Karl Schwarzschild: astronomy, artillery, black holes. Schwarzschild space-time Schwarzschild equation

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Metrik Schwarzschild- this is the only spherically symmetric exact solution of the Einstein equations without a cosmological constant in empty space by virtue of the Birkhoff theorem. In particular, this metric quite accurately describes the gravitational field of a solitary non-rotating and uncharged black hole and the gravitational field outside of a solitary spherically symmetric massive body. It is named after Karl Schwarzschild, who first discovered it in 1916.

This solution is necessarily static, so that spherical gravitational waves are not possible.

Metric type

Schwarzschild coordinates

In the so-called Schwarzschild coordinates $(t,\; \backslash ;\; r,\; \backslash ;\; \backslash \; theta,\; \backslash ;\; \backslash \; varphi)$, of which the last 3 are similar to spherical, the metric tensor of the most physically important part of the Schwarzschild space-time with the topology $R\; ^\; 2\; \backslash \; times\; S\; ^\; 2$(the product of a region of two-dimensional Euclidean space and a two-dimensional sphere) has the form

Coordinate $r$ is not the length of the radius vector, but is introduced so that the area of ​​the sphere $t\; =\; \backslash \; mathrm\; (const),\; \backslash ;\; r\; =\; r\_0$ in this metric was equal to $4\; \backslash \; pi\; r\_0\; ^\; 2$... In this case, the "distance" between two events with different $r$(but with the same other coordinates) is given by the integral

$\backslash \; int\; \backslash \; limits\_\; (r\_1)\; ^\; (r\_2)\; \backslash \; frac\; (dr)\; (\backslash \; sqrt\; (1-\; \backslash \; displaystyle\; \backslash \; frac\; (r\_s)\; (r)))>\; r\_2-r\_1,\; \backslash \; qquad\; r\_2,\; \backslash ;\; r\_1>\; r\_s.$

At $M\; \backslash \; to\; 0$ or $r\; \backslash \; to\; \backslash \; infty$ the Schwarzschild metric tends (componentwise) to the Minkowski metric in spherical coordinates, so that far from a massive body $M$ space-time turns out to be approximately pseudo-Euclidean signature $(1,3)$... Because $g\_\; (0\; 0)\; =\; 1-\; \backslash \; frac\; (r\_s)\; (r)\; \backslash \; leqslant\; 1$ at $r>\; r\_s$ and $g\_\; (0\; 0)$ monotonically increases with growth $r$, then the proper time at points near the body "flows more slowly" than far from it, that is, a peculiar gravitational time dilation massive bodies.

Differential characteristics

We denote

$g\_\; (0\; 0)\; =\; e\; ^\; \backslash \; nu,\; \backslash \; quad\; g\_\; (1\; 1)\; =\; -\; e\; ^\; \backslash \; lambda.$

Then the non-zero independent Christoffel symbols have the form

$\backslash \; Gamma\; ^\; 1\_\; (1\; 1)\; =\; \backslash \; frac\; (\backslash \; lambda\; ^\; \backslash \; prime\_r)\; (2),\; \backslash \; quad\; \backslash \; Gamma\; ^\; 0\_\; (1\; 0)\; =\; \backslash \; frac\; (\backslash \; nu\; ^\; \backslash \; prime\_r)\; (2),\; \backslash \; quad\; \backslash \; Gamma\; ^\; 2\_\; (3\; 3)\; =\; -\; \backslash \; sin\; \backslash \; theta\; \backslash \; cos\; \backslash \; theta,$ $\backslash \; Gamma\; ^\; 0\_\; (1\; 1)\; =\; \backslash \; frac\; (\backslash \; lambda\; ^\; \backslash \; prime\_t)\; (2)\; e\; ^\; (\backslash \; lambda-\; \backslash \; nu),\; \backslash \; quad\; \backslash \; Gamma\; ^\; 1\_\; (2\; 2)\; =\; -\; re\; ^\; (-\; \backslash \; lambda)\; ,\; \backslash \; quad\; \backslash \; Gamma\; ^\; 1\_\; (0\; 0)\; =\; \backslash \; frac\; (\backslash \; nu\; ^\; \backslash \; prime\_r)\; (2)\; e\; ^\; (\backslash \; nu-\; \backslash \; lambda),$ $\backslash \; Gamma\; ^\; 2\_\; (1\; 2)\; =\; \backslash \; Gamma\; ^\; 3\_\; (1\; 3)\; =\; \backslash \; frac\; (1)\; (r),\; \backslash \; quad\; \backslash \; Gamma\; ^\; 3\_\; (2\; 3)\; =\; \backslash \; operatorname\; (ctg)\; \backslash ,\; \backslash \; theta,\; \backslash \; quad\; \backslash \; Gamma\; ^\; 0\_\; (0\; 0)\; =\; \backslash \; frac\; (\backslash \; nu\; ^\; \backslash \; prime\_t)\; (2),$ $\backslash \; Gamma\; ^\; 1\_\; (1\; 0)\; =\; \backslash \; frac\; (\backslash \; lambda\; ^\; \backslash \; prime\_t)\; (2),\; \backslash \; quad\; \backslash \; Gamma\; ^\; 1\_\; (3\; 3)\; =\; -\; r\; \backslash \; sin\; ^\; 2\; \backslash \; theta\; \backslash ,\; e\; ^\; (-\; \backslash \; lambda)\; ...$ $I\_1\; =\; \backslash \; left\; (\backslash \; frac\; (r\_s)\; (2r\; ^\; 3)\; \backslash \; right)\; ^\; 2,\; \backslash \; quad\; I\_2\; =\; \backslash \; left\; (\backslash \; frac\; (r\_s)\; (2r\; ^\; 3)\; \backslash \; right)\; ^\; 3.$

The curvature tensor is of the type $\backslash \; mathbf\; (D)$ according to Petrov.

Mass defect

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

$m\; =\; \backslash \; frac\; (4\; \backslash \; pi)\; (c\; ^\; 2)\; \backslash \; int\; \backslash \; limits\_0\; ^\; a\; T\_0\; ^\; 0\; r\; ^\; 2\; \backslash ,\; dr.$

In particular, for the static distribution of matter $T\_0\; ^\; 0\; =\; \backslash \; varepsilon$, where $\backslash \; varepsilon$- energy density in space. Considering that the volume of the spherical layer in the coordinates chosen by us is

$dV\; =\; 4\; \backslash \; pi\; r\; ^\; 2\; \backslash \; sqrt\; (g\_\; (1\; 1))\; \backslash ,\; dr>\; 4\; \backslash \; pi\; r\; ^\; 2\; \backslash ,\; dr,$

we get that

This distinction expresses itself gravitational defect in body weight... 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: when $r\; =\; 0$ and at $r\; =\; r\_s$... Indeed, in Schwarzschild coordinates, a particle falling on a body will take an infinitely long time $t$ to reach the surface $r\; =\; r\_s$, however, the transition, for example, to the Lemaitre coordinates in the accompanying frame of reference shows that from the point of view of the falling observer, there is no peculiarity of space-time on a given surface, and both the surface itself and the region $r\; \backslash \; approx\; 0$ will be reached in a finite proper time.

A real feature of the Schwarzschild metric is observed only for $r\; \backslash \; to\; 0$, where the scalar invariants of the curvature tensor tend to infinity. This feature (singularity) cannot be eliminated by changing the coordinate system.

Event horizon

Surface $r\; =\; r\_s$ called event horizon... With a better choice of coordinates, for example, in the coordinates of Lemaitre or Kruskal, it can be shown that no signals can escape from the black hole through the event horizon. In this sense, it is not surprising that the field outside the Schwarzschild black hole depends only on one parameter - the total mass of the body.

Coordinates of Kruskal

One can try to introduce coordinates that do not give a singularity at $r\; =\; r\_s$... Many such coordinate systems are 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). it more space-time $\backslash \; tilde\; (\backslash \; mathcal\; M)$ is usually called the (maximally extended) Schwarzschild space or (less often) the Kruskal space (Kruskal - Szekeres diagram). The metric in Kruskal coordinates is

$ds\; ^\; 2\; =\; -F\; (u,\; v)\; ^\; 2\; \backslash ,\; du\; \backslash ,\; dv\; +$

r ^ 2 (u, v) (d \ theta ^ 2 + \ sin ^ 2 \ theta \, d \ varphi ^ 2), \ qquad \ qquad (2)

where $F\; =\; \backslash \; frac\; (4\; r\_s\; ^\; 3)\; (r)\; e\; ^\; (-\; r\; /\; r\_s)$ and the function $r\; (u,\; v)$ is defined (implicitly) by the equation $(1-r\; /\; r\_s)\; e\; ^\; (r\; /\; r\_s)\; =\; uv$.

Space $\backslash \; tilde\; (\backslash \; mathcal\; M)$ maximally, that is, it can no longer be isometrically embedded in a larger space-time, and the region $r>\; r\_s$ in Schwarzschild coordinates ( $\backslash \; mathcal\; M$) is just a part of $\backslash \; tilde\; (\backslash \; mathcal\; M)$(this is the area $v>\; 0,\; \backslash \; r>\; r\_s$- area 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 $45\; ^\; \backslash \; circ$, see curve $\backslash \; gamma$ in the picture - can leave $\backslash \; mathcal\; M$... Moreover, it falls into region II, where

V $\backslash \; tilde\; (\backslash \; mathcal\; M)$ there is one more asymptotically flat region III, in which Schwarzschild coordinates can also be introduced. However, this area is not causally related to area 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 by a non-empty space-time filled with collapsing matter.

Let us point out several remarkable properties of the maximally extended Schwarzschild space $\backslash \; tilde\; (\backslash \; mathcal\; M)$:

1. It is singular: coordinate $r$ the observer falling below the horizon decreases and tends to zero when his own time $\backslash \; tau$ tends to some finite value $\backslash \; tau\_0$... However, his world line cannot be extended to the area $\backslash \; tau\; \backslash \; geqslant\; \backslash \; tau\_0$ since points with $r\; =\; 0$ in this space, no. Thus, the fate of the observer is known to us only up to a certain moment of his (own) time.
2. Although the space $\backslash \; mathcal\; M$ static (it is clear that the metric (1) does not depend on time), space $\backslash \; tilde\; (\backslash \; mathcal\; M)$ it is not. This is formulated more rigorously as follows: the Killing vector, which is from timelike to $\backslash \; mathcal\; M$, in areas II and IV of the extended space $\backslash \; tilde\; (\backslash \; mathcal\; M)$ becomes spacelike.
3. Region III is also isometric $\backslash \; mathcal\; M$... Thus, the maximally extended Schwarzschild space contains two "universes" - "our" (this $\backslash \; mathcal\; M$) and one more the same. Region II inside the black hole, connecting them, is called Einstein-Rosen bridge... An observer starting from I and moving slower than light will not be able to enter the second universe (see Fig. 1), but in the time interval between crossing the horizon and hitting the singularity he will be able see her. This structure of space-time, which is preserved and even becomes more complicated when considering more complex black holes, has given rise to numerous discussions on the topic of possible "other" universes and travel in them through black holes in both scientific literature and science fiction (see Mole burrows).

