INTRODUCTION
Black holes are absolutely fantastic objects in their properties. “Of all the inventions of the human mind, from unicorns and chimeras to the hydrogen bomb, perhaps the most fantastic is the image of a black hole, separated from the rest of space by a certain boundary that nothing can cross; a hole with a gravitational field so strong that even light is held back by its stranglehold; a hole that bends space and slows down time. Like unicorns and chimeras, a black hole seems more appropriate in fantasy novels or ancient myths than in the real universe. And yet, the laws of modern physics actually require black holes to exist. Perhaps only our Galaxy contains them,” said the American physicist K. Thorn about black holes.
To this it should be added that inside the black hole, the properties of space and time change surprisingly, twisting into a kind of funnel, and in the depths there is a boundary beyond which time and space decay into quanta ... Inside the black hole, beyond the edge of this kind of gravitational abyss, from where there is no exit, amazing physical processes flow, new laws of nature are manifested.
Black holes are the most grandiose sources of energy in the universe. We probably see them in distant quasars, in exploding galactic nuclei. They also arise after the death of large stars. Perhaps black holes in the future will become sources of energy for humanity.
FORMATION OF BLACK HOLES. GRAVITATIONAL COLLAPSE. GRAVITY RADIUS
Scientists have found that black holes must arise as a result of a very strong compression of any mass, in which the gravitational field increases so much that it does not release any light or any other radiation, signals or bodies.
Back in 1798, P. Laplace, studying the propagation of light in the gravitational field of an object whose large mass is concentrated inside a small region of space, came to the conclusion that bodies absolutely black for an external observer can occur in nature. The gravitational field of such bodies is so great that it does not emit light rays (in the language of astronautics, this means that the second space velocity would be greater than the speed of light c). For this, it is only necessary that the mass of the object M be concentrated in a region with a radius smaller than the so-called gravity radius body R g . Radius
R g \u003d 2GM / cІ1.5 * 10 -28 M, where G is the constant of gravity;
M-mass (measured in grams),
R g - in centimeters.
Laplace's conclusion was based on classical mechanics and Newton's theory of gravity.
Therefore, for the emergence of a black hole, it is necessary that the mass shrink to such a size that the second cosmic velocity becomes equal to the speed of light. This size is called the gravitational radius and depends on the mass of the body. Its value is very small even for the masses of celestial bodies. So, for the Earth, the gravitational radius is approximately equal to 1 cm, for the Sun - 3 km.
In order to overcome gravity and escape from a black hole, a second cosmic velocity, greater than the speed of light, would be required. According to the theory of relativity, no body can accelerate faster than the speed of light. That is why nothing can fly out of a black hole, no information can come out. After any bodies, any substance or radiation fall under the influence of gravity into a black hole, the observer will never know what happened to them in the future. Near black holes, according to scientists, the properties of space and time should change dramatically.
If a black hole arises as a result of compression of a rotating body, then near its boundary all bodies are involved in rotational motion around it.
Scientists believe that black holes may appear at the end of the evolution of sufficiently massive stars. After the exhaustion of nuclear fuel reserves, the star loses its stability and under the influence of its own gravity begins to rapidly shrink. The so-called gravitational collapse(such a compression process in which the forces of gravity increase uncontrollably).
Namely, by the end of their lives, stars lose mass as a result of a number of processes: stellar wind, mass transfer in binary systems, supernova explosions, etc.; however, it is known that there are many stars with a mass of 10, 20 and even 50 times greater than the sun. It is unlikely that all these stars will somehow get rid of the "excessive" mass in order to enter the indicated limits (2-3M). According to the theory, if a star or its core with a mass above the specified limit begins to collapse under the influence of its own gravity, then nothing is able to stop its collapse. The matter of the star will shrink indefinitely, in principle, until it shrinks into a point. In the course of compression, the force of gravity on the surface steadily increases - finally, there comes a moment when even light cannot overcome the gravitational barrier. The star disappears: what we call a BLACK HOLE is formed.
GRAVITY RADIUS
radius, in the general theory of relativity (see. Gravitation) the radius of the sphere on which the gravitational force created by the mass m, which lies entirely inside this sphere, tends to infinity. G. r. is determined by the mass of the body m and is equal to r g 2 G m / c 2, where G is the gravitational constant, c is the speed of light. G. r. ordinary astrophysical objects are negligible compared to their actual size; so, for the Earth r g " 0.9 cm, for the Sun r g " 3 km.
