Karl Schwarzschild: astronomy, artillery, black holes. Schwarzschild space-time Schwarzschild metric in Cartesian coordinates

The objects were called “collapsed stars” or “collapsers” (from the English. collapsed stars), as well as “frozen stars” (eng. frozen stars).

The question of the real existence of black holes in accordance with the definition given above is largely related to how correct the theory of gravity is, from which the existence of such objects follows. In modern physics, the standard theory of gravity, best confirmed experimentally, is the general theory of relativity (GTR), although the existence of black holes is also possible within the framework of other (not all) theoretical models of gravity (see: Theories of gravity). Therefore, observational data are analyzed and interpreted, first of all, in its context, although, strictly speaking, this theory is not experimentally confirmed for conditions corresponding to the region of space-time in the immediate vicinity of a black hole. Therefore, statements about direct evidence of the existence of black holes, including in this article below, should, strictly speaking, be understood in the sense of confirmation of the existence of objects so dense and massive, as well as having some other observable properties, that they can be interpreted as black holes 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 in their properties to such a general relativity black hole, for example, 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” black hole are almost the same.

History of ideas about black holes

In the history of ideas about black holes, three periods are distinguished:

The beginning of the first period is associated with the work of John Michell, published in 1784, which outlined the calculation of mass for an object inaccessible to observation.
The second period is associated with the development of the general theory of relativity, the stationary solution of the equations of which was obtained by Karl Schwarzschild in 1915.
The publication in 1975 of Stephen Hawking's work, in which he proposed the idea of radiation from black holes, begins the third period. The boundary between the second and third periods is rather arbitrary, since all the consequences of Hawking’s discovery did not immediately become clear, the study of which is still ongoing.

"Black Star" Michell

"Black Hole" Michell

In the Newtonian gravitational field for particles at rest at infinity, taking into account the law of conservation of energy:

Let the gravitational radius be the distance from the gravitating mass at which the particle speed becomes equal to the speed of light. Then .

The concept of a massive body whose gravitational pull is so great that the speed required to overcome that pull (second escape velocity) is equal to or greater than the speed of light, was first proposed in 1784 by John Michell in a letter he sent to the Royal Society. The letter contained a calculation from which it followed that for a body with a radius of 500 solar radii and with the density of the Sun, the second escape velocity on its surface will be equal to the speed of light. Thus, light will not be able to leave this body and it will be invisible. Michell suggested that there could be many such inaccessible objects in space. In 1796, Laplace included a discussion of this idea in his Exposition du Systeme du Monde, but this section was omitted in subsequent editions.

After Laplace, before Schwarzschild

Throughout the 19th century, the idea of bodies invisible due to their massiveness did not attract much interest among scientists. This was due to the fact that within the framework of classical physics, the speed of light has no fundamental meaning. However, at the end of the 19th - beginning of the 20th century, it was established that the laws of electrodynamics formulated by J. Maxwell, on the one hand, are satisfied in all inertial frames of reference, and on the other hand, do not have invariance under Galilean transformations. This meant that the prevailing ideas in physics about the nature of the transition from one inertial reference system to another needed significant adjustment.

In the course of further development of electrodynamics, G. Lorentz proposed a new system of transformations of space-time coordinates (known today as Lorentz transformations), with respect to which Maxwell's equations remained invariant. Developing Lorentz's ideas, A. Poincaré assumed that all other physical laws are also invariant with respect to these transformations.

Curvature of space

(Pseudo)Riemannian spaces are spaces that, on small scales, behave “almost” like ordinary (pseudo)Euclidean ones. Thus, in small areas of the sphere, the Pythagorean theorem and other facts of Euclidean geometry are fulfilled with very high accuracy. At one time, this circumstance made it possible to construct Euclidean geometry based on observations of the Earth’s surface (which in reality is not flat, but close to spherical). The same circumstance also determined the choice of pseudo-Riemannian (and not any other) spaces as the main object of consideration in General Relativity: the properties of small sections of space-time should not differ much from those known from Special Relativity.

However, on large scales, Riemannian spaces can be very different from Euclidean spaces. One of the main characteristics of such a difference is the concept of curvature. Its essence is as follows: Euclidean spaces have the property absolute parallelism: vector X" , obtained as a result of parallel translation of the vector X along any closed path, coincides with the original vector X. For Riemannian spaces this is no longer always the case, which can be easily shown in the following example. Suppose that the observer stood at the intersection of the equator with the prime meridian, facing east and began to move along the equator. Having reached a point with a longitude of 180°, he changed the direction of movement and began to move along the meridian to the north, without changing the direction of his gaze (that is, now he is looking to the right along the way). When he thus crosses the north pole and returns to his starting point, he will find himself facing the west (and not the east, as originally). In other words, the vector, parallelly transferred along the observer’s route, “scrolled” relative to the original vector. The characteristic of the magnitude of such “scrolling” is curvature.

Solutions of Einstein's equations for black holes

Stationary solutions for black holes within the framework of general relativity are characterized by three parameters: mass ( M), angular momentum ( L) and electric charge ( Q), which consist of the corresponding characteristics of the bodies and radiation that fell into it. Any black hole tends to become stationary in the absence of external influences, which has been proven by the efforts of many theoretical physicists, of whom especially noteworthy is the contribution of Nobel laureate Subramanian Chandrasekhar, who wrote the monograph “Mathematical Theory of Black Holes”, fundamental for this direction.

Solutions of Einstein's equations for black holes with the corresponding characteristics:

The solution for a spinning black hole is extremely difficult. It is interesting that the most complex type of solution was “guessed” by Kerr from “physical considerations”. The first consistent derivation of Kerr's solution was first made by S. Chandrasekhar more than fifteen years later. It is believed that Kerr's solution is of greatest importance for astrophysics, since charged black holes should quickly lose charge, attracting and absorbing oppositely charged ions and dust from outer space. There is also a theory connecting gamma-ray bursts with the process of explosive neutralization of charged black holes through the birth of electron-positron pairs from a vacuum and the fall of one of the particles onto the hole with the second going to infinity (R. Ruffini and co-workers).

Schwarzschild solution

Objects whose size is closest to their Schwarzschild radius, but which are not yet black holes, are neutron stars.

You can introduce the concept of “average density” of a black hole by dividing its mass by the volume contained under the event horizon:

The average density drops as the black hole's mass increases. So, if a black hole with a mass on the order of the Sun has a density exceeding the nuclear density, then a supermassive black hole with a mass of 10 9 solar masses (the existence of such black holes is suspected in quasars) has an average density of the order of 20 kg/m³, which is significantly less than the density of water !

Thus, a black hole can be obtained not only by compressing the existing volume of matter, but also in an extensive way, by accumulating a huge amount of material.

To accurately describe real black holes, it is necessary to take into account quantum corrections, as well as the presence of angular momentum. Near the event horizon, quantum effects associated with material fields (electromagnetic, neutrino, etc.) are strong. Taking this into account, the theory (that is, general relativity, in which the right-hand side of Einstein’s equations is the average over the quantum state of the energy-momentum tensor) is usually called “semiclassical gravity.”

Reissner-Nordström solution

This is a static solution to Einstein's equations for a spherically symmetric black hole with charge but no rotation.

Reissner-Nordström black hole metric:

c− speed of light, m/s, t− time coordinate (time measured on an infinitely distant clock), in seconds, r− radial coordinate (length of the “equator” divided by 2π), in meters, θ − geographic latitude (angle from north), in radians, − longitude, in radians, r s− Schwarzschild radius (in meters) of a body with mass M , r Q− length scale (in meters) corresponding to electric charge Q(analogue of the Schwarzschild radius, only not for mass, but for charge) defined as

where is the Coulomb constant.

The parameters of a black hole cannot be arbitrary. The maximum charge that a Reissner-Nordström black hole can have is , where e- electron charge. This is a special case of the Kerr-Newman constraint for a black hole with zero angular momentum ( J= 0, that is, without rotation).

However, it should be noted that in realistic situations (see: The principle of cosmic censorship) black holes should not be charged to any significant extent.

Kerr's solution

The Kerr black hole has a number of remarkable properties. Around the event horizon there is a region called the ergosphere, inside which it is impossible for relatively distant observers to rest, but only to rotate around the black hole in the direction of its rotation. This effect is called “drag of the inertial reference frame” (eng. frame-dragging) and is observed around any rotating massive body, such as the Earth or the Sun, but in much to a lesser extent. However, the ergosphere itself can still be left; this area is not exciting. The dimensions of the ergosphere depend on the angular momentum of rotation.

