Gravity. Movement of bodies under the influence of gravity Bodies under the influence of gravity

DEFINITION

The law of universal gravitation was discovered by I. Newton:

Two bodies attract each other with , directly proportional to their product and inversely proportional to the square of the distance between them:

Description of the law of universal gravitation

The coefficient is the gravitational constant. In the SI system, the gravitational constant has the meaning:

This constant, as can be seen, is very small, therefore the gravitational forces between bodies with small masses are also small and practically not felt. However, the movement of cosmic bodies is completely determined by gravity. The presence of universal gravitation or, in other words, gravitational interaction explains what the Earth and planets are “supported” by, and why they move around the Sun along certain trajectories, and do not fly away from it. The law of universal gravitation allows us to determine many characteristics of celestial bodies - the masses of planets, stars, galaxies and even black holes. This law makes it possible to calculate the orbits of planets with great accuracy and create a mathematical model of the Universe.

Using the law of universal gravitation, cosmic velocities can also be calculated. For example, the minimum speed at which a body moving horizontally above the Earth’s surface will not fall on it, but will move in a circular orbit is 7.9 km/s (first escape velocity). In order to leave the Earth, i.e. to overcome its gravitational attraction, the body must have a speed of 11.2 km/s (second escape velocity).

Gravity is one of the most amazing natural phenomena. In the absence of gravitational forces, the existence of the Universe would be impossible; the Universe could not even arise. Gravity is responsible for many processes in the Universe - its birth, the existence of order instead of chaos. The nature of gravity is still not fully understood. Until now, no one has been able to develop a decent mechanism and model of gravitational interaction.

Gravity

A special case of the manifestation of gravitational forces is the force of gravity.

Gravity is always directed vertically downward (toward the center of the Earth).

If the force of gravity acts on a body, then the body does . The type of movement depends on the direction and magnitude of the initial speed.

We encounter the effects of gravity every day. , after a while he finds himself on the ground. The book, released from the hands, falls down. Having jumped, a person does not fly into outer space, but falls down to the ground.

Considering the free fall of a body near the Earth's surface as a result of the gravitational interaction of this body with the Earth, we can write:

where does the acceleration of free fall come from:

The acceleration of gravity does not depend on the mass of the body, but depends on the height of the body above the Earth. The globe is slightly flattened at the poles, so bodies located near the poles are located a little closer to the center of the Earth. In this regard, the acceleration of gravity depends on the latitude of the area: at the pole it is slightly greater than at the equator and other latitudes (at the equator m/s, at the North Pole equator m/s.

The same formula allows you to find the acceleration of gravity on the surface of any planet with mass and radius.

Examples of problem solving

EXAMPLE 1 (problem about “weighing” the Earth)

Exercise	The radius of the Earth is km, the acceleration of gravity on the surface of the planet is m/s. Using these data, estimate approximately the mass of the Earth.
Solution	Acceleration of gravity at the Earth's surface: where does the Earth's mass come from: In the C system, the radius of the Earth m. Substituting numerical values ​​of physical quantities into the formula, we estimate the mass of the Earth:
Answer	Earth mass kg.

EXAMPLE 2

Exercise

An Earth satellite moves in a circular orbit at an altitude of 1000 km from the Earth's surface. At what speed is the satellite moving? How long will it take the satellite to complete one revolution around the Earth?

Solution

According to , the force acting on the satellite from the Earth is equal to the product of the mass of the satellite and the acceleration with which it moves: The force of gravitational attraction acts on the satellite from the side of the earth, which, according to the law of universal gravitation, is equal to: where and are the masses of the satellite and the Earth, respectively. Since the satellite is at a certain height above the Earth's surface, the distance from it to the center of the Earth is: where is the radius of the Earth.

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 all other bodies around them.

Earth gravity

If gravity is a general concept and quality that all objects in the Universe possess, then gravity is a special case of this comprehensive phenomenon. The earth attracts to itself all material objects located on it. Thanks to this, people and animals can safely move across the earth, rivers, seas and oceans can remain within their shores, and the air can not fly across the vast expanses of space, 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 us, it’s just that, compared to its force of attraction, our gravity is negligible. Secondly, the force of gravity depends directly on 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: the greater the distance, the less the effect of gravity. Thanks also to this, the planets move in their orbits and do not fall on each other.

It is noteworthy that the Earth, Moon, 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.

Earth's gravitational field

The Earth's gravitational field is a force energy field that is formed around our planet due to the action of two forces:

• gravity;
• centrifugal force, which owes its appearance to the rotation of the Earth around its axis (diurnal rotation).