If a body is compressed to the size of a G. R., then no forces will be able to stop its further compression under the influence of gravitational forces. Such a process, called relativistic gravitational collapse, can occur with fairly massive stars (as calculations show, with a mass of more than two solar masses) at the end of their evolution: if, having exhausted the nuclear "fuel", the star does not explode and does not lose mass, then, shrinking up to the size of a G. R., it must experience a relativistic gravitational collapse. During gravitational collapse, no radiation, no particles can escape from under the sphere of radius r g. From the point of view of an external observer located far from the star, as the size of the star approaches rg, time slows down the rate of its flow indefinitely. Therefore, for such an observer, the radius of the collapsing star approaches the G. r. asymptotically, never getting smaller than it.
I. D. Novikov.
Great Soviet Encyclopedia, TSB. 2012
See also interpretations, synonyms, meanings of the word and what is GRAVITATIONAL RADIUS in Russian in dictionaries, encyclopedias and reference books:
- GRAVITY RADIUS
- GRAVITY RADIUS
in the theory of gravitation, the radius rgr of a sphere on which the gravitational force created by the mass m lying inside this sphere tends to infinity; … - RADIUS in the Big Encyclopedic Dictionary:
(lat. radius letters. - wheel spoke, beam), a segment connecting any point of a circle or sphere with a center, as well as the length of this ... - RADIUS
circles (or spheres) (lat. radius, literally - the spoke of a wheel, a ray), a segment connecting a point of a circle (or sphere) with the center. R. is also called ... - RADIUS
[from the Latin radius spoke in a wheel, beam] in geometry, the radius of a circle (or ball) is a straight line segment connecting the center of a circle (or ... - GRAVITATIONAL in the Encyclopedic Dictionary:
[see gravity] based on the law... - RADIUS in the Encyclopedic Dictionary:
a, m. 1. geom. A line segment connecting the center of a circle or ball with some point on the circle (or surface of the ball), as well as ... - RADIUS in the Encyclopedic Dictionary:
, -a, m. 1^ In mathematics: a straight line segment connecting the center of a ball or circle with any point of a sphere or circle, a ... - RADIUS
RADIUS OF INERTIA, the value of r, which has the dimension of length, with the help of which the moment of inertia of the body relative to a given axis is expressed by f-loy: I \u003d ... - RADIUS in the Big Russian Encyclopedic Dictionary:
RADIUS (lat. radius, lit. - wheel spoke, beam), segment connecting c.-l. point of a circle or sphere with a center, as well as the length ... - GRAVITATIONAL in the Big Russian Encyclopedic Dictionary:
GRAVITATIONAL TRANSPORT, a method of transporting goods under the influence of own. weight (e.g. on an inclined conveyor chute, screw descent, gravity roller … - GRAVITATIONAL in the Big Russian Encyclopedic Dictionary:
GRAVITATIONAL RADIUS, in the theory of gravitation, the radius r gr of a sphere, on which the gravitational force created by the mass m lying inside this ... - GRAVITATIONAL in the Big Russian Encyclopedic Dictionary:
GRAVITATIONAL COLLAPSE, catastrophically fast compression of massive bodies under the influence of gravity. forces. G.K. the evolution of stars with a mass of St. two... - GRAVITATIONAL in the Big Russian Encyclopedic Dictionary:
Gravity logging, the study of the acceleration of gravity in boreholes to determine the cf. hearth density values. rocks in their natures. occurrence. … - RADIUS
ra"dius, ra"dius, ra"dius, ra"dius, ra"dius, ra"diusam, ra"dius, ra"dius, ra"dius, ra"dius, ra"dius, ... - GRAVITATIONAL in the Full accentuated paradigm according to Zaliznyak:
gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, gravitational, ... - RADIUS
(lat. radius spoke in a wheel, beam) 1) geom. R. circle (or ball) - a straight line segment connecting the center of a circle (or ball) ... - GRAVITATIONAL in the New Dictionary of Foreign Words:
(lat.; see gravity) physical. associated with the forces of gravity; i-th field - the field of gravitational forces; g-th radiation - ... - RADIUS
[ 1. geom. R. circle (or ball) - a straight line segment connecting the center of a circle (or ball) with some. point of a circle (or ball), ... - GRAVITATIONAL in the Dictionary of Foreign Expressions:
[phys. associated with the forces of gravity; i-th field - the field of gravitational forces; r-th radiation - radiation of gravity waves (r-th waves) ... - RADIUS in the dictionary of Synonyms of the Russian language.