The parameters of a black hole cannot be arbitrary (see: The principle of cosmic censorship). At J max = M 2 the metric is called the Kerr limit solution. This is a special case of the Kerr-Newman constraint, for a black hole with zero charge ( Q = 0 ).

This and other black hole solutions give rise to amazing space-time geometry. However, an analysis of the stability of the corresponding configuration is required, which can be disrupted due to interaction with quantum fields and other effects.

For Kerr spacetime, this analysis was carried out by Subramanian Chandrasekhar and it was found that the Kerr black hole - its outer region - is stable. Similarly, as special cases, the Schwarzschild and Reissner-Nordström holes turned out to be stable. However, the analysis of Kerr-Newman space-time has not yet been carried out due to great mathematical difficulties.

Kerr-Newman solution

The three-parameter Kerr-Newman family is the most general solution corresponding to the final equilibrium state of a black hole. In Boyer - Lindquist coordinates, the Kerr - Newman metric is given by:

From this simple formula it easily follows that the event horizon is located at the radius: .

And therefore the parameters of a black hole cannot be arbitrary. Electric charge and angular momentum cannot be greater than the values corresponding to the disappearance of the event horizon. The following restrictions must be met:

- This Kerr-Newman constraint.

If these restrictions are violated, the event horizon will disappear, and the solution, instead of a black hole, will describe the so-called “naked” singularity, but such objects, according to popular belief, should not exist in the real universe. (see: The principle of cosmic censorship, but it has not yet been proven).

The Kerr-Newman metric can be analytically extended to connect infinitely many “independent” spaces in a black hole. These can be both “other” Universes and distant parts of our Universe. In the resulting spaces there are closed time-like curves: the traveler can, in principle, get into his past, that is, meet himself. There is also a region called the ergosphere around the event horizon of a rotating black hole, practically equivalent to the ergosphere from Kerr's solution; a stationary observer located there must rotate with a positive angular velocity (in the direction of rotation of the black hole).

Thermodynamics and evaporation of black holes

The idea of a black hole as an absolutely absorbing object was corrected by S. Hawking in 1975. By studying the behavior of quantum fields near a black hole, he predicted that the black hole necessarily radiates particles into outer space and thereby loses mass. This effect is called Hawking radiation (evaporation). To put it simply, the gravitational field polarizes the vacuum, as a result of which the formation of not only virtual, but also real particle-antiparticle pairs is possible. One of the particles, just below the event horizon, falls into the black hole, and the other, just above the horizon, flies away, carrying away the energy (that is, part of the mass) of the black hole. The radiation power of a black hole is equal to

The composition of the radiation depends on the size of the black hole: for large black holes it is mainly photons and neutrinos, and heavy particles begin to be present in the spectrum of light black holes. The spectrum of Hawking radiation turned out to strictly coincide with the radiation of an absolutely black body, which made it possible to assign a temperature to the black hole

where is the reduced Planck constant, c- speed of light, k- Boltzmann constant, G- gravitational constant, M- the mass of the black hole.

On this basis, the thermodynamics of black holes was built, including the introduction of the key concept of entropy of a black hole, which turned out to be proportional to the area of its event horizon:

Where A- area of the event horizon.

The rate of evaporation of a black hole is greater, the smaller its size. The evaporation of black holes of stellar (and especially galactic) scales can be neglected, however, for primary and especially quantum black holes, evaporation processes become central.

Due to evaporation, all black holes lose mass and their lifetime turns out to be finite:

In this case, the intensity of evaporation increases like an avalanche, and the final stage of evolution has the character of an explosion, for example, a black hole weighing 1000 tons will evaporate in about 84 seconds, releasing energy equal to the explosion of approximately ten million atomic bombs of average power.

At the same time, large black holes, whose temperature is lower than the temperature of the cosmic microwave background radiation of the Universe (2.7 K), at the present stage of the development of the Universe can only grow, since the radiation they emit has less energy than the radiation they absorb. This process will last until the photon gas of the cosmic microwave background radiation cools down as a result of the expansion of the Universe.

Without a quantum theory of gravity, it is impossible to describe the final stage of evaporation, when black holes become microscopic (quantum). According to some theories, after evaporation there should be a “cinder” left - a minimal Planck black hole.

"No hair" theorems

Theorems about the "no hair" of a black hole No hair theorem) say that a stationary black hole cannot have external characteristics other than mass, angular momentum and certain charges (specific to various material fields), and detailed information about matter will be lost (and partially emitted outward) during collapse. Brandon Carter, Werner Israel, Roger Penrose, Piotr Chruściel, and Markus Heusler made a major contribution to the proof of similar theorems for various systems of physical fields. It now appears that this theorem is true for currently known fields, although in some exotic cases, for which no analogues have been found in nature, it is violated.

Falling into a black hole

Let's imagine what falling into a Schwarzschild black hole would look like. A body freely falling under the influence of gravity is in a state of weightlessness. A falling body will experience tidal forces, stretching the body in the radial direction and compressing it in the tangential direction. The magnitude of these forces grows and tends to infinity at . At some point in its own time, the body will cross the event horizon. From the point of view of an observer falling along with the body, this moment is not highlighted by anything, but now there is no return. The body finds itself in a throat (its radius at the point where the body is located), compressing so quickly that it is no longer possible to fly away from it before the moment of final collapse (this is the singularity), even moving at the speed of light.

Let us now consider the process of a body falling into a black hole from the point of view of a remote observer. Let, for example, the body be luminous and, in addition, send signals back with a certain frequency. At first, a remote observer will see that the body, being in the process of free fall, is gradually accelerating under the influence of gravity towards the center. The body color does not change, the frequency of detected signals is almost constant. However, as the body begins to approach the event horizon, photons coming from the body will experience greater and greater gravitational redshift. In addition, due to the gravitational field, both light and all physical processes from the point of view of a remote observer will go slower and slower. It will appear that the body - in an extremely flattened form - will be slow down, approaching the event horizon and, in the end, will practically stop. The signal frequency will drop sharply. The wavelength of the light emitted by the body will rapidly increase, so that the light will quickly turn into radio waves and then into low-frequency electromagnetic vibrations, which will no longer be possible to detect. The observer will never see the body crossing the event horizon, and in this sense, the fall into the black hole will last indefinitely. There is, however, a moment from which a remote observer will no longer be able to influence the falling body. A ray of light sent after this body will either never catch up with it, or will catch up with it already beyond the horizon.

The process of gravitational collapse will look similar to a distant observer. At first, the matter will rush towards the center, but near the event horizon it will begin to slow down sharply, its radiation will go into the radio range, and, as a result, a distant observer will see that the star has gone out.

String theory model

Samir Mathur's group calculated the sizes of several black hole models using their own method. The results obtained coincided with the dimensions of the “event horizon” in traditional theory.

In this regard, Mathur suggested that the event horizon is actually a foaming mass of strings, rather than a rigidly defined boundary.

Therefore, according to this model, a black hole does not actually destroy information because there is no singularity in black holes. The mass of the strings is distributed throughout the volume up to the event horizon, and information can be stored in the strings and transmitted by outgoing Hawking radiation (and therefore go beyond the event horizon).

Another option was proposed by Gary Horowitz from the University of California at Santa Barbara and Juan Maldacena from the Princeton Institute for Advanced Study. According to these researchers, a singularity at the center of a black hole exists, but information simply does not get into it: matter goes into the singularity, and information - through quantum teleportation - is imprinted on Hawking radiation.

Black holes in the Universe

Since the theoretical prediction of black holes, the question of their existence has remained open, since the presence of a “black hole” type solution does not guarantee that there are mechanisms for the formation of such objects in the Universe. However, there are known mechanisms that can lead to the fact that some region space-time will have the same properties (the same geometry) as the corresponding region at a black hole. For example, as a result of the collapse of a star, the space-time shown in the figure can be formed.

