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.

Relativistic law addition of speeds.

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 relating the velocities of a body in different systems reference (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 passing from one inertial system reference to another, the invariance condition follows 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 in the following way: 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 the most important conclusion special theory relativity comes down to the fact 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, universal dependence total energy particles from 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 not available, since the law of conservation of matter requires that total number baryons (so called elementary particles– neutrons and protons) remained constant in any closed system. 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 have, in addition to rest energy, great energy interactions 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 a source of energy nuclear reactors and atomic bombs.

The correctness of Einstein's relation can be proven using the example of decay free neutron to 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 general theory relativity is gravitational interaction, or gravity. Newton's law of universal gravitation implies that the force of gravity acts instantaneously. Such a statement contradicts one of the basic principles of the theory of relativity, namely: neither energy nor signal can propagate faster speed Sveta. Thus, Einstein faced the problem of the relativistic theory of gravity. To solve this problem, it was also necessary to answer the question: do gravitational mass (included in the law Universal gravity) and inertial mass (included in Newton's second law)? 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 more related complex relationships 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 on long 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 derived from general relativity (the theory of gravity) predicted the presence of new physical phenomena near massive bodies: changes in the passage 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 towards sufficient 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 spectral lines 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.

Example. Let's go back to example (1.13):

x = 1 + 12t 3t2

(coordinate is measured in meters, time in seconds). Consistently differentiating twice, we get:

vx = x = 12 6t;

ax = vx = 6:

As we can see, the acceleration is constant in absolute value and equal to 6 m/s2. The acceleration is directed in the direction opposite to the X axis.

The given example is the case of uniformly accelerated motion, in which the magnitude and direction of acceleration are unchanged. Uniformly accelerated motion is one of the most important and frequently occurring types of motion in mechanics.

From this example it is not difficult to understand that with uniformly accelerated motion the projection of velocity is linear function time, and the coordinate quadratic function. We will talk about this in more detail in the corresponding section on uniformly accelerated motion.

Example. Let's consider a more exotic case:

x = 2 + 3t 4t2 + 5t3 :

Let's differentiate:

vx = x = 3 8t + 15t2 ;

ax = vx = 8 + 30t:

This movement is not uniformly accelerated: acceleration depends on time.

Example. Let the body move along the X axis according to the following law:

We see that the body coordinate changes periodically, ranging from 5 to 5. This movement is an example harmonic vibrations, when the coordinate changes over time according to the sine law.

Let's differentiate twice:

vx = x = 5 cos 2t 2 = 10 cos 2t;

ax = vx = 20 sin 2t:

The velocity projection changes according to the cosine law, and the acceleration projection again according to the sine law. The quantity ax is proportional to the x coordinate and opposite in sign (namely, ax = 4x); in general, a relation of the form ax = !2 x is characteristic of harmonic oscillations.

1.2.8 Law of addition of speeds

Let there be two reference systems. One of them is associated with a stationary reference body O. We will denote this reference system by K and call it stationary.

The second reference system, denoted by K0, is associated with the reference body O0, which moves relative to the body O with a speed of ~u. We call this reference system moving. Additionally

we assume that coordinate axes systems K0 move parallel to themselves (there is no rotation of the coordinate system), so the vector ~u can be considered the speed of the moving system relative to the stationary one.

The fixed reference frame K is usually related to the ground. If a train moves smoothly along the rails with a speed of ~u, then the reference frame associated with the train car will be a moving reference frame K0.

Note that the speed of any point in car3 is ~u. If a fly sits motionless at some point in the carriage, then relative to the ground the fly moves with a speed of ~u. The fly is carried by the carriage, and therefore the speed ~u of the moving system relative to the stationary one is called the portable speed.

Now suppose that a fly crawled along the carriage. Then there are two more speeds that need to be considered.

The speed of the fly relative to the car (that is, in the moving system K0) is denoted by ~v0 and

called relative speed.

The speed of the fly relative to the ground (that is, in a stationary K frame) is denoted by ~v and

called absolute speed.

Let's find out how these three speeds - absolute, relative and portable - are related to each other.

