(The terms gravity and gravitation are equivalent).
Acceleration, which experiences the body m 2 , located at a distance r from this body m 1 is equal to:
.
This value does not depend on the nature (composition) and mass of the body receiving acceleration. This ratio expresses an experimental fact, known even to Galilai, according to which all bodies fall into gravity. Earth's field with the same acceleration.
Newton established that acceleration and force are inversely proportional by comparing the acceleration of bodies falling near the surface of the Earth with the acceleration with which the Moon moves in its orbit. (The radius of the Earth and the approximate distance to the Moon were known by that time.) Further, it was shown that Kepler's laws follow from the law of universal gravitation, which were found by I. Kepler by processing numerous observations of the movements of the planets. This is how celestial mechanics arose. A brilliant confirmation of the Newtonian theory of T. was the prediction of the existence of a planet beyond Uranus (English astronomer J. Adams, French astronomer W. Le Verrier, 1843-45) and the discovery of this planet, which was called Neptune (German astronomer I. Galle , 1846).
The f-ly describing the motion of the planets includes the product G and the mass of the Sun, it is known with great accuracy. To define the same constant G required laboratory experiments by measuring the force of gravity. interaction of two bodies with a known mass. The first such experiment was set up by the English. scientist G. Cavendish (1798). Knowing G, it is possible to determine abs. the value of the mass of the Sun, the Earth and other celestial bodies.
The law of gravity in the form (1) is directly applicable to point bodies. It can be shown that it is also valid for extended bodies with a spherically symmetric mass distribution, and r is the distance between the centers of symmetry of bodies. For spherical bodies located far enough from each other, law (1) is valid approximately.
In the course of the development of the theory of thermodynamics, the idea of the direct force interaction of bodies gradually gave way to the idea of a field. Gravity field in Newton's theory is characterized by the potential , where x,y,z- coordinates, t- time, as well as field strength , i.e.
.
Gravity potential. field created by a set of resting masses does not depend on time. Gravity several potentials. bodies satisfy the principles of superposition, i.e. potential k.-l. the point of their common field is equal to the sum of the potentials of the bodies under consideration.
It is assumed that the gravitational the field is described in the inertial coordinate system, i.e. in a coordinate system, relative to which the body maintains a state of rest or uniform rectilinear motion if no forces act on it. In gravitational field the force acting on a particle of matter is equal to the product of its mass and the field strength at the location of the particle: F=mg. The acceleration of a particle relative to the inertial coordinate system (the so-called abs. acceleration) is, obviously, g.
Point body with mass dm creates gravity. potential
.
A continuous medium distributed in space with a density (may depend on time) creates gravity. potential equal to the sum of the potentials of all elements of the environment. In this case, the field strength is expressed as the vector sum of the strengths created by all particles.
Gravity the potential obeys the Poisson equation:
. (2)
It is clear that the potential of an isolated spherically symmetric body depends only on r. Outside such a body, the potential coincides with the potential of a point body located at the center of symmetry and having the same mass m. If at r>R, then at r>R. This justifies the approximation material points in celestial mechanics, where they usually deal with almost spherical. bodies that are, moreover, quite far apart. The exact Poissnoa equation, taking into account the real, asymmetric distribution of masses, is used, for example, in studying the structure of the Earth by gravimetric methods. T.'s law in the form of the Poisson equation is applied in the theoretical. study of the structure of stars. In stars, the force of gravity, which varies from point to point, is balanced by a pressure gradient; in rotating stars, centrifugal force is added to the pressure gradient.
Let us note some fundamental features of the classical theories T.
1) In the equation of motion of a material body - the second law of Newton's mechanics, ma=F(where F- acting force, a- the acceleration acquired by the body), and Newton's law of gravitation includes the same characteristic of the body - its mass. This implies that the inertial mass of the body and its gravitational mass are equal (for more details, see section 3).
2) Instantaneous value of gravity. potential is completely determined by the instantaneous mass distribution throughout space and the limiting conditions for the potential at infinity. For limited distributions of matter, the condition of vanishing at infinity (at ) is accepted. Adding a constant term to the potential violates the condition at infinity, but does not change the field strength g and does not change the ur-tion of the movement of material bodies in a given field.
3) Transition in accordance with Galilean transformations ( x"=x-vt, t"=t) from one inertial coordinate system to another, moving relative to the first at a constant speed v, does not change the Poisson equation and does not change the equation of movement of material bodies. In other words, mechanics, including Newton's theory of theory, is invariant under Galilean transformations.
4) Transition from an inertial coordinate system to an acceleration moving with acceleration a(t)(without rotation) does not change the Poisson equation, but leads to the appearance of an additional term that does not depend on the coordinates ma in ur-tions of movement. Exactly the same shuttle in the equations of motion occurs if in the inertial coordinate system to gravitational. to the potential, add a term that depends linearly on the coordinates, , i.e. add a uniform T. T. field, a homogeneous T. field can be compensated under conditions of accelerated movement.
The most important task of Newtonian celestial mechanics is yavl. the problem of motion of two point material bodies interacting gravitationally. To solve it, using Newton's law of gravity, they make up the equations of motion of bodies. Holy Islands solutions of these ur-tions are known with exhaustive completeness. According to a well-known solution, it can be established that certain quantities characterizing the system remain constant in time. They are called integrals of motion. Main integrals of motion (conserved quantities) yavl. energy, momentum, angular momentum of the system. For a two-body system, a complete mechanical energy E, equal to the sum kinetic energy ( T) and potential energy ( U), is saved:
E=T+U= const ,
where is the kinetic the energy of two bodies.