- RADIUS
m. 1) A straight line segment connecting the center of a circle or ball with some. a point on a circle or on the surface of a sphere. 2) trans. Distribution area... - GRAVITATIONAL in the New explanatory and derivational dictionary of the Russian language Efremova:
adj. 1) Related by value. with noun: gravity associated with it. 2) Inherent to gravity, characteristic of it. 3) Serving for ... - RADIUS in the Dictionary of the Russian Language Lopatin:
r`radius, ... - GRAVITATIONAL in the Dictionary of the Russian language Lopatin.
- RADIUS in the Complete Spelling Dictionary of the Russian Language:
radius... - GRAVITATIONAL in the Complete Spelling Dictionary of the Russian Language.
- RADIUS in the Spelling Dictionary:
r`radius, ... - GRAVITATIONAL in the Spelling Dictionary.
- RADIUS in the Dictionary of the Russian Language Ozhegov:
coverage, the area of \u200b\u200bdistribution of something R. aviation action. radius! In mathematics: a line segment connecting the center of a ball or circle with any ... - RADIUS in Dahl's Dictionary:
husband. , lat. the half-diameter of the circle, the half-axis of the ball, the beam, the leg with which the circle is outlined; line or measure from the awn (center, center) to ... - RADIUS in the Modern Explanatory Dictionary, TSB:
(lat. radius, lit. - wheel spoke, beam), a segment connecting any point of a circle or sphere with a center, as well as the length of this ... - RADIUS in the Explanatory Dictionary of the Russian Language Ushakov:
radius, m. (Latin radius - beam, spoke). 1. A straight line connecting the center point with any point of the circle or the surface of the ball (mat.). … - RADIUS
radius m. 1) A straight line segment connecting the center of a circle or ball with some. a point on a circle or on the surface of a sphere. 2) trans. Region… - GRAVITATIONAL in the Explanatory Dictionary of Efremova:
gravity adj. 1) Related by value. with noun: gravity associated with it. 2) Inherent to gravity, characteristic of it. 3) Employee ... - RADIUS
- GRAVITATIONAL in the New Dictionary of the Russian Language Efremova:
- RADIUS
m. 1. A line segment connecting the center of a circle or ball with any point on the circle or surface of the ball. 2. trans. Distribution area... - GRAVITATIONAL in the Big Modern Explanatory Dictionary of the Russian Language:
adj. 1. ratio with noun. gravity associated with it 2. Inherent to gravity, characteristic of it. 3. Servant for study... - COLLAPSE GRAVITATIONAL in the Big Encyclopedic Dictionary:
see gravitational... - GRAVITATIONAL COLLAPSE in the Big Encyclopedic Dictionary:
catastrophically fast compression of massive bodies under the influence of gravitational forces. The evolution of stars with a mass of more than two solar masses can end with a gravitational collapse ... - COLLAPSE GRAVITATIONAL in the Great Soviet Encyclopedia, TSB:
gravitational (in astronomy), catastrophically fast compression of a star under the influence of gravitational forces (gravitation). According to existing astronomical concepts, K. g. plays a decisive ... - GRAVITY GRADIENTOMETER in the Great Soviet Encyclopedia, TSB:
gravity horizontal, a device for gravimetric exploration, measuring only the horizontal components of the gravity gradient (without measuring the curvature of the level surface). G. g. ... - GRAVITATIONAL COLLAPSE in the Great Soviet Encyclopedia, TSB:
collapse, see Collapse gravitational ... - GRAVITY VARIOMETER in the Great Soviet Encyclopedia, TSB:
variometer, a device for measuring the second derivatives of the potential of gravity, characterizing the curvature of the surface of an equal potential of gravity and the change (gradient) of the force ... - VARIOMETER GRAVITY in the Great Soviet Encyclopedia, TSB:
gravity, see gravity variometer ... - GRAVITATIONAL COLLAPSE in Collier's Dictionary:
the rapid contraction and disintegration of an interstellar cloud or star under the influence of its own gravitational force. Gravitational collapse is a very important astrophysical phenomenon; … - COLLAPSE GRAVITATIONAL in the Modern Explanatory Dictionary, TSB:
see gravitational...
What is the difference between Einstein's theory of gravitation and Newton's theory? Let's start with the simplest case. Suppose that we are on the surface of a spherical non-rotating planet and we measure the force of attraction of a body by this planet with the help of spring balances. We know that according to Newton's law, this force is proportional to the product of the mass of the planet and the mass of the body and is inversely proportional to the square of the radius of the planet. Radius of a planet: can be determined, for example, by measuring the length of its equator and dividing by 2n.
