In simple terms: The speed of movement of a body relative to a fixed frame of reference is equal to vector sum the speed of this body relative to the moving frame of reference and the speed of the moving frame of reference itself relative to the stationary frame.

Examples

1. The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed with which the record carries it due to its rotation.
2. If a person walks along the corridor of a carriage at a speed of 5 kilometers per hour relative to the carriage, and the carriage moves at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of the train, and at a speed of 50 - 5 = 45 kilometers per hour when he goes to reverse direction. If a person in the carriage corridor moves relative to the Earth at a speed of 55 kilometers per hour, and a train at a speed of 50 kilometers per hour, then the speed of the person relative to the train is 55 - 50 = 5 kilometers per hour.
3. If the waves move relative to the shore at a speed of 30 kilometers per hour, and the ship also moves at a speed of 30 kilometers per hour, then the waves move relative to the ship at a speed of 30 - 30 = 0 kilometers per hour, that is, they become motionless.

Relativistic mechanics

In the 19th century, classical mechanics was faced with the problem of extending this rule for adding velocities to optical (electromagnetic) processes. Essentially, there was a conflict between two ideas of classical mechanics, transferred to new area electromagnetic processes.

For example, if we consider the example with waves on the surface of water from the previous section and try to generalize to electromagnetic waves, then there will be a contradiction with observations (see, for example, Michelson’s experiment).

The classic rule for adding velocities corresponds to the transformation of coordinates from one system of axes to another system moving relative to the first without acceleration. If with such a transformation we preserve the concept of simultaneity, that is, we can consider two events simultaneous not only when they are registered in one coordinate system, but also in any other inertial system, then the transformations are called Galilean. In addition, with Galilean transformations, the spatial distance between two points - the difference between their coordinates in one inertial frame - is always equal to their distance in another inertial frame.

The second idea is the principle of relativity. Being on a ship moving uniformly and rectilinearly, its movement cannot be detected by any internal mechanical effects. Does this principle apply to optical effects? Is it not possible to detect the absolute motion of a system by the optical or, what is the same thing, electrodynamic effects caused by this motion? Intuition (related quite clearly to the classical principle of relativity) says that absolute motion cannot be detected by any kind of observation. But if light propagates at a certain speed relative to each of the moving inertial systems, then this speed will change when moving from one system to another. This follows from the classical rule of adding velocities. In mathematical terms, the speed of light will not be invariant under Galilean transformations. This violates the principle of relativity, or rather, does not allow the principle of relativity to be extended to optical processes. Thus, electrodynamics destroyed the connection between two seemingly obvious provisions classical physics- rules for adding velocities and the principle of relativity. Moreover, these two provisions in relation to electrodynamics turned out to be incompatible.

The theory of relativity provides the answer to this question. It expands the concept of the principle of relativity, extending it to optical processes. The rule for adding velocities is not canceled completely, but is only refined for high velocities using the Lorentz transformation:

It can be noted that in the case when, the Lorentz transformations turn into Galilean transformations. The same thing happens when . This suggests that special relativity coincides with Newtonian mechanics either in a world with infinite speed of light or at speeds small compared to the speed of light. The latter explains how these two theories are combined - the first is a refinement of the second.

see also

Literature

• B. G. Kuznetsov Einstein. Life, death, immortality. - M.: Science, 1972.
• Chetaev N. G. Theoretical mechanics. - M.: Science, 1987.

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See what the “Rule of Addition of Velocities” is in other dictionaries:

When considering complex motion (that is, when a point or body moves in one reference system, and it moves relative to another), the question arises about the connection between velocities in 2 reference systems. Contents 1 Classical mechanics 1.1 Examples ... Wikipedia

Geometric construction expressing the law of addition of velocities. Rule P. s. is that during complex movement (see. Relative motion) absolute speed points are represented as the diagonal of a parallelogram built on... ...