In the classic celestial mechanics potential energy due to gravity. phone interaction. For a pair of bodies, the gravitational (potential) energy is:
,
where is the gravitational mass potential m 2 at the location of the mass m 1 , a is the potential created by the mass m 1 at the location of the mass m 2. Zero value U have bodies spaced at an infinite distance. Because when the bodies approach each other, their kinetic energy increases and potential energy decreases, then, therefore, the sign U negative.
For stationary gravitating systems, cf. abs value gravity values. twice the energy cf. kinetic values. the energies of the particles that make up the system (see). So, for example, for a small mass m, rotating in a circular orbit around the central body, the condition for the equality of the centrifugal force mv2 /r gravitational force leads to , i.e. kinetic energy while . Hence, U=-2T and E=U+T=-T= const
In Newton's theory of gravity, a change in the position of a particle instantly leads to a change in the field throughout space (gravitational interaction occurs at infinite speed). In other words, in the classic theory T. the field serves the purpose of describing the instantaneous interaction at a distance, it does not have its own. degrees of freedom, cannot propagate and radiate. It is clear what the idea of gravitation is. the field is valid only approximately for sufficiently slow motions of the sources. Accounting for the finite velocity of propagation of gravity. interaction is produced in the relativistic theory of thermodynamics (see below).
In the nonrelativistic theory of gravity, the total mechanical energy of a system of bodies (including the energy of gravitational interactions) must remain unchanged for an infinitely long time. Newton's theory admits a systematic a decrease in this energy only in the presence of dissipation associated with the conversion of part of the energy into heat, for example. in inelastic collisions of bodies. If the bodies are viscous, then their deformations and oscillations when moving in gravity. the field also reduce the energy of the system of bodies due to the conversion of energy into heat.
inertial body mass ( m i) is a value that characterizes its ability to acquire one or another acceleration under the action of a given force. Inertial mass is included in Newton's second law of mechanics. Gravity weight ( m g) characterizes the body's ability to create one or another field T. Gravity. mass is included in the law of T.
From the experiments of Galileo, with the accuracy with which they were set, it followed that all bodies fall with the same acceleration, regardless of their nature and inertial mass. This means that the force with which the Earth acts on these bodies depends only on their inertial mass, and the force is proportional to the inertial mass of the body in question. But according to Newton's third law, the body under study acts on the Earth with exactly the same force with which the Earth acts on the body. Consequently, the force created by the falling body depends only on one of its characteristics - the inertial mass - and is proportional to it. At the same time, the falling body acts on the Earth with a force determined by gravity. body weight. Thus, for all bodies gravitational. mass is proportional to inertial. Counting m i and m g simply coinciding, find from experiments a specific numerical value of the constant G.
Proportionality of inertial and gravitational. the masses of the bodies different nature was the subject of research in the experiments of Hung. physicist R. Eötvös (1922), Amer. physicist R. Dicke (1964) and Soviet physicist V.B. Braginsky (1971). It is tested in the laboratory with high accuracy (with an error
The high accuracy of these experiments makes it possible to estimate the influence on the mass of various types of bond energy between body particles (see ). Proportionality of inertial and gravitational. mass means that physical. interactions within the body are equally involved in the creation of its inertial and gravitational forces. wt.
Relative to a coordinate system moving with acceleration a, all free bodies acquire the same acceleration - a. Due to the equality of the inertial and gravitational. masses, they all acquire the same acceleration relative to the inertial coordinate system under the influence of gravitational forces. fields with intensity g=-a. That is why we can say that from the point of view of the laws of mechanics, homogeneous gravity. field is indistinguishable from the acceleration field. In an inhomogeneous gravitational field compensation of the field intensity by acceleration at once in all space is impossible. However, the field strength can be compensated by accelerating a specially selected coordinate system along the entire trajectory of a body freely moving under the action of forces T. Such a coordinate system is called. freely falling. The phenomenon of weightlessness takes place in it.
Space movement. spacecraft (satellite) in the Earth's T. field can be considered as the motion of a falling coordinate system. The acceleration of astronauts and all objects on the ship relative to the Earth is the same and equal to the acceleration of free fall, and relative to each other is practically zero, so they are in weightlessness.
In free fall in non-uniform gravity. Field compensation of the field strength by acceleration cannot be ubiquitous, since the acceleration of neighboring freely falling particles is not exactly the same, i.e. particles have relative acceleration. In space on a ship, relative accelerations are practically imperceptible, since in order of magnitude they are cm / s 2, where r is the distance from the ship to the center of the earth, is the mass of the earth, x- the size of the ship. These accelerations can be neglected and the gravitational force can be used. Earth's field in the distance r from its center homogeneous in volume with a characteristic size x. In any given volume of space, the inhomogeneity of gravitational fields can be set by observations enough high precision, but for any given accuracy of observations, you can specify the amount of space in which the field will look homogeneous.
Relative accelerations manifest themselves, for example, on Earth in the form of ocean tides. The force with which the Moon pulls the Earth is different at different points on the Earth. The parts of the water surface closest to the Moon are attracted more strongly than the center of gravity of the Earth, and it, in turn, is stronger than the most distant parts of the oceans. Along the line connecting the Moon and the Earth, relative accelerations are directed from the center of the Earth, and in orthogonal directions - towards the center. As a result, the water shell of the Earth is deformed so that it is extended in the form of an ellipsoid along the Earth-Moon line. Due to the rotation of the Earth, tidal humps roll over the surface of the ocean twice a day. A similar but smaller tidal deformation is caused by gravity inhomogeneities. fields of the sun.