Since both gravity and centrifugal force act constantly, the gravitational field is a constant phenomenon.

The field is slightly affected by the gravitational forces of the Sun, Moon and some other celestial bodies, as well as the atmospheric masses of the Earth.

The law of universal gravitation and Sir Isaac Newton

The English physicist, Sir Isaac Newton, according to a famous legend, one day while 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 rotates in orbit around the Earth.

And then the brilliant scientist, illuminated by insight, came up with the idea that perhaps the apple falls to the ground, obeying the same force thanks to which the Moon is in its orbit, and not rushing 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/D 2 ,

Where F- the force of mutual gravity between two bodies;

M- mass of the first body;

m- mass of the second body;

D 2- the distance between two bodies;

G- gravitational constant equal to 6.67x10 -11.

(the terms gravity and gravitation are equivalent).

Acceleration experienced by a body m 2 located at a distance r from this body m 1 equals:
.
This value does not depend on the nature (composition) and mass of the body receiving acceleration. This relationship expresses the experimental fact, known to Galileo, 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 Earth's surface 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.) It was further 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 Newton's 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 named Neptune (German astronomer I. Galle , 1846).

The formulas describing the motion of the planets include the product G and the mass of the Sun, it is known with great accuracy. To determine the constant G laboratory experiments are required to measure the force of gravity. interaction of two bodies with known mass. The first such experiment was carried out by English. scientist G. Cavendish (1798). Knowing G, it is possible to determine the abs. the value of the mass of the Sun, Earth and other celestial bodies.

The law of gravitation 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 the bodies. For spherical bodies located sufficiently far from each other, law (1) is approximately valid.

In the course of the development of T.'s theory, the idea of ​​direct force interaction between bodies gradually gave way to the idea of ​​a field. Gravity the field in Newton's theory is characterized by potential, where x,y,z- coordinates, t- time, as well as field strength, i.e.
.
Gravitational potential the field created by a set of masses at rest does not depend on time. Gravity several potentials bodies satisfy the principles of superposition, i.e. potential of k.-l. point of their common field is equal to the sum of the potentials of the bodies under consideration.

It is assumed that gravitational the field is described in an inertial coordinate system, i.e. in a coordinate system, a relative 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 absolute acceleration) is, obviously, g.

Point body with mass dm creates gravitational potential
.
A continuous medium distributed in space with density (which may also depend on time) creates gravitational force. potential equal to the sum of the potentials of all elements of the medium. In this case, the field strength is expressed as the vector sum of the intensities 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 when r>R. This justifies the approximation of material points in celestial mechanics, where they usually deal with almost spherical. bodies located, moreover, quite far from each other. The exact Poisnois equation, taking into account the real, asymmetrical distribution of masses, is used, for example, when studying the structure of the Earth using gravimetric methods. T.'s law in the form of the Poisson equation is applied theoretically. study of the structure of stars. In stars, the force of pressure, which varies from point to point, is balanced by the pressure gradient; in rotating stars, centrifugal force is added to the pressure gradient.

Let us note some fundamental features of the classical. theories of T.
1) In the equation of motion of a material body - Newton’s second law of mechanics, ma=F(Where F- active force, a- acceleration acquired by a body), and Newton’s law of gravitation includes the same characteristic of a body - its mass. This implies that the inertial mass of a body and its gravity. mass are equal (for more details, see section 3).

2) Instantaneous gravity value. potential is completely determined by the instantaneous distribution of masses throughout space and the limiting conditions for the potential at infinity. For limited distributions of matter, we accept the condition that it vanishes at infinity (at ). 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 level of motion of material bodies in a given field.

3) Transition in accordance with Galileo’s transformations ( x"=x-vt, t"=t) from one inertial coordinate system to another, moving relative to the first with constant speed v, does not change the Poisson equation and does not change the equation of motion 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 accelerating one 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 the levels of movement. Exactly the same shuttle in the equations of motion arises if in the inertial coordinate system to gravitational. add a term to the potential that linearly depends on the coordinates, i.e. add a uniform field T. Thus, a uniform field T. can be compensated under conditions of accelerated movement.

2. Movement of bodies under the influence of gravity

The most important task of Newtonian celestial mechanics is the phenomenon. the problem of the movement of two point material bodies interacting gravitationally. To solve it, using Newton's law of gravitation, they compose the equation of motion of bodies. The properties of the solutions to these equations are known with complete completeness. Using a well-known solution, it can be established that certain quantities characterizing the system remain constant over time. They are called integrals of motion. Basic integrals of motion (conserved quantities) yavl. energy, impulse, angular momentum of the system. For a two-body system, complete mechanical energy E, equal to the sum of kinetic. energy ( T) and potential energy ( U), is saved:
E=T+U=const,
where is the kinetic energy of two bodies.