Postage stamp with the formula E = mc2, dedicated to Albert Einstein, one of the creators of SRT. Special theory ... Wikipedia

A physical theory that considers space-time patterns that are valid for any physical. processes. The universality of spatio-temporal svs, considered by O.t., allows us to speak of them simply as svs of space... ... Physical encyclopedia

- [from Greek. mechanike (téchne) the science of machines, the art of building machines], the science of mechanical movement material bodies and the interactions between bodies that occur during this process. Under mechanical movement understand change over time... ... Great Soviet Encyclopedia Mathematical Encyclopedia

A; m. 1. Normative act, resolution supreme body state power, adopted in accordance with the established procedure and having legal force. Labor Code. Z. o social security. Z. o military duty. Z. about the market valuable papers.… … encyclopedic Dictionary

Lorentz transformations give us the opportunity to calculate the change in the coordinates of an event when moving from one reference system to another. Let us now pose the question of how, when the reference system changes, the speed of the same body will change?

IN classical mechanics, as is known, the speed of a body is simply added to the speed of the reference system. Now we will see that in the theory of relativity, speed is transformed according to a more complex law.

We will again limit ourselves to considering the one-dimensional case. Let two reference systems S and S` “observe” the motion of some body, which moves uniformly and rectilinearly parallel to the axes X And x` both reference systems. Let the speed of the body, measured by the reference system S, There is And; the speed of the same body, measured by the system S`, will be denoted by and` . Letter v We will continue to denote the speed of the system S` regarding S.

Let us assume that two events occur with our body, the coordinates of which in the system S essence x 1 ,t 1 , AndX 2 , t 2 . Coordinates of the same events in the system S` let them be x` 1, t` 1 ; x` 2 , t` 2 . But the speed of a body is the ratio of the distance traveled by the body to the corresponding period of time; therefore, to find the speed of a body in one and the other frame of reference, you need the difference spatial coordinates divide both events by the difference in time coordinates

which can, as always, be obtained from the relativistic one if the speed of light is considered infinite. The same formula can be written as

For small, “ordinary” speeds, both formulas—relativistic and classical—give almost identical results, which the reader can easily verify if desired. But at speeds close to the speed of light, the difference becomes very noticeable. So, if v=150,000 km/sec, u`=200 000 km/Withek, km/sec the relativistic formula gives u = 262 500 km/Withek.

S at speed v = 150,000 km/sec. S` gives the result u =200 000 km/sec. km/Withek.


km/sec, and the second - 200,000 km/sec, km.

With. It is not difficult to prove this statement quite strictly. It's really easy to check.

For small, “ordinary” speeds, both formulas—relativistic and classical—give almost identical results, which the reader can easily verify if desired. But at speeds close to the speed of light, the difference becomes very noticeable. So, if v=150,000 km/sec, u`=200 000 km/Withek, then instead of the classical result u = 350,000 km/sec the relativistic formula gives u = 262 500 km/Withek. According to the meaning of the formula for adding speeds, this result means the following.

Let the reference system S` move relative to the reference system S at speed v = 150,000 km/sec. Let a body move in the same direction, and its speed is measured by the reference system S` gives results u` =200 000 km/sec. If we now measure the speed of the same body using the reference frame S, we get u=262,500 km/Withek.


It should be emphasized that the formula we obtained is intended specifically for recalculating the velocity of the same body from one reference system to another, and not at all for calculating the “speed of approach” or “removal” of two bodies. If we observe two bodies moving towards each other from the same reference frame, and the speed of one body is 150,000 km/sec, and the second - 200,000 km/sec, then the distance between these bodies will decrease by 350,000 every second km. The theory of relativity does not abolish the laws of arithmetic.

The reader has already understood, of course, that by applying this formula to speeds not exceeding the speed of light, we will again obtain a speed not exceeding With. It is not difficult to prove this statement quite strictly. Indeed, it is easy to check that the equality holds

Because u` ≤ с And v < c, then on the right side of the equality the numerator and denominator, and with them the entire fraction, are non-negative. That's why square bracket less than one, and therefore and ≤ c .
If And` = With, then and and=With. This is nothing more than the law of the constancy of the speed of light. One should not, of course, consider this conclusion as “proof” or at least “confirmation” of the postulate of the constancy of the speed of light. After all, from the very beginning we proceeded from this postulate and it is not surprising that we came to a result that does not contradict it, in otherwise this postulate would be refuted by proof by contradiction. At the same time, we see that the law of addition of velocities is equivalent to the postulate of the constancy of the speed of light; each of these two statements logically follows from the other (and the remaining postulates of the theory of relativity).