A. Einstein, based on the equivalence of homogeneous fields T. and accelerated coordinate systems in mechanics, suggested that such an equivalence applies in general to all, without exception, physical. phenomena. This postulate is called the principle of equivalence: all physical processes proceed in exactly the same way (under the same conditions) in an inertial frame of reference located in a uniform gravitational field, and in a frame of reference moving forward with acceleration in the absence of gravitational forces. fields. The equivalence principle played an important role in the construction of Einstein's theory of T.
The study of el.-mag. phenomena by M. Faraday and D. Maxwell in the second half of the 19th century. led to the creation of the theory of el.-magn. fields. The conclusions of this theory have been confirmed experimentally. Maxwell's equations are not invariant with respect to Galilean transformations, but are invariant with respect to Lorentz transformations, i.e. the laws of electromagnetism are formulated in the same way in all inertial coordinate systems connected by Lorentz transformations.
If the inertial coordinate system x", y", z", t" moves relative to the inertial coordinate system x, y, z, t at a constant speed v in axis direction x, then the Lorentz transformations have the form:
y"=y, z"=z, .
At low speeds () and neglecting the terms ( v/c) 2 and vx/c 2 these transformations go over to the Galilean transformations.
Logic analysis of the contradictions that arose when comparing the conclusions of the theory of el.-mag. phenomena from the classical ideas about space and time, led to the construction of a private (special) theory of relativity. A decisive step was taken by A. Einstein (1905), a huge role in its construction was played by the works of the Dutch physicist G. Lorenz and the French. mathematician A. Poincare. Particular relativity requires a revision of the classical ideas about space and time. In the classic In physics, the time interval between two events (for example, between two flashes of light), as well as the concept of the simultaneity of events, have an absolute meaning. They do not depend on the motion of the observer. This is not so in the theory of relativity: judgments about the time intervals between events and about length segments depend on the motion of the observer (the coordinate system associated with him). These quantities turn out to be relative in approximately the same sense as they are relative, depending on the location of the observers, yavl. their judgments about the angle under which they see the same pair of objects. Invariant, absolute, independent of the coordinate system, yavl. only 4-dimensional interval ds between events, including as a period of time dt, and the distance element between them:
ds 2 =c 2 dt 2 -dx 2 -dy 2 -dz 2
. (3)
The transition from one inertial frame to another, preserving ds 2 unchanged, is carried out exactly in accordance with the Lorentz transformations.
Invariance ds 2 means that space and time are combined into a single 4-dimensional world - space-time. Expression (3) can also be written as: 
, (4)
where the indices and run through the values 0, 1, 2, 3 and summation is performed over them, x 0 =ct, x 1 =x,
x 2 =y, x 3 =z, , other quantities are equal to zero. The set of quantities is called the metric tensor of the flat space-time or the Minkowski world [in general relativity (GR) it was shown that space-time has curvature, see below].
In the term "metric tensor" the word "metric" indicates the role of these quantities in determining distances and time intervals. In general, metric tensor is a collection of ten functions depending on x 0 , x 1 , x 2 , x 3 in the selected coordinate system. Metric tensor (or simply metric) allows you to determine the distance and time interval between events separated by .
Specialist. the theory of relativity establishes the limiting speed of the movement of material bodies and the propagation of interactions in general. This speed coincides with the speed of light in vacuum. Together with a change in ideas about space and time, special. the theory of relativity clarified the concept of mass, momentum, force. In relativistic mechanics, i.e. in mechanics invariant under Lorentz transformations, the inertial mass of a body depends on the velocity: , where m 0 - bodies. The energy of the body and its momentum are combined into a 4-component energy-momentum vector. For a continuous medium, you can enter the energy density, momentum density and momentum flux density. These quantities are combined into a 10-component quantity - the energy-momentum tensor. All components undergo a joint transformation when moving from one coordinate system to another. Relativistic theory of el.-magnet. fields (electrodynamics) are much richer than electrostatics, which is valid only in the limit of slow motions of charges. In electrodynamics, there is a combination of electric. and magnetic fields. Accounting for the finite rate of propagation of field changes and delay in the transfer of interaction leads to the concept of el.-magnet. waves, to-rye carry away energy from the radiating system.
Similarly, the relativistic theory of thermodynamics turned out to be more complicated than Newton's. Gravity the field of a moving body has a number of sv-in, similar to sv-you el.-mag. fields of a moving charged body in electrodynamics. Gravity field on long distance from bodies depends on the position and movement of bodies in the past, since gravitational. the field propagates at a finite speed. Radiation and distribution of gravitats becomes possible. waves (see). The relativistic theory of thermodynamics, as might be expected, turned out to be non-linear.
According to the principle of equivalence, no observations, using any laws of nature, can distinguish the acceleration created by homogeneous field T., from the acceleration of a moving coordinate system. In homogeneous gravity. field, it is possible to achieve zero acceleration of all particles placed in given area space, if we consider them in a coordinate system freely falling along with the particles. Such a coordinate system is mentally represented in the form of a laboratory with rigid walls and clocks located in it. The situation is different in non-uniform gravity. field in which neighboring free particles have relative accelerations. They will move with acceleration, albeit small, relative to the center of the laboratory (coordinate system), and such a coordinate system should be recognized as only locally inertial. It is possible to consider the coordinate system as inertial only in the region where it is permissible to neglect the relative accelerations of particles. Therefore, in a non-uniform gravitational field only in a small region of space-time and with limited accuracy, space-time can be considered as flat and f-loy (3) can be used to determine the interval between events.