What does Einstein's theory say about the force of attraction? According to her, the force will be a little more than that calculated by Newton's formula. We will later clarify what this “a little more” means.
Imagine now that we can gradually reduce the radius of the planet, squeezing it while maintaining its total mass. The gravitational force will increase (after all, the radius decreases). According to Newton, when you double the force, the force quadruples. According to Einstein, the increase in force will again occur a little faster. The smaller the radius of the planet, the greater this difference.
If we compress the planet so much that the gravitational field becomes superstrong, then the difference between the magnitude of the force calculated according to Newton's theory and its true value, given by Einstein's theory, grows enormously. According to Newton, the force of gravity tends to infinity when we compress a body into a point (the radius is close to zero). According to Einstein, the conclusion is quite different: the force tends to infinity when the radius of the body becomes equal to the so-called gravitational radius. This gravitational radius is determined by the mass of the celestial body. It is the smaller, the smaller the mass. But even for gigantic masses it is very small. So, for the Earth it is equal to only one centimeter! Even for the Sun, the gravitational radius is only 3 kilometers. The dimensions of celestial bodies are usually much larger than their gravitational radii.
owls. For example, the average radius of the Earth is 6400 kilometers, the radius of the Sun is 700 thousand kilometers. If the true radii of the bodies are much greater than their gravitational ones, then the difference between the forces calculated according to Einstein's theory and Newton's theory is extremely small. So, on the surface of the Earth, this difference is one billionth of the magnitude of the force itself.
Only when the radius of the body during its compression approaches the gravitational radius, in such a strong field cha At the same time, the differences grow noticeably, and, as already mentioned, when the radius of the body is equal to the gravitational one, the true value of the strength of the gravitational field becomes infinite.
Before discussing what consequences this leads to, let's look at some other implications of Einstein's theory.
Its essence lies in the fact that it inextricably linked the geometric properties of space and the passage of time with the forces of gravity. These relationships are complex and varied. Let us note only two important facts.
According to Einstein's theory, time in a strong gravitational field flows more slowly than time measured away from gravitating masses (where gravity is weak). The fact that time can flow in different ways, the modern reader, of course, heard. And yet it is difficult to get used to this fact. How can time flow differently? After all, according to our intuitive ideas, time is duration, something common that is inherent in all processes. It is like a river flowing unchangingly. Separate processes can flow both faster and slower, we can influence them by placing them in different conditions. For example, it is possible to speed up the course of a chemical reaction by heating or to slow down the vital activity of an organism by freezing, but the movement of electrons in atoms will proceed at the same pace. All processes, as it seems to us, are immersed in the river of absolute time, which, it would seem, cannot be influenced by anything. It is possible, according to our ideas, to remove all processes from this river in general, and still time will flow like an empty duration.
So it was considered in science both at the time of Aristotle, and at the time of I. Newton, and later - up to A. Einstein. Here is what Aristotle writes in his book “Physics”: “Time passing in two similar and simultaneous motions is one and the same. If both periods of time did not flow simultaneously, they would still be the same ... Consequently, the movements can be different and independent of each other. In both cases, the time is exactly the same.”
I. Newton wrote even more expressively, believing that he was talking about the obvious: “Absolute, true, mathematical time, taken by itself, without relation to any body, flows uniformly, in accordance with its own nature.”
Guesses that ideas about absolute time are by no means so obvious were sometimes expressed even in ancient times. So, Lucretius Carus in the 1st century BC wrote in the poem “On the Nature of Things”: “Time does not exist by itself ... You cannot understand time by itself, regardless of the state of rest and movement of bodies”
But only A. Einstein proved that there is no absolute time. The passage of time depends on movement and, what is especially important for us now, on the gravitational field. In a strong gravitational field, all processes, absolutely everything, being of a very different nature, slow down for an outside observer. This means that time - that is, that which is common to all processes - slows down.
The delay is usually small. Thus, on the surface of the Earth, time passes more slowly than in deep space, by only one part in a billion, as in the case of calculating the force of gravity.
I would like to especially emphasize that such an insignificant time dilation in the Earth's gravitational field has been directly measured. Time dilation has also been measured in the gravitational field of stars, although usually it is also extremely small. In a very strong gravitational field, the deceleration is noticeably greater and becomes infinitely greater when the radius of the body becomes equal to the gravitational one.