In classic In celestial mechanics, potential energy is determined by gravity. interaction between bodies For a pair of bodies, gravitational (potential) energy is equal to:
,
where is gravitational potential created by mass m 2 at the point where the mass is located m 1, a is the potential created by the mass m 1 at the mass location m 2. Zero value U possess bodies separated by an infinitely large distance. Since when 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 gravitational magnitude energy is twice as much avg. kinetic values energy of the particles that make up the system (see). So, for example, for low mass m, rotating in a circular orbit around a central body, the condition of equality of centrifugal force mv 2 /r the force of gravity leads to, i.e. kinetic energy, whereas 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 of T. the field serves the purpose of describing instantaneous interaction at a distance; it does not have its own. degrees of freedom, cannot propagate and emit. It is clear what the concept of gravity is. the field is only approximately valid for sufficiently slow movements of the sources. Taking into account the final speed of gravitational propagation. interactions are carried out in the relativistic theory of T. (see below).

In the nonrelativistic theory of physics, the total mechanical energy of a system of bodies (including the energy of gravitational interaction) must remain unchanged indefinitely. Newton's theory allows for 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. during inelastic collisions of bodies. If bodies are viscous, then their deformations and vibrations when moving in gravity. fields also reduce the energy of a system of bodies by converting energy into heat.

3. Acceleration and gravity

Inert body mass ( m i) is a quantity that characterizes its ability to acquire a particular acceleration under the influence of a given force. Inertial mass is part of Newton's second law of mechanics. Gravity weight ( m g) characterizes the body’s ability to create one or another T. Gravitation field. mass is included in the law of T.

From Galileo's experiments, with the precision with which they were carried out, 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 a 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 the mass is proportional to the inert mass. Counting m i And m g simply coinciding, they find from experiments a specific numerical value of the constant G.

Proportionality of inertia and gravity. masses of bodies of various natures was the subject of research in the experiments of Weng. physics R. Eotvos (1922), American. physicist R. Dicke (1964) and Soviet physicist V.B. Braginsky (1971). It has been tested in the laboratory with high accuracy (with an error

The high accuracy of these experiments makes it possible to evaluate the effect on the mass of various types of binding energy between particles of a body (see). Proportionality of inertia and gravity. mass means that physical. interactions within a body are equally involved in creating its inertia and gravity. wt.

Relative to a coordinate system moving with acceleration a, all free bodies acquire the same acceleration - a. Due to the equality of inertia and gravity. masses, they all acquire the same acceleration relative to the inertial coordinate system under the influence of gravity. fields with intensity g=-a. That is why we can say that from the point of view of the laws of mechanics, a homogeneous gravitational force. the field is indistinguishable from the acceleration field. In a non-uniform gravitational field, compensation of field strength by acceleration in all space at once is impossible. However, the field strength can be compensated by the acceleration of a specially selected coordinate system along the entire trajectory of a body freely moving under the influence of forces T. Such a coordinate system is called. free falling. The phenomenon of weightlessness takes place in it.

Space movement spacecraft (AES) in the Earth's T field can be considered as the movement of a falling coordinate system. The acceleration of the 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 it is practically zero, so they are in weightlessness.

In free fall in inhomogeneous gravity. Field compensation of field strength by acceleration cannot be universal, since the acceleration of neighboring freely falling particles is not exactly the same, i.e. particles have relative acceleration. In space ship, relative accelerations are practically unnoticeable, since their order of magnitude is cm/s 2, where r- distance from the ship to the center of the Earth, - mass of the Earth, x- size of the ship. These accelerations can be neglected and gravity can be assumed. 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 The field can be established by observations of sufficiently high accuracy, but for any given accuracy of observation it is possible to indicate the volume of space in which the field will appear homogeneous.

Relative accelerations manifest themselves, for example, on Earth in the form of ocean tides. The force with which the Moon attracts the Earth is different at different points on the Earth. The parts of the water surface closest to the Moon are attracted stronger than the Earth's center of gravity, and it, in turn, is stronger than the most distant parts of the world's 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 Earth's water shell is deformed so that it stretches out in the form of an ellipsoid along the Earth-Moon line. Due to the rotation of the Earth, tidal humps roll across the surface of the ocean twice a day. A similar but smaller tidal deformation is caused by gravitational inhomogeneity. fields of the Sun.