Inability to enter inertial system coordinates in inhomogeneous gravity. the field makes all conceivable coordinate systems more or less equal. Ur-niya gravitats. fields must be written in such a way that they are valid in all coordinate systems, without giving preference to c.-l. of them. Hence the name for the relativistic theory of thermodynamics - the general theory of relativity.
Gravity the fields produced by real bodies, such as the Sun or the Earth, are always non-uniform. They are called true or nonremovable fields. In such a gravity field, no local-inertial coordinate system can be extended to the entire space-time. This means that the interval ds 2 cannot be reduced to the form (3) in the entire space-time continuum, i.e. space-time cannot be flat. Einstein came up with the radical idea of identifying non-uniform gravities. fields with space-time curvature. From these positions, gravitational the field of any body can be considered as a distortion of the geometry of space-time by this body.
Fundamentals of Mathematics. apparatus for the geometry of a space with curvature (non-Euclidean geometry) were laid down in the works of N.I. Lobachevsky, Hung. mathematics J. Bolyai, German. mathematicians K. Gauss and G. Riemann. In non-Euclidean geometry, curved space-time is characterized by metric. tensor included in the expression for the invariant interval: 
, (5)
a special case of this expression yavl. formula (4). Having a set of functions , one can raise the question of the existence of such coordinate transformations, which would translate (5) into (3), i.e. would make it possible to check whether space-time is flat. The desired transformations are realizable if and only if a certain tensor, composed of f-tions, the squares of their first derivatives and second derivatives, is equal to zero. This tensor is called the curvature tensor. In the general case, of course, it is not equal to zero.
A set of values is used for an invariant, independent of the choice of coordinate system, description of the geometric. st-in curved space-time. With physical point of view of the curvature tensor, expressed in terms of the second derivatives of gravitational. potentials, describes tidal accelerations in non-uniform gravity. field.
The curvature tensor is a dimensional quantity, its dimension is the square of the reciprocal length. Curvature at each point of space-time corresponds to characteristic lengths - radii of curvature. In a small space-time region surrounding a given point, a curved space-time is indistinguishable from a flat one up to small terms , where l is the characteristic size of the region. In this sense, the curvature of the world has the same properties as, say, the curvature the globe: it is insignificant in small regions. The curvature tensor at a given point cannot be "destroyed" by any coordinate transformations. However, in a certain system of coordinates and with a predetermined accuracy, the field T can be considered absent in a small region of space-time. In this area, all the laws of physics acquire the form that is consistent with the special. theory of relativity. This is how the principle of equivalence manifests itself, which was the basis of the theory of thermodynamics during its construction.
Metric the space-time tensor, and in particular the curvature of the world, are available for experimental determination. To prove the curvature of the globe, it is necessary to have a small "ideal" scale and use it to measure the distance between sufficiently distant points on the surface. A comparison of the measured distances will indicate the difference between the real geometry and the Euclidean one. Similarly, the geometry of space-time can be established by measurements made with "ideal" rulers and clocks. It is natural to assume, following Einstein, that the properties of a small "ideal" atom do not depend on where in the world it is placed. Therefore, having made, for example, a measurement of the shift in the frequency of light (by determining the gravitational redshift), it is possible in principle to determine the metric. space-time tensor and its curvature.
By summing the curvature tensor with metric. tensor can form a symmetric tensor 
, which has the same number of components as the energy momentum tensor of matter, which serves as a source of gravity. fields.
Einstein suggested that the equations of gravity should establish a relationship between and . In addition, he took into account that in gravitational field, the continuity equation for matter must be fulfilled in the same way as the current continuity equation is performed in electrodynamics. Such ur-tions are performed automatically if the ur-niya gravitats. write fields like this:
. (6)
This is Einstein's equations, obtained by him in 1916. These equations also follow from the variations. principle that independently showed him. mathematician D. Hilbert.
Einstein's equations express the connection between the distribution and motion of matter, on the one hand, and geometric. St. you space-time - on the other.
In equations (6) on the left side are the components of the tensor describing the geometry of space-time, and on the right - the components of the energy-momentum tensor describing the physical. Holy Islands of matter and fields (sources of gravitational fields). Quantities are not just functions that describe the gravitational field, but at the same time are components of the metric space-time tensor.
Einstein wrote that most of his work (special relativity theory, quantum nature of light) was in line with the actual problems of his time. They would have been made by other scientists with a delay of no more than 2-3 years if these works had not been done Einstein made an exception for general relativity and wrote that the relativistic theory of thermodynamics might have been delayed by 50 years. common methods field theory, and another approach to the nonlinear theory of thermodynamics arose, starting from the concept of a field given in flat space-time. It was shown that such a path leads to the same equations, to which Einstein came on the basis of the geometric. interpretations of T.
It should be emphasized that it is precisely in astronomy and cosmology that questions are encountered in which geometrical approach yavl. preferred. An example is the cosmological the theory of a spatially closed universe, as well as the theory. Therefore, Einstein's theory, based on the geometric concept retains its full meaning.
In the geometric interpretation of the motion of a material point in gravity. the field is a movement along a 4-dimensional trajectory - geodetic. lines of space-time. In a world with curvature, the geodesic. line generalizes the concept of a straight line in Euclidean geometry. The ur-tions of the motion of matter contained in the ur-nies of Einstein are reduced to the ur-tions of the geodesic. lines for point bodies. Bodies (particles), which cannot be considered pointlike, deviate in their motion from the geodesic. lines and experience the action of tidal forces.