When deriving the law of addition of velocities, we assumed that the speed of the body is parallel relative speed reference systems. This assumption could not be made, but then our formula would relate only to that component of the velocity that is directed along the x axis, and the formula should be written in the form

Using these formulas we will analyze the phenomenon aberrations(see § 3). Let's limit ourselves to the simplest case. Let some luminary in the reference system S motionless, let, further, the reference system S` moves relative to the system S with speed v and let the observer, moving with S`, receive rays of light from the star just at the moment when it is exactly above his head (Fig. 21). Velocity components of this beam in the system S will
u x = 0, u y = 0, u x = -c.

For the reference frame S` our formulas give
u` x = -v, u` y = 0,
u` z = -c√ (1 - v 2 /c 2 )
We get the tangent of the angle of inclination of the beam to the z` axis if we divide and`X on u` z:
tan α = and`X / and`z = (v/c) / √(1 - v 2 /c 2)

If the speed v is not very large, then we can apply the approximate formula known to us, with the help of which we obtain
tan α = v/c + 1/2*v 2 /c 2 .
The first term is a well-known classical result; the second term is the relativistic correction.

The Earth's orbital speed is approximately 30 km/sec, So (v/ c) = 1 0 -4 . For small angles, the tangent is equal to the angle itself, measured in radians; since a radian contains in round 200,000 arcseconds, we obtain for the aberration angle:
α = 20°
The relativistic correction is 20,000,000 times smaller and lies far beyond the accuracy of astronomical measurements. Due to aberration, stars annually describe ellipses in the sky with a semi-major axis of 20".

When we look at a moving body, we see it not where it is in this moment, but where it was a little earlier, because the light takes some time to reach our eyes from the body. From the point of view of the theory of relativity, this phenomenon is equivalent to aberration and is reduced to it when passing to the frame of reference in which the body in question is motionless. Based on this simple consideration, we can obtain the aberration formula in a completely elementary way, without resorting to the relativistic law of addition of velocities.

Let our luminary move in parallel earth's surface from right to left (Fig. 22). When it arrives at the point A, an observer located exactly below him at point C sees him still at point IN. If the speed of the star is equal v, and the period of time during which it passes the segment AIN, equals Δt, That

AB =Δt ,
B.C. = cΔt ,

sinα = AB/BC = v/c.

But then, according to trigonometry formula,

Q.E.D. Note that in classical kinematics these two points of view are not equivalent.

Also interesting next question. As is known, in classical kinematics velocities are added according to the parallelogram rule. We replaced this law with another, more complex one. Does this mean that in the theory of relativity speed is no longer a vector?

Firstly, the fact that u≠u`+ v (we denote vectors by bold letters), in itself does not provide grounds to deny the vector nature of speed. From two given vectors, the third vector can be obtained not only by adding them, but, for example, by vector multiplication, and in general in countless ways. It does not follow from anywhere that when the reference system changes, the vectors and` And v must exactly add up. Indeed, there is a formula expressing And through and` And v using vector calculus operations:

In this regard, it should be admitted that the name “law of addition of velocities” is not entirely apt; it is more correct to speak, as some authors do, not about addition, but about the transformation of speed when changing the reference system.

Secondly, in the theory of relativity it is possible to indicate cases when the velocities still add up vectorially. Let, for example, the body move for a certain period of time Δt with speed u 1, and then - the same period of time at a speed u 2. This complex movement can be replaced by movement with constant speed u = u 1+ u 2 . Here's the speed u 1 and u 2 add up like vectors, according to the parallelogram rule; the theory of relativity does not make any changes here.
In general, it should be noted that most of the “paradoxes” of the theory of relativity are connected in one way or another with a change in the frame of reference. If we consider phenomena in the same frame of reference, then the changes in their patterns introduced by the theory of relativity are far from being as dramatic as is often thought.