A. Einstein, based on the equivalence of homogeneous fields of technology and accelerated coordinate systems in mechanics, assumed that such equivalence generally applies to all physical objects without exception. 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 reference frame located in a uniform gravitational field, and in a reference frame moving translationally with acceleration in the absence of gravity. fields. The principle of equivalence played an important role in the construction of Einstein's theory of T.

4. Relativistic mechanics and field theory

Study of el.-magn. phenomena by M. Faraday and D. Maxwell in the second half of the 19th century. led to the creation of the theory of electric magnetism. fields. The conclusions of this theory were confirmed experimentally. Maxwell's equations are non-invariant under Galilean transformations, but are invariant under Lorentz transformations, i.e. the laws of electromagnetism are formulated identically 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 constant speed v in the direction of the axis x, then the Lorentz transformations have the form:
y"=y, z"=z, .
At low speeds () and neglecting members ( v/c) 2 and vx/c 2 these transformations turn into Galilean transformations.

Logical analysis of the contradictions that arose when comparing the conclusions of the electric-magnetic theory. phenomena with classical ideas about space and time, led to the construction of a particular (special) theory of relativity. The decisive step was taken by A. Einstein (1905); the works of the Dutch physicist G. Lorentz and the French played a huge role in its construction. mathematician A. Poincare. The partial theory of relativity requires a revision of classical ideas about space and time. In classic In physics, the time interval between two events (for example, between two flashes of light), as well as the concept of simultaneity of events, have absolute meaning. They do not depend on the movement of the observer. In the partial theory of relativity this is not the case: judgments about time intervals between events and about length segments depend on the movement of the observer (the coordinate system associated with him). These quantities turn out to be relative in approximately the same sense in which they are relative, depending on the location of observers, phenomena. 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 both a period of time dt, and the element of the distance between them:
ds 2 =c 2 dt 2 -dx 2 -dy 2 -dz 2 . (3)
Transition from one inertial system to another, preserving ds 2 unchanged, is carried out precisely 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, , the remaining quantities are equal to zero. The set of quantities is called the metric tensor of flat space-time or the Minkowski world [in the general theory of relativity (GTR) 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 set of ten functions depending on x 0 , x 1 , x 2 , x 3 in the selected coordinate system. Metric. a tensor (or simply a metric) allows you to determine the distance and time interval between events separated by .

Specialist. the theory of relativity establishes the limiting speed of movement of material bodies and, in general, the propagation of interactions. This speed coincides with the speed of light in vacuum. Along with the change in ideas about space and time, special The theory of relativity clarified the concept of mass, momentum, and force. In relativistic mechanics, i.e. in mechanics, invariant under Lorentz transformations, the inertial mass of a body depends on the speed: , where m 0 - bodies. The energy of a body and its momentum are combined into a 4-component energy-momentum vector. For a continuum, you can enter 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.-magn. fields (electrodynamics) are much richer than electrostatics, which is valid only in the limit of slow charge movements. In electrodynamics, electrical energy is combined. and magnetic fields. Taking into account the finite speed of propagation of field changes and the delay in the transfer of interaction leads to the concept of electric magnet. waves, which carry away energy from the radiating system.

Similarly, the relativistic theory of T. turned out to be more complicated than Newton’s. Gravity the field of a moving body has a number of saints similar to the saints of el.-magnetic. fields of a moving charged body in electrodynamics. Gravity the field at a great distance from the bodies depends on the position and movement of the bodies in the past, since gravitational the field propagates at a finite speed. Emission and propagation of gravity becomes possible. waves (see). The relativistic theory of T., as one might have expected, turned out to be nonlinear.

5. Curvature of space-time in general relativity

According to the principle of equivalence, no observations, using any laws of nature, can distinguish the acceleration created by a uniform field T. from the acceleration of a moving coordinate system. In a homogeneous gravitational field, it is possible to achieve zero acceleration of all particles placed in a given region of 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 a clock located in it. The situation is different in non-uniform gravity. a 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 considered only locally inertial. The coordinate system can be considered inertial only in the region where it is permissible to neglect the relative accelerations of particles. Consequently, in an inhomogeneous gravitational field only in a small region of space-time and with limited accuracy can one consider space-time as flat and use f-loy (3) to determine the interval between events.

The impossibility of introducing an inertial coordinate system in an inhomogeneous gravitational system. the field makes all conceivable coordinate systems more or less equal. Gravity level fields must be written so that they are valid in all coordinate systems, without giving preference to any one. of them. Hence the name for the relativistic theory of T. - general theory of relativity.