In the weak field approximation, the laws of the Newtonian theory of gravitation and mechanics follow from the equations of general relativity. The effects of general relativity under such conditions represent only minor corrections.
The simplest effect, although difficult to observe, yavl. slowing down the flow of time in gravity. field, or, in a more common formulation, the effect of shifting the frequency of light. If a light signal with a frequency is emitted at a point with a value of gravitational potential and is accepted with a frequency at a point with a potential value (where there is exactly the same transducer for frequency comparison), then the equality must hold. Gravity effect. frequency shift of light was predicted by Einstein back in 1911 on the basis of the law of conservation of photon energy in gravity. field. It is reliably established in the spectra of stars, measured with an accuracy of up to 1% in the laboratory and with an accuracy of up to 1% under space conditions. flight. In the most accurate experiment, a hydrogen-maser frequency standard was used; a rocket that has risen to a height of 10 thousand km. Another similar standard has been set on Earth. Their frequencies were compared on different heights. The results confirmed the predicted frequency change.
When passing near a gravitating body, an el.-mag. the signal experiences a relativistic delay in propagation time. According to its physical nature, this effect is similar to the previous one. According to radio observations of the planets and especially interplanetary space. ships, the delay effect coincides with the calculated value within 0.1% (see).
The most important from the point of view of verification of general relativity yavl. rotation of the orbit of a body revolving around a gravitating center (it is also called the perihelion shift effect). This effect makes it possible to reveal the nonlinear nature of the relativistic gravity. fields. According to Newtonian celestial mechanics, the motion of the planets around the Sun is described by the ellipse equation: , where p=a(1-e 2) - orbit parameter, a- big semi-axle, e- eccentricity (see). Taking into account relativistic corrections, the trajectory has the form:
.
For each revolution of the planet around the Sun, its major axis is elliptical. orbit is rotated in the direction of motion by an angle . For Mercury, the relativistic rotation angle is a century. The fact that the angle of rotation accumulates over time makes it easier to observe this effect. During one revolution, the angle of rotation of the major axis of the orbit is so insignificant ~ 0.1" that its detection is significantly complicated by the curvature of the light rays within solar system. Nevertheless, modern radar data confirm the relativistic effect of the shift of Mercury's perihelion with an accuracy of 1%.
The listed effects are classic. It is also possible to check other predictions of general relativity (for example, the precession of the gyroscope axis) in weak gravitation. field of the solar system. Relativistic effects are used not only to test the theory, but also to refine astrophysical parameters, for example, to determine the mass of the components of binary stars. Thus, in a binary system, including the pulsar PSR 1913+16, the perihelion shift effect is observed, which made it possible to determine the total mass of the system components with an accuracy of 1%.
Einstein's equations include classical gravity. field characterized by the components of the metric. tensor , and the matter energy-momentum enzor . To describe the motion of gravitating bodies, the quantum nature of matter, as a rule, is not important. This is because they usually deal with gravity. macroscopic interaction. bodies, consisting of a huge number of atoms and molecules. The quantum mechanical description of the motion of such bodies is practically indistinguishable from the classical one. Science does not yet have experimental data on gravity. interaction under conditions when quantum properties of particles interacting with gravity become essential. field, and quantum properties of the gravity itself. fields.
Quantum processes involving gravity. The fields are certainly important in space (see , ) and, possibly, will become available for study also in laboratory conditions. The unification of the theory of thermodynamics with quantum theory is one of the most important problems in physics, and the solution to it has already begun.
Under normal conditions, the influence of gravity. field on quantum systems is extremely small. To excite the atom ext. gravitational field, the relative acceleration created by gravity. field at a distance of "radius of the hydrogen atom" cm and equal to , should be comparable with the acceleration with which the electron moves in the atom, . (Here - the radius of curvature of the gravitational field of the Earth, equal to: 
see) In gravitational field of the Earth with a margin of 10 19 this ratio is not fulfilled, therefore, atoms under terrestrial conditions under the influence of gravity are not excited and do not experience energy shifts. levels.
Nevertheless, under certain conditions, the probability of transitions in a quantum system under the action of gravitational forces. margins may be noticeable. It is on this principle that some modern assumptions on the detection of gravity. waves.
In specially designed (macroscopic) quantum systems a transition between neighboring quantum levels can occur even under the influence of a very weak alternating gravitational field. waves. An example of such a system is an el.-magnet. field in a cavity with highly reflective walls. If initially the system had N field quanta (photons) (), then under the influence of gravity. waves, their number can change with a noticeable probability and become equal to N+2 or N-2. In other words, transitions with energy are possible. level , and they are, in principle, discoverable.
The role of intense gravitations is especially important. fields. Such fields probably existed at the beginning of the expansion of the Universe, near the cosmological singularities and can occur in the later stages of gravity. collapse. The high intensity of these fields is manifested in the fact that they are capable of leading to observable effects (the production of pairs of particles) even in the absence of atoms, real particles or photons. These fields have an effective impact on physical. vacuum - physical. fields in the lowest energy state. In vacuum, due to the fluctuations of quantized fields, the so-called. virtual, actually unobservable particles. If the intensity of the external gravitational field is so large that at distances characteristic of quantum fields and particles, it is able to produce work that exceeds the energy of a pair of particles, then as a result, a pair of particles can be born - their transformation from a virtual pair into a real one. Necessary condition this process should be comparability of the characteristic radius of curvature describing the intensity of gravity. field, with the Compton wavelength , associated with particles with a rest mass m. A similar condition must be satisfied for massless particles so that the process of production of a pair of quanta with energy is possible. In the above example of a cavity containing an el.-mag. field, this process is similar to a transition with a probability comparable to unity from a vacuum state N=0 to a state describing two quanta, N=2. In ordinary gravity fields, the probability of such processes is negligible. However, in space, they could lead to the birth of particles in the very early Universe, as well as to the so-called. quantum "evaporation" of black holes of small mass (according to) works of English. scientist S. Hawking).