Star collapse. The metric outside the shaded area is unknown to us (or uninteresting)

The region depicted in dark color is filled with the matter of the star and its metric is determined by the properties of this matter. But the light gray region coincides with the corresponding region of Schwarzschild space, see Fig. higher. It is precisely such situations that are spoken of in astrophysics as the formation of black holes, which formal point of view is some freedom of speech. From the outside, however, very soon this object will become practically indistinguishable from a black hole in all its properties, so this term is applied to the resulting configuration with a very high degree of accuracy.

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

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Stellar-mass black holes

Stellar-mass black holes are formed as the final stage in the life of a star; after the thermonuclear fuel has completely burned out and the reaction has stopped, the star should theoretically begin to cool, which will lead to a decrease in internal pressure and compression of the star under the influence of gravity. Compression can stop at a certain stage, or it can turn into rapid gravitational collapse. Depending on the star's mass and angular momentum, the following final states are possible:

An extinct, very dense star composed primarily, depending on mass, of helium, carbon, oxygen, neon, magnesium, silicon, or iron (the major elements are listed in order of increasing mass of the remnant star).
A white dwarf whose mass is limited above by the Chandrasekhar limit.
A neutron star whose mass is limited by the Oppenheimer–Volkoff limit.
Black hole.

As the mass of the stellar remnant increases, the equilibrium configuration moves downward along the described sequence. The torque increases the maximum mass at each stage, but not qualitatively, but quantitatively (maximum 2-3 times).

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. Modeling stars at late stages of their evolution presents additional difficulties due to the complexity of the emerging chemical composition and the sharp decrease in the characteristic time of processes. Suffice it to mention that some of the largest cosmic catastrophes, supernova explosions, occur precisely at these stages of stellar evolution. 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.

Subsequently, the black hole can grow due to the absorption of matter - as a rule, this is the gas of a neighboring star in binary star systems (a collision of a black hole with any other astronomical object is very unlikely due to its small diameter). The process of gas falling onto any compact astrophysical object, including a black hole, is called

In 1916, just a few months after Einstein published his gravitational field equations in general relativity, German astronomer Karl Schwarzschild found a solution to these equations that described a simple black hole. A Schwarzschild black hole is "simple" in the sense that it is spherically symmetric (that is, it has no "preferred" direction, say an axis of rotation) and is characterized only by mass. Therefore, the complications introduced by rotation, electric charge and magnetic field are not taken into account.

Beginning in 1924, physicists and mathematicians began to realize that there was something unusual in Schwarzschild's solution of the gravitational field equations. In particular, this solution has a mathematical feature at the event horizon. Sir Arthur Eddington was the first to select a new coordinate system in which this effect is absent. In 1933, Georges Lemaître took this research further. However, only John Lighton Synge revealed (in 1950) the true essence of the geometry of the Schwarzschild black hole, thereby opening the way for subsequent important work by M. D. Kruskal and G. Szekeres in 1960.

To understand the details, let's first select three guys - Borya, Vasya and Masha - and imagine that they are floating in space (Fig. 9.1). You can always take an arbitrary point in space and determine the positions of all three by measuring the distances from them to this point. For example, Borya is 1 km away from this arbitrary starting point, Vasya is 2 km away, and Masha is 4 km away. The position characteristic in this case is usually denoted by the letter r and is called the radial distance. In this way you can express the distance to any object in the Universe.

Let us now note that our three friends are motionless in space, but “moving” in time, because they are getting older and older. This feature can be depicted on a space-time diagram (Fig. 9.2). The distance from an arbitrary initial reference point ("beginning") to another point in space is plotted here along the horizontal axis, and time - along the vertical. In addition, as in the partial theory of relativity, it is convenient to take such scales on the coordinate axes of this graph that the light rays are described by a straight line with a slope of 45°. In such a space-time diagram, the world lines of all three guys go vertically upward. They always remain at the same distances from the starting point ( r = 0), but are gradually getting older and older.

It is important to realize that to the left of the point r = 0 in Fig. 9.2 there is nothing at all. This area corresponds to something that can be called "negative space". Since it is impossible to be “at a distance of minus 3 m” from any point (the origin), distances from the origin are always expressed in positive numbers.

Let us now move on to the Schwarzschild black hole. As discussed in the previous chapter, such a hole consists of a singularity surrounded by an event horizon at a distance of 1 Schwarzschild radius. An image of such a black hole in space is shown in Fig. 9.3 on the left. When depicting a black hole on a space-time diagram, an arbitrary point of origin is, for convenience, compatible with the singularity. Then the distances are measured directly from the singularity along the radius. The resulting space-time diagram is shown in Fig. 9.3 on the right. Just as our friends Borya, Vasya and Masha are depicted in Fig. 9.2 by vertical world lines, the world line of the event horizon goes vertically upward exactly 1 Schwarzschild radius to the right of the singularity world line, which in Fig. 9.3 is shown as a sawtooth line.

Although in Fig. 9.3, depicting a Schwarzschild black hole in space-time as if there was nothing mysterious, by the early 1950s physicists began to understand that this diagram does not exhaust the essence of the matter. A black hole has different regions of spacetime: the first between the singularity and the event horizon and the second outside the event horizon. We failed fully expressed on the right side of Fig. 9.3, how exactly these areas are connected to each other.

To understand the relationship between the regions of space-time inside and outside the event horizon, imagine a black hole with a mass of 10 solar masses. Let an astronomer fly out of the singularity, fly out through the event horizon, rise to a maximum height of 1 million kilometers above the black hole, and then fall back through the event horizon and fall back into the singularity. The astronomer's flight is shown in Fig. 9.4.

To an attentive reader this may seem impossible - after all, it is impossible to jump out of the singularity at all! Let us limit ourselves to referring to purely mathematical the possibility of such a trip. As will be seen below, the complete Schwarzschild solution contains both black and So and a white hole. Therefore, the reader's patience and attention will be required over the next few sections. Here and in subsequent chapters we will illustrate the story using the journeys of astronomers or cosmonauts to black holes. For convenience, we will simply refer to the astronaut as “he.”

The traveling astronomer carries a watch with him to measure his own time. The stay-at-home scientists monitoring its flight from a distance of 1 million kilometers from the black hole also have watches. Space there is flat, and the clock measures coordinate time. Upon reaching the highest point of the trajectory (at a distance of a million kilometers from the black hole) All The clocks are set to the same moment (synchronized) and now show 12 noon. Then we can calculate at what moment (both according to the traveler’s own time and coordinate time) the astronomer will reach each point of his trajectory that interests us.

Let us recall that the astronomer's watch measures his own time. Therefore, it is impossible to notice from them the “slowdown of time” caused by the effect of gravitational redshift. For given values of the mass of the black hole and the height above it of the highest point of the path, calculations lead to the following result:

In the astronomer's own time

The astronomer flies out of the singularity at 11:40 a.m. (according to his own clock).
1/10,000 sec after 11:40 a.m., it flies through the event horizon into the outside world.
At 12 noon it reaches a maximum altitude of 1 million kilometers above the black hole.
In one 1/10,000 second before 12:20 p.m., it crosses the event horizon, moving inward.
The astronomer returns to the singularity at 12:20 p.m.

In other words, it takes the same time to move from the singularity to the event horizon and back - 1/10,000 s, while to move from the event horizon to the highest point of its trajectory and vice versa it spends 20 minutes each time (in 20 minutes it travels 1 million kilometers). It should be borne in mind that proper time flows in a standard way during flight.

Scientists conducting observations from afar measure coordinate time using their clocks; their calculations give the following results:

In coordinate time

Of course, everyone agrees that the traveling astronomer reaches his maximum flight altitude at 12 noon, i.e. the moment at which all clocks are synchronized. Everyone will also agree on when the astronomer flies out of the singularity and when he returns to it. But in other respects, Schwarzschild geometry is clearly abnormal. Having departed from the singularity, the astronomer moves in coordinate time back in time up to a year It then rushes forward in time again, reaching its maximum flight altitude at noon, and descending below the event horizon of the year. After that it moves again back in time and falls into the singularity at 12:20 p.m. In the space-time diagram, its world line has the form shown in Fig. 9.5.

Some of these strange findings can be understood intuitively. Let us remember that from the point of view of a remote observer (whose clock measures coordinate time), time stops at the event horizon. Let us also remember that a stone or any other body falling on the event horizon never will not reach a point with a height of the Schwarzschild radius in the view of a distant observer. Therefore, an astronomer falling into a black hole cannot cross the event horizon until a year, i.e. in the infinitely distant future. Since the entire journey is symmetrical relative to the moment of 12 noon (i.e., take-off and fall take the same time), then distant scientists must observe that the astronomer has been rising, moving towards them, for billions of years. It must move outward to the event horizon per year.