Orbital motion

History of receipt and interpretation

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

In mid-1915, Einstein published the preliminary equations of the theory of gravity $R\_\; (ij)\; =\; T\_\; (ij)$... These were not Einstein's equations yet, but they already coincided with the final ones in the vacuum case $T\_\; (ij)\; =\; 0$... Schwarzschild integrated the spherically symmetric equations for the vacuum in the period from November 18, 1915 until the end of the year. On January 9, 1916, Einstein, to whom Schwarzschild approached about the publication of his article in the Berliner Berichte, wrote to him that he “read his work with great passion” and “was stunned that the true solution to this problem could be expressed so easily” - Einstein initially doubted whether it would be possible to obtain a solution to such complex equations at all.

Schwarzschild completed his work in March, obtaining also a spherically symmetric static internal solution for a constant density fluid. At this time, a disease (pemphigus) fell on him, which in May brought him to the grave. Since May 1916, I. Droste, a student of GA Lorentz, conducting research within the framework of the final Einstein field equations, obtained a solution to the same problem by 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, stars, the radius of which 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 an object of controversy, in which Einstein joined.

During the 1922 Paris colloquium, organized in connection with the arrival of Einstein, they discussed not only the idea that the Schwarzschild radius would not be singular, but also a hypothesis that anticipated what is today called gravitational collapse.

Schwarzschild's ingenious development had only relative success. Neither his method nor his interpretation was adopted. Almost nothing has been preserved from his work, except for the "bare" result of the metric, with which the name of its creator was linked. But questions of interpretation and, above all, the question of "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, which consists in the fact that "this does not exist in nature." This point of view spread and became dominant in all the special literature of that time.

The next stage is associated with an intensive study of the issues of gravity at the beginning of the "golden age" of the theory of relativity.

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Literature

• K. Schwarzschild// Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1. - 1916. - 189-196.
  Rus. per .: Schwarzschild K. On the gravitational field of a point mass in Einstein's theory // Albert Einstein and the theory of gravitation. M .: Mir, 1979.S. 199-207.
• Landau, L. D., Lifshits, E. M. Field theory. - Edition 7, revised. - M .: Nauka, 1988 .-- 512 p. - ("Theoretical Physics", volume II). - ISBN 5-02-014420-7.
• Droste 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.
• Einstein A. In memory of Karl Schwarzschild // Einstein A. Collection of scientific papers. Moscow: Nauka, 1967.Vol. 4.P. 33-34.
• S. M. Blinder Centennial of General Relativity (1915-2015); The Schwarzschild Solution and Black Holes. - 2015 .-- arXiv: 1512.02061.

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Excerpt characterizing the Schwarzschild metric

Moscow, October 3, 1812.
Napoleon. ]

"Je serais maudit par la posterite si l" on me regardait comme le premier moteur d "un accommodement quelconque. Tel est l "esprit actuel de ma nation", [I would be damned if they looked at me as the first instigator of any deal; this is the will of our people.] - Kutuzov answered and continued to use all his strength for this to keep troops from advancing.
In the month of the plundering of the French army in Moscow and the quiet stay of the Russian army near Tarutino, a change took place in the ratio of the strength of both troops (spirit and number), as a result of which the advantage of strength turned out to be on the side of the Russians. Despite the fact that the position of the French army and its numbers were unknown to the Russians, how soon the attitude changed, the need for an offensive immediately manifested itself in countless signs. These signs were: the sending of Loriston, and the abundance of provisions in Tarutino, and information that came from all sides about the inaction and disorder of the French, and the recruiting of our regiments, and good weather, and prolonged rest of Russian soldiers, and usually arising in the troops as a result of rest impatience to carry out the work for which everyone was gathered, and curiosity about what was done in the French army, so long lost from sight, and the courage with which the Russian outposts were now prowling around the French stationed in Tarutino, and the news of easy victories over the French men and partisans, and the envy generated by this, and the feeling of revenge that lay in the soul of every person as long as the French were in Moscow, and (most importantly) the vague, but arising in the soul of every soldier, the consciousness that the attitude of force has changed now and the advantage is on our side. The essential relationship of forces changed and an offensive became necessary. And immediately, just as surely as the chimes begin to strike and play in the clock, when the hand has completed a full circle, in the higher spheres, in accordance with a significant change in forces, an intensified movement, hissing and playing of chimes were reflected.

The Russian army was ruled by Kutuzov with his headquarters and the sovereign from St. Petersburg. In St. Petersburg, even before the news of the abandonment of Moscow was received, a detailed plan of the entire war was drawn up and sent to Kutuzov for leadership. Despite the fact that this plan was drawn up on the assumption that Moscow was still in our hands, this plan was approved by the headquarters and accepted for execution. Kutuzov only wrote that distant sabotage is always difficult to execute. And to resolve the difficulties encountered, new instructions were sent and persons who were supposed to monitor his actions and report them.
In addition, now the entire headquarters of the Russian army has been transformed. The places of the murdered Bagration and the offended, retired Barclay were replaced. They very seriously considered what would be better: put A. in place of B., and B. in place of D., or, on the contrary, D. in place of A., etc., as if something other than A.'s pleasure and B., could have depended on it.
At the army headquarters, on the occasion of Kutuzov's hostility with his chief of staff, Bennigsen, and the presence of the emperor's confidants and these movements, a more than usual game of parties was going on: A. undermined B., D. under S., etc. ., in all possible displacements and combinations. With all these undermining, the subject of intrigue was, for the most part, the military affairs that all these people thought to lead; but this military business proceeded independently of them, exactly the way it was supposed to go, that is, it never coincided with what people invented, but proceeded from the essence of the attitude of the masses. All these inventions, interbreeding, entangling, represented in the higher spheres only a true reflection of what was to be accomplished.
“Prince Mikhail Ilarionovich! - wrote the sovereign on October 2 in a letter received after the Battle of Tarutino. - Since September 2, Moscow has been in the hands of the enemy. Your last reports from the 20th; and during all this time, not only has nothing been done to act against the enemy and liberate the capital of the capital, but even, according to your latest reports, you have retreated. Serpukhov is already occupied by an enemy detachment, and Tula, with its famous and so necessary for the army its plant, is in danger. According to reports from General Vintsingerode, I see that the enemy's 10,000th corps is advancing along the St. Petersburg road. Another, in several thousand, is also served to Dmitrov. The third moved forward along the Vladimir road. The fourth, rather significant, stands between Ruza and Mozhaisk. Napoleon himself was in Moscow on the 25th. According to all this information, when the enemy split his forces with strong detachments, when Napoleon was still in Moscow himself, with his guards, is it possible that the enemy forces in front of you were significant and did not allow you to act offensively? Probably, on the contrary, you should believe that he is pursuing you in detachments, or at least in a corps, much weaker than the army entrusted to you. It seemed that, taking advantage of these circumstances, you could profitably attack the enemy weaker than you and destroy him or, at least, forcing him to retreat, retain in our hands a noble part of the provinces now occupied by the enemy, and thereby avert danger from Tula and our other inner cities. It will remain your responsibility if the enemy is able to dispatch a significant corps to Petersburg to threaten this capital, in which many troops could not remain, for with the army entrusted to you, acting with determination and activity, you have all the means to ward off this new misfortune. Remember that you still owe a response to the insulted fatherland in the loss of Moscow. You have experienced my willingness to reward you. This readiness will not weaken in me, but I and Russia have the right to expect from you all the zeal, firmness and success that your mind, your military talents and the bravery of the troops you lead, portend to us. "
But while this letter, proving that a significant relationship of forces was already reflected in Petersburg, was on the way, Kutuzov could no longer keep the army under his command from the offensive, and the battle had already been given.
On October 2, the Cossack Shapovalov, while on the road, killed one hare with a gun and shot another. Chasing a shot hare, Shapovalov wandered far into the forest and came across the left flank of Murat's army, standing without any precautions. The Cossack, laughing, told his comrades how he almost fell for the French. The cornet, hearing this story, reported it to the commander.
The Cossack was summoned, questioned; The Cossack commanders wanted to take advantage of this opportunity to repulse the horses, but one of the commanders, familiar with the highest ranks of the army, reported this fact to the staff general. Of late, the situation at Army Headquarters has been extremely tense. Ermolov, a few days before this, having come to Bennigsen, begged him to use his influence on the commander-in-chief in order to make an offensive.
“If I didn’t know you, I would think that you don’t want what you ask for. It is worth advising me one thing, so that His Serene Highness would probably do the opposite, - answered Bennigsen.
The news of the Cossacks, confirmed by the sent trips, proved the final maturity of the event. The stretched string came off, and the clock hissed, and the chimes began to play. Despite all his imaginary power, in his intelligence, experience, knowledge of people, Kutuzov, taking into account the note of Bennigsen, who personally sent reports to the sovereign, expressed by all the generals the same desire, the sovereign's supposed desire and the bringing of the Cossacks, could no longer restrain inevitable movement and gave the order for what he considered useless and harmful - blessed the accomplished fact.

The note submitted by Bennigsen about the need for an offensive, and the information of the Cossacks about the uncovered left flank of the French were only the last signs of the need to order an offensive, and the offensive was scheduled for October 5th.
On the morning of October 4, Kutuzov signed the disposition. Toll read it to Yermolov, inviting him to take up further orders.
“Okay, okay, I don’t have time now,” said Yermolov and left the hut. Tol's disposition was very good. Just as in the Austerlitz disposition, it was written, although not in German:
“Die erste Colonne marschiert [The first column goes (German)] this and that, die zweite Colonne marschiert [the second column goes (German)] this and that,” etc. And all these columns are on paper came at the appointed time to their place and destroyed the enemy. Everything was, as in all dispositions, perfectly thought out, and, as with all dispositions, not a single column came in its time and in its place.
When the disposition was ready in the proper number of copies, an officer was summoned and sent to Yermolov to give him the papers for execution. A young cavalry officer, Kutuzov's orderly, pleased with the importance of the assignment given to him, went to Yermolov's apartment.
“We’re gone,” answered Yermolov’s orderly. The cavalry officer went to the general, whom Ermolov often visited.
- No, and there is no general.
The cavalry officer sat on horseback and rode to another.
- No, they left.
“How would I not be responsible for the delay! What a shame! " - thought the officer. He traveled all over the camp. Someone said that they saw how Yermolov drove with other generals somewhere, who said that he was probably at home again. The officer, without having dinner, searched until six o'clock in the evening. Ermolov was nowhere to be found and no one knew where he was. The officer had a quick bite to eat at his comrade's and went back to the vanguard to Miloradovich. Miloradovich was not at home either, but then he was told that Miloradovich was at the ball at General Kikin's, that Ermolov must be there.
- But where is it?
“And over there, in Echkin,” said the Cossack officer, pointing to a distant landlord’s house.
- But what about there, behind the chain?
- They sent two of our regiments into the chain, there is such a revelry nowadays, trouble! Two music, three choirs of songwriters.
The officer went by the chain to Echkin. From a distance, still driving up to the house, he heard the friendly, cheerful sounds of a soldier's dancing song.
"In the oluzya ah ... in the oluzi! .." - with a whistle and with a torban he heard him, occasionally drowned out by the shout of voices. The officer felt cheerful in his soul from these sounds, but at the same time it was also scary for the fact that he was guilty, for so long not having given the important order entrusted to him. It was already past nine. He dismounted from his horse and entered the porch and the hallway of a large, intact manor house, located between the Russians and the French. In the pantry and in the hall, footmen were bustling about with wines and food. There were songbooks under the windows. The officer was led through the door, and he suddenly saw all together the most important generals of the army, including the large, noticeable figure of Yermolov. All the generals were in unbuttoned coats, with red, lively faces and were laughing loudly, standing in a semicircle. In the middle of the room, a handsome, short general with a red face was smartly and deftly making a trepak.
- Ha, ha, ha! Ah yes Nikolai Ivanovich! ha, ha, ha! ..
The officer felt that, entering at that moment with an important order, he was doubly guilty, and he wanted to wait; but one of the generals saw him and, learning why he was, told Ermolov. Ermolov, with a frowning face, went out to the officer and, having listened, took the paper from him, without saying anything to him.
- Do you think he left by accident? - That evening the staff comrade said to the officer of the cavalry guard about Yermolov. - These are things, this is all on purpose. Give Konovnitsyn a ride. Look, what porridge will be tomorrow!