In Fig. 1.11 the fly is indicated by point M. Next:

~r radius vector of point M in a fixed system K; ~r0 radius vector of point M in the moving system K0 ;

~ radius vector of the body of reference 0 in a stationary system.

~r 0

Rice. 1.11. To the conclusion of the law of addition of velocities

As can be seen from the figure,

~ 0 ~r = R + ~r:

Differentiating this equality, we get:

					d~r 0

The derivative d~r=dt is the speed of point M in the K system, that is absolute speed:

d~r dt = ~v:

Similarly, the derivative d~r 0 =dt is the speed of point M in the K0 system, that is, the relative

speed:

d~r dt 0 = ~v0 :

3 In addition to rotating wheels, but we do not take them into account.

What is ~? This is the speed of point0 in a stationary system, that is, portable dR=dt O

speed ~u of a moving system relative to a stationary one:

dR dt = ~u:

As a result, from (1.28) we obtain:

~v = ~u + ~v 0 :

The law of addition of speeds. The speed of a point relative to a fixed frame of reference is equal to vector sum the speed of the moving system and the speed of the point relative to the moving system. In other words, absolute speed is the sum of portable and relative speeds.

Thus, if a fly crawls along a moving carriage, then the speed of the fly relative to the ground is equal to the vector sum of the speed of the carriage and the speed of the fly relative to the carriage. Intuitively obvious result!

1.2.9 Types of mechanical movement

The simplest types mechanical movement of a material point are uniform and rectilinear motion.

The movement is called uniform if the magnitude of the velocity vector remains constant (the direction of the velocity can change).

Movement is called rectilinear if it occurs along a certain straight line (the magnitude of the speed may change). In other words, the trajectory of rectilinear motion is a straight line.

For example, a car traveling at a constant speed along a winding road makes uniform (but not linear) motion. A car accelerating on a straight section of highway moves in a straight line (but not uniformly).

But if, during the movement of a body, both the magnitude of the velocity and its direction remain constant, then the motion is called uniform rectilinear. So:

uniform motion, j~vj = const;

uniform rectilinear movement, ~v = const.

The most important special case uneven movement is uniformly accelerated motion, at which they remain constant module and direction of the acceleration vector:

uniformly accelerated motion, ~a = const.

Along with the material point, another idealization is considered in mechanics - a rigid body.

A rigid body is a system of material points, the distances between which do not change over time. Model solid is used in cases where we cannot neglect the size of the body, but can not take into account the change in the size and shape of the body during movement.

The simplest types of mechanical motion of a solid body are translational and rotational motion.

The movement of a body is called translational if any straight line connecting any two points of the body moves parallel to its original direction. During translational motion, the trajectories of all points of the body are identical: they are obtained from each other by a parallel shift.

So, in Fig. 1.12 shown forward movement gray square. An arbitrarily chosen green segment of this square moves parallel to itself. The trajectories of the ends of the segment are depicted with blue dotted lines.

Rice. 1.12. Forward movement

The motion of a body is called rotational if all its points describe circles lying in parallel planes. In this case, the centers of these circles lie on one straight line, which is perpendicular to all these planes and is called the axis of rotation.

In Fig. 1.13 shows a ball rotating around vertical axis. This is how they usually draw Earth in corresponding problems of dynamics.

Rice. 1.13. Rotational movement

Let two photons 1 and 2 move towards each other with speeds equal to v 1 = c and v 2 = c (c is the speed of light) relative to the conventionally “stationary” reference frame Earth K (see figure). Let's find the speed 1st photon in the K reference frame associated with the 2nd photon, using classic formula for adding speeds:

Table 3

Thus, the speed of one photon in the reference frame associated with the 2nd one turned out to be equal to 2c, but according to STR, not a single particle can move at a speed greater than the speed of light.

When bodies move at speeds comparable to the speed of light in the STR, another formula was obtained, which is called the relativistic formula for adding velocities. Let us write down formulas for the simplest case of systems moving in one direction.

u - body speed in a stationary reference frame K

u is the speed of the body in the moving reference frame K

v - speed of system K relative to system K

(we have replaced the letters from previous formulas to avoid using subscripts and further cluttering the formulas)

Let's get these formulas.