The second important conclusion of Einstein's theory is that in the strong field of gravity the geometric properties of space change. Euclidean geometry, so familiar to us, turns out to be already unfair. This means, for example, that the sum of the angles in a triangle is not equal to two right angles, and the circumference of a circle is not equal to its distance from the center multiplied by 2pi. The properties of ordinary geometric figures become the same as if they were drawn not on a plane, but on a curved surface. That is why they say that the space
"curves" in the gravitational field. Of course, this curvature is noticeable only in a strong gravitational field, if the size of the body approaches its gravitational radius.
Of course, the notion of the curvature of space itself is just as incompatible with our deep-seated intuitions as the notion of the different flow of time.
Just as definitely as about time, I. Newton wrote about space: "Absolute space, by its own nature, independent of any relation to external objects, remains unchanged and motionless." The space was presented to him as a kind of endless "scene" on which "events" are played out that do not affect this "scene" in any way.
Even the discoverer of non-Euclidean, “curved” geometry - N. Lobachevsky expressed the idea that in some physical situations his - N. Lobachevsky - geometry, and not the geometry of Euclid, may appear. A. Einstein showed with his calculations that space really “curves” in a strong gravitational field.
This conclusion of the theory is also confirmed by direct experiments.
Why do we have such difficulty accepting the conclusions of the general theory of relativity about space and time?
Yes, because the daily experience of mankind, and even the experience of exact science, over the centuries has dealt only with conditions when changes in the properties of time and space are completely imperceptible and therefore completely neglected. All our knowledge is based on daily experience. Here we are accustomed to the thousand-year-old dogma about absolutely unchanging space and time.
Our era has come. Mankind in its knowledge has encountered conditions when the influence of matter on the properties of space and time cannot be neglected. Despite the inertia of our thinking, we must get used to such unusualness. And now a new generation of people is already much easier to perceive the truths of the theory of relativity (the foundations of the special theory of relativity are now being studied at school!), Than it was several decades ago, when even the most advanced minds could hardly perceive Einstein's theory
Let us make one more remark about the conclusions of the theory of relativity. Its author showed that the properties of space and time not only can change, but that space and time are united together into a single whole - a four-dimensional “space-time” It is this single variety that is bent. Of course, visual representations in such a four-dimensional supergeometry are even more difficult and we will not dwell on them here.
Let's return to the gravitational field around a spherical mass. Since the geometry in a strong gravitational field is non-Euclidean, curved, it is necessary to clarify what is the radius of a circle, for example, the planet's equator. In ordinary geometry, the radius can be defined in two ways: firstly, it is the distance of the points of the circle from the center, and secondly, it is the circumference divided by 2pi. But in non-Euclidean geometry, these two quantities do not coincide due to the “curvature” of space.
The use of the second method for determining the radius of a gravitating body (and not the distance from the center to the circle itself) has a number of advantages. To measure such a radius, it is not necessary to approach the center of gravitating masses. The latter is very important, for example, to measure the radius of the Earth it would be very difficult to penetrate into its center, but it is not very difficult to measure the length of the equator.
For the Earth, there is no need to directly measure the distance to the center, because the Earth's gravitational field is small, and Euclid's geometry is valid for us with greater accuracy, and the length of the equator divided by 2pi, equal to the distance to the center. In superdense stars with a strong gravitational field, however, this is not so:
the difference in the “radii” determined in different ways can be quite noticeable. Moreover, as we will see below, in some cases it is fundamentally impossible to reach the center of gravity. Therefore, we will always understand the radius of a circle as its length divided by 2pi.
The gravitational field we are considering around a spherical non-rotating body is called the Schwarzschild field, after the scientist who, immediately after Einstein created the theory of relativity, solved its equations for this case
The German astronomer K Schwarzschild was one of the creators of modern theoretical astrophysics, he performed a number of valuable works in the field of practical astrophysics and other branches of astronomy At a meeting of the Prussian Academy of Sciences dedicated to the memory of K. Schwarz
Schild, who died at the age of only 42, A. Einstein assessed his contribution to science as follows:
“In the theoretical works of Schwarzschild, what is especially striking is the confident mastery of mathematical methods of research and the ease with which he comprehends the essence of an astronomical or physical problem. Rarely do you find such deep mathematical knowledge combined with common sense and such flexibility of thinking as his. It was these talents that allowed him to carry out important theoretical work in those areas that scared away other researchers with mathematical difficulties. The motive for his inexhaustible creativity, apparently, can be considered to a much greater extent the joy of the artist, who discovers the subtle connection of mathematical concepts, than the desire to know the hidden dependencies in nature.