Gravity fields generated by real bodies, such as the Sun or the Earth, are always inhomogeneous. These are called true or irreducible fields. In such gravity. field, no locally inertial coordinate system can be extended to the entire space-time. This means that the interval ds 2 cannot be reduced to form (3) throughout the space-time continuum, i.e. spacetime cannot be flat. Einstein came up with the radical idea of ​​identifying inhomogeneous gravitational forces. fields with space-time curvature. From these positions, gravity. the field of any body can be considered as a distortion of the geometry of space-time by this body.

Fundamentals of mathematics apparatus of the geometry of 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 is f-la (4). Having a set of functions, we can raise the question of the existence of such coordinate transformations that would translate (5) into (3), i.e. would allow us to check whether space-time is flat. The desired transformations are feasible if and only if a certain tensor, composed of functions, the squares of their first derivatives and second derivatives, is equal to zero. This tensor is called the curvature tensor. In the general case, it is, of course, not equal to zero.

A set of quantities 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, the curvature tensor, expressed through the second derivatives of gravity. potentials, describes tidal accelerations in inhomogeneous gravity. field.

The curvature tensor is a dimensional quantity, its dimension is the square of the reciprocal length. The curvature at each point in space-time corresponds to characteristic lengths - radii of curvature. In a small space-time region surrounding a given point, curved space-time is indistinguishable from flat space-time up to small terms, where l- characteristic size of the area. In this sense, the curvature of the world has the same properties as, say, the curvature of the globe: in small areas it is insignificant. The curvature tensor at a given point cannot be “destroyed” by any coordinate transformations. However, in a certain coordinate system and with a previously known accuracy, the T field in a small region of space-time can be considered absent. In this area, all the laws of physics take on the form that is consistent with the special ones. the theory of relativity. This is how the principle of equivalence, which formed the basis of the theory of theory during its construction, manifests itself.

Metric. The space-time tensor, and in particular the curvature of the world, are accessible to experimental determination. To prove the curvature of the globe, you need to have a small “ideal” scale and use it to measure the distance between fairly distant points on the surface. A comparison of the measured distances will indicate the difference between the real geometry and the Euclidean one. Likewise, the geometry of spacetime 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, by, for example, measuring the frequency shift of light (by determining the gravitational red shift), it is possible in principle to determine the metric. space-time tensor and its curvature.

6. Einstein's equations

By summing the curvature tensor with the metric. a tensor can form a symmetric tensor , which has as many components as the momentum energy tensor of matter, serves as a source of gravity. fields.

Einstein suggested that the equations of gravity should establish a connection between and. In addition, he took into account that in gravity. field, the continuity equation for matter must be satisfied in the same way as the current continuity level in electrodynamics is satisfied. Such equations are carried out automatically if the gravity equation is write the fields like this:
. (6)
This is Einstein’s equation, 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 movement of matter, on the one hand, and geometric. sacred space-time - on the other.

In equations (6) on the left side there are components of the tensor describing the geometry of space-time, and on the right side there are components of the energy-momentum tensor describing the physical. sacred properties of matter and fields (sources of gravitational fields). Quantities are not just functions that describe the gravitational field, but at the same time they are components of the metric space-time tensor.

Einstein wrote that most of his works (special theory of relativity, quantum nature of light) were in line with pressing problems of his time. They would have been done by other scientists with a delay of no more than 2-3 years, if these works had not been done he himself. For GTR, Einstein made an exception and wrote that the relativistic theory of T. might have been delayed for 50 years. This forecast was essentially justified, since it was in the 60s of the 20th century that new general methods of field theory and another approach to the nonlinear theory of thermodynamics arose, based on the concept of a field defined in flat space-time.It was shown that this path leads to the same equations that Einstein arrived at on the basis of geometrics. interpretations by T.

It should be emphasized that it is in astronomy and cosmology that questions arise, in which geometric. approach yavl. preferred. As an example, we can point out cosmological. the theory of a spatially closed Universe, as well as the theory of . Therefore, Einstein's theory, based on geometric. concept, retains its full meaning.

In geometric interpretation of the movement of a material point in gravitational the field represents movement along a 4-dimensional trajectory - geodesic. lines of space-time. In a world with curvature, geodesic. line generalizes the concept of a straight line in Euclidean geometry. The equations of motion of matter contained in Einstein’s equations are reduced to the geodesic equations. lines for point bodies. Bodies (particles), which cannot be considered pointlike, deviate in their motion from the geodesic. lines and experience tidal forces.