The next day, early in the morning, the decrepit Kutuzov got up, prayed to God, dressed, and with the unpleasant consciousness that he should lead a battle, which he did not approve of, got into a carriage and drove out of Letashevka, five miles behind Tarutin, to that place, where the advancing columns were to be assembled. Kutuzov rode, falling asleep and waking up and listening to see if there were any shots on the right, was the case starting? But it was still quiet. The dawn of a damp and cloudy autumn day was just beginning. Approaching Tarutin, Kutuzov noticed the cavalrymen leading the horses to the watering hole across the road along which the carriage was traveling. Kutuzov looked at them closely, stopped the carriage and asked which regiment? The cavalrymen were from the column that should have been already far ahead in ambush. "A mistake, maybe," thought the old commander-in-chief. But, having driven even further, Kutuzov saw infantry regiments, guns in the box, soldiers with porridge and firewood, in underpants. An officer was called. The officer reported that there was no order to march.
- How not ... - began Kutuzov, but immediately fell silent and ordered to call the senior officer. Climbing out of the carriage, head bowed and breathing heavily, silently waiting, he walked up and down. When the demanded officer of the General Staff, Eichen, appeared, Kutuzov turned purple not because this officer was the fault of a mistake, but because he was a worthy subject for expressing anger. And, shaking, gasping for breath, the old man, having come to that state of fury, in which he was able to come when he was lying on the ground from anger, he attacked Eichen, threatening with his hands, shouting and swearing with square words. Another who turned up, Captain Brozin, who was not guilty of anything, suffered the same fate.
- What kind of canalya is this? Shoot the scoundrels! He shouted hoarsely, waving his arms and staggering. He was in physical distress. He, the commander-in-chief, the most luminous, whom everyone assures that no one has ever had such power in Russia as he is, he is put in this position - made fun of the whole army. “In vain did I bother so much to pray for the present day, in vain I did not sleep at night and thought everything over! - he thought of himself. “When I was a boy as an officer, no one would have dared to laugh at me like that ... But now!” He experienced physical suffering, as from corporal punishment, and could not help expressing it with angry and suffering cries; but soon his strength weakened, and he, looking around, feeling that he had said a lot of bad things, got into the carriage and silently drove back.

A hundred years ago, a full member of the Royal Academy of Sciences of Prussia, Karl Schwarzschild, sent his fellow at the Academy Albert Einstein an article with a mathematical description of the gravitational field outside and inside a sphere filled with a stationary fluid of constant density. This work was the beginning of theoretical research into exotic objects that we call black holes.

John Michell's inspiration

The history of the creation of the modern theory of black holes and their discovery in outer space is too extensive and complex to fit into an article of reasonable size without omissions and simplifications. Therefore, I will only bring the story to the first examples of the use of the Schwarzschild mathematical model in real astrophysics, which took place almost a quarter of a century after the publication of his wonderful article. However, in the opposite direction, I will go into history much further - at the end of the 18th century. Just then, in 1784, an article appeared in the official journal of the Royal Society of London with an unusually (at least for us) long headline: On the Means of Discovering the Distance, Magnitude, & c. of the Fixed Stars, in Consequence of the Diminution of the Velocity of Their Light, in Case Such a Diminution Should be Found to Take Place in any of Them, and Such Other Data Should be Procured from Observations, as Would be Farther Necessary for That Purpose. By the Rev. John Michell, B. D. F. R. S. In a Letter to Henry Cavendish, Esq. F. R. S. and A.S. Its author, the Reverend John Michell, was already able to calculate the physical quantity, which now bears the name of the Schwarzschild radius. Although this work in no sense can be considered a predecessor of the modern concept of black holes, for the sake of historical completeness, it is necessary to start with it.

There is every reason to call John Michell (1724–1793) the most brilliant English scientist of the 18th century to graduate from the University of Cambridge. He was educated at Queens "College, where he taught from 1751 to 1763. Having married, he began to look for a church office for a decent income, and from 1767 until his death he was rector (rector) of the parish of St. Michael in the village of Thornhill near Leeds, where he continued to study science for the rest of his life.

Michell was a remarkable and eminently original explorer. He is deservedly considered the founding father of two sciences at once - seismology and stellar statistics. Michell was the first to discover that the repulsive force between the poles of the same name of permanent magnets decreases in inverse proportion to the square of the distance, and long before Charles-Augustin de Coulomb invented and made "in iron" a torsion balance, which he wanted, but did not have time to use for gravimetric experiments ... After Michell's death, his friend Henry Cavendish, who received this device and independently built a modified version of it, made precision measurements of the force of gravity, the results of which already at the beginning of the 19th century allowed calculating the gravitational constant with an error of only about one percent. (It may be worth recalling that this fundamental physical constant, as is commonly believed, first appeared in the first volume of the famous monograph Traité de mécanique by Siméon Denis Poisson, and became widely used by physicists only in the second half of the 19th century.) By the way. , Michell's article in question was sent to Cavendish, who read it at several meetings of the Royal Society in late 1783 and early 1784. Michell, himself an active member of the Society since 1760, was then unable or unwilling to come to London (why exactly is unknown).

Unfortunately, Michell was an unimportant communicator. He often included his most remarkable results in the texts of long journal articles, where descriptions of discoveries were almost lost against a rather truistic background. Because of this, Michell, neither during his life nor after death, did not receive the recognition that he undoubtedly deserved.

In the introductory letter to Cavendish, which precedes the main article, Michell articulates the purpose of the new study very clearly. He, like other British scientists of the time, following Newton, considered light to be a stream of the smallest particles. Michell also, following Joseph Priestley, suggested that these particles, like ordinary matter, obey the laws of mechanics and, in particular, should be slowed down by the forces of gravity. Michell decided that using this effect, in principle, it was possible to measure distances to stars, magnitudes and stellar masses (p. 35). He also expressed the hope that astronomers will be able to make fruitful use of this never-before-seen method of observation (pp. 35–36).

The essence of the matter is as follows. Considering that the speed of light at the moment of its emission is always the same, Michell proposed to determine the speed of light coming to Earth from various stars, and using the laws of celestial mechanics to squeeze information about the stars themselves from these measurements. For example, if we assume that all stars (or some group of stars) are at approximately the same distance from the Earth, such measurements will make it possible to estimate the stellar mass ratios: the heavier the star, the more its gravity will slow down the light corpuscles.

Michell explained in great detail the details of his method, and, in the spirit of Newton's "Mathematical Principles of Natural Philosophy", did not give a single formula - his presentation is strictly geometric. In his article there are many witty conclusions, especially since, in addition to mechanics, he uses optics and astronomy for his reasoning. Of course, this work was wasted: the speed of light in a vacuum is constant. Therefore, Michell's article would most likely have been firmly forgotten, if not for one conclusion - by the way, made quite casually. Developing his deductions, he eventually concludes that a very massive star must slow down light particles so much that they can never go to infinity. All of her light, under the influence of her own attraction, "will be forced to return back to the star" (p. 42). It follows that such a star will be invisible - at least from very large distances. Michell noted that, according to his calculations, in order for the light of a star with the same density as that of the Sun to escape to infinity, its diameter must be about 500 times greater than that of the Sun. Thus, Michell concludes, if stars of the same (and even more) massive exist very far away from us, we will never be able to obtain any information about them through their light (p. 50). Interestingly, he uses precisely the word information, which was by no means in the same way back then as it is today.

It is easy to see that the analogy between black holes in the modern sense and Michellian exotic stars is very superficial and approximate. A classical black hole does not emit any light at all (the hypothetical Hawking radiation is a purely quantum effect) and in this sense is indeed black. Light corpuscles in Michell's model, on the contrary, leave the surface of the star in any scenario, but they do not always go to infinity. Therefore, Michell does not and cannot have any absolutely black stars, they are all visible from various distances. There are many other obvious differences as well.

Michell also wondered if a star could somehow be found from Earth if its light did not reach our planet. And he offered (I can't help but admire his insight!) Not only a feasible, but also an absolutely modern solution. Suppose that such a star is part of a binary system, and the light of the second star is visible through our telescopes. Then we will be able to judge the presence and even the properties of an invisible star by observing the "rocking" of its partner. It is well known that this method has long been used in the search for exoplanets.

How right was Michell in his calculation of the parameters of a star that cannot be seen from an infinitely large distance? It is very easy to obtain the corresponding formula; this is a task for a student. It is necessary to take the well-known mathematical expression for the second cosmic speed and substitute the speed of light in its place. As a result, we get that a star with mass M will send light particles to finite distances if its radius R does not exceed the value \ (R_ (cr) = \ frac (2GM) (c ^ 2) \), where G is the Newtonian constant of gravitation, and c is the speed of light. For a star with the mass of the Sun, this is about 3 kilometers. Consequently, the critical radius of any star in the Michellian model is equal to three kilometers multiplied by its mass in solar units (in other words, by the ratio of its mass to the mass of the Sun). Of course, Michell could not own the algebraic formula for the critical radius, if only because of the absence in the then physical language of the concept of a gravitational constant. Michell (again in the spirit of Newton) estimated it with the help of geometric constructions, and very witty ones.

Let's go back to Michell's example. The mass of a star of solar density, whose diameter is 500 times that of the Sun, is 125 million solar masses. The critical radius of a body with such a mass, according to the above formula, is 375 million kilometers. The average radius of the Sun is about 700 thousand kilometers, and if you multiply it by 500, you get 350 million. So Michell was quite a bit wrong.

John Michell trusted his logic and intuition and therefore admitted that the depths of space hide many stars that cannot be seen from Earth through any telescope. Three years after his death, the great French mathematician, astronomer and physicist Pierre-Simon Laplace, who at that time did not yet have either the title of count from Napoleon, or the title of marquis, which was awarded to him by the Bourbons, came to the same conclusion. In the first (1796) edition of his popular treatise Exposition du Système du Monde, he very briefly mentioned bodies luminous, but invisible from the Earth (corps obscurs). In the 19th century, this work withstood many re-editions during its lifetime, which no longer mentioned this hypothesis. This is understandable, since most physicists at that time already considered light to be oscillations of the ether. The existence of "dark" stars contradicted the wave concept of light, and Laplace thought it best to forget about them. In later times, this idea was considered a curiosity, worthy of mention only in works on the history of science.

And one more important detail. Both Michell and Laplace attributed invisibility at long distances only to the most gigantic and, automatically, the most massive stars (at that time it was believed that the densities of all stars were approximately equal to the density of the Sun). Neither one nor the other noticed that within the framework of Newton's theory of light, a small luminous body of extremely high density can have the same property. However, no one thought about the possibility of such compact space objects at that time.