K. Schwarzschild obtained the solution of Einstein's equations for the gravitational field of a spherical body in December 1915, a month after A. Einstein completed the publication of his theory. As we have already said, this theory is very complex due to completely new, revolutionary concepts, but it turns out that its equations are still very complex, so to speak, purely technically. If the formula of I. Newton's law of gravitation is famous for its classical simplicity and brevity, then in the case of a new theory, to determine the gravitational field, it is necessary to solve a system of ten equations, each of which contains hundreds (!) Terms And these are not just algebraic equations, but differential equations in partial second order derivatives
In our time, to operate with such tasks, the entire arsenal of electronic computers is used. In the time of K. Schwarzschild, of course, there was nothing of the kind and the only tools were pen and paper.
But it must be said that even today work in the field of the theory of relativity sometimes requires long and painstaking mathematical transformations by hand (without an electronic machine), which are often tedious and monotonous due to the huge number of terms in the formulas. But you can't do without hard work. I often suggest that students (and sometimes graduate students and scientists), who are captivated by the fantastic nature of the general theory of relativity, who got to know it from textbooks and wish to work in it, concretely calculate with their own hands at least one relatively simple quantity in the problems of this theory. Not everyone, after many days (and sometimes much longer!) Calculations, continues to strive so ardently to devote his life to this science.
To justify such a “hard” love test, I will say that I myself went through a similar test. (By the way, according to the legends in the old days, even ordinary human love was tested by feats.) In my student years, my teacher in the theory of relativity was a well-known specialist and a very modest person A. Zelmanov. For my thesis, he set me a task related to the amazing property of the gravitational field - the ability to “destroy” it anywhere you want. "How? the reader will exclaim. “After all, the textbooks say that, in principle, no screens can be fenced off from gravity, that the “key-vorit” substance invented by the science fiction writer G. Wells is pure fiction, impossible in reality!”
All this is true, and if you remain motionless, for example, relative to the Earth, then its gravitational force cannot be destroyed. But the action of this force can be completely eliminated by starting to fall freely! Then weightlessness sets in. There is no gravity in the cabin of a spacecraft with its engines turned off, flying in orbit around the Earth, things and the astronauts themselves float in the cabin without feeling any gravity. We have all seen this on TV screens many times in reports from orbit. Note that no other field, except for the gravitational field, allows such a simple “annihilation”. The electromagnetic field, for example, cannot be removed in this way.
The property of “removability” of gravitation is connected with the most difficult problem of the theory - the problem of the energy of the gravitational field. It, according to some physicists, has not been solved to this day. The formulas of the theory make it possible to calculate for any mass the total energy of its gravitational field in all space. But it is impossible to indicate exactly where this energy is located, how much of it is in one or another place in space. As physicists say, there is no concept of the density of gravitational energy at points in space.
In my thesis, I had to show by direct calculation that the mathematical expressions known at that time for the energy density of the gravitational field are meaningless even for observers, not
experiencing free fall, say, for observers standing on the Earth and clearly feeling the force with which the planet attracts them. The mathematical expressions with which I had to work were even more cumbersome than the equations of the gravitational field, which we talked about above. I even asked A. Zelmanov to give me someone else to help me who would do the same calculations in parallel, because I could make a mistake. A. Zelmanov quite definitely refused me. “You have to do it yourself,” was his reply.
When everything was over, I saw that I had spent several hundred hours on this routine work. Almost all of the calculations had to be done twice, and some more. By the day of graduation, the pace of work increased rapidly, like the speed of a freely falling body in a gravitational field. True, it should be noted that the essence of the work consisted not only in direct calculations. Along the way, it was still necessary to think and solve fundamental questions.
This was my first publication on general relativity.
But back to the work of K. Schwarzschild. With the help of elegant mathematical analysis, he solved the problem for a spherical body and sent it to A. Einstein for transfer to the Berlin Academy. The solution struck A. Einstein, since by that time he himself had obtained only an approximate solution, valid only in a weak gravitational field. The solution of K. Schwarzschild was exact, that is, fair for an arbitrarily strong gravitational field around a spherical mass; this was its importance. But neither A. Einstein nor K. Schwarzschild himself knew then that this solution contained something much more. It later turned out to contain a description of a black hole.