Let us also note that a natural generalization of the usual 3D vectors in the theory of relativity, vectors are four-dimensional; when the reference system changes, they are transformed according to the Lorentz formulas. In addition to three spatial components, they have a temporal component. In particular, one can consider four dimensional vector speed. The spatial “part” of this vector, however, does not coincide with the usual three-dimensional speed, and in general, four-dimensional speed is noticeably different in its properties from three-dimensional. In particular, the sum of two four-dimensional velocities will not, generally speaking, be a velocity.

12.2. Postulates of SRT

12.2.1. Relativistic law of addition of velocities

Relativistic theory is also called special theory of relativity and is based on two postulates formulated by A. Einstein in 1905.

First postulate special theory relativity (SRT) is called the principle of relativity: all laws of physics are invariant with respect to transition from one inertial system reference to another, i.e. no experiments (mechanical, electrical, optical) carried out inside a given ISO make it possible to detect whether this ISO is at rest or moves uniformly and in a straight line.

The first postulate extends Galileo's mechanical principle of relativity to any physical processes.

The second postulate of the special theory of relativity (STR) is called principle of invariance of the speed of light: the speed of light in a vacuum does not depend on the speed of the light source or observer and is the same in all ISOs.

The second postulate states that the constancy of the speed of light is a fundamental property of nature.

Lorentz transformations(1904) allow us to obtain the values ​​of three spatial and one time coordinates when moving from one inertial frame (x, y, z, t) to another (x′, y′, z′, t′) moving in the positive direction coordinate axis Ox with relativistic speed u → :

x = x ′ + u t ′ 1 − β 2 , y = y ′, z = z ′, t = t ′ + u x ′ / c 2 1 − β 2 ,

where β = u/c; c is the speed of light in vacuum, c = 3.0 ⋅ 10 8 m/s.

Practical value has to solve problems law of addition of speeds, written as

v ′ x = v x − u x 1 − u x v x c 2 ,

where the values ​​v ′ x, u x, v x are projections of velocities onto the selected coordinate axis Ox:

• v ′ x - relative speed of relativistic particles;
• u x - particle speed, chosen for the reference system, relative to a stationary observer;
• v x - the speed of another particle relative to the same stationary observer.

For calculation relative speed of motion of two relativistic particles It is advisable to use the following algorithm:

1) choose the direction of the coordinate axis Ox along the movement of one of the relativistic particles;

2) associate the frame of reference with one of the particles, designate its speed u → ; the speed of the second particle relative to a stationary observer is denoted by v → ;

3) write down the projections of the velocities u → and v → onto the selected coordinate axis:

• when a particle moves in the positive direction of the Ox axis, the sign of the velocity projection is considered positive;
• when a particle moves in the negative direction of the Ox axis, the sign of the velocity projection is considered negative;

v ′ x = v x − u x 1 − u x v x c 2 ;

5) write the module of the relative velocity of relativistic particles in the form

v rel = | v ′ x | .

Example 1. A rocket moving away from the Earth at a speed of 0.6c (c is the speed of light) sends a light signal in the direction opposite to its speed. The signal is recorded by an observer on Earth. Find the speed of this signal relative to an observer on earth.

Solution . According to the second postulate of STR, the speed of light in a vacuum does not depend on the speed of the light source or the observer.

Therefore, the speed of the signal sent by the rocket relative to an observer on earth is equal to the speed of light:

vrel = c,

where c is the speed of light in vacuum, c = 3.0 ⋅ 10 8 m/s.

Example 2. At the moment of departure from the accelerator, a radioactive nucleus ejected an electron in the direction of its movement. The magnitudes of the velocities of the nucleus and electron relative to the accelerator are 0.40c and 0.70c, respectively (c is the speed of light in vacuum, c ≈ 3.00 ⋅ 10 8 m/s). Determine the velocity modulus of the nucleus relative to the electron. How will the velocity modulus of the nucleus relative to the electron change if the nucleus ejects an electron in the opposite direction?

Solution . In the first case, the nucleus ejects an electron in the direction of its motion. In Fig. a shows a nucleus that has ejected an electron along the direction of its movement, and the directions of the coordinate axis Ox, the velocity of the nucleus v → poison, the velocity of the electron v → el are indicated.