Karl Schwarzschild and his formulas

On November 25, 1915, Albert Einstein presented a written report to the Prussian Academy of Sciences containing a system of fully covariant equations of the relativistic theory of the gravitational field, also known as general theory of relativity (GR). A week earlier, he gave a lecture at a meeting of the Academy in which he demonstrated in his work an earlier version of these equations, which did not have full covariance (he had presented it to the Academy two weeks earlier). However, these equations already gave Einstein the opportunity, using the method of successive approximations, to correctly calculate the anomalous rotation of the orbit of Mercury and predict the magnitude of the angular deviation of starlight in the gravitational field of the Sun , 25.11.2015).

This speech found a grateful listener in the person of Einstein's colleague at the Academy Karl Schwarzschild (Karl Schwarzschild, 1873-1916), who served in the army of the German Empire as a lieutenant of artillery and just then came on vacation. Returning to his duty station, Schwarzschild in December found the exact solution of the first version of Einstein's equations, which he published through him in the "Meeting Records" ( Sitzungsberichte) Academy. In February, having already familiarized himself with the final version of the equations of general relativity, Schwarzschild sent Einstein a second article, in which for the first time the gravitational, or Schwarzschild, radius appeared explicitly. On February 24, Einstein submitted this work to the press.

Like John Michell, Schwarzschild was not only brilliant but also a very versatile scientist. He left a deep mark in observational astronomy, where he became one of the pioneers of equipping telescopes with photographic equipment and its use for photometry purposes. He owns deep and original works in the field of electrodynamics, stellar astronomy, astrophysics and optics. Schwarzschild even managed to make an important contribution to the quantum mechanics of atomic shells, having constructed in his last scientific work the theory of the Stark effect (K. Schwarzschild, 1916. Zur Quantenhypothese). In 1900, fifteen years before the creation of general relativity, he not only seriously considered the possibility that the geometry of the Universe differs from the Euclidean one (Lobachevsky still admitted it), but also estimated the lower limits of the radius of curvature of space for the spherical and pseudospherical geometry of the cosmos. Before reaching even thirty, he became a professor at the University of Göttingen and director of the university observatory. In 1909 he was elected a member of the London Astronomical Society and headed the Potsdam Astrophysical Observatory, and four years later became a member of the Prussian Academy.

Schwarzschild's scientific career was cut short by the First World War. Not subject to conscription by age, he volunteered for the army and eventually ended up on the Russian front at the headquarters of an artillery unit, where he calculated the trajectories of long-range guns. There he became a victim of pemphigus, a very severe autoimmune skin disease to which he had a hereditary tendency. This pathology does not respond well to drugs in our time, and then it was incurable. In March 1916, Schwarzschild was discharged and returned to Potsdam, where he died on May 11. Schwarzschild and the English physicist Henry Moseley, who died in the Dardanelles operation, became the largest scientists whose lives were taken by the First World War.

The famous space-time Schwarzschild metric historically became the first exact solution of the equations of general relativity. It describes a static gravitational field, which is created in a vacuum by a motionless spherically symmetric body of mass M... In standard notation in Schwarzschild coordinates t, r, θ, φ and when choosing a signature (+, -, -, -) it is given by the formula

\ [\ mathrm (d) s ^ 2 = \ left (1- \ frac (r_s) (r) \ right) c ^ 2 \ mathrm (d) t ^ 2- \ left (1- \ frac (r_s) ( r) \ right) ^ (- 1) \ mathrm (d) r ^ 2- r ^ 2 (\ sin ^ 2 \ theta \, \ mathrm (d) \ varphi ^ 2 + \ mathrm (d) \ theta ^ 2 ), \ quad \ quad \ quad \ text ((1)) \]

By the end of the first quarter of the twentieth century, astronomers have learned with decent accuracy to determine intergalactic distances in the vicinity of the Milky Way. After that, it became clear that some of the new stars emit thousands of times more energy than others. In 1925, the Swedish astronomer Knut Emil Lundmark proposed to single them out as a special group of new stars of the highest class, but this name somehow did not take root. In the early 1930s, Caltech physics professor Fritz Zwicky began calling extremely bright flares supernova in lectures for graduate students. This term took root, although over time it lost the hyphen.

In December 1933, Zwicky and the astronomer from the Mount Wilson Observatory Walter Baade (both emigrants from Europe) presented at the session of the American Physical Society the report "On supernovae", which soon appeared in print (WA Baade and F. Zwicky , 1934 On Super-Novae). The report was seen outside the physical community and featured in the American media. Baade and Zwicky calculated that a typical supernova sends out as much light into space during a month as our sun emits in 10 million years. They came to the conclusion that this is possible only with a partial transformation of the mass of a star into ray energy in accordance with Einstein's formula. Therefore, they suggested that a supernova explosion is a transformation of an ordinary star into a new type of star, consisting mainly of neutrons. A neutron star must have a very small radius and, therefore, consist of extremely high density matter, many orders of magnitude greater than the density of white dwarfs. This hypothesis was formulated in Cosmic Rays from Super-Novae, published in the same issue. Proceedings of the National Academy of Sciences immediately following the first message. In the same work, they put forward a truly prophetic hypothesis: supernova explosions can be a source of cosmic rays.

Most experts considered the assumption about the birth of neutron stars at the final stage of supernova explosions, to put it mildly, poorly substantiated - especially since Zwicky and Baade could not offer a physical mechanism for the birth of such strange space objects. At first, even Chandrasekhar did not accept him, although in 1939, speaking at a conference in Paris, he nevertheless admitted that this hypothesis has a right to exist. Its validity finally became clear only after the discovery of radio pulsars in 1967. It should be noted that the term "pulsar" at the end of the same year was invented not by a scientist, but by a journalist, scientific columnist for the newspaper Daily telegraph Anthony Michaelis.

Baade and Zwicky were not the first to admit the existence of space objects consisting of superdense matter. Earlier, Lev Davidovich Landau came up with a similar idea, who suggested that stellar cores consisting of such matter can serve as a source of gravitational energy that stars spend on their radiation. His article was written in early 1931, that is, even before the discovery of the neutron by the deputy director of the Cavendish Laboratory, James Chadwick, in 1932 (of course, this particle is not mentioned in Landau's article), but it was published a year later (LD Landau, 1932 . On the theory of stars). In the first part of the article, Landau not only independently rediscovered the formula for the Chandrasekhar limit (which he, no doubt, did not have time to find out about), but also calculated for him a completely acceptable value of 1.5 M s... Landau turned out to be closer to the truth, since he used a completely realistic estimate of the mass per electron, considering it to be equal to twice the mass of the proton (Chandrasekhar in his first article considered it to be equal to two and a half proton masses).

In the second part, Landau, in a sense, gave free rein to fantasy. He made a very exotic assumption, according to which ordinary stars have compact superdense cores, in fact, giant atomic nuclei, which serve as their energy sources. Since it was impossible to substantiate this idea in the context of the then (however, as well as today's) fundamental physical theories, Landau even admitted that the law of conservation of energy may be violated in such stellar depths. In doing so, he referred to the authority of Niels Bohr, who tried in the same way to explain the mysterious spread of energies and momenta of beta-decay electrons (as you know, Wolfgang Pauli "saved" the energy conservation law with the help of a hypothetical neutral particle, later called neutrino).

In general, the "neutronization" of stellar matter as the cause of the phenomenal power of supernovae is entirely the idea of ​​Baade and Zwicky. True, Baade never returned to her and, most likely, did not take it too seriously. But Zwicky launched a whole program of searching for supernovae using an 18-inch telescope with a camera, purchased from the Rockefeller Foundation. By the fall of 1937, in just a year of observations, he discovered three supernovae. This program was curtailed after the Japanese attack on Pearl Harbor.

In retrospect, it is clear that the hypothesis of Baade and Zwicky pointed to the very transition from a degenerate electron gas to a substance of a different nature, which logically followed from the works of Frenkel, Anderson, Stoner and Chandrasekhar. It is not surprising that Landau was very interested in it, who a few years later returned to his model and published its modified version in the magazine Nature(L. D. Landau, 1938. Origin of Stellar Energy). In this note, Landau already wrote directly not about nuclear matter in general, but about neutron matter that arose during the fusion of electrons with atomic nuclei at ultrahigh pressures inside the stellar depths (it is interesting that he referred not to Baad and Zwicky, but to a professor at Leipzig University Friedrich Hund, who was very active in astrophysics in the mid-1930s). Landau argued that normal stars can have stable neutron cores with a mass in excess of one thousandth (in other assumptions, one-twentieth) of the mass of the Sun, the compression of which provides the energy going into their radiation.

However, in this case, Landau was changed by his illustrious intuition. His hypothesis was disproved that same year by Julius Robert Oppenheimer and his postdoc Robert Serber (J. R. Oppenheimer and R. Serber, 1938. On the Stability of Stellar Neutron Cores). They showed that an adequate account of nuclear forces practically excludes the possibility of the existence of neutron nuclei in stars whose masses are comparable to the mass of the Sun. Oppenheimer and Serber also came to the absolutely correct, as time has shown, conclusion that no neutron nucleus can arise before the star has completely exhausted all sources of nuclear energy (and thus, although the article does not say this directly, it will main sequence). In their short message, it was also noted (albeit without proof) that the mass of such a core, in any case, cannot be less than one tenth of the mass of the Sun. This estimate was obtained on the basis of energy considerations alone and turned out to be completely correct. According to modern concepts, with a core mass of less than 0.1 M s neutrons would be converted to protons through beta decay. Newborn protons would merge with neutrons, forming highly neutron-rich and therefore extremely unstable atomic nuclei. As a result, if the neutron star somehow lost so much weight that its mass dropped below 0.1 M s, she would disappear in a nuclear explosion. For this information I am very grateful to Dr. F.-m. Sciences A. Yu. Potekhin.

Landau shortly after the publication of the article in Nature was arrested and spent a year in prison. He never returned to his model of the neutron core as a source of stellar energy - most likely because by the time of its release in April 1939 it was already clear that main sequence stars are powered by thermonuclear fusion energy. Perhaps it will be useful to recall that during the war years Serber became one of the main participants in the Manhattan project led by Oppenheimer, and it was he who came up with the names for the atomic bombs "Little Boy" and "Fat Man", dropped 6 and 9 August 1945 to Hiroshima and Nagasaki.

Back to Schwarzschild: First Steps

Since the hypothesis of Zwicky and Baade still did not go anywhere, a natural question arose: is there an upper mass limit for those supernovae that supposedly leave behind neutron stars (recall that Landau spoke not about the upper, but about the lower limit of the mass of neutron nuclei of ordinary stars )? In other words, is there an upper limit on the mass of hypothetical neutron stars similar to the one for white dwarfs? At the same time, it was clear that neutron stars, if they really are born in outer space, immeasurably surpass white dwarfs in density. In 1937, Georgy Gamov estimated the maximum density of neutron matter at 10 17 kg / m 3 (G. Gamow, 1937. Structure of Atomic Nuclei and Nuclear Transformations; G. Gamov, 1939. Physical Possibilities of Stellar Evolution), which is 9 orders of magnitude higher the mass density of a typical white dwarf. Its result quite stood the test of observations: the measured densities of neutron stars vary in the range (4–6) · 10 17 kg / m 3. In the same monograph, Gamow, recalling Landau's hypothesis published in 1932, noted that neutron nuclei could provide active life for a star "for a very long time," although at that time this point of view was already an anachronism.