Let us introduce an intermediate variable t

Let's find the derivative using Lorentz transformations

Let's multiply the derivatives, taking into account that

having produced algebraic operations, we find from this equation u or u

Let us now calculate the speed of the photon from the previous example using the relativistic formula.

v 1 = u 1 = c-speed of the 1st photon in K, v 1 = u 1 = c- speed of the 1st in K, v 2 = v - speed of the 2nd photon, i.e. speed K in K. Thus, according to the relativistic formula, the speed of a photon does not exceed the speed of light c.

The concept of relativistic dynamics

When using Lorentz transformations, the basic law of dynamics m(dp/dt) = F turns out to be invariant provided that the particle momentum is written in the form:

Relativistic momentum of a particle

Basic law of relativistic dynamics

Then the fundamental law of relativistic dynamics formally retains the same form as Newton’s II law, but between them there is fundamental difference. (see below)

The quantity m is called relativistic mass; it depends on the speed of the body and is not an invariant, i.e. It has different meaning in different ISOs.

m 0 - body mass, also called rest mass, is an invariant and has the same value in any ISO.

In classical mechanics, the acceleration of a particle and the force that caused this acceleration are always directed in the same direction. At a particle speed comparable to the speed of light, i.e. in the relativistic case, the direction of acceleration and force coincide only in two cases: 1) when the force is parallel to the speed at each moment of time and 2) when the force is perpendicular to the speed. IN general case the directions of acceleration and force do not coincide (see figure)

The relationship between mass and energy in the theory of relativity.

Let us introduce new notations for energy, which are most often used in SRT.

total energy

kinetic energy (we will use the notation T)

Let's find an expression for kinetic energy in SRT, considering that the increase in kinetic energy occurs due to the work of some force. Body in starting moment motionless and free, i.e. does not interact with other bodies and thus does not have potential energy.

to integrate and obtain, you need to reduce m to one variable, while there are two of them, and all equalities are dot products vectors,

instead of the variable p, variables appeared

no longer here vector products because , but two variables remain

square it, express it, substitute in and get

Now you can integrate, because there is only one variable left m

integrating, we obtain the expression for kinetic energy in STR

Relativistic kinetic energy

Rest energy

Total relativistic energy, i.e. energy of a moving body

Thus, from SRT it follows that any motionless body has a reserve of energy equal to. For example, a body weighing 1 kg contains energy E 0 = 1910 16 J. This energy can heat a reservoir with dimensions of 1 km 20 km 20 m by 100 o C. The problem is how to release this energy. Even with thermonuclear reaction less than 1% of the total energy corresponding to the entire rest mass is released. In classical mechanics the concept of “rest energy” was absent.

The expression is called Einstein's law of the relationship between mass and energy

According to this law, total stock The energy of a body (or system of bodies), whatever types of energy it consists of (kinetic, potential, thermal, electrical, etc.) is related to the mass of the body (system of bodies) by this ratio. In other words, if the mass of a body changes, its energy will change, and vice versa.

Let a piece of iron weighing 1 kg be heated by 1000 o C. Let us calculate how much the mass of the piece should change.

a change in the energy of a body must change its mass by

Q - heat during heating, C - specific heat heated substance

There are no devices that can detect such a small change in a mass of 1 kg

All SRT formulas become classical at v<< c.Например, найдем кинетическую энергию тела при малых скоростях. Приближенное выражение, известное из математики

relativistic expression turns into classical

From SRT it follows that particles with zero mass can exist, but they cannot be stationary, but must move continuously, and only at the speed of light c - these are photons and, possibly, neutrinos.

relationship between energy and momentum for particles with zero mass (photons) m 0 =0

Some formulas from SRT that can be derived from the above expressions

Relationship between the kinetic energy of a particle and its momentum

Relationship between the total energy of a particle and its momentum

Relationship between total energy and rest energy with momentum

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.

The first postulate of the special theory of relativity (STR) is called the principle of relativity: all laws of physics are invariant with respect to the transition from one inertial frame of 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 reference frame (x, y, z, t) to another (x′, y′, z′, t′) moving in the positive direction of the 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.

Has practical value for solving 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 velocity 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 projection of the relative velocity has a negative sign, therefore the modulus of the velocity of the nucleus relative to the electron is equal to the modulus of the 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.