And now let's continue talking about the second cosmic velocity. What speed, according to Einstein's equations, must be given to a rocket starting from the surface of the planet so that it, having overcome the force of gravity, flies into space?
The answer turned out to be extremely simple. The same formula is valid here as in the Newtonian theory. Hence, P. Laplace's conclusion about the impossibility for light to escape from a compact gravitating mass was confirmed by Einstein's theory of gravitation, according to which the second space velocity should be equal to the speed of light just at the gravitational radius.
A sphere with a radius equal to the gravitational one is called the Schwarzschild sphere.
If it were distributed spherically symmetrically, it would be motionless (in particular, it would not rotate, but radial motions are allowed), and would lie entirely inside this sphere.
The gravitational radius is proportional to the mass of the body m and equal to , where G- gravitational constant, with is the speed of light in vacuum. This expression can be written as , where it is measured in meters, and - in kilograms. For astrophysics, it is convenient to write km, where is the mass of the Sun.
In magnitude, the gravitational radius coincides with the radius of a spherically symmetric body, for which, in classical mechanics, the second cosmic velocity on the surface would be equal to the speed of light. John Michell first drew attention to the importance of this quantity in his letter to Henry Cavendish, published in 1784. Within the framework of the general theory of relativity, the gravitational radius (in other coordinates) was first calculated in 1916 by Karl Schwarzschild (see the Schwarzschild metric).
The gravitational radius of ordinary astrophysical objects is negligible compared to their actual size: for example, for the Earth = 0.884 cm, for the Sun = 2.95 km. The exceptions are neutron stars and hypothetical bosonic and quark stars. For example, for a typical neutron star, the Schwarzschild radius is about 1/3 of its own radius. This determines the importance of the effects of the general theory of relativity in the study of such objects.
If the body is compressed to the size of the gravitational radius, then no forces can stop its further compression under the influence of gravitational forces. Such a process, called relativistic gravitational collapse, can occur with fairly massive stars (as the calculation shows, with a mass of more than two or three solar masses) at the end of their evolution: if, having exhausted the nuclear “fuel”, the star does not explode and does not lose mass, then , shrinking to the size of the gravitational radius, it must experience a relativistic gravitational collapse. During gravitational collapse, no radiation, no particles can escape from under the sphere of radius. From the point of view of an external observer, located far from the star, as the size of the star approaches the proper time of the particles of the star, the rate of its flow indefinitely slows down. Therefore, for such an observer, the radius of the collapsing star approaches the gravitational radius asymptotically, never getting smaller than it.
A physical body that has experienced gravitational collapse, like a body whose radius is less than its gravitational radius, is called a black hole. Sphere Radius r g coincides with the event horizon of a non-rotating black hole. For a spinning black hole, the event horizon is ellipsoidal, and the gravitational radius gives an estimate of its size. The Schwarzschild radius for a supermassive black hole at the center of the galaxy is about 16 million kilometers. The Schwarzschild radius of a sphere uniformly filled with matter with a density equal to the critical density coincides with the radius of the observable Universe [ not in source] .
Literature
- Mizner C., Thorne K., Wheeler J. Gravity. - M .: Mir, 1977. - T. 1-3.
- Shapiro S.L., Tjukolsky S.A. Black holes, white dwarfs and neutron stars / Per. from English. ed. Ya. A. Smorodinsky. - M .: Mir, 1985. - T. 1-2. - 656 p.