Even more incomprehensible is the fact that remote observers see two moving astronomers. So, for example, at 3 o'clock in the afternoon they see one astronomer falling on the event horizon (moving forward in time). However, according to their own calculations, must there could be another astronomer inside the event horizon, falling into the singularity (and moving backwards in time).

Of course this is nonsense. More precisely, this strange behavior of coordinate time means that the one shown in Fig. 9.3, the picture of a Schwarzschild black hole simply cannot be correct. We have to look for other - and there may be many of them - true space-time diagrams for a black hole. In the simple diagram shown in Fig. 9.5, the same regions of space-time turn out to be overlapped twice, which is why two astronomers are observed at once, while in fact there is only one. This means that you need to expand or transform this simple picture in such a way as to reveal the true one, or global the structure of the entire space-time associated with the Schwarzschild black hole.

To better understand what this global picture should look like, consider the event horizon. In a simplified two-dimensional space-time diagram (see the right side of Fig. 9.3), the event horizon is a line running from moment (distant past) to moment (distant future) and located exactly 1 Schwarzschild radius from the singularity. Such a line, of course, correctly depicts the location of the surface of the sphere in ordinary three-dimensional space. But when physicists tried to calculate the volume of this sphere, they, to their amazement, discovered that it was equal to zero. If the volume of a certain sphere is zero, then it is, of course, just a point. In other words, physicists began to suspect that this “line” in the simplified diagram should actually be a point in the global picture of the black hole!

Imagine, in addition, an arbitrary number of astronomers jumping out of the singularity, flying up to different maximum heights above the event horizon and falling back again. Regardless of exactly when they were thrown out of the singularity, and regardless of exactly what height above the event horizon they took off, All of them will cross the event horizon at moments of coordinate time (on the way out) and (on the way back). As a result, astute physicists will also suspect that these two “points”, and , must necessarily be represented in the global picture of the black hole in the form of two segments of world lines!

To move from a simplified picture of a black hole to a global picture of it, we need to transform our simplified picture into a much more complex space-time diagram. And yet our end result will be a new space-time diagram! In this diagram, space-like quantities will be directed horizontally (from left to right), and time-like quantities will be directed vertically (from bottom to top). In other words, the transformation should work so that old spatial and temporal coordinates were replaced by new spatial and temporal coordinates that would reflect the completely true nature of the black hole.

To try to understand how the old and new coordinate systems can be related to each other, consider an observer near a black hole. To avoid falling into a black hole and remain at a constant distance from it, it must have powerful rocket engines that eject streams of gases downward. In flat space-time, far from gravitating masses, a spacecraft with the engines running would acquire acceleration and would move faster and faster, because the thrust of the rocket engines would provide it with a constant increase in speed. The world line of such a ship is depicted in the space-time diagram in Fig. 9.6. This line gradually approaches a straight line with a slope of 45º as, due to the continuous operation of the engines, the speed of the ship approaches the speed of light. A curve depicting such a world line is called hyperbole. An observer who is near a black hole and tries to remain at a constant distance from it will constantly experience acceleration caused by the work of the ship's rocket engines. Astute physicists will therefore suspect that the "constant height" lines in the revised and improved diagram of space-time near a black hole will be branches of hyperbolas.

Finally, the observer who is trying to stay on the event horizon must have incredibly powerful rocket engines. To prevent him from falling inside a black hole, these engines must operate with such power that the observer, if he were in a flat world, would move at the speed of light. This means that the world lines of the event horizon should be inclined at exactly 45º in the revised and improved space-time diagram.

In 1960, independently of each other, Kruskal and Szekeres found the required transformations, transforming the old space-time diagram for the Schwarzschild black hole into a new diagram - revised and improved. This new one Kruskal-Szekeres diagram correctly covers all spacetime and fully reveals the global structure of the black hole. At the same time, all previously noted suspicions are confirmed and some new surprising and unexpected details are discovered. However, although the Kruskal and Szekeres transformations immediately transform the old picture into a new one, it is better to visualize them in the form of a sequence of transformations, schematically depicted in Fig. 9.7. The end result is again a diagram of space-time (the spatial direction is horizontal and the time direction is vertical), with the rays of light going to and from the black hole depicted, as usual, as straight lines with an inclination of 45º.

The end result of the transformation is striking and at first arouses disbelief: you see that there are actually two singularities depicted there, one in the past and the other in the future; in addition to this, there are two outer universes far from the black hole.

But in fact, the Kruskal-Szekeres diagram is correct, and to understand this, we will again consider the flight of an astronomer thrown out of a singularity, crossing the event horizon and falling back again. We already know that its world line in a simplified space-time diagram is unusual. This line is again shown on the left in Fig. 9.8. In the Kruskal-Szekeres diagram (Fig. 9.8, right), such a line looks much more meaningful. The observer actually jumps out of a singularity in the past and ends up in a singularity in the future. Consequently, such an “analytically complete” description of the Schwarzschild solution includes How black, So and a white hole. Our astronomer actually flies out of a white hole and ends up falling into a black hole. Please note that its world line is everywhere inclined to the vertical by less than 45º, i.e. this line is everywhere time-like and therefore admissible. Comparing the left and right parts of Fig. 9.8, you will find that the "points" of the instants and on the event horizon have now stretched into two straight lines with a slope of 45°, which confirms our previous suspicions.

When moving to the Kruskal-Szekeres diagram, the true nature of all spacetime in the vicinity of a Schwarzschild black hole is revealed. In a simplified diagram, different regions of spacetime overlapped with each other. That is why remote scientists, observing the fall of an astronomer into a black hole (or his departure from it), mistakenly assumed that there were two astronomer The Kruskal-Szekeres diagram properly disentangles these overlapping regions. In Fig. Figure 9.9 shows how these different areas are related to each other in both types of diagrams. There are actually two outer universes (regions I and III), as well as the inner parts of the black hole (regions II and IV) between the singularities and the event horizon.

It is also useful to analyze how individual parts of the space-time grid are transformed when moving from a simplified diagram to a Kruskal-Szekeres diagram. In a simplified representation (Fig. 9.10), the dashed lines of constant heights above the singularity are simply straight lines directed vertically. The dotted lines of constant coordinate time are also straight, but horizontal. The space-time grid looks like a piece of ordinary graph paper.

In the Kruskal-Szekeres diagram (Figure 9.11), the constant time lines (dashed lines) remain straight, but now they diverge at different angles. The lines of constant distance from the black hole (dashed lines) are hyperbolas, as we suspected before.

Analyzing Fig. 9.11, one can understand why, when passing through the event horizon, space and time change roles, as already mentioned in the previous chapter. Recall that in a simplified diagram (see Fig. 9.10) lines of constant distance are directed vertically. So, a specific dashed line can represent a point that is constantly located at an altitude of 10 km above the black hole. Such a line should be parallel to the event horizon on the simplified diagram, i.e. it must be vertical; since it depicts something stationary at all times, the line of constant distance must have a time-like direction (in other words, up) in this simplified diagram.

In Fig. Figure 9.11 shows the Kruskal-Szekeres diagram; here the dashed lines of constant distance have a general upward direction if taken far enough from the black hole. There they are still time-like. However, within the event horizon, the dashed lines of constant distance are oriented generally horizontally. This means that under the event horizon, lines of constant distance have a space-like direction! Consequently, what is usually (in the external Universe) associated with distance behaves like time inside the event horizon.

Similarly, in a simplified diagram (see Fig. 9.10), the constant time lines are horizontal and have a space-like direction. For example, a specific dotted line may mean the moment "3 o'clock in the afternoon for all points in space." Such a line should be parallel to the spatial axis in the simplified diagram, i.e. it should be horizontal.

In Fig. In Figure 9.11, which shows the Kruskal-Szekeres diagram, the dotted constant-time lines generally have a spacelike direction when taken far from the black hole, i.e. they are almost horizontal there. But inside the event horizon, the dotted lines of constant time are generally directed from bottom to top, i.e. oriented in a time-like direction. So, under the event horizon, constant time lines have a time-like direction! Consequently, what is usually (in the external Universe) associated with time behaves like distance inside the event horizon. When crossing the event horizon, space and time change roles.