To calculate the relative speed of movement of two relativistic particles, we will use an algorithm.

1. Let us choose the direction of the coordinate axis Ox in the direction of the velocity of the electron and nucleus.

u → = v → el;

v → = v → poison.

u x = 0.40c ; v x = 0.70c.

v ′ x = v x − u x 1 − u x v x c 2 = 0.70 c − 0.40 c 1 − 0.40 c ⋅ 0.70 c c 2 = 0.30 c 1 − 0.40 c ⋅ 0.70 c c 2 = 1.25 ⋅ 10 8 m/s.

5. The projection of the relative velocity has a positive sign, therefore the magnitude of the velocity of the nucleus relative to the electron is equal to the found projection:

v rel = v ′ x = 1.25 ⋅ 10 8 m/s.

In the second case, the nucleus ejects an electron in the direction opposite to the speed of its movement. In Fig. b shows a nucleus that has ejected an electron opposite to the direction of its movement, and the directions of the coordinate axis Ox, the speed of the nucleus v → poison, the speed of the electron v → electron are indicated.

We will also use the algorithm for the calculation.

1. Let us choose the direction of the coordinate axis Ox in the direction of the electron velocity.

2. Let us associate the reference frame with the electron, and denote its speed relative to the accelerator

u → = v → el;

speed of the core relative to the accelerator -

v → = v → poison.

3. Let us write down the projections of the velocities u → and v → onto the selected coordinate axis:

u x = 0.40s; v x = −0.70c .

4. Calculate the projection of the relative velocity of particles using the formula

v ′ x = v x − u x 1 − u x v x c 2 = − 0.70 c − 0.40 c 1 − 0.40 c ⋅ (− 0.70) c c 2 =

= − 1.1 ⋅ 3.00 ⋅ 10 8 1 − 0.40 s ⋅ (− 0.70) s c 2 = − 2.58 ⋅ 10 8 m/s.

5. The relative velocity projection has negative sign, therefore the modulus of the velocity of the nucleus relative to the electron equal to modulus found projection:

v rel = | v ′ x | = 2.58 ⋅ 10 8 m/s.

The modulus of the relative velocity of particles increases by 2.58 times.

We said that the speed of light is maximum possible speed signal propagation. But what happens if light is emitted by a moving source in the direction of its speed? V? According to the law of addition of speeds, following from Galileo's transformations, the speed of light should be equal to c + V. But in the theory of relativity this is impossible. Let's see what law of velocity addition follows from the Lorentz transformations. To do this, we write them for infinitesimal quantities:

By determining the speed, its components in the reference frame K are found as the ratio of the corresponding movements to time intervals:

The speed of an object in a moving reference frame is determined similarly K", only spatial distances and time intervals must be taken relative to this system:

Therefore, dividing the expression dx to the expression dt, we get:

Dividing the numerator and denominator by dt", we find a connection x-velocity component in different systems reference, which differs from the Galilean rule for adding velocities:

In addition, unlike classical physics, the velocity components orthogonal to the direction of motion also change. Similar calculations for other velocity components give:

Thus, formulas for the transformation of velocities in relativistic mechanics are obtained. Formulas inverse conversion are obtained by replacing primed values ​​with unprimed ones and vice versa and replacing V on –V.

Now we can answer the question posed at the beginning this section. Let at the point 0" moving reference frame K" a laser is installed that sends a pulse of light in the positive axis direction 0"x". What will be the speed of the impulse for a stationary observer in the reference frame TO? In this case the speed light pulse in the reference system TO" has components

Applying the law of relativistic addition of velocities, we find for the components of the momentum velocity relative to the stationary system TO :

We find that the speed of the light pulse in the stationary reference frame relative to which the light source is moving is equal to

The same result will be obtained in any direction of propagation of the pulse. This is natural, since the independence of the speed of light from the movement of the source and observer is inherent in one of the postulates of the theory of relativity. The relativistic law of addition of velocities is a consequence of this postulate.