see also
Links
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See what "Gravity Radius" is in other dictionaries:
In the general theory of relativity (see GRAVITY), the radius of a sphere, for which the gravitational force created by a spherical, non-rotating mass m, lying entirely inside this sphere, tends to infinity. G. p. (rg) is determined by body weight: rg= 2Gm/c2 … Physical Encyclopedia
In the theory of gravitation, the radius rgr of a sphere on which the gravitational force created by the mass m lying inside this sphere tends to infinity; rgr = 2mG/c2, where G is the gravitational constant, c is the speed of light in vacuum. The gravitational radii of ordinary ... ... Big Encyclopedic Dictionary
In the theory of gravitation, the radius rgr of a sphere on which the gravitational force created by the mass m lying inside this sphere tends to infinity; rgr=2mG/c2, where G is the gravitational constant, c is the speed of light in vacuum. The gravitational radii of ordinary ... ... encyclopedic Dictionary
gravity radius- gravitacinis spindulys statusas T sritis fizika atitikmenys: angl. gravitational radius vok. Gravitationsradius, m rus. gravity radius, m pranc. rayon gravitationnel, m … Fizikos terminų žodynas
In the general theory of relativity (see. Gravitation) the radius of the sphere on which the gravitational force created by the mass m, which lies entirely inside this sphere, tends to infinity. G. r. is determined by the body mass m and is equal to rg = 2G m/c2, where G… … Great Soviet Encyclopedia
In the theory of gravitation, the radius rgr of a sphere, on which the gravitational force created by the mass m lying inside this sphere tends to infinity; rgr = 2mG/c2, where G is gravitational. constant, with the speed of light in a vacuum. G. r. ordinary celestial bodies are negligible ... ... Natural science. encyclopedic Dictionary
Gravity radius- (see Gravity) the radius to which a celestial body (usually a star) can shrink as a result of gravitational collapse. So, for the Sun it is 1.48 km, for the Earth 0.443 cm ... Beginnings of modern natural science
Circles This term has other meanings, see Radius (disambiguation). Radius (lat. ... Wikipedia
The gravitational radius (or Schwarzschild radius) in the General Theory of Relativity (GR) is a characteristic radius defined for any physical body with mass: this is the radius of the sphere on which the event horizon would be, ... ... Wikipedia
Created by this mass (from the point of view of general relativity), if it were distributed spherically symmetrically, it would be motionless (in particular, it would not rotate, but radial motions are permissible), and would lie entirely inside this sphere. Introduced into scientific use by the German scientist Karl Schwarzschild in 1916.
The gravitational radius is proportional to the mass of the body M and is equal to r g = 2 G M / c 2 , (\displaystyle r_(g)=2GM/c^(2),) where G- gravitational constant, with is the speed of light in vacuum. This expression can be rewritten as r g≈ 1.48 10 −25 cm ( M/ 1 kg). For astrophysicists it is convenient to write r g ≈ 2 .95 (M / M ⊙) (\displaystyle r_(g)\approx 2(,)95(M/M_(\odot ))) km, where M ⊙ (\displaystyle M_(\odot )) is the mass of the sun.
The gravitational radius of ordinary astrophysical objects is negligible compared to their actual size: for example, for the Earth r g≈ 0.887 cm, for the Sun r g≈ 2.95 km. The exceptions are neutron stars and hypothetical bosonic and quark stars. For example, for a typical neutron star, the Schwarzschild radius is about 1/3 of its own radius. This determines the importance of the effects of the general theory of relativity in the study of such objects. The gravitational radius of an object with the mass of the observable universe would be about 10 billion light years.
With sufficiently massive stars (as the calculation shows, with a mass of more than two or three solar masses), at the end of their evolution, a process called relativistic gravitational collapse can occur: if, having exhausted the nuclear “fuel”, the star does not explode and does not lose mass, then, experiencing relativistic gravitational collapse, it can shrink to the size of a gravitational radius. During the gravitational collapse of a star to a sphere, no radiation, no particles can escape. From the point of view of an external observer located far from the star, as the size of the star approaches r g (\displaystyle r_(g)) the proper time of the particles of a star slows down the rate of its flow indefinitely. Therefore, for such an observer, the radius of the collapsing star approaches the gravitational radius asymptotically, never becoming equal to it. But it is possible, however, to indicate the moment from which an external observer will no longer see the star and will not be able to find out any information about it. So from now on, all the information contained in the star will actually be lost to an outside observer.
A physical body that has experienced gravitational collapse and reached a gravitational radius is called a black hole. Sphere Radius r g coincides with the event horizon of a non-rotating black hole. For a spinning black hole, the event horizon is ellipsoidal, and the gravitational radius gives an estimate of its size. The Schwarzschild radius for the supermassive black hole at the center of our galaxy is about 16 million kilometers.
The Schwarzschild radius of an object with satellites can in many cases be measured with much higher accuracy than the mass of that object. This somewhat paradoxical fact is connected with the fact that when passing from the measured period of the satellite's revolution T and the major semiaxis of its orbit a(these quantities can be measured with very high accuracy) to the mass of the central body M it is necessary to separate the gravitational parameter of the object μ = GM= 4π 2 a 3 /T 2 to the gravitational constant G, which is known to a much worse accuracy (about 1 in 7000 as of 2018) than the accuracy of most other fundamental constants. At the same time, the Schwarzschild radius is, up to the coefficient 2/ with 2 , the gravitational parameter of the object.