In connection with the discussion of the properties of space and time, it is important to note that in the Kruskal-Szekeres diagram (Fig. 9.11) both singularities (both past and future) are oriented horizontally. Both hyperbolas depicting a "point" r= 0, have a slope everywhere less than 45º k verticals. These lines are spacelike, and therefore the Schwarzschild singularity is said to be spacelike.

The fact that the Schwarzschild singularity is spacelike will lead to important conclusions. As in the special theory of relativity (see Fig. 1.9), here it is impossible to move at superluminal speeds, so space-like world lines are prohibited as “paths” of movement. It is impossible to move along world lines with an inclination of more than 45° to the vertical (time-like) direction. Therefore, it is impossible to get from our Universe (in the Kruskal-Szekeres diagram on the right) to another Universe (in the same diagram on the left). Any path connecting both Universes with each other must be space-like in at least one place, and such paths are prohibited for movement. In addition, since the event horizon is tilted exactly at an angle of 45º, an astronomer from our Universe who descends below this horizon will never be able to emerge from under it again. For example, if someone infiltrates area II in Fig. 9.9, then All valid timelike worldlines would lead it straight to the singularity. A Schwarzschild black hole is a trap with no way out.

To get a fuller sense of the nature of Kruskal-Szekeres geometry, it is instructive to consider the space-like slices of the space-time diagram made by these authors. These will be nesting diagrams curved space near a black hole. This method of obtaining slices of space-time using space-like hypersurfaces was used by us previously (see Fig. 5.9, 5.10 and 5.11) and facilitated the understanding of the properties of space in the vicinity of the Sun.

In Fig. Figure 9.12 shows a Kruskal-Szekeres diagram “sliced” along characteristic space-like hypersurfaces. Slice A refers to an early point in time. Initially, the two Universes located outside the black hole are in no way connected with each other. On the way from one Universe to another, the space-like slice encounters a singularity. Therefore the nesting diagram for the slice A describes two separate Universes (depicted as two asymptotically flat sheets parallel to each other), each of which has a singularity. Later, with the further evolution of these Universes, the singularities are connected and a bridge arises in which there are no longer any singularities. This corresponds to the slice B, where the singularity does not enter. Over time, this bridge, or "mole Hole", expands and reaches a maximum diameter equal to two Schwarzschild radii (moment corresponding to the cut IN). Later, the bridge begins to tighten again (cut G) and finally breaks (cut D), so that we again have two separate Universes. This evolution of a wormhole (Fig. 9.12) takes less than 1/10,000 s if the black hole has the mass of the Sun.

The discovery by Kruskal and Szekeres of a similar global structure of space-time around a black hole was a decisive breakthrough on the front of theoretical astrophysics. For the first time, it was possible to construct diagrams that completely depict all regions of space and time. But after 1960 further successes were achieved, most notably by Roger Penrose. Although the Kruskal-Szekeres diagram represents the whole story, the diagram extends to the right and left indefinitely. For example, our Universe extends infinitely to the right on the Kruskal-Szekeres diagram, while the space-time of the “other” asymptotically flat Universe, which is parallel to ours, extends to infinity on the same diagram. Penrose was the first to realize how useful and instructive it would be to use a “map” that mapped these endless expanses into some finite areas from which it would be possible to accurately judge what was happening far from the black hole. To implement this idea, Penrose used the so-called methods conformal mapping, with the help of which the entire space-time, including both Universes in its entirety, is depicted on one finite diagram.

To introduce Penrose's methods, let's look at an ordinary flat spacetime of the type shown in Fig. 9.2. All spacetime there is concentrated on the right side of the diagram simply because it is impossible to be at a negative distance from an arbitrary origin. You can be, say, 2 m from him, but certainly not minus 2 m. Let's return to Fig. 9.2. The world lines of Bory, Vasya and Masha are depicted there only in a limited area of space-time due to the limited size of the page. If you want to see where Borya, Vasya and Masha will be in a thousand years, or where they were a billion years ago, you will need a much larger sheet of paper. It would be much more convenient to depict all these positions (events) far from the point “here and now” on a compact, small diagram.

We have already seen that the “outermost” regions of space-time are called infinities. These areas are extremely far from the here and now in space or time (the latter meaning that they can be in the very distant future or the very distant past). As can be seen from Fig. 9.13, there can be five types of infinities. First of all this 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 into I+ - 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, then nothing (except perhaps tachyons) 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 inclined by 45º on the space-time diagram. This allows you to enter " - the light infinity of the past, where all light rays come from. There is finally and - light infinity of the future(where all the "light rays" go.) Every remote region of space-time belongs to one of these five infinities; I -, , I 0 , or I+.

Rice. 9.13. Infinity. The most distant "outskirts" of space-time (infinity) are divided into five types. Timelike infinity of the past ( I -) is the region from which all material bodies come, and the time-like infinity of the future ( I+) is the area where they all go. The light infinity of the past () is the region from which light rays come, and the light infinity of the future is that region ( I+), where they go. Nothing (except tachyons) can get into spacelike infinity ( I 0).

Rice. 9.14. Penrose conformal mapping. There is a mathematical technique with the help of which it is possible to “pull together” the most distant outskirts of space-time (all five infinities) into a completely visible finite region.

The Penrose method comes down to a mathematical technique of contracting all these infinities onto the same sheet of paper. The transformations that carry out such contraction act like bulldozers (see the figurative representation of these transformations in Fig. 9.14), raking the most distant sections of space-time to where they can be better seen. The result of this transformation is shown in Fig. 9.15. It should be kept in mind that lines of constant distance from an arbitrary reference point are mostly vertical and always indicate a timelike direction. Constant time lines are mostly horizontal and always indicate a space-like direction.

On conformal map of the entire flat space-time (Fig. 9.15), space-time as a whole fits into a triangle. All timelike infinity is in the past ( I -) is collected into one single point at the bottom of the diagram. All timelike world lines of all material objects emerge from this point, representing the extremely distant past. All timelike infinity in the future ( I+) is collected into a single point at the top of the diagram. The time-like world lines of all material objects in the Universe eventually end up at this point, representing the distant future. Spacelike infinity ( I 0) is collected at the point on the right in the diagram. Nothing (except tachyons) can ever get into I 0 . Light infinities in the past and in the future and turned into straight lines with a slope of 45º, limiting the diagram at the top right and bottom right along the diagonals. Light rays always follow world lines with an inclination of 45°, so that light coming from the distant past begins its path somewhere at , and one who goes into the distant future ends his journey somewhere on . The vertical line bounding the diagram on the left is simply the timelike world line of our chosen arbitrary starting point ( r = 0).


Rice. 9.15. Penrose diagram for flat spacetime. All space-time is collected inside a triangle using the conformal mapping method invented by Penrose. Of the five infinities, three ( I -, I 0 , I+ ) are compressed to individual points, and two are light infinities And- became straight lines with a slope of 45º.	Rice. 9.16. An example of a conformal Penrose diagram. This diagram shows essentially the same thing as Fig. 9.2. However, in a conformal diagram, the world lines of objects are represented completely (from the distant past I - until the distant future I+).

To finish with the description of the conformal Penrose diagram of flat space-time, we depicted it in Fig. 9.16 completely world lines of Bori, Vasya and Masha. Compare this diagram with Fig. 9.2 - after all, this is one and the same thing, only on the conformal diagram the world lines can be traced along their entire length (from the distant past I -љ until the distant future I+)

Penrose's depiction of ordinary flat space-time does not produce anything sensational. However, Penrose's method also applies to black holes! In particular, the Kruskal-Szekeres diagram (see Fig. 9.11) can be displayed conformally in such a way that the physicist sees All the space-time of all Universes depicted on a single piece of paper. As this is clearly depicted in Fig. 9.17, the Penrose conformal transformations here again work like bulldozers “raking” space-time. The final result is shown in Fig. 9.18.

In the Penrose diagram of a Schwarzschild black hole (Figure 9.18), we again notice that the constant time lines and constant distance lines behave essentially the same as in the Kruskal-Szekeres diagram. The event horizon retains its slope of 45º, and singularities (both past and future) remain spacelike. The exchange of roles between space and time, as before, occurs when crossing the event horizon. However, now the most distant parts of both universes associated with the black hole are before our eyes. All five infinities of our Universe ( I -, , I 0 , , I+ ) are visible on the right in the diagram, and on the left on it you can see all five infinities of another Universe ( I -, , I 0 , , I+ ).

