Consequences from the Lorentz transformations. Particles with zero mass

The creators of SRT are: Lorentz, Poincaré, Einstein. Representations of SRT are valid only for processes occurring in inertial frames of reference.

Einstein's principle of relativity was preceded by Galileo's principle of relativity, formulated only for mechanical processes (i.e., only for classical mechanics- Newtonian mechanics).

Galileo's principle of relativity represent in two equivalent forms:

Inside a uniformly moving laboratory (reference frame), all mechanical processes proceed in the same way as inside a laboratory at rest.

The uniform movement of the laboratory (reference frame associated with the reference body - the laboratory) cannot be detected by any mechanical experiments carried out inside it

Let us explain this principle using the following example: if a passenger (observer) of an electric train (moving uniformly) drops an object (for example, a watch), then for him they will fall vertically downward, and for a person (observer) standing on the ground, the object will fall along a parabola , because the train is moving while the object is falling. Each observer has his own frame of reference. But, although the descriptions of events change when moving from one frame of reference to another, there are universal things that remain unchanged. If, instead of describing the fall of an object, we ask about the nature of the law that causes it to fall, then the answer to it will be the same for an observer in a fixed coordinate system and for an observer in a moving coordinate system. In other words, while the description of events depends on the observer, the laws of mechanics (later Poincare and Einstein generalized this to all physical laws) do not depend on him, i.e. are invariant.

The principle of relativity (both in classical mechanics and in SRT) is closely related to privileged frames of reference, the so-called inertial frames of reference.

Inertial reference systems are called, with respect to which a material point (body) without external influences (or if external influences are compensated):

rests

Moves evenly and in a straight line

Any frame of reference that is stationary or moving uniformly and rectilinearly with respect to inertial system reference is also inertial (i.e. all inertial frames of reference are equal)

The initial principles of classical mechanics are based on the formulas for transforming coordinates and time, the so-called Galilean transformation . Using these transformations, it is possible to transfer the consideration of the motion of a body (particle) from one inertial frame of reference to another, as, for example, the example considered earlier with the fall of an object in an electric train.

All laws of classical mechanics are invariant with respect to the transition from one inertial frame of reference to another, carried out with the help of Galilean transformations. Galileo's transformations are based on the sameness (invariance) of time in different inertial frames of reference and the classical law of addition of velocities.

From Galileo's transformations (i.e. from classical mechanics) it follows that during the transition from one frame of reference to another, the following remain unchanged (invariant):

- time

- body dimensions

- body mass

Let's move on to the special theory of relativity. SRT is based on two postulates (principles) of Einstein:

The principle of relativity (Einstein's first postulate, which is a generalization of Galileo's principle to all physical processes): all physical processes in all inertial frames of reference proceed in the same way.

We formulate this principle in another equivalent form: the laws of nature are invariant in all inertial frames of reference.

Principle of invariance (permanence) speed of light (Einstein's second postulate): the speed of light in vacuum is constant in all inertial frames of reference and does not depend on the motion of light sources and receivers.

The postulate of the constancy of the speed of light causes the greatest misunderstanding, because it is in clear conflict with classical rule addition of speeds. That the speed of light has such unusual property, can be felt when considering the following thought experiment: let the astronaut be in spaceship, the ship is moving away from the Earth with constant speed 200,000 km/s, and an observer on Earth directs a beam of light propagating at a speed of 300,000 km/s towards the spacecraft. The light, catching up with the spacecraft, passes through the spacecraft through small holes and goes further into space. Since the astronaut (together with the ship) is moving at a speed of 200,000 km / s relative to the Earth, then, on the basis of classical law adding the velocities, it should have seemed that relative to it, light propagates at a speed of 300,000 km / s - 200,000 km / s \u003d 100,000 km / s. But as follows from the principle of the constancy of the speed of light, if such an experiment is really set up, then it will seem to an astronaut (i.e. an observer in a moving inertial frame of reference) that light propagates, relative to him, at a speed of 300,000 km/s. Based on the same principle, an observer on Earth will also consider that light propagates relative to him also at a speed of 300,000 km/s.