In 1939, Robert Oppenheimer and his Canadian graduate student George Michael Volkoff, a Muscovite by birth and in his former life, Georgy Mikhailovich, tried to solve this problem. Their joint article (J. R. Oppenheimer and G. M. Volkoff, 1939. On Massive Neutron Cores) is deservedly considered one of the brightest achievements of theoretical astrophysics in the first half of the twentieth century. And this is despite the fact that the estimate of the upper limit of the mass of neutron remnants of massive stars obtained in it turned out to be greatly underestimated.

One might expect that Oppenheimer, in posing this problem, wanted to clarify the applicability of the Baade and Zwicky hypothesis. However, if he had such an intention, he did everything to hide it. In the article in question, there are no references at all to any publication of these researchers. Which is not surprising. Oppenheimer was then a professor of physics at the University of California, Berkeley, but regularly visited Caltech, where Zwicky worked. It is no secret that Oppenheimer could not stand Zwicky as a person and did not trust him as a scientist (and this attitude was shared by many contemporaries in both plans). So Oppenheimer and Volkov limited themselves to a neutral phrase: “The possibility was suggested that highly compressed neutron nuclei were formed in the central regions of sufficiently massive stars that had depleted thermonuclear energy sources” (p. 475). As one of the sources of this hypothesis, they named the recent publication of Landau in Nature, while Baade and Zwicky only pass through the category “and others” (Ibid). They also referred to the aforementioned report by Oppenheimer and Serber, more precisely, to their estimate of the minimum mass of a neutron nucleus at 0.1 M s.

And then the fun begins. Oppenheimer and Volkov worked with a model of a degenerate cold neutron Fermi gas with a spherically symmetric particle distribution. In this regard, their approach is quite similar to the approach of Anderson, Stoner, Chandrasekhar and Landau, who made calculations based on the model of a degenerate relativistic electron gas. Oppenheimer and Volkov specifically emphasized that if we directly take from Landau's 1932 article the formula for the maximum mass of a star consisting of such gas (remember that this is an exact analogue of Chandrasekhar's formula) and simply replace electrons there with neutrons, the upper limit of the star's mass will be about 6 solar masses, which, in fact, is calculated quite simply. However, the co-authors further point out that such an approach would be erroneous, and for two reasons. To obtain the correct result, it is necessary to take into account the non-Newtonian nature of the gravitation of the hypothetical neutron nucleus with its gigantic gravity. In addition, it cannot be assumed in advance that the neutron gas will be relativistically degenerate throughout the entire volume of the star. "This study aims to find out what differences in the results of calculations will make the use of both general theory of relativity instead of Newton's theory of gravity, and a more accurate equation of state" (p. 575).

To solve this problem, Oppenheimer and Volkov performed calculations based on the general static solution of Einstein's field equations for a spherically symmetric distribution of matter and, in particular, the Schwarzschild solution, which describes the metric of empty space surrounding this matter. They also suggested that matter is composed of quantum particles obeying Fermi-Dirac statistics, whose thermal energy and non-gravitational interactions can be neglected. Equating the mass of the particles of this cold Fermi gas to the mass of neutrons and carrying out an approximate numerical integration of the obtained equations, Oppenheimer and Volkov came to the conclusion that the masses of neutron cores of stars that fully used their thermonuclear energy resources cannot exceed 70% of the solar mass.

It has long been known that this first estimate of the maximum mass of neutron nuclei turned out to be greatly underestimated. Later modeling showed that the masses of neutron stars should lie in the interval (1.5–3) · M s; the masses of actually observed neutron stars range from one and a half to two solar masses. The reason for this error is also understandable. At the end of the 1930s, there was still no detailed theory of nuclear forces that would allow one to write at least approximate equations of state of matter at ultrahigh densities and pressures. It is now known that powerful nuclear repulsive forces act in this region, which increase the lower mass limit for neutron stars in comparison with the Oppenheimer-Volkov model.

Comparing the Oppenheimer-Volkov estimate with the Chandraksekhar limit obviously created an unpleasant problem, which they themselves perfectly understood and commented on. If the pressure of a degenerate relativistic electron gas is capable of resisting the gravitational collapse of stars with a mass of up to almost one and a half times the mass of the Sun, then it is completely incomprehensible how a neutron star could arise, since its mass cannot exceed 0.7 M s... Oppenheimer and Volkov got around this difficulty by suggesting that neutron nuclei can be arbitrarily massive if the difference between the density of matter and its threefold pressure takes large negative values ​​(p. 381). Now we know that this assumption was not justified, and the upper limit of the masses of neutron stars still exists. Oppenheimer and Volkov also expressed almost certainty that taking into account the nuclear forces of mutual repulsion would not significantly increase the upper limit of the masses of neutron nuclei calculated by them - and in this they also turned out to be wrong.

Of course, all this in no way diminishes the significance of the work of Oppenheimer and Volkov. They operated in completely uncharted territory, and practically alone, except for the informal assistance of Caltech Professor Richard Tolman. The demonstration, albeit on a simplified model, of the existence of an upper mass limit for neutron stars was a result of paramount importance. This result suggested that the most massive descendants of supernovae do not become neutron stars, but go into some other state.

It is worth dwelling on this in more detail. Oppenheimer, Volkov and Tolman obtained an equation for the radial pressure gradient of matter inside a contracting star. Figuratively speaking, it shows how the star resists compression by increasing its internal pressure. However, in general relativity, in contrast to Newtonian mechanics, pressure itself serves as a factor in the curvature of space-time and thus a source of the gravitational field. Therefore, the gravity inside the star can build up so quickly that the collapse becomes irreversible. This consequence of the Tolman - Oppenheimer - Volkov equation now seems very transparent, but the authors did not trace it.

In the same 1939, Oppenheimer and another of his graduate students, Hartland Snyder, came close to describing such an ending (J. R. Oppenheimer and H. Snyder, 1939. On Continued Gravitational Contraction). They considered the process of gravitational contraction of a strictly spherical, non-rotating dust cloud with constant density - again, with explicit use of the Schwarzschild metric. Of course, this was the most simplified model of cosmic matter. Particles of dusty matter, by definition, interact with each other exclusively through mutual attraction (hence, the pressure in such a cloud is equal to zero) and therefore move along geodesic world lines; moreover, such a system has no thermodynamic characteristics. However, more realistic calculations based on the general theory of relativity were then simply beyond the reach, which the authors of the article admitted to. However, they noted that the solution they found most likely roughly reflects the main features of the process of gravitational contraction of a real star of sufficiently large mass, which completely burned up its thermonuclear fuel (p. 457).

To obtain an analytical solution to the equations of general relativity, Oppenheimer and Snyder went over to the accompanying coordinates, in which the energy-momentum tensor in this case has a single nonzero component \ (T_4 ^ 4 \), equal to the density of the substance. On the basis of their - I repeat, highly idealized - model, they came to the conclusion that a rather massive star that had time to burn fusion fuel, in the course of subsequent compression, contracts to its gravitational radius. This process takes infinitely long time from the point of view of a distant observer, but can be very short for an observer who moves with the collapsing stellar matter. For example, according to their calculations, the gravitational collapse of a cloud with an initial density of 1 g / cm 3 and a total mass of 10 33 g (hence, with a radius of the order of a million kilometers) from the point of view of such an observer will take only one Earth day. Approaching the gravitational radius, “the star completely isolates itself from any contact with a distant observer; only its gravitational field is preserved ”(p. 456).

It follows almost unambiguously from the Oppenheimer and Snyder equations that the star, upon reaching the gravitational radius, does not stop and continues to contract to a state with an infinitely small volume and an infinitely high density. The co-authors nevertheless refrained from such a radical conclusion and did not even offer it as a hypothesis. Unfortunately, then their remarkable work did not arouse much interest - perhaps in part because its publication exactly coincided with the date of the outbreak of World War II (September 1, 1939). In addition, at that time physicists and astronomers had little interest in general relativity and knew little about it. It seems that the only extra-class theoretical physicist who appreciated it without delay was Landau.

A little earlier Oppenheimer and Snyder paid attention to the problem of gravitational collapse of a spherically symmetric system of noninteracting particles (Albert Einstein, 1939. Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses). This article, which he submitted for publication two months earlier, was unsuccessful. Einstein did not believe in the Schwarzschild singularity, which arises near the gravitational radius, and therefore tried to prove that it is physically unattainable. He used the Schwarzschild metric (albeit in a non-standard notation), but made a completely artificial assumption that all particles move around the center of symmetry in circular orbits. His calculations showed that an increase in the mass of such a system leads to an increase in centrifugal forces, and this does not allow it to contract beyond a certain limit. As a result, Einstein stated with obvious satisfaction that "the Schwarzschild singularity does not exist in physical reality" (p. 936). He believed that this conclusion is of a general nature, not limited by the specifics of the model, in which he was greatly mistaken. Some historians of science generally consider this article to be the worst of Einstein's scientific work. As far as I know, history is silent about whether Einstein got acquainted with the Oppenheimer-Snyder model, and if so, how he appreciated it.

The remarkable studies of Oppenheimer - Volkov and Oppenheimer - Snyder stand at the beginning of a long and glorious history of the application of the Schwarzschild solution of the equations of general relativity to the analysis of specific astrophysical models. New steps in this direction were made already in the post-war period, and their description is beyond the scope of my article.

Therefore, I will limit myself to an extremely brief summary. The physical reality of black holes began to be gradually recognized after the discovery of quasars in the late 1950s and early 1960s. The final solution to the problem of the total collapse of very massive stars that have exhausted their nuclear fuel was found in the second half of the twentieth century through the efforts of a galaxy of brilliant theoretical physicists, including Soviet ones, mainly from the group of Ya. B. Zel'dovich. It turned out that a similar collapse always compresses the star "all the way", completely destroying its substance and giving rise to a black hole. A singularity appears inside the hole, a "superconcentrate" of the gravitational field, closed in an infinitely small volume. For a static hole, this is a point, for a rotating one, a ring. The curvature of space-time and, consequently, the gravitational force near the singularity tends to infinity (of course, we are talking about a description based on general relativity, which does not take into account quantum effects). The mathematical theory of black holes is well developed and very beautiful - and it all goes back historically to Schwarzschild's solution.

Supplement: author, author!

Princeton University professor John Archibald Wheeler is considered the official father of the term "black hole". In the early 1950s, he switched from nuclear physics to general relativity and did a lot to turn this research into a serious and rapidly growing field at the intersection of fundamental physics, astrophysics and cosmology. It is reliably known that he spoke about black holes on December 29, 1967, speaking at the annual conference of the American Association for Science (it is possible that this expression has already slipped several times in his public lectures). Soon his performance appeared in print (John Archibald Wheeler, 1968. Our Universe: The Known and the Unknown). The spectacular and memorable name came up at the right time, since it almost coincided in time with the first message about the discovery of radio pulsars (A. Hewish et al.,). It fell in love with physicists and delighted journalists who spread it around the world.

Although Wheeler undoubtedly coined the term "black hole" both into the language of physics and into mass circulation, it was still invented by others. Its etymology is detailed in a new book by MIT professor Marcia Bartusiak (2015. Black Hole: How an Idea Abandoned by Newtonians, Hated by Einstein, and Gambled on by Hawking Became Loved, pp. 137-141). According to her research, already in 1960, Wheeler's colleague from the Physics Department of Princeton University, Robert Dicke, who at the beginning of the second half of the last century also took up gravity, speaking at a colloquium at the Institute for Advanced Research, jokingly compared the collapse of a massive star to the "Calcutta black hole "(Black Hole of Calcutta). In the middle of the 18th century, this became the name for a small prison cell at Fort William, which was built in Calcutta by the British East India Company. In June 1756, the new ruler of Bengal, Bihar and Orissa, Siraj-ud-Daud, captured Fort William and killed several dozen British prisoners in this cell, who died from suffocation or heatstroke. Since that time, the expression black hole has stuck in the English language as a symbol of something from where there is no return. In this sense, it was used by Robert Dicke.