Intense gravitational fields that can significantly affect the zero fluctuations of other physical. fields should equally effectively affect their own zero fluctuations. If the process of birth of quantum physical. fields, then with the same probability (and in some cases with even greater probability) the process of the birth of gravitational quanta itself should be possible. fields - gravions. A rigorous and exhaustive consideration of such processes is possible only on the basis of the quantum theory of T. Such a theory has not yet been created. Application to gravity. field of the same ideas and methods, which led to the successful construction of quantum electrodynamics, encounters serious difficulties. It is not yet clear what paths the development of the quantum theory of T will take. One thing is certain - the most important way to test such theories will be the search for the phenomena predicted by the theory in space.
Based on the interpretation of Newton's second law, we can conclude that the change in motion occurs through force. Mechanics considers the forces of various physical nature. Many of them are determined by the action of gravitational forces.
In 1862, the law of universal gravitation was discovered by I. Newton. He suggested that the forces holding the moon are of the same nature as the forces that make the apple fall to the earth. The meaning of the hypothesis is the presence of the action of attractive forces directed along the line and connecting the centers of mass, as shown in Figure 1. ten . one . A spherical body has a center of mass coinciding with the center of the ball.
Picture 1 . 10 . 1 . Gravitational forces attraction between bodies. F 1 → = - F 2 →.
Definition 1
With known directions of motion of the planets, Newton tried to find out what forces act on them. This process has been named inverse problem of mechanics.
The main task of mechanics is to determine the coordinates of a body of known mass with its speed at any time using famous forces, acting on the body, and a given condition (direct problem). The reverse is performed with the determination of the acting forces on the body with its known direction. Such tasks led the scientist to the discovery of the definition of the law of universal gravitation.
Definition 2All bodies are attracted to each other with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them.
F = G m 1 m 2 r 2 .
The value of G determines the coefficient of proportionality of all bodies in nature, called the gravitational constant and denoted by the formula G \u003d 6, 67 10 - 11 N m 2 / k g 2 (C I) .
Most phenomena in nature are explained by the presence of the force of universal gravitation. The movement of planets, artificial satellites of the Earth, the flight paths of ballistic missiles, the movement of bodies near the surface of the Earth - everything is explained by the law of gravity and dynamics.
Definition 3
The manifestation of the force of gravity is characterized by the presence gravity. This is the name of the force of attraction of bodies to the Earth and near its surface.
When M is denoted as the mass of the Earth, R З is the radius, m is the mass of the body, then the gravity formula takes the form:
F = G M R Z 2 m = m g .
Where g is the free fall acceleration equal to g = G M R З 2 .
Gravity is directed towards the center of the Earth, as shown in the Moon-Earth example. In the absence of the action of other forces, the body moves with the acceleration of free fall. Its average value is 9.81 m / s 2. With known G and radius R 3 \u003d 6, 38 10 6 m, the mass of the Earth M is calculated using the formula:
M \u003d g R 3 2 G \u003d 5.98 10 24 k
If the body moves away from the surface of the Earth, then the action of the force of gravity and the acceleration of free fall change inversely with the square of the distance r to the center. Picture 1 . ten . 2 shows how the gravitational force acting on the astronaut of the ship changes with distance from the Earth. It is obvious that F of its attraction to the Earth is 700 N.
Picture 1 . 10 . 2 . Change in the gravitational force acting on the astronaut when moving away from the Earth.
Example 1
The Earth-Moon is suitable as an example of the interaction of a two-body system.
The distance to the Moon is r L = 3, 84 10 6 m. It is 60 times greater than the radius of the Earth R З. Hence, in the presence of gravity, the free fall acceleration α L of the Moon's orbit will be α L = g R З r L 2 = 9.81 m/s 2 60 2 = 0.0027 m/s 2.
It is directed towards the center of the Earth and is called centripetal. The calculation is made according to the formula a L \u003d υ 2 r L \u003d 4 π 2 r L T 2 \u003d 0, 0027 m / s 2, where T \u003d 27, 3 days is the period of the Moon's revolution around the Earth. Results and calculations performed different ways, suggest that Newton was right in his assumption of the same nature of the force that keeps the Moon in orbit, and the force of gravity.
The moon has its own gravitational field, which determines the free fall acceleration g L on the surface. The mass of the Moon is 81 times less than the mass of the Earth, and the radius is 3.7 times. This shows that the acceleration g L should be determined from the expression:
g L \u003d G M L R L 2 \u003d G M W 3, 7 2 T 3 2 \u003d 0, 17 g \u003d 1, 66 m / s 2.
Such weak gravity is typical for astronauts on the Moon. Therefore, you can make huge jumps and steps. A jump up a meter on Earth corresponds to a jump of seven meters on the Moon.
The movement of artificial satellites is fixed outside the Earth's atmosphere, so they are affected by the Earth's gravitational forces. The trajectory of a space body can change depending on initial speed. The movement of an artificial satellite in near-Earth orbit is approximately taken as the distance to the center of the Earth, equal to the radius R 3. They fly at altitudes of 200 - 300 km.
Definition 4
It follows that the centripetal acceleration of the satellite, which is reported by the forces of gravity, is equal to the free fall acceleration g. The speed of the satellite will take the designation υ 1 . They call her first cosmic speed.