7. Weak gravitational fields and observed effects

The T. field of most astronomical objects of phenomena weak. An example would be gravity. Earth's field. In order for a body to leave the Earth forever, it must be given a speed of 11.2 km/s at the Earth’s surface, i.e. speed, small compared to the speed of light. In other words, gravitational The Earth's potential is small compared to the square of the speed of light, which is the phenomenon. criterion of gravitational weakness. fields.

In the weak field approximation, the laws of Newton's 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, is 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 gravitational value. potential and accepted with a frequency at a point with a potential value (where there is exactly the same emitter for frequency comparison), then the equality must be satisfied. Gravity effect The frequency shift of light was predicted by Einstein back in 1911 on the basis of the law of conservation of photon energy in gravitational forces. field. It is reliably established in the spectra of stars, measured with an accuracy of 1% in the laboratory and with an accuracy of up to 1% in space conditions. flight. In the most accurate experiment, a hydrogen maser frequency standard was used, which was installed on the cosmic maser. a rocket that rose to a height of 10 thousand km. Another similar standard was established on Earth. Their frequencies were compared at different altitudes. The results confirmed the predicted frequency change.

When passing near a gravitating body, an electric magnet. the signal experiences a relativistic delay in propagation time. According to its physical In nature, this effect is similar to the previous one. Based on radio observations of planets and especially interplanetary cosmic ones. 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 is the phenomenon. 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 flow. fields. According to Newtonian celestial mechanics, the motion of planets around the Sun is described by the equation of an ellipse: , where p=a(1-e 2) - orbit parameter, a- semi-major axis, 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. The orbit rotates in the direction of motion by an angle. For Mercury, the relativistic rotation angle is per century. The fact that the rotation angle accumulates over time makes it easier to observe this effect. During one revolution, the rotation angle of the major axis of the orbit is so insignificant ~ 0.1", that its detection is significantly complicated by the bending of light rays within the Solar System. Nevertheless, modern radar data confirm the relativistic effect of the shift of the perihelion of Mercury with an accuracy of 1%.

The listed effects are called. classic. It is also possible to check other predictions of general relativity (for example, precession of the gyroscope axis) in weak gravity. 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 double stars. Thus, in a binary system including the pulsar PSR 1913+16, the effect of a perihelion shift is observed, which made it possible to determine the total mass of the system components with an accuracy of 1%.

8. Gravity and quantum physics

Einstein's equations include classical gravity. field characterized by metric components. tensor, and the energy-momentum tensor of matter. To describe the motion of gravitating bodies, the quantum nature of matter, as a rule, is not important. This happens because they usually deal with gravitational forces. 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 in conditions when the quantum properties of particles interacting with gravity become significant. field, and the quantum properties of gravity itself. fields.

Quantum processes involving gravitational forces. fields are certainly important in space (see,) and, perhaps, will become available for study also in laboratory conditions. The unification of the theory of quantum theory with quantum theory is one of the most important problems in physics, which has already begun to be solved.

Under normal conditions, the influence of gravity. fields on quantum systems are extremely small. To excite an atom externally. gravitational field, relative acceleration created by gravitational field at a distance of “hydrogen atom radius” cm and equal to , should be comparable to the acceleration with which an electron moves in an atom, . (Here is the radius of curvature of the Earth’s gravitational field, equal to: cm.) In gravitational field of the Earth with a margin of 10 19 this relationship is not satisfied, therefore atoms in 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 influence of gravity. margins may be noticeable. It is on this principle that certain modern ones are based. assumptions for gravitational detection. waves

In specially created (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 electric 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 with a noticeable probability can change and become equal N+2 or N-2. In other words, energy transitions are possible. level, and they are, in principle, detectable.

The role of intense gravitational forces is especially important. fields. Such fields probably existed at the beginning of the expansion of the Universe, near the cosmological. singularities and can arise in the later stages of gravity. collapse. The high intensity of these fields is manifested in the fact that they are capable of producing observable effects (the creation of pairs of particles) even in the absence of atoms, real particles or photons. These fields have an effective effect on the physical. vacuum - physical fields in the lowest energy state. In a vacuum, due to fluctuations of quantized fields, the so-called. virtual, really unobservable particles. If the intensity is ext. gravitational field is so great that at distances characteristic of quantum fields and particles, it is capable of producing work that exceeds the energy of a pair of particles, then as a result, the birth of a pair of particles can occur - their transformation from a virtual pair into a real one. A necessary condition for this process should be the comparability of the characteristic radius of curvature, which describes the intensity of gravitational forces. fields, with Compton wavelength, comparable to particles with rest mass m. A similar condition must be satisfied for massless particles so that the process of birth of a pair of quanta with energy is possible. In the above example of a cavity containing an el.-magn. field, this process is similar to a transition with a probability comparable to unity from the vacuum state N=0 into a state describing two quanta, N=2. In ordinary gravitational In 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 low-mass black holes (according to) the works of English. scientist S. Hawking).