As they say, dashing trouble is the beginning. Dicke's humorous expression was destined for a long and honorable life in a completely new meaning. The name "black hole" sounded several times on the sidelines of the First Texas Symposium on Relativistic Astrophysics, which took place in Dallas in December 1963. Soon it was used by the scientific editor of the journal Life Albert Rosenfeld, who published a report on this meeting. Its first appearance in the scientific press took place on January 18, 1964, when in the journal Science News Letters a note was posted on the meeting of astronomers at the annual session of the American Association for Science, which was held in late December in Cleveland. According to the author of the note, Anne Ewing, this expression was used more than once by a physicist from the Goddard Institute Hong-Yee Chiu, who admitted that he first heard it from Dikke a couple of years earlier. So the palm in naming completely collapsed stars as black holes most likely belongs to Robert Dicke. Interestingly, Chiu himself coined a new astrophysical term in 1964, namely "quasar".

In general, the expression "black hole" as the name of the final stage of the gravitational collapse of the most massive stars was occasionally used even before Wheeler. This is the real story.

Supplement: post-solar dwarf

If our Galaxy were doomed to a solitary journey through Space, this forecast would have one hundred percent reliability. However, in 4 billion years, the Milky Way will meet and merge with neighboring Andromeda, forming a new giant galaxy. In an even more distant future, it is destined to unite with the galaxy M33, which is also the Galaxy of the Triangle. It cannot be ruled out in advance that in this stellar association, the Sun, which has become a white dwarf, will turn out to be a member of a close binary system, having a main sequence star or a red giant as a partner. If its matter begins to flow to the surface of the Sun, it may happen that the Sun either becomes a new star, or even turns into a type Ia supernova and completely disappears in a monstrous explosion. However, as far as can be judged, the likelihood of such an outcome is very small, so that the standard scenario has every chance of being implemented.

Alexey Levin

SCHWARZSHILD SPACE-TIME-space-time outside a massive non-rotating body in (Ricci tensor R ik= 0). Length element ds defined by the expression

where r, q, f - spherical coordinates centered in the center of the massive body, M- body mass. This is the solution to ur-ny Einstein general relativity was found by K. Schwarzschild (K. Schwarzschild, 1916). The magnitude r q = 2GM / s 2 called the Schwarzschild radius or gravitational radius... Sh. P - v. is asymptotically flat for r and has the correct Newtonian asymptotics there:, where is the Newtonian gravitational potential.

On the surface of a massive body, the metric Ш.п - в. (1) must be continuously stitched with a metric describing space-time inside the body. In this case, the radial coordinate of the body surface in Sh. P - in. should be more r q otherwise the balance of the body is impossible. Sh. P - v. makes sense in the absence of a central body. Then it can be analytically continued under the gravitational radius, into the region r using other reference systems [D. Finkelstein (D. Finkelstein), 1958]. Surface r = r q is isotropic, so that all massive or massless particles can cross it in only one direction (because of this, it is also called the horizon). If the boundary conditions at r = r q are such that the particles cross the gravitational radius in the direction of decreasing r, then Ш.п - в. describes black hole formed as a result of the collapse of the initially regular distribution of matter (for example, a star), and then the surface r = r q is the event horizon. Otherwise, Sh.p - v. contains white hole... In the area under the gravitational radius, particles can move either only in the direction of decreasing r in the case of a black hole, or only in the opposite direction in the case of a white hole. The maximum analytic continuation of Sh.p. - v. in the absence of matter contains both black and white holes (inside each of which there is a surface r = 0) ,

as well as two unconnected spatial asymptotically flat infinities r... However, such a maximum extension of the Sh. N - c. is not physical in the sense that it cannot arise as a result of the dynamic evolution of a regular distribution of matter. Its curvature tensor is finite and regular for r 0. Two unconnected surfaces r = 0, on which it diverges, there are 3-dimensional spacelike hypersurfaces. Therefore, it cannot be said that r = 0 is the "center" r 0 > r q.

It can be proved that Sh.p. is the only static vacuum asymptotically flat solution of the equations of the general theory of relativity. The SH.P - V. Describing a black hole is stable: small perturbations of the metric (1) of a general form decay according to a power law for t(the exponent is determined by the multipolarity of the disturbance). Gravitational binding energy of bodies with mass T<<М moving in stable circular orbits in the Sh.p. - in., can reach 6% of (S. A. Kaplan, 1949). Particles falling into a black hole reach the surface of the event horizon in a finite proper time ~ r q / s, but for an infinite time interval t from the point of view of any external. an observer who does not fall into a black hole. This statement remains true in the case of a nonstationary black hole, the mass of which increases due to absorption ( accretion) it of the surrounding matter [in this case, however, it should be remembered that in the case of accretion onto a black hole, the radius of the surface of the event horizon r h, (t) is always slightly larger than the current gravitational radius r q (t)]. After crossing the event horizon, the particles reach a singularity r= 0 also for a finite interval of proper time. Ext. the observer will never see this.

Lit .: Landau L. D., Lifshits E. M., Theory of the field, 7th ed., M., 1988; Hawking S., Ellis J., Large-scale structure of space-time, trans. from English, M., 1977.

A. A. Starobinsky.

In 1916, just a few months after Einstein published his gravitational field equations in general relativity, the German astronomer Karl Schwarzschild found a solution to these equations, describing the simplest 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 here.

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

In order to understand the details, let's first select three guys - Borya, Vasya and Masha - and imagine that they are soaring in space (Fig. 9.1). You can always take an arbitrary point in space and determine the positions of all three, measuring the distance 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 "move" in time, for they are getting older and older. This feature can be depicted on a space-time diagram (Fig. 9.2). The distance from an arbitrary starting point of reference ("beginning") to another point in space is plotted here along the horizontal axis, and time - along the vertical. In addition, as in the special theory of relativity, it is convenient to take on the coordinate axes of this graph such scales that the rays of light are described by a straight line with an inclination of 45њ. On such a space-time diagram, the world lines of all three guys go vertically up. They all the time remain at the same distances from the starting point ( r = 0), but gradually get 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 what might be called "negative space". Since it is impossible to be "at a distance of minus 3 m" from any point (origin), the distances from the origin are always expressed in positive numbers.

We now turn to the Schwarzschild black hole. As mentioned 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 of coordinates is compatible with a singularity for convenience. 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 vertical world lines, the world line of the event horizon goes vertically up exactly 1 Schwarzschild radius to the right of the world singularity line, which in Fig. 9.3 is depicted with a sawtooth line.

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

To understand the relationship between the regions of spacetime 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 through the event horizon outward, rise to a maximum height of 1 million kilometers above the black hole, and then fall back through the event horizon, and again fall into the singularity. The flight of the astronomer is shown in Fig. 9.4.

To the attentive reader, this may seem impossible - after all, you cannot jump out of the singularity at all! We will restrict ourselves to referring to purely mathematical the possibility of such a trip. As will be seen from what follows, the complete Schwarzschild solution contains both black, So and a white hole. Therefore, patience and attention will be required of the reader throughout the next few sections. Here and in subsequent chapters, we will illustrate the presentation using the travels of astronomers or astronauts to black holes. For convenience, we will simply speak of an astronaut as "he".

The traveling astronomer carries a watch with him to measure his own time. The couch potato scientists who follow its flight from a distance of 1 million kilometers from the black hole also have a watch. 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 clock is set at the same moment (synchronized) and now shows 12 noon. Then it is possible to calculate at what moment (both by the traveler's own time and by the coordinate time) the astronomer will get to each point of interest on his trajectory.

Recall that the astronomer's clock measures his own time. Therefore, it is impossible to notice the "slowing down of time" due to 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, the calculations lead to the following result:

In an astronomer's own time

1. The astronomer leaves the singularity at 11:40 am (according to his own clock).
2. In 1 / 10,000 s after 11 h 40 min, it flies over the event horizon to the outside world.
3. At 12 noon, it reaches a maximum height of 1 million kilometers above the black hole.
4. In one 1 / 10,000 s to 12 h 20 min of the day, it crosses the event horizon, moving inward.
5. The astronomer returns to the singularity at 12.20 pm.

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

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

In coordinate time

Of course, everyone agrees that the astronomer-traveler reaches its maximum flight altitude at 12 noon, i.e. at the moment at which all clocks are synchronized. Everyone will also agree on when the astronomer leaves the singularity and when he returns to it. But otherwise the Schwarzschild geometry is clearly abnormal. After flying out of the singularity, the astronomer moves in coordinate time back in time up to a year. It then zooms forward in time again, reaches its maximum flight altitude at noon, and descends below the event horizon in a year. After that it moves again back in time and falls into the singularity at 12.20 pm. On the space-time diagram, its world line has the form shown in Fig. 9.5.

Some of these strange findings are intuitive. Recall that from the point of view of a distant observer (whose clock measures coordinate time), time stops at the event horizon. Let us also recall that a stone or any other body falling on the event horizon never do not reach a point with the 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 up to a year, that is, in the infinitely distant future. Since the entire trip is symmetrical about 12 noon (i.e. takeoff and fall take the same time), then distant scientists must observe that the astronomer has risen, moving towards them, for billions of years. It must move outward the event horizon in a year.

Even more confusing is the fact that remote observers can see two moving astronomers. So, for example, at 3 pm they see one astronomer falling onto the event horizon (moving forward in time). However, according to their own calculations, should there is another astronomer within the event horizon, falling on the singularity (and moving backward in time).

Of course, this is nonsense. More precisely, such a strange behavior of the 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 - 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, therefore, two astronomers are observed at once, while in fact there is only one. This means that you need to unfold 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 Figure 9.3), the event horizon is a line going from moment (distant past) to moment (far future) and located Exactly at a distance of 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, found that it is equal to zero. If the volume of some sphere is equal to zero, then it is, of course, just a point. In other words, physicists began to suspect that this "line" in a simplified diagram must be in fact a point in the global picture of a black hole!

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

To move from a simplified depiction of a black hole to its global picture, our simplified depiction must be reworked into a much more complex space-time diagram. And yet our end result will be a new space-time diagram! In this diagram, spacelike quantities will be directed horizontally (from left to right), and timelike quantities will be directed vertically (from bottom to top). In other words, the transformation should work in such a way that old spatial and temporal coordinates have been replaced with new spatial and temporal coordinates that would reflect the fully 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 to remain at a constant distance from it, he must have powerful rocket engines, throwing streams of gases downward. In flat space-time, far from gravitating masses, a spacecraft, with its engines running, would acquire acceleration and would move faster and faster, for the thrust of the rocket engines would provide him with a constant increase in speed. The world line of such a ship is shown on the space-time diagram in Fig. 9.6. This line gradually approaches the straight line, which has a slope of 45њ, as the speed of the ship approaches the speed of light due to the continuous operation of the engines. A curve depicting such a world line is called hyperbole. An observer who is near the black hole and tries to stay at a constant distance from it will constantly experience acceleration caused by the operation of the ship's rocket engines. Astute physicists will therefore suspect that the "constant height" lines in the revised and improved spacetime diagram near the black hole will be hyperbolic branches.