Applying the kinematic formula for centripetal acceleration, we obtain
a n \u003d υ 1 2 R З \u003d g, υ 1 \u003d g R З \u003d 7, 91 10 3 m / s.
At this speed, the satellite was able to fly around the Earth in a time equal to T 1 = 2 πR W υ 1 = 84 m and n 12 s.
But the period of revolution of the satellite in a circular orbit near the Earth is much longer than indicated above, since there is a difference between the radius of the real orbit and the radius of the Earth.
The satellite moves according to the principle of free fall, vaguely similar to the trajectory of a projectile or ballistic missile. The difference lies in the high speed of the satellite, and the radius of curvature of its trajectory reaches the length of the radius of the Earth.
Satellites that move in circular trajectories over long distances have a weakened earth gravity that is inversely proportional to the square of the radius r of the trajectory. Then finding the speed of the satellite follows the condition:
υ 2 k \u003d g R 3 2 r 2, υ \u003d g R 3 R Z r \u003d u 1 R 3 r.
Therefore, the presence of satellites in high orbits indicates a lower speed of their movement than from near-Earth orbit. The formula for the period of revolution is:
T \u003d 2 πr υ \u003d 2 πr υ 1 r R Z \u003d 2 πR Z υ 1 r R 3 3 / 2 \u003d T 1 2 π R Z.
T 1 takes the value of the period of revolution of the satellite in near-Earth orbit. T increases with the size of the orbit radius. If r is 6 , 6 R 3 then the T of the satellite is 24 hours. When it is launched in the equatorial plane, it will be observed as it hangs over a certain point on the earth's surface. The use of such satellites is known in the space radio communication system. An orbit with a radius r = 6 , 6 R З is called geostationary.
Picture 1 . 10 . 3 . Satellite movement model.
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Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with free fall acceleration. Consequently, a force directed towards the Earth acts on the stone from the side of the Earth.
According to Newton's third law, the stone also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.
Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of the Universe. Here is the course of his reasoning, given in Newton's main work "Mathematical Principles of Natural Philosophy": "A stone thrown horizontally will deviate from a straight path under the influence of gravity and, having described a curved trajectory, will finally fall to the Earth. If you throw it with greater speed, then it will fall further” (Fig. 3.2). Continuing this reasoning, Newton comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from high mountain with a certain speed, could become such that it would never reach the surface of the Earth at all, but would move around it “just as the planets describe their orbits in the sky”.
Rice. 3.2
Now we have become so accustomed to the movement of satellites around the Earth that there is no need to explain Newton's thought in more detail.
So, according to Newton, the movement of the Moon around the Earth or the planets around the Sun is also a free fall, but only a fall that lasts without stopping for billions of years. The reason for such a “fall” (whether we are really talking about the fall of an ordinary stone on the Earth or the movement of the planets in their orbits) is the force of universal gravitation. What does this force depend on?
In § 1.23 we talked about the free fall of bodies. Galileo's experiments were mentioned, which proved that the Earth communicates the same acceleration to all bodies in a given place, regardless of their mass. This is possible only if the force of attraction to the Earth is directly proportional to the mass of the body. It is in this case that the acceleration of free fall, equal to the ratio of the force of gravity to the mass of the body, is a constant value.
Indeed, in this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of force also by a factor of two, and the acceleration, which is equal to the ratio, will remain unchanged.
Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts. But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body.
So The gravitational force between two bodies is directly proportional to the product of their masses.:
What else determines the gravitational force acting on a given body from another body?
It can be assumed that the force of gravity should depend on the distance between the bodies. To test the correctness of this assumption and to find the dependence of the force of gravity on the distance between bodies, Newton turned to the motion of the Earth's satellite - the Moon. Its motion was studied in those days much more accurately than the motion of the planets.
The revolution of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula
where R is the radius of the lunar orbit, equal to about 60 radii of the Earth, T \u003d 27 days 7 h 43 min \u003d 2.4 10 6 s is the period of the Moon's revolution around the Earth. Given that the radius of the Earth R 3 = 6.4 10 6 m, we get that the centripetal acceleration of the Moon is equal to:
The found value of acceleration is less than the acceleration of free fall of bodies near the surface of the Earth (9.8 m/s 2) by approximately 3600 = 60 2 times.
Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration imparted by the earth's gravity, and, consequently, the force of attraction itself by 60 2 times (1).
This leads to an important conclusion: the acceleration imparted to bodies by the force of attraction to the earth decreases in inverse proportion to the square of the distance to the center of the earth:
where C 1 is a constant coefficient, the same for all bodies.
The study of the motion of the planets showed that this motion is caused by the force of gravity towards the Sun. Using careful long-term observations of the Danish astronomer Tycho Brahe, the German scientist Johannes Kepler at the beginning of the 17th century. established the kinematic laws of planetary motion - the so-called Kepler's laws.
All planets move in ellipses with the Sun at one of the foci.
An ellipse (Fig. 3.3) is a flat closed curve, the sum of the distances from any point of which to two fixed points, called foci, is constant. This sum of distances is equal to the length of the major axis AB of the ellipse, i.e.
where F 1 and F 2 are the foci of the ellipse, and b = is its major axis; O is the center of the ellipse. The point of the orbit closest to the Sun is called perihelion, and the point farthest from it is called aphelion. If the Sun is in focus F 1 (see Fig. 3.3), then point A is perihelion, and point B is aphelion.