Indeed, when the speed of movement of the moving frame of reference V<<c, Lorentz transformations turn into Galilean transformations, we obtain the usual law of addition of velocities

In this case, the passage of time and the length of the ruler will be the same in both reference systems. Thus, the laws of classical mechanics apply if the speed of objects is much less than the speed of light. The theory of relativity did not erase the achievements of classical physics, it established the framework of their validity.

Example. Body with speed v 0 collides perpendicularly with a wall moving towards it at speed v. Using formulas for relativistic addition of velocities, we find the speed v 1 body after the rebound. The impact is absolutely elastic, the mass of the wall is much greater than the mass of the body.

Let us use formulas expressing the relativistic law of addition of velocities.

Let's direct the axis X along the initial speed of the body v 0 and connect the reference system K" with a wall. Then v x= v 0 and V= –v. In the reference frame associated with the wall, the initial velocity v" 0 body is equal

Let us now return back to the laboratory frame of reference TO. Substituting into the relativistic law of addition of velocities v" 1 instead v"x and considering again V = –v, we find after transformations:

Relativistic law of addition of velocities.

Let us consider the movement of a material point in the K’ system with speed u. Let us determine the speed of this point in the system K if the system K’ moves with speed v. Let us write down the projections of the point’s velocity vector relative to the systems K and K’:

K: u x =dx/dt, u y =dy/dt, u z =dz/dt; K’: u x ’=dx’/dt’, u y ’ =dy’/dt’, u’ z =dz’/dt’.

Now we need to find the values ​​of the differentials dx, dy, dz and dt. Differentiating the Lorentz transformations, we obtain:

, , , .

Now we can find the velocity projections:

, ,
.

From these equations it is clear that the formulas connecting the velocities of a body in different reference systems (the laws of addition of velocities) differ significantly from the laws of classical mechanics. At speeds small compared to the speed of light, these equations turn into classical equations for adding speeds.

6. 5. The basic law of the dynamics of a relativistic particle. @

The mass of relativistic particles, i.e. particles moving at speeds v ~ c is not constant, but depends on their speed: . Here m 0 is the rest mass of the particle, i.e. mass measured in the frame of reference relative to which the particle is at rest. This dependence has been confirmed experimentally. Based on it, all modern charged particle accelerators (cyclotron, synchrophasotron, betatron, etc.) are calculated.

From Einstein's principle of relativity, which asserts the invariance of all laws of nature when moving from one inertial frame of reference to another, follows the condition for the invariance of physical laws with respect to Lorentz transformations. Newton's fundamental law of dynamics F=dP/dt=d(mv)/dt also turns out to be invariant with respect to Lorentz transformations if it contains the time derivative of the relativistic momentum on the right.

The basic law of relativistic dynamics has the form: ,

and is formulated as follows: the rate of change of the relativistic momentum of a particle moving at a speed close to the speed of light is equal to the force acting on it. At speeds much lower than the speed of light, the equation we obtained becomes the fundamental law of dynamics of classical mechanics. The basic law of relativistic dynamics is invariant with respect to Lorentz transformations, but it can be shown that neither acceleration, nor force, nor momentum are invariant quantities in themselves. Due to the homogeneity of space in relativistic mechanics, the law of conservation of relativistic momentum is satisfied: the relativistic momentum of a closed system does not change over time.

In addition to all the listed features, the main and most important conclusion of the special theory of relativity is that space and time are organically interconnected and form a single form of existence of matter.

6. 6. Relationship between mass and energy. Law of conservation of energy in relativistic mechanics. @

Exploring the consequences of the fundamental law of relativistic dynamics, Einstein came to the conclusion that the total energy of a moving particle is equal to . From this equation it follows that even a stationary particle (when b = 0) has energy E 0 = m 0 c 2, this energy is called rest energy (or self-energy).

So, the universal dependence of the total energy of a particle on its mass: E = mс 2. This is a fundamental law of nature - the law of the relationship between mass and energy. According to this law, a mass at rest has a huge supply of energy and any change in mass Δm is accompanied by a change in the total energy of the particle ΔE=c 2 Δm.