Finally, the observer trying to stay on the horizon must have incredibly powerful rocket engines at his disposal. To prevent it from falling into the 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 worlds of the event horizon should be tilted at exactly 45њ 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 Kruskal-Szekeres diagram correctly covers all spacetime and fully reveals the global structure of the black hole. This confirms all the previously noted suspicions and reveals some new surprising and unexpected details. However, although the transformations of Kruskal and Szekeres immediately translate the old picture into a new one, it is better to visualize them in the form of a sequence of transformations, schematically shown in Fig. 9.7. The end result is again a space-time diagram (the spatial direction is horizontal and the temporal direction is vertical), and the rays of light coming to and from the black hole are depicted, as usual, straight with an inclination of 45њ.

The final 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 in order to understand this, we will consider again the flight of an astronomer thrown from the singularity, crossing the event horizon and falling back again. We already know that its world line on a simplified space-time diagram is unusual. This line is again depicted on the left in Fig. 9.8. On 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 eventually hits 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 the white hole and eventually falls into the black hole. Note that its worldline is tilted less than 45њ to the vertical throughout, i.e. this line is everywhere temporal and therefore permissible. Comparing the left and right parts of Fig. 9.8, you will find that the "points" of the moments in time and on the event horizon have now stretched into two straight lines with a slope of 45њ, which confirms our previous suspicions.

The transition to the Kruskal-Szekeres diagram reveals the true nature of all space-time near the Schwarzschild black hole. In a simplified diagram, different sections of space-time overlapped with each other. That is why remote scientists, observing the fall of an astronomer into a black hole (or his departure from it), erroneously assumed that there are two astronomer. In the Kruskal-Szekeres diagram, these overlapping areas are properly unraveled. In fig. 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 the individual parts of the space-time grid are transformed when moving from a simplified diagram to a Kruskal-Szekeres diagram. In a simplified view (Figure 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) remain straight, but now they diverge at different angles. The lines of constant distance from the black hole (dashed) are hyperbolas, as we suspected earlier.

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

In fig. 9.11 shows the Kruskal-Szekeres diagram; here the dashed lines of constant distance are generally upward if we take them far enough from the black hole. There they are still temporary. However, within the event horizon, dashed lines of constant distance are oriented generally horizontally. This means that below the event horizon, the lines of constant distance have a spacelike direction! Therefore, what is usually (in the outer universe) associated with distance behaves like time inside the event horizon.

Similarly, in a simplified diagram (see Figure 9.10), the constant time lines are horizontal and spacelike. For example, some 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. 9.11, which depicts the Kruskal-Szekeres diagram, the dotted lines of constant time generally have a spacelike direction when taken far from the black hole, i.e. they are almost horizontal there. But within 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, below the event horizon, the constant time lines have a time-like direction! Therefore, what is usually (in the outer universe) associated with time behaves like distance inside the event horizon. When the event horizon is crossed, space and time change roles.

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

The fact that the Schwarzschild singularity is spacelike leads to important conclusions. As in the special theory of relativity (see Fig. 1.9), it is impossible to move with superluminal speed, so space-like world lines as "paths" of motion are prohibited. It is impossible to move along world lines with an inclination of more than 45њ to the vertical (time-like) direction. Therefore, it is impossible to get from our Universe (on the Kruskal-Szekeres diagram on the right) to another Universe (on the same diagram on the left). Any path connecting both Universes with each other must be spacelike at least in one place, and such paths are prohibited for movement. In addition, since the event horizon is tilted exactly at an angle of 45њ, an astronomer from our Universe who has descended under this horizon will never be able to get out from under it again. For example, if someone enters area II in fig. 9.9, then all admissible time-like worldlines will bring him right into the singularity. The Schwarzschild black hole is a trap with no way out.

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

In fig. 9.12 shows the Kruskal-Szekeres diagram, "sliced" along the characteristic space-like hypersurfaces. Slice A refers to an early point in time. Initially, the two universes outside the black hole are not connected in any way. On its way from one universe to another, the spacelike 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 merge and a bridge appears, in which there are no singularities anymore. This corresponds to the cut B, where the singularity is not included. Over time, this bridge, or "mole Hole", expands and reaches the largest diameter equal to two Schwarzschild radii (the moment corresponding to the shear V). Later, the bridge begins to contract again (cut G) and finally breaks apart (slice D), so that we have again 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 such a global space-time structure near a black hole was a decisive breakthrough at the front of theoretical astrophysics. For the first time, it was possible to construct diagrams that fully depict all areas of space and time. But after 1960, new successes were also achieved, primarily by Roger Penrose. Although the Kruskal-Szekeres diagram represents the whole story, this diagram extends to the right and left infinitely far. For example, our Universe stretches infinitely to the right on the Kruskal-Szekeres diagram, while to the left on the same diagram, the space-time of the "other" asymptotically flat Universe, which is parallel to ours, extends to infinity. Penrose was the first to realize how useful and instructive it would be to use a "map" that maps these endless expanses to some finite regions, by which it would be possible to accurately judge what is happening far from the black hole. To implement this idea, Penrose drew on the so-called methods conformal mapping, with the help of which all space-time, including completely and both Universes, is depicted on one final diagram.

To acquaint you with Penrose's methods, let us turn to ordinary flat space-time of the type shown in Fig. 9.2. All space-time there is concentrated on the right side of the diagram, simply because it is impossible to be at a negative distance from an arbitrary beginning. You can be from it, say, 2 m, but certainly not minus 2 m. Let's return to fig. 9.2. The world lines of Bori, 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 met with the fact that the "most remote" regions of space-time are called infinities. These areas are extremely far from "here and now" in space or in time (the latter means that they can be in the very distant, future or very distant past). As seen from Fig. 9.13, there can be five types of infinities. First of all it is I - -timelike infinity in the past. It is the "place" from which all material objects originated (Borya, Vasya, Masha, Earth, galaxies and everything else). All such objects move along time-like world lines and must go to I + - the time-like infinity of the future, somewhere in the 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 none of the objects known to physics moves faster than light, then the photons move exactly at the speed of light along the world lines, inclined by 45њ on the space-time diagram. This makes it possible to enter " - the luminous infinity of the past, where all the light rays come from. There is finally and - light infinity of the future(where all the "light rays go.") Any distant 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 area where all material bodies come from, and the time-like infinity of the future ( I +) is the area where they all go. The light infinity of the past () is the area where the light rays come from, and the light infinity of the future is that area ( I +) where they go. Nothing (except tachyons) can fall into spacelike infinity ( I 0).

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

Penrose's method is reduced to a mathematical trick of contracting all these infinities onto one and 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), shoveling 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. Note that lines of constant distance from an arbitrary datum are generally vertical and always indicate a time-like direction. Constant time lines are generally horizontal and always indicate spacelike direction.

On conformal the map of the entire flat space-time (Fig. 9.15) space-time as a whole fit into the triangle. All timelike infinity in the past ( I -) is collected in a single point at the bottom of the diagram. All timelike world lines of all material objects emerge from this point, depicting an extremely distant past. All time-like infinity in the future ( I +) is collected in a single point at the top of the diagram. The time-like world lines of all material objects in the Universe end up at this point, which depicts 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 45њ, bounding the diagram at the top right and bottom right along the diagonals. Light rays always follow the world lines with an inclination of 45њ, so that the light coming from the distant past starts its way somewhere on , and those who go into the distant future ends their journey somewhere on . The vertical line that bounds the chart on the left is just a time-like world line of our chosen arbitrary starting point ( r = 0).

Rice. 9.15. Penrose diagram for flat space-time. All spacetime is wrapped inside a triangle using Penrose's conformal mapping method. Of the five infinities, three ( I -, I 0 , I + ) are compressed to individual points, and two are light infinities and- become straight lines with a slope of 45њ.	Rice. 9.16. An example of a conformal Penrose diagram. This diagram depicts virtually the same as Fig. 9.2. However, on the conformal diagram, the world lines of objects are completely represented (from the distant past I - until the distant future I +).

To complete the description of the conformal Penrose diagram of flat space-time, we have drawn in Fig. 9.16 completely world lines of Bori, Vasya and Masha. Compare this diagram with fig. 9.2 - after all, this is the same thing, only on the conformal diagram world lines are traced throughout their entire length (from the distant past I -љ until the distant future I +)

Penrose's depiction of ordinary flat space-time does nothing sensational. However, Penrose's method applies to black holes too! In particular, the Kruskal-Szekeres diagram (see Figure 9.11) can be displayed conformally in such a way that the physicist sees all space-time of all Universes depicted on a single piece of paper. As illustrated in Fig. 9.17, 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 (Fig. 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 45њ tilt, and the singularities (both past and future) remain spacelike. The exchange of roles between space and time, as before, occurs when the event horizon is crossed. However, now the most distant parts of both universes connected with a 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 Schwarzschild black hole exercise - figuring out what the desperately inquisitive kamikaze astronomers will see falling onto 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, that is, at our outer Universe. Note that the spacecraft now has no rocket motors to slow down 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) first passes 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 must be timelike, i.e. everywhere have a slope of less than 45њ to the vertical. During the journey, astronomers take four pairs of photographs at different stages of the journey - one from each porthole. The first pair (pictures A) taken when they were still very far from the black hole. In fig. 9.21, A the black hole is seen as a small speck in the center of the nose window's field of view. Although the sky is distorted in the immediate vicinity of the black hole, the rest of the sky looks normal. As the speed at which astronomers fall onto the black hole increases, the light from objects in the distant universe, as viewed through the aft window, experiences more and more redshift.

Rice. 9.21. Photo A. Far from the black hole. From a distance, the black hole looks like a small black speck in the center of the nose window's field of view. Astronomers falling into the hole observe through the aft window an undistorted view of the universe from which they flew. Photo B. Not an event horizon. Due to the effect of aberration, the image of the black hole is compressed towards the center of the field of view of the nose window. An astronomer observing through the aft window sees only the Universe from which the ship arrived. Photo V. Between the event horizon and the singularity. Having descended below the event horizon, an astronomer observing through a nasal window can see a different universe. The light coming from the area of ​​another Universe fills the central part of his field of view. Photo G. Directly above the singularity. As astronomers approach the singularity, the other universe becomes increasingly visible through the nose window. The image of the black hole itself (in the form of a ring) becomes thinner and thinner, rapidly approaching the edge of the field of view of the nose window.

Although, according to remote observers, the fall of the spacecraft slows down to a complete stop on the event horizon, astronomers at the very spacecraft will not notice anything like it. In their opinion, the speed of the ship increases all the time and when crossing the event horizon, it makes up an appreciable fraction of the speed of light. This is significant for the reason that as a result, falling astronomers observe the phenomenon of stellar light aberration, very similar to that discussed by us in Chap. 3 (see fig. 3.9, 3.11). Recall that when you move at near light speed, you will notice severe distortions in the sky. In particular, images of celestial bodies are, as it were, collected in front of the moving observer. As a result of this effect, the image of the black hole is concentrated closer to the middle of the forward window of the falling spacecraft.

The picture observed by falling astronomers from the event horizon is shown in Fig. 9.21, B... This and the following figures are based on calculations done by Cunningham at the California Institute of Technology in 1975. If the astronomers were at rest, the image of the black hole would occupy the entire field of view of the nasal window (Fig. 8.15, D). But since they are moving at high speed, the image is concentrated in the middle of the nose window. Its angular diameter is approximately 80њ. 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 flew.