Rice. 3.3
The radius vector of the planet for the same time intervals describes equal areas . So, if the shaded sectors (Fig. 3.4) have the same area, then the paths s 1, s 2, s 3 will be traversed by the planet in equal time intervals. It can be seen from the figure that s 1 > s 2 . Hence, line speed the motion of the planet at different points in its orbit is not the same. At perihelion, the speed of the planet is greatest, at aphelion - the smallest.
Rice. 3.4
The squares of the periods of revolution of the planets around the Sun are related as the cubes of the semi-major axes of their orbits. Denoting the semi-major axis of the orbit and the period of revolution of one of the planets through b 1 and T 1 and the other - through b 2 and T 2, Kepler's third law can be written as follows:
Based on Kepler's laws, certain conclusions can be drawn about the accelerations imparted to the planets by the Sun. For simplicity, we will assume that the orbits are not elliptical, but circular. For the planets of the solar system, this replacement is not a very rough approximation.
Then the force of attraction from the side of the Sun in this approximation should be directed for all planets to the center of the Sun.
If through T we denote the periods of revolution of the planets, and through R the radii of their orbits, then, according to Kepler's third law, for two planets we can write
Normal acceleration when moving in a circle a \u003d ω 2 R. Therefore, the ratio of the accelerations of the planets
Using equation (3.2.4), we get
Since Kepler's third law is valid for all planets, the acceleration of each planet is inversely proportional to the square of its distance from the Sun:
The constant C 2 is the same for all planets, but does not coincide with the constant C 1 in the formula for the acceleration imparted to bodies by the globe.
Expressions (3.2.2) and (3.2.6) show that the gravitational force in both cases (attraction to the Earth and attraction to the Sun) gives all bodies an acceleration that does not depend on their mass and decreases inversely with the square of the distance between them:
The existence of dependences (3.2.1) and (3.2.7) means that the force of universal gravitation
In 1667, Newton finally formulated the law of universal gravitation:
The force of mutual attraction of two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them. The coefficient of proportionality G is called the gravitational(2) constant.
The law of universal gravitation (3.2.8) is valid only for such bodies, the dimensions of which are negligible compared to the distance between them. In other words, it is valid only for material points. At the same time, the forces gravitational interaction directed along the line connecting these points (Fig. 3.5). Such forces are called central.
Rice. 3.5
To find the gravitational force acting on a given body from another, in the case when the size of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into such small elements that each of them can be considered a point. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3.6). Having performed such an operation for each element of a given body and summing up the resulting forces, they find the total gravitational force acting on this body. This task is difficult.
Rice. 3.6
There is, however, one practically important case when formula (3.2.8) is applicable to extended bodies. It can be proved that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, attract with forces whose modules are determined by formula (3.2.8). In this case, R is the distance between the centers of the balls.
And finally, since the sizes of bodies falling to the Earth are many smaller sizes Earth, then these bodies can be considered as point bodies. Then under R in the formula (3.2.8) one should understand the distance from the given body to the center of the Earth.
Rice. 3.7
(1) Interestingly, as a student, Newton realized that the Moon moves under the influence of gravity towards the Earth. But at that time the radius of the Earth was not known exactly, and the calculations did not lead to correct result. Only 16 years later, new, corrected data appeared, and the law of universal gravitation was published.
(2) From the Latin word gravitas - heaviness.
Gravity, also known as attraction or gravitation, is a universal property of matter that all objects and bodies in the Universe possess. The essence of gravity is that all material bodies attract to themselves all other bodies that are around.
If gravity is general concept and the quality that all objects in the Universe possess, then the earth's gravity is a special case of this all-encompassing phenomenon. The earth attracts to itself all the material objects that are on it. Thanks to this, people and animals can safely move around the earth, rivers, seas and oceans can remain within their shores, and air can not fly through the vast expanses of the Cosmos, but form the atmosphere of our planet.
A fair question arises: if all objects have gravity, why does the Earth attract people and animals to itself, and not vice versa? Firstly, we also attract the Earth to ourselves, it's just that compared to its force of attraction, our gravity is negligible. Secondly, the force of gravity is directly proportional to the mass of the body: the smaller the mass of the body, the lower its gravitational forces.
The second indicator on which the force of attraction depends is the distance between objects: than more distance, the smaller the effect of gravity. Including due to this, the planets move in their orbits, and do not fall on each other.
It is noteworthy that the Earth, the Moon, the Sun and other planets owe their spherical shape precisely to the force of gravity. It acts in the direction of the center, pulling towards it the substance that makes up the "body" of the planet.
The gravitational field of the Earth is a force energy field that is formed around our planet due to the action of two forces:
Since both gravity and centrifugal force act constantly, the gravitational field is also a constant phenomenon.
The gravitational forces of the Sun, the Moon and some other celestial bodies, as well as the atmospheric masses of the Earth, have an insignificant effect on the field.
English physicist Sir Isaac Newton famous legend, one day walking in the garden during the day, he saw the moon in the sky. At the same time, an apple fell from the branch. Newton was then studying the law of motion and knew that an apple falls under the influence of a gravitational field, and the Moon revolves in an orbit around the Earth.
And then the thought came to the mind of a brilliant scientist, illuminated by insight, that perhaps the apple falls to the earth, obeying the same force due to which the Moon is in its orbit, and does not rush randomly throughout the galaxy. This is how the law of universal gravitation, also known as Newton's Third Law, was discovered.
In the language of mathematical formulas, this law looks like this:
F=GMm/D2 ,
where F- force of mutual gravitation between two bodies;
M- mass of the first body;
m- mass of the second body;
D2- distance between two bodies;
G- gravitational constant, equal to 6.67x10 -11.