Intense gravitational fields that can significantly influence the zero fluctuations of other physics. fields, should equally effectively influence their own zero fluctuations. If the process of birth of physical quanta is possible. fields, then with the same probability (and in some cases with even greater probability) the process of birth of quanta of the gravitational force itself should be possible. fields - gravions. A rigorous and exhaustive examination of such processes is possible only on the basis of quantum theory T. Such a theory has not yet been created. Application to gravity The field of the same ideas and methods that led to the successful construction of quantum electrodynamics is encountering serious difficulties. It is not yet clear what path the development of quantum theory of T will take. One thing is certain - the most important way to test such theories will be to search for phenomena predicted by the theory in space.

The motion of a body under the influence of gravity is one of the central topics in dynamic physics. Even an ordinary school student knows that the dynamics section is based on three. Let's try to analyze this topic thoroughly, and an article describing each example in detail will help us make the study of the movement of a body under the influence of gravity as useful as possible.

A little history

People watched with curiosity various phenomena occurring in our lives. For a long time, humanity could not understand the principles and structure of many systems, but a long journey of studying the world around us led our ancestors to a scientific revolution. Nowadays, when technology is developing at an incredible speed, people hardly think about how certain mechanisms work.

Meanwhile, our ancestors were always interested in the mysteries of natural processes and the structure of the world, looked for answers to the most complex questions and did not stop studying until they found answers to them. For example, the famous scientist Galileo Galilei asked the questions back in the 16th century: “Why do bodies always fall down, what force attracts them to the ground?” In 1589, he carried out a series of experiments, the results of which turned out to be very valuable. He studied in detail the patterns of free fall of various bodies, dropping objects from the famous tower in the city of Pisa. The laws he derived were improved and described in more detail by formulas by another famous English scientist, Sir Isaac Newton. It is he who owns the three laws on which almost all modern physics is based.

The fact that the patterns of body movement described more than 500 years ago are still relevant today means that our planet is subject to unchanging laws. Modern man needs to at least superficially study the basic principles of the world.

Basic and auxiliary concepts of dynamics

In order to fully understand the principles of such a movement, you should first become familiar with some concepts. So, the most necessary theoretical terms:

• Interaction is the influence of bodies on each other, during which a change occurs or the beginning of their movement relative to each other. There are four types of interaction: electromagnetic, weak, strong and gravitational.
• Speed ​​is a physical quantity that indicates the speed with which a body moves. Speed ​​is a vector, meaning it not only has a value, but also a direction.
• Acceleration is the quantity that shows us the rate of change in the speed of a body over a period of time. She is also
• The trajectory of the path is a curve, and sometimes a straight line, which the body outlines when moving. With uniform rectilinear motion, the trajectory can coincide with the displacement value.
• The path is the length of the trajectory, that is, exactly as much as the body has traveled in a certain amount of time.
• An inertial reference frame is a medium in which Newton's first law is satisfied, that is, the body retains its inertia, provided that all external forces are completely absent.

The above concepts are quite enough to correctly draw or imagine in your head a simulation of the movement of a body under the influence of gravity.

What does strength mean?

Let's move on to the main concept of our topic. So, force is a quantity, the meaning of which is the impact or influence of one body on another quantitatively. And gravity is the force that acts on absolutely every body located on the surface or near our planet. The question arises: where does this very power come from? The answer lies in the law of universal gravitation.

What is gravity?

Any body from the Earth is influenced by the gravitational force, which imparts some acceleration to it. The force of gravity always has a vertical direction downwards, towards the center of the planet. In other words, gravity pulls objects toward the Earth, which is why objects always fall down. It turns out that gravity is a special case of the force of universal gravitation. Newton derived one of the main formulas for finding the force of attraction between two bodies. It looks like this: F = G * (m 1 x m 2) / R 2.

What is the acceleration due to gravity?

A body that is released from a certain height always flies down under the influence of gravity. The movement of a body under the influence of gravity vertically up and down can be described by equations, where the main constant will be the acceleration value "g". This value is due solely to the force of gravity, and its value is approximately 9.8 m/s 2 . It turns out that a body thrown from a height without an initial speed will move down with an acceleration equal to the “g” value.