To understand what will be seen when the ship is inside the event horizon, let's go back to the Penrose diagram of the Schwarzschild black hole (see Fig. 9.18 or 9.20). Recall that the light rays going into the black hole have an inclination of 45њ in this diagram. Therefore, once under the event horizon, astronomers will be able to see another universe. Light rays from distant parts of another universe (i.e. from its infinity on the left side of the Penrose diagram) will now be able to reach astronomers. As shown in fig. 9.21, V, in the center of the field of view of the spacecraft's nose window, 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 the falling observers approach the singularity, the black ring becomes thinner, pressing against the very edge of the field of view of the nose window. A view of the sky from a point directly above the singularity is shown in Fig. 9.21, G... The other Universe is getting better and better visible through the nasal window, and right at the singularity, its view completely fills the field of view of the nasal window. The astronomer, who makes observations through the aft window, sees only our outer Universe during 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... Recall that light escaping from the vicinity of the event horizon into the distant Universe undergoes a strong redshift. This phenomenon is called gravitational redshift, we discussed in chap. 5 and 8. The redshift of light coming from an area with a strong gravitational field corresponds to the loss of energy. Conversely, when light "hits" a black hole, it experiences purple offset and gains energy. Weak radio waves coming from the distant universe there 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 for billions of years near the event horizon. This incident light acquires a monstrous energy, and when astronomers descend below the event horizon, they therefore encounter an unexpected burst of X-rays and gamma rays. The light that comes from the region - Schwarzschild solution - Kerr solution - white hole - singularity

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This metric is written as

ds 2 = (1 - rsr) c 2 dt 2 - dr 2 (1 - rsr) - r 2 (sin 2 ⁡ θ d φ 2 + d θ 2), (\ displaystyle ds ^ (2) = \ left (1 - (\ frac (r_ (s)) (r)) \ right) c ^ (2) dt ^ (2) - (\ frac (dr ^ (2)) (\ left (1- \ displaystyle (\ frac ( r_ (s)) (r)) \ right))) - r ^ (2) \ left (\ sin ^ (2) \ theta \, d \ varphi ^ (2) + d \ theta ^ (2) \ right ),)

where r s = 2 G M c 2 (\ displaystyle r_ (s) = (\ frac (2GM) (c ^ (2))))- so-called Schwarzschild radius, or gravitational radius, M (\ displaystyle M)- the mass that creates the gravitational field (in particular, the mass of the black hole), G (\ displaystyle G)- gravitational constant, c (\ displaystyle c) is the speed of light. In this case, the area of ​​change of coordinates − ∞ < t < ∞ , r s < r < ∞ , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2 π {\displaystyle -\infty with identification of points (t, r, θ, φ = 0) (\ displaystyle (t, r, \ theta, \ varphi = 0)) and (t, r, θ, φ = 2 π) (\ displaystyle (t, r, \ theta, \ varphi = 2 \ pi)) as in normal spherical coordinates.

Coordinate r (\ displaystyle r) is not the length of the radius vector, but is introduced so that the area of ​​the sphere t = c o n s t, r = r 0 (\ displaystyle t = \ mathrm (const), \; r = r_ (0)) in this metric was equal to 4 π r 0 2 (\ displaystyle 4 \ pi r_ (0) ^ (2))... In this case, the "distance" between two events with different r (\ displaystyle r)(but with the same other coordinates) is given by the integral

∫ r 1 r 2 d r 1 - r s r> r 2 - r 1, r 2, r 1> r s. (\ displaystyle \ int \ limits _ (r_ (1)) ^ (r_ (2)) (\ frac (dr) (\ sqrt (1- \ displaystyle (\ frac (r_ (s)) (r)))) )> r_ (2) -r_ (1), \ qquad r_ (2), \; r_ (1)> r_ (s).)

At M → 0 (\ displaystyle M \ to 0) or r → ∞ (\ displaystyle r \ to \ infty) the Schwarzschild metric tends (componentwise) to the Minkowski metric in spherical coordinates, so that far from a massive body M (\ displaystyle M) space-time turns out to be approximately pseudo-Euclidean signature (1, 3) (\ displaystyle (1,3))... Because g 00 = 1 - r s r ⩽ 1 (\ displaystyle g_ (00) = 1 - (\ frac (r_ (s)) (r)) \ leqslant 1) at r> r s (\ displaystyle r> r_ (s)) and g 00 (\ displaystyle g_ (00)) monotonically increases with growth r (\ displaystyle r), then the proper time at points near the body "flows more slowly" than far from it, that is, there is gravitational time dilation massive bodies.

Differential characteristics

For a centrally symmetric gravitational field in a vacuum (and this is the case of the Schwarzschild metric), we can put:

g 00 = e ν, g 11 = - e λ; λ + ν = 0, e - λ = e ν = 1 - r s r. (\ displaystyle g_ (00) = e ^ (\ nu), \ quad g_ (11) = - e ^ (\ lambda); \ quad \ lambda + \ nu = 0, \ quad e ^ (- \ lambda) = e ^ (\ nu) = 1 - (\ frac (r_ (s)) (r)).)

Then the non-zero independent Christoffel symbols have the form

Γ 11 1 = λ r ′ 2, Γ 10 0 = ν r ′ 2, Γ 33 2 = - sin ⁡ θ cos ⁡ θ, (\ displaystyle \ Gamma _ (11) ^ (1) = (\ frac (\ lambda _ (r) ^ (\ prime)) (2)), \ quad \ Gamma _ (10) ^ (0) = (\ frac (\ nu _ (r) ^ (\ prime)) (2)), \ quad \ Gamma _ (33) ^ (2) = - \ sin \ theta \ cos \ theta,) Γ 11 0 = λ t ′ 2 e λ - ν, Γ 22 1 = - re - λ, Γ 00 1 = ν r ′ 2 e ν - λ, (\ displaystyle \ Gamma _ (11) ^ (0) = ( \ frac (\ lambda _ (t) ^ (\ prime)) (2)) e ^ (\ lambda - \ nu), \ quad \ Gamma _ (22) ^ (1) = - re ^ (- \ lambda) , \ quad \ Gamma _ (00) ^ (1) = (\ frac (\ nu _ (r) ^ (\ prime)) (2)) e ^ (\ nu - \ lambda),) Γ 12 2 = Γ 13 3 = 1 r, Γ 23 3 = ctg θ, Γ 00 0 = ν t ′ 2, (\ displaystyle \ Gamma _ (12) ^ (2) = \ Gamma _ (13) ^ (3 ) = (\ frac (1) (r)), \ quad \ Gamma _ (23) ^ (3) = \ operatorname (ctg) \, \ theta, \ quad \ Gamma _ (00) ^ (0) = ( \ frac (\ nu _ (t) ^ (\ prime)) (2)),) Γ 10 1 = λ t ′ 2, Γ 33 1 = - r sin 2 ⁡ θ e - λ. (\ displaystyle \ Gamma _ (10) ^ (1) = (\ frac (\ lambda _ (t) ^ (\ prime)) (2)), \ quad \ Gamma _ (33) ^ (1) = - r \ sin ^ (2) \ theta \, e ^ (- \ lambda).) I 1 = (r s 2 r 3) 2, I 2 = (r s 2 r 3) 3. (\ displaystyle I_ (1) = \ left ((\ frac (r_ (s)) (2r ^ (3))) \ right) ^ (2), \ quad I_ (2) = \ left ((\ frac ( r_ (s)) (2r ^ (3))) \ right) ^ (3).)

The curvature tensor is of the type D (\ displaystyle \ mathbf (D)) according to Petrov.

Mass defect

If there is a spherically symmetric distribution of matter "radius" (in terms of coordinates) a (\ displaystyle a), then the total mass of the body can be expressed through its energy-momentum tensor by the formula

m = 4 π c 2 ∫ 0 a T 0 0 r 2 d r. (\ displaystyle m = (\ frac (4 \ pi) (c ^ (2))) \ int \ limits _ (0) ^ (a) T_ (0) ^ (0) r ^ (2) \, dr. )

In particular, for the static distribution of matter T 0 0 = ε (\ displaystyle T_ (0) ^ (0) = \ varepsilon), where ε (\ displaystyle \ varepsilon)- energy density in space. Considering that the volume of the spherical layer in the coordinates chosen by us is

d V = 4 π r 2 g 11 dr> 4 π r 2 dr, (\ displaystyle dV = 4 \ pi r ^ (2) (\ sqrt (g_ (11))) \, dr> 4 \ pi r ^ ( 2) \, dr,)

we get that

m = ∫ 0 a ε c 2 4 π r 2 d r< ∫ V ε c 2 d V . {\displaystyle m=\int \limits _{0}^{a}{\frac {\varepsilon }{c^{2}}}4\pi r^{2}\,dr<\int \limits _{V}{\frac {\varepsilon }{c^{2}}}\,dV.}

This distinction expresses itself gravitational defect in body weight... 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: when r = 0 (\ displaystyle r = 0) and at. Indeed, in Schwarzschild coordinates, a particle falling on a body will take an infinitely long time t (\ displaystyle t) to reach the surface r = r s (\ displaystyle r = r_ (s)), however, the transition, for example, to the Lemaitre coordinates in the accompanying frame of reference shows that from the point of view of the falling observer, there is no peculiarity of space-time on a given surface, and both the surface itself and the region r ≈ 0 (\ displaystyle r \ approx 0) will be reached in a finite proper time.

A real feature of the Schwarzschild metric is observed only for r → 0 (\ displaystyle r \ to 0), where the scalar invariants of the curvature tensor tend to infinity. This feature (singularity) cannot be eliminated by changing the coordinate system.

Event horizon

Surface r = r s (\ displaystyle r = r_ (s)) called event horizon... With a better choice of coordinates, for example, in the coordinates of Lemaitre or Kruskal, it can be shown that no signals can escape from the black hole through the event horizon. In this sense, it is not surprising that the field outside the Schwarzschild black hole depends only on one parameter - the total mass of the body.

Coordinates of Kruskal

One can try to introduce coordinates that do not give a singularity at r = r s (\ displaystyle r = r_ (s))... Many such coordinate systems are 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). it more space-time M ~ (\ displaystyle (\ tilde (\ mathcal (M)))) is usually called the (maximally extended) Schwarzschild space or (less often) the Kruskal space (Kruskal - Szekeres diagram). The metric in Kruskal coordinates is

ds 2 = - F (u, v) 2 dudv + r 2 (u, v) (d θ 2 + sin 2 ⁡ θ d φ 2), (2) (\ displaystyle ds ^ (2) = - F (u , v) ^ (2) \, du \, dv + r ^ (2) (u, v) (d \ theta ^ (2) + \ sin ^ (2) \ theta \, d \ varphi ^ (2) ), \ qquad \ qquad (2))

where F = 4 r s 3 r e - r / r s (\ displaystyle F = (\ frac (4r_ (s) ^ (3)) (r)) e ^ (- r / r_ (s))) and the function r (u, v) (\ displaystyle r (u, v)) is defined (implicitly) by the equation (1 - r / r s) e r / r s = u v (\ displaystyle (1-r / r_ (s)) e ^ (r / r_ (s)) = uv).

Schwarzschild's ingenious development had only relative success. Neither his method nor his interpretation was adopted. Almost nothing has been preserved from his work, except for the "bare" result of the metric, with which the name of its creator was linked. But questions of interpretation and, above all, the question of "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, which consists in the fact that "this does not exist in nature." This point of view spread and became dominant in all the special literature of that time.

The next stage is associated with an intensive study of the issues of gravity at the beginning of the "golden age" of the theory of relativity.