Einstein realized that the only explanation that allows two observers moving relative to each other to get same values the speed of light lies in the fact that their perception of time and space is not the same, that the clock of the spacecraft does not run the same way as on earth, the same rulers for both observers have different sizes, etc. That is, based on SRT, the speed of light in a spacecraft is 300,000 cosmic kilometers per cosmic second, and on Earth - 300,000 terrestrial kilometers per terrestrial second. The above example clearly shows that if the speeds of other objects are relative, since they depend on the speed of the measuring observer, then the speed of light is not relative - it is absolute. The same example shows the relativity of time and space. The speed of light corresponds to the maximum possible signal transmission speed in nature.

The principle of constancy of the speed of light was first confirmed in the experiments of Michelson-Morley. The authors themselves tried to confirm or disprove the existence of the world ether by this experiment. The world ether was presented as a mechanical medium (an invisible weightless substance) transmitting the "push" of action from one point to another, i.e. transmitting the wave process of light propagation. The Michelson-Morley experiments compared the speeds of light when a beam of light was directed along and across the Earth's orbital motion. In this case, no difference was found, which indicates the constancy of the speed of light, regardless of the inertial frame of reference in which the propagation of light is considered (for a beam of light propagating along the direction of the Earth's motion, the frame of reference is mobile, for propagating across it, it is motionless).

It follows from the SRT postulates that the spatial interval and the time interval (duration of the event) are relative, i.e. dependent on the movement of the observer. However, the objectivity of the description of nature requires that the phenomenon under study can be characterized by quantities that do not depend on the choice of reference frame. An invariant quantity in SRT is the so-called space-time interval between events , which includes the temporal and spatial characteristics of material processes. Those. SRT makes the world four-dimensional: time is added to three spatial dimensions. All four dimensions are inseparable, so we are no longer talking about the spatial distance between objects, as is the case in the three-dimensional world, but about the space-time intervals between events that unite their distance from each other, both in time and in space. Those. space and time are viewed as a four-dimensional space-time continuum, or simply space-time. On this continuum, observers moving relative to each other may disagree about whether two events happened at the same time, or one preceded the other, but the space-time interval for both observers will be the same.

SRT shows that it is impossible to transmit an impact (light, information, etc.) at a speed exceeding the speed of light, and this makes it impossible to violate causal relationships (because it is the transmission of an impact at a superluminal speed that would lead to a violation of causal investigative links). The inviolability of causal relationships can be called invariance of causal relationships .

The law of the relationship between energy and mass also follows from SRT: there is an unambiguous relationship between the total energy of a body isolated from external influences and its mass: . This law is also valid for a body at rest:
, showing that even bodies at rest have very great energy, including the energy of interactions and thermal motion of atoms and molecules, the energy of nuclear interaction, and other energies. This law shows: no matter what mutual transformations different types matter did not occur, the change in energy in the system corresponds to an equivalent change in mass. Those. energy and mass are two uniquely related characteristics of matter. This law reveals the source of the energy used nuclear power. The mass of radioactive decay products occurring in nuclear reactor, less mass original substance. The difference between the masses of the initial and final (called mass defect) multiplied by the square of the speed of light (
) shows the energy produced in nuclear reactors.

The transition from one inertial frame of reference to another, in SRT, is carried out with the help of Lorentz transformations.

From the Lorentz transformations (i.e. from SRT) it follows that with an increase in the speed of a moving inertial frame of reference relative to a fixed one:

- the length of the segment in the direction of motion decreases relative to the segment in a stationary system

- the course of time in a moving frame, relative to time in a fixed frame of reference, slows down

The above consequences explain the thought experiment we considered earlier: an astronaut, determining the speed of light, divides his small kilometers into small seconds and gets the same result as an earthly observer who divides large kilometers into large seconds.

The consequences of SRT are relative nature :

- distances (length of the segment), i.e. space

- simultaneity of events, i.e. time

- body weight

The consequences of SRT are:

- space and time exist as a single four-dimensional space-time structure and are described by Euclidean geometry