We can now move on to the final exercise with the Schwarzschild black hole - finding out what the desperately curious kamikaze astronomers will see as they fall into the black hole and crossing event horizon.

The spacecraft of these astronomers is shown in Fig. 9.19. The bow porthole is always directed directly at the singularity, and the stern porthole is always directed in the opposite direction, i.e., towards our outer Universe. Note that the spacecraft now does not have rocket engines to slow its fall. Starting from a great height above the black hole, astronomers simply fall vertically at an ever-increasing (according to their measurements) speed. Their world line (Fig. 9.20) passes first through the event horizon and then leads to the singularity. Since their speed is always less than the speed of light, the world line of the ship on the Penrose diagram should be time-like, i.e. everywhere have an inclination to the vertical of less than 45º. During the journey, astronomers take four pairs of photographs at different stages of the journey - one from each porthole. First couple (photos A) made when they were still very far from the black hole. In Fig. 9.21, A the black hole is visible as a small speck in the center of the field of view of the bow window. Although the sky in the immediate vicinity of the black hole is distorted, the rest of the sky appears completely normal. As the speed at which astronomers fall into the black hole increases, light from objects in the distant universe seen through the aft window is increasingly redshifted.

Rice. 9.21.

Photo A. Far from the black hole. From a great distance, a black hole appears as a small black speck in the center of the bow window's field of view. Astronomers falling into the hole observe through the aft window an undistorted view of the Universe from which they arrived.

Photo B. Neither the event horizon. Due to the aberration effect, the image of the black hole is compressed towards the center of the field of view of the bow window. An astronomer observing through the aft porthole sees only the Universe from which the ship arrived.

Photo B. Between the event horizon and the singularity. Dropping below the event horizon, an astronomer looking out the bow window can see another Universe. Light coming from a region of another Universe fills the central part of his field of view.

Photo G. Directly above the singularity. As astronomers approach the singularity, another Universe becomes increasingly visible through the bow window. The image of the black hole itself (which looks like a ring) becomes thinner and thinner, quickly approaching the edge of the field of view of the bow window.

Although, according to remote observers, the fall of the spacecraft slows to a complete stop at the event horizon, astronomers on himself nothing like this will be noticed on a spaceship. In their opinion, the speed of the ship is increasing all the time and when crossing the event horizon it amounts to a noticeable fraction of the speed of light. This is significant for the reason that as a result, falling astronomers observe the phenomenon of aberration of star light, very similar to the one we discussed in Chapter. 3 (see Fig. 3.9, 3.11). Remember that when moving at near the speed of light, you will notice severe distortions in the sky. In particular, images of celestial bodies seem to be collected in front of a moving observer. As a result of this effect, the image of the black hole is concentrated closer to the middle of the bow window of the falling spacecraft.

The picture observed by falling astronomers from the event horizon is shown in Fig. 9.21, B. This and subsequent figures are based on calculations made by Cunningham at the California Institute of Technology in 1975. If astronomers were at rest, the image of the black hole would occupy the entire field of view of the bow window (Fig. 8.15, D). But since they move at high speed, the image is concentrated in the middle of the bow window. Its angular diameter is approximately 80º. The view of the sky next to the black hole is very distorted, and the astronomer observing through the aft window sees only the Universe from which they came.

To understand what will be visible when the ship is inside event horizon, let's return to the Penrose diagram of a Schwarzschild black hole (see Fig. 9.18 or 9.20). Let us remember that the light rays going into the black hole have a slope of 45° in this diagram. Therefore, once under the event horizon, astronomers will be able to see another Universe. Rays of light from distant parts of another Universe (i.e. from its infinity on the left side of the Penrose diagram) can now reach astronomers. As shown in Fig. 9.21, IN, in the center of the field of view of the bow window of the spacecraft, located between the event horizon and the singularity, another Universe is visible. The black part of the hole is now represented as rings, separating the image of our Universe from the image of another Universe. As falling observers approach the singularity, the black ring becomes thinner, pressing closer to the very edge of the bow porthole's field of view. The view of the sky from a point directly above the singularity is shown in Fig. 9.21, G. Through the bow window, the other Universe becomes better and better visible, and right at the singularity, its view completely fills the field of view of the bow window. An astronomer conducting observations through the aft window sees only our outer Universe throughout the entire flight, although its image becomes more and more distorted.

Falling astronomers will note another important effect that is not reflected in the “snapshots” 9.21, A-G. Let us remember that light leaving the vicinity of the event horizon into the distant Universe undergoes a strong red shift. This phenomenon is called gravitational redshift, we discussed in Chap. 5 and 8. The red shift of light coming from a region with a strong gravitational field corresponds to its loss of energy. Conversely, when light "falls" on a black hole, it experiences purple shift and gains energy. Weak radio waves arriving from the distant Universe are transformed, for example, into powerful X-rays or gamma rays directly above the event horizon. If described by Penrose diagrams of the type shown in Fig. 9.18 black holes really exist in nature, then the light falling on them from , accumulates over billions of years near the event horizon. This incident light acquires tremendous energy, and when astronomers descend below the event horizon, they are therefore greeted with an unexpected, sharp burst of X-rays and gamma rays. That light that comes from the region - Schwarzschild solution - Kerr solution - white hole - singularity

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The expressions for the tensor components and through the functions v and A are as follows

Just the expression for the component is cumbersome, but it happens that its exact expression is rarely necessary to use.

The important point is that the divergence of this tensor must be equal to zero. If we have an expression for other components, then requiring the divergence to vanish often helps avoid using the exact expression for .

The following exercises may be suggested at this point.

1) Prove that if there is no matter inside a sphere of radius b and the distribution of matter outside this sphere is spherically symmetric, then the space inside the sphere is flat with the metric .

2) Prove that if the energy-momentum tensor is known everywhere inside the sphere of radius, then whatever it is outside this sphere, this will not affect the physics inside the sphere of radius (It is assumed that outside this sphere the energy-momentum tensor is characterized by a spherically symmetric distribution.)

The solution outside the spherically symmetric mass distribution is obtained if we solve the resulting differential equations.

We start by noting that depends only on A. Since is equal to zero, we get

The factor 2 is taken for convenience, so that the constant is the total mass of the star multiplied by the Newtonian gravitational constant. If there are no singularities inside a sphere of radius , where all the mass is located, then the constant must be equal to

(11.3.3)

We are confident that there is no time dependence because

so A does not depend on time at all. The last task is to obtain an expression for . We do this by equating since both of these quantities are equal to zero. From here we come to the conclusion that

Which can only happen if the function v has the following form:

(11.3.5)

where is an arbitrary function of time. However, since the function v appears in the coefficient of the magnitude in the metric as follows:

we can eliminate the multiplier by changing the scale of the time coordinate. Other elements of the metric tensor do not change with such a replacement, since only the function is included in them. The result obtained is known as the Schwarzschild metric

Interestingly, the resulting metric does not depend on time, although we never said that we were looking for a static solution. The absence of time dependence of the Schwarzschild metric follows from the assumption of spherical symmetry and the fact that we are considering the metric in a region with zero pressure density.

For the case of a real star such as the Sun, there is no exact spherical symmetry, since there is rotation and since there is a thickening () at the equator. However, these differences cause slight deviations from the case of spherical symmetry. If there is a luminous flux from the star, then other corrections will appear, since the energy density will not be equal to zero in space outside the star. However, the Schwarzschild solution describes the situation with the Sun quite accurately, so that the precession of Mercury's perihelion is given correctly within the limits of measurement errors.

Metric type

Schwarzschild coordinates

In the so-called Schwarzschild coordinates, of which the last 3 are analogous to spherical ones, the metric tensor of the most physically important part of Schwarzschild space-time with topology (the product of a region of two-dimensional Euclidean space and a two-dimensional sphere) has the form

The coordinate is not the length of the radius vector, but is introduced so that the area of the sphere in a given metric is equal to . In this case, the “distance” between two events with different (but identical other coordinates) is given by the integral

When or the Schwarzschild metric tends (componentwise) to the Minkowski metric in spherical coordinates, so that far from a massive body, spacetime turns out to be approximately pseudo-Euclidean in signature. Since at and monotonically increases with increasing , the proper time at points near the body “flows slower” than far from it, that is, a peculiar gravitational time dilation massive bodies.