For example, 1 kg of river sand should contain 1×(3.0∙10 8 m/s) 2 =9∙10 16 J of energy. This is double the weekly energy consumption in the United States. However, most of this
energy is inaccessible, since the law of conservation of matter requires that the total number of baryons (the so-called elementary particles - neutrons and protons) in any closed system remains constant. It follows that the total mass of baryons does not change and, accordingly, it cannot be converted into energy.

But inside atomic nuclei, neutrons and protons, in addition to rest energy, have a large interaction energy with each other. In a number of processes such as nuclear fusion and fission, part of this potential interaction energy can be converted into additional kinetic energy of particles obtained in reactions. This transformation serves as the source of energy for nuclear reactors and atomic bombs.

The correctness of Einstein's relation can be proven using the example of the decay of a free neutron into a proton, electron and neutrino (with zero rest mass): n → p + e - + ν. In this case, the total kinetic energy of the final products is equal to 1.25∙10 -13 J. The rest mass of the neutron exceeds the total mass of the proton and electron by 13.9∙10 -31 kg. This decrease in mass should correspond to the energy ΔE=c 2 Δm=(13.9∙10 -31)(3.0∙10 8) 2 =1.25∙10 -15 J. It coincides with the observed kinetic energy of the decay products.

In relativistic mechanics, the law of conservation of rest mass is not observed, but the law of conservation of energy is satisfied: the total energy of the closed system is conserved, i.e. does not change over time.

6.7. General theory of relativity. @

A few years after the publication of the special theory of relativity, Einstein developed and finally formulated in 1915 the general theory of relativity, which is the modern physical theory of space, time and gravity.

The main subject of the general theory of relativity is gravitational interaction, or gravitation. Newton's law of universal gravitation implies that the force of gravity acts instantaneously. This statement contradicts one of the basic principles of the theory of relativity, namely: neither energy nor a signal can travel faster than the speed of light. Thus, Einstein faced the problem of the relativistic theory of gravity. To solve this problem, it was also necessary to answer the question: are gravitational mass (included in the law of universal gravitation) and inertial mass (included in Newton’s second law) different? The answer to this question can only be given by experience. The entire set of experimental facts indicates that the inertial and gravitational masses are identical. It is known that the forces of inertia are similar to the forces of gravity: being inside a closed cabin, no experiments can establish what causes the action of the force mg on the body - whether the cabin is moving with acceleration g, or the fact that the stationary cabin is located near the surface of the Earth. The above represents the so-called equivalence principle: the gravitational field in its manifestation is identical to the accelerating reference frame. This statement was used by Einstein as the basis for the general theory of relativity.

In his theory, Einstein found that the properties of space and time are connected by more complex relations than the Lorentz relations. The type of these connections depends on the distribution of matter in space; it is often figuratively said that matter bends space and time. If there is no matter at large distances from the observation point or the curvature of space-time is small, then the Lorentz relations can be used with satisfactory accuracy.

Einstein explained the phenomenon of gravity (the attraction of bodies with mass) by the fact that massive bodies bend space in such a way that the natural movement of other bodies by inertia occurs along the same trajectories, as if attractive forces existed. Thus, Einstein solved the problem of the coincidence of gravitational and inertial mass by refusing to use the concept of gravitational forces.

Consequences obtained from the general theory of relativity (theory of gravity) predicted the presence of new physical phenomena near massive bodies: changes in the course of time; changes in the trajectories of other bodies that are not explained in classical mechanics; deflection of light rays; changing the frequency of light; irreversible attraction of all forms of matter to sufficiently massive stars, etc. All these phenomena were discovered: a change in the clock rate was observed during an airplane flight around the Earth; the trajectory of movement of the planet closest to the Sun, Mercury, is explained only by this theory, the deviation of light rays is observed for rays coming from stars to us near the Sun; a change in the frequency or wavelength of light is also detected, this effect is called gravitational redshift, it is observed in the spectral lines of the Sun and heavy stars; The irreversible attraction of matter to stars explains the presence of “black holes” - cosmic stellar objects that absorb even light. In addition, many cosmological questions are explained in the general theory of relativity.