later he will cross the event horizon again, emerging into another Universe. He will again be faced with an alternative - to stay in this future Universe and fly to some planets, or to 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 future Universes. The full picture is shown in Fig. 1.13. As before, the diagram repeats an infinite number of times into the past and into the future, like a tape with a repeating stencil pattern.

From a mathematical point of view, a black hole with a huge charge M is also acceptable<|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 departed from the Earth, rushes towards the “bare” singularity, he does not have to descend below the event horizon. He remains in our Universe all the time. 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 into a singularity - a “naked” 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 represented by a vertical on the Penrose diagram. Since there are now no other Universes besides our Universe, the Penrose diagram for a “bare” singularity looks quite simple. From Fig. 1.14 it is clear that our Universe, as usual, is depicted by a triangle with five infinities, bounded on the left by a singularity. Whatever is to the left of the singularity is completely cut off from us. 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 is described above is only of academic interest. However, we have eventually established trouble-free rules for constructing complex Penrose diagrams.

Chapter 2. Development of the Reissner-Nordström model of a charged black hole in the Delphi programming environment

2.1 Mathematical description of the model

The Reissner-Nordström metric is defined by the expression:

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 happens on the external and internal horizons r + and r-:

From the point of view of the location of the horizon r ±, the metric coefficient B(r) is defined as follows:

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

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

Rice. 2.1. Reissner-Nordström space diagram

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

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

Rice. 2.2. Scheme of Finkelstein space for Reissner-Nordström geometry

As usual, the radial Finkelstein coordinate r is the radius of the circle, defined so that the corresponding circle of the ball at radius r is 2рr, while the time Finkelstein coordinate is defined so that radially incident rays of light (yellow lines) move at an angle of 45 o 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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The coloring of the lines, as in the case of a Schwarzschild black hole: the red horizon line, the blue line is the line at zero radius, the yellow and ocher lines are respectively world lines for radially incident and outgoing light rays, while the dark purple and cyan lines are respectively lines of Schwarzschild constant time and constant radius of a circle.

Let's consider the waterfall model of the Reissner-Nordström space. The waterfall model works well for a charged black hole of 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 Gullstrand-Pineliv metric as for the Schwarzschild metric, but the mass M for escape velocity is replaced by the mass M(r) of the internal radius r:

Figure 2.3. Reissner-Nordström Falls.

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

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

The speed of 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 to 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 now enters the white hole, where space moves outward faster than light. Rice. Figure 2.3 shows a white hole in the same location as a black hole, but in fact, as can be seen from the Penrose diagram, the white hole and black hole are different regions of spacetime. 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 pull of the mass. The outgoing space slows down to the speed of light at the outer horizon of the r+ white hole. This space emerges into a new region of space-time, possibly a new universe.

2.2 Results of modeling a charged Reissner-Nordström black hole in the Delphi programming environment

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

1. View the structure of a black hole. Allows you to simulate changes in the position of the inner and outer horizons depending on the charge of the black hole. At 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 internal horizon appears. In this case, the outer horizon contracts 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.

When 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 when the charge value is equal to the mass of the black hole.

When the charge value of the black hole mass is exceeded, the horizons disappear and a naked singularity opens.

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

Rice. 2.7. Space diagram of 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 inwards from bottom to top, ocher lines are world lines of light rays moving radially outwards also from bottom to top.

The change in direction (from top to bottom) 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 horizons and do not have asymptotes on them.

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

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