Differential characteristics

Let's denote

Then the non-zero independent Christoffel symbols have the form

The curvature tensor is of Petrov type.

Mass defect

If there is a spherically symmetric distribution of matter of “radius” (in terms of coordinates), then the total mass of the body can be expressed in terms of its energy-momentum tensor using the formula

In particular, for the static distribution of matter, where is the energy density in space. Considering that the volume of the spherical layer in the coordinates we have chosen is equal to

we get that

This difference expresses gravitational body mass defect. We can say that part of the total energy of the system is contained in the energy of the gravitational field, although it is impossible to localize this energy in space.

Feature in the metric

At first glance, the metric contains two features: at and at . Indeed, in Schwarzschild coordinates, a particle falling on a body will require an infinitely long time to reach the surface, but the transition, for example, to Lemaître coordinates in the accompanying reference frame shows that from the point of view of the falling observer there is no feature of space-time on this surface, and both the surface itself and the region will be reached in a finite proper time.

The real feature of the Schwarzschild metric is observed only at , where the scalar invariants of the curvature tensor tend to infinity. This feature (singularity) cannot be eliminated by changing the coordinate system.

Event Horizon

The surface is called event horizon. With a better choice of coordinates, such as Lemaître or Kruskal coordinates, it can be shown that no signals can exit the black hole through the event horizon. In this sense, it is not surprising that the field outside a Schwarzschild black hole depends on only one parameter - the total mass of the body.

Kruskal coordinates

You can try to introduce coordinates that do not give a singularity at . There are many such coordinate systems known, and the most common of them is the Kruskal coordinate system, which covers with one map the entire maximally extended manifold that satisfies Einstein’s vacuum equations (without the cosmological constant). This more spacetime is usually called (maximally extended) Schwarzschild space or (less commonly) Kruskal space. The metric in Kruskal coordinates has the form

where , and the function is defined (implicitly) by the equation .

Rice. 1. Section of Schwarzschild space. Each point in the figure corresponds to a sphere with area . Light-like geodesics (that is, world lines of photons) are straight lines at an angle to the vertical, in other words, these are straight or

Space maximum, that is, it can no longer be isometrically embedded in a larger space-time, and the region in Schwarzschild coordinates () is just a part (this region is region I in the figure). A body moving slower than light - the world line of such a body will be a curve with an angle of inclination to the vertical less than , see the curve in the figure - can leave. In this case, it falls into region II, where . As can be seen from the figure, it will no longer be able to leave this area and return to it (to do this it would have to deviate more than from the vertical, that is, exceed the speed of light). Region II is thus a black hole. Its boundary (broken line, ) is accordingly the event horizon.

There is another asymptotically flat region III, in which Schwarzschild coordinates can also be introduced. However, this region is not causally connected with region I, which does not allow us to obtain any information about it, remaining outside the event horizon. In the case of a real collapse of an astronomical object, regions IV and III simply do not arise, since the left side of the presented diagram must be replaced with a non-empty space-time filled with collapsing matter.

Let us note several remarkable properties of a maximally extended Schwarzschild space:

Orbital movement

Main article: Kepler's problem in general relativity

History of acquisition and interpretation

The Schwarzschild metric, while acting as an object of significant theoretical interest, is also a kind of tool for theorists, seemingly simple, but nevertheless immediately leading to difficult questions.

In mid-1915, Einstein published preliminary equations for the theory of gravity. These were not yet Einstein’s equations, but they already coincided with the final ones in the vacuum case. Schwarzschild integrated the spherically symmetric equations for vacuum in the period from November 18, 1915 to the end of the year. On January 9, 1916, Einstein, whom Schwarzschild had approached about the publication of his paper in the Berliner Berichte, wrote to him that he had “read his work with great passion” and was “stunned that the true solution of this problem could be expressed so easily” - Einstein initially doubted whether it was even possible to obtain a solution to such complex equations.

Schwarzschild completed his work in March, also obtaining a spherically symmetric static internal solution for a fluid with constant density. At this time, a disease (pemphigus) fell upon him, which brought him to his grave in May. Since May 1916, I. Droste, a student of G. A. Lorentz, conducting research within the framework of Einstein's final field equations, obtained a solution to the same problem using a simpler method than Schwarzschild. He also made the first attempt to analyze the divergence of a solution when tending to the Schwarzschild sphere.

Following Droste, most researchers began to be satisfied with various considerations aimed at proving the impenetrability of the Schwarzschild sphere. At the same time, theoretical considerations were supported by a physical argument, according to which “this does not exist in nature,” since there are no bodies, atoms, or stars whose radius would be less than the Schwarzschild radius.

For K. Lanczos, as well as for D. Gilbert, the Schwarzschild sphere became a reason to think about the concept of “singularity”; for P. Painlevé and the French school, it was the object of a controversy in which Einstein became involved.

During the 1922 Paris colloquium organized in connection with Einstein's visit, they discussed not only the idea that the Schwarzschild radius would not be singular, but also a hypothesis anticipating what is now called gravitational collapse.

Schwarzschild's skillful development was only a relative success. Neither his method nor his interpretation were adopted. Almost nothing has been preserved from his work except the “bare” result of the metric, with which the name of its creator was associated. But the questions of interpretation and, above all, the question of the “Schwarzschild singularity” were nevertheless not resolved. The point of view began to crystallize that this singularity does not matter. Two paths led to this point of view: on the one hand, theoretical, according to which the “Schwarzschild singularity” is impenetrable, and on the other hand, empirical, consisting in the fact that “this does not exist in nature.” This point of view spread and became dominant in all the specialized literature of that time.

The next stage is associated with intensive research into issues of gravity at the beginning of the “golden age” of the theory of relativity.

Literature

K. SchwarzschildÜber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie // Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1. - 1916. - 189-196.
Rus. translation: Schwarzschild K. On the gravitational field of a point mass in Einstein’s theory // Albert Einstein and the theory of gravity. M.: Mir, 1979. pp. 199-207.
Landau, L. D., Lifshits, E. M. Field theory. - 7th edition, revised. - M.: Nauka, 1988. - 512 p. - (“Theoretical Physics”, volume II). - ISBN 5-02-014420-7
Drost J. Het van een enkel centrum in Einstein s theorie der zwaartekracht en de beweging van een stoffelijk punt in dat veld // Versl. gev Vergad. Akad. Amsterdam. - 1916. - D.25. - Biz.163-180.

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Backgroundpublications

On November 25, 1915, Albert Einstein, a professor at the University of Berlin, presented a written report to the Royal Prussian Academy of Sciences containing a system of completely covariant (not changing form when changing the coordinate system) equations of the relativistic theory of gravitational field, also known as the General Theory of Relativity (GR).

A week earlier, Einstein gave a lecture at a meeting of the Academy, where he demonstrated an earlier and still incomplete version of these equations, which did not have complete covariance. However, these equations already gave Einstein the opportunity, using the method of successive approximations, to correctly calculate the anomalous rotation of Mercury's orbit and predict the magnitude of the angular deviation of starlight in the gravitational field of the Sun. Karl Schwarzschild This speech found a grateful listener - Karl Schwarzschild, Einstein's colleague at the Academy. He served as an artillery lieutenant in the active army of the German Empire and just then came on leave. In December, upon returning to the front, Schwarzschild found an exact solution to the first version of Einstein’s equations, which he published through him in “Reports on the Meetings” ( Sitzungsberichte) Academy. In February, having already familiarized himself with the final version of the general relativity equations, Schwarzschild sent Einstein a second article, in which the gravitational, also known as Schwarzschild, radius appears for the first time. In the modern interpretation, this is the radius of the horizon of the black hole, from under which signal transmission to the outside is impossible. On February 24, when Einstein sent this work to press, the battle of Verdun had already lasted three days.