Body motion under the influence of gravity: formulas for solving problems

The basic formula for finding the force of gravity is as follows: F gravity = m x g, where m is the mass of the body on which the force acts, and “g” is the acceleration of gravity (to simplify problems, it is usually considered equal to 10 m/s 2) .

There are several more formulas used to find one or another unknown when a body moves freely. So, for example, in order to calculate the path traveled by a body, it is necessary to substitute known values ​​into this formula: S = V 0 x t + a x t 2 / 2 (the path is equal to the sum of the products of the initial speed multiplied by time and acceleration by the square of time divided on 2).

Equations for describing the vertical motion of a body

The vertical movement of a body under the influence of gravity can be described by an equation that looks like this: x = x 0 + v 0 x t + a x t 2 / 2. Using this expression, you can find the coordinates of the body at a known moment in time. You just need to substitute the quantities known in the problem: initial location, initial speed (if the body was not just released, but pushed with some force) and acceleration, in our case it will be equal to acceleration g.

In the same way, you can find the speed of a body that moves under the influence of gravity. The expression for finding an unknown quantity at any moment of time: v = v 0 + g x t (the value of the initial speed can be equal to zero, then the speed will be equal to the product of the acceleration of gravity and the time value during which the body moves).

The movement of bodies under the influence of gravity: problems and methods for solving them

When solving many problems related to gravity, we recommend using the following plan:

1. To determine a convenient inertial reference system for yourself, it is usually customary to choose the Earth, because it meets many of the requirements for ISO.
2. Draw a small drawing or picture that shows the main forces acting on the body. The motion of a body under the influence of gravity involves a sketch or diagram that shows in which direction the body moves when subjected to an acceleration equal to g.
3. The direction for projecting the forces and the resulting accelerations must then be selected.
4. Write down unknown quantities and determine their direction.
5. Finally, using the problem solving formulas above, calculate all the unknown quantities by substituting the data into the equations to find the acceleration or distance traveled.

Ready solution to an easy task

When we are talking about such a phenomenon as the movement of a body under the influence of what is the most practical way to solve a given problem, it can be difficult. However, there are several tricks, using which you can easily solve even the most difficult task. So, let's look at live examples of how to solve this or that problem. Let's start with an easy to understand problem.

A certain body was released from a height of 20 m without an initial speed. Determine how long it will take it to reach the surface of the earth.

Solution: we know the path traveled by the body, we know that the initial speed was equal to 0. We can also determine that only the force of gravity acts on the body, it turns out that this is the movement of the body under the influence of gravity, and therefore we should use this formula: S = V 0 x t + a x t 2 /2. Since in our case a = g, then after some transformations we obtain the following equation: S = g x t 2 / 2. Now all that remains is to express time through this formula, we find that t 2 = 2S / g. Let's substitute the known values ​​(we assume that g = 10 m/s 2) t 2 = 2 x 20 / 10 = 4. Therefore, t = 2 s.

So, our answer: the body will fall to the ground in 2 seconds.

The trick to quickly solving the problem is as follows: you can notice that the described movement of the body in the above problem occurs in one direction (vertically downward). It is very similar to uniformly accelerated motion, since no force acts on the body except gravity (we neglect the force of air resistance). Thanks to this, you can use an easy formula to find the path during uniformly accelerated motion, bypassing the images of drawings with the arrangement of forces acting on the body.

An example of solving a more complex problem

Now let's see how best to solve problems on the movement of a body under the influence of gravity, if the body does not move vertically, but has a more complex nature of movement.

For example, the following task. An object of mass m moves with unknown acceleration down an inclined plane whose coefficient of friction is equal to k. Determine the value of acceleration that occurs during the movement of a given body if the angle of inclination α is known.

Solution: You should use the plan described above. First of all, draw a drawing of an inclined plane depicting the body and all the forces acting on it. It turns out that three components act on it: gravity, friction and the support reaction force. The general equation of resultant forces looks like this: Friction F + N + mg = ma.

The main highlight of the problem is the condition of inclination at an angle α. When ox and axis oy it is necessary to take into account this condition, then we get the following expression: mg x sin α - F friction = ma (for the ox axis) and N - mg x cos α = F friction (for the oy axis).

Friction F is easy to calculate using the formula for finding the friction force, it is equal to k x mg (friction coefficient multiplied by the product of body mass and gravitational acceleration). After all the calculations, all that remains is to substitute the found values ​​into the formula, and you will get a simplified equation for calculating the acceleration with which a body moves along an inclined plane.