The science And war

Karl Schwarzschild (1873−1916) was not only a brilliant, but also a versatile scientist. He left a deep mark on observational astronomy, being one of the pioneers of equipping telescopes with photographic equipment and using it for photometry purposes. He owns profound and original works in the field of electrodynamics, stellar astronomy, astrophysics and optics. Schwarzschild even managed to make a significant contribution to the quantum mechanics of atomic shells, building in his last scientific work the theory of the Stark effect - the displacement and splitting of atomic levels in an electric field. In 1900, fifteen years before the creation of General Relativity, he not only seriously considered the paradoxical possibility that the geometry of the Universe differs from Euclidean (Lobachevsky had already assumed this), but also assessed the lower limits of the radius of curvature of space for spherical and pseudospherical geometry of space. Before reaching thirty years of age, he became a professor at the University of Göttingen and director of the university observatory, in 1909 he was elected a member of the Royal Astronomical Society of London and headed the Potsdam Astrophysical Observatory, and four years later became a full member of the Prussian Academy of Sciences. News of the death of a German soldier who fell at Verdun. Schwarzschild's slender scientific career was cut short by the First World War. He was not subject to conscription due to his age, but he volunteered to join the army and eventually ended up on the Russian front at the headquarters of an artillery unit, where he was involved in calculating the trajectories of long-range gun projectiles. There he became a victim of pemphigus, or pemphigus, a very severe autoimmune skin disease to which he had a hereditary tendency. This pathology is difficult to treat in our time, and then it was completely incurable.

In March 1916, Schwarzschild was commissioned and returned to Potsdam, where he died on May 11. He was one of the most prominent physicists whose lives were claimed by the First World War. You can also remember Henry Moseley, one of the founders of X-ray spectroscopy. He served as a liaison officer and died at age 27 during the Dardanelles operation on August 10, 1915.

Schwarzschild metric

The famous space-time metric (or four-tensor) of Schwarzschild historically became the first exact solution of the general relativity equations. It describes a static gravitational field that is created in a vacuum by a stationary spherically symmetric body of mass M. In the standard notation in Schwarzschild coordinates, t, r, θ, φ has two singular points (in formal language - singularities), near which one of the elements of the metric tends to zero , and the other to infinity. One of the singularities arises at r = 0, that is, at the same place where the Newtonian gravitational potential turns to infinity. The second singularity corresponds to the value r = 2GM/c 2, where G is the gravitational constant, M is the gravitating mass and c is the speed of light. This parameter is usually denoted r s and is called the Schwarzschild radius or gravitational radius. This is already a non-Newtonian singularity, arising from the equations of general relativity, over the meaning of which several generations of physicists agonized. The gravitational radius of a body with the mass of the Sun is approximately 3 km. As is known, this parameter plays a key role in the theory of black holes.

It is worth recalling that the Schwarzschild angular coordinates θ and φ are completely analogous to the polar and azimuthal angles in ordinary spherical coordinates, however, the value of the radial coordinate r is by no means equal to the length of the radius vector. In the Schwarzschild metric, the length of a circle with a center at the origin is expressed by the Euclidean formula 2πr, but the distance between two points with radii r 1 and r 2 located on the same radius vector always exceeds the arithmetic difference r 2 -r 1. From this it is immediately clear that Schwarzschild space is non-Euclidean - the ratio of the circumference of a circle to the length of its radius is less than 2π.

First bridgeTo black holes

Now comes the fun part. The Schwarzschild metric, as given above, is completely absent in both of his articles. The first of his publications, “On the gravitational field of a point mass resulting from Einstein’s theory,” presents a space-time metric corresponding to the gravitational field of a point mass, which is not at all equivalent to the standard metric, although it is superficially similar. In the metric that Schwarzschild himself wrote, the radial coordinate has a lower positive limit, so there is no Newtonian-type singularity in it. All that remains is the singularity, which arises when the radius takes its minimum value, which appears as a constant of integration. For this constant in Schwarzschild’s article there is neither a formula nor a numerical estimate, only the designation α. The informal meaning of this singularity is that the point center of mass is surrounded by a sphere of radius α and something strange and incomprehensible is happening on this spherical surface. Schwarzschild does not go into details.

Karl Schwarzschild obtained his metric as a result of solving Einstein's equations in their first version, which he read on November 18. On its basis, he confirmed the magnitude of the anomalous rotation of Mercury's orbit calculated by Einstein. He also derived a relativistic analogue of Kepler's third law - but only for circular orbits. Specifically, he showed that the square of the angular velocity of test bodies orbiting in such orbits around a central point is given by the simple formula n 2 = α/2R 3 (the letter n Schwarzschild denotes the angular velocity; R is the radial coordinate). Since R cannot be less than α, the angular velocity has an upper limit n 0 = 1/(√2α).

Let me remind you that in Newtonian mechanics the angular velocity of bodies revolving around a point mass can be arbitrarily large, so the specificity of general relativity is clearly visible here.

The formula for n 0 looks unusual because of its dimension. This is due to the fact that Schwarzschild takes the speed of light to be unity. To obtain the angular velocity with the usual dimension of 1/sec, you need to multiply the right-hand side of the formula for n 0 by the speed of light c.

Schwarzschild saved the highlight for the end. At the end of the article, he noted that if the value of the point mass at the origin is equal to the mass of the Sun, then the maximum rotation frequency is approximately 10 thousand revolutions per second. It immediately follows that α = 10 -4 s/2π√2. Since c = 3×10 5 km/sec, the parameter α turns out to be approximately equal to 3 km, that is, the gravitational radius of the Sun! Without appearing explicitly in Schwarzschild's article, this number entered through the back door and without any justification (Schwarzschild did not specify how he obtained the numerical value of the limiting frequency). In general, Schwarzschild’s first paper already lays a very thin bridge to the theory of black holes, although it is not so easy to detect. Noticing this, I was quite surprised, since it is generally accepted that the gravitational radius appears only in Schwarzschild’s second paper.

Second bridgeTo black holes

Schwarzschild's second paper is entitled "On the gravitational field of a sphere filled with an incompressible fluid, calculated in accordance with Einstein's theory." In it (let me remind you, already on the basis of the complete system of general relativity equations) two metrics are calculated: for external space and for space inside the sphere. At the end of this article, the gravitational radius 2GM/s 2 appears for the first time, only expressed in other units and not specifically named. As Schwarzschild notes, in the case of a body with the mass of the Sun it is equal to 3 km, and for a mass of 1 g it is equal to 1.5 × 10 -28 cm.

But these numbers are not the most interesting thing. Schwarzschild also points out that the radius of a spherical body, measured by an external observer, cannot be less than its gravitational radius. It follows that the point mass, which was discussed in Schwarzschild’s first article, also appears from the outside as a sphere. Physically, this is due to the fact that no light beam can approach this mass closer than its gravitational radius and then return to the external observer. Schwarzschild's article does not contain these statements, but they follow directly from its logic. This is the second bridge to the concept of black holes, which can be found in Schwarzschild himself.

Epilogue

After Schwarzschild, pure mathematicians, physicists, and cosmologists studied spherically symmetric solutions of the general relativity equations. In the spring of 1916, the Dutchman Johannes Droste, who was completing his doctoral dissertation at the University of Leiden under the direction of Hendrik Lorenz, submitted to his boss for publication a work in which he calculated the space-time metric for a point mass more simply than Schwarzschild did (Droste had not yet learned about his results ). It was Droste who first published the version of the metric that later became considered standard.

During the subsequent refinement of Schwarzschild's solution, a completely different nature of singularities was also discovered: one, arising in the standard form of the metric at r = rs, as it turned out, can be eliminated by replacing coordinates, the other, arising at r = 0, turned out to be irremovable and physically corresponds to the infinity of the gravitational field .

All this is very interesting, but completely beyond the scope of my article. Suffice it to say that the mathematical theory of black holes has long been well developed and very beautiful - and it all historically goes back to Schwarzschild’s solution. As for the physical reality of black holes resulting from the collapse of the most massive stars, astronomers began to believe in it only in the early 1960s, after the discovery of the first quasars. But that's a completely different story.

1. Schwarzschild K. Zur Quantenhypothese / Sitzungsberichte der Preussischen Akademie der Wissenschaften. I (1916). P. 548−568.

2. Schwarzschild K. Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie / Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin. Phys.-Math. Klasse 1916. P. 189−196.

3. Schwarzschild K. Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie / Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin. Phys.-Math. Class. 1916. P. 424−434.

4. Droste J. The Field of a Single Center in EINSTEIN’s Theory of Gravitation, and the Motion of a Particle in that Field.Proc. K.Ned. Akad. Wet. Ser. A 19.197 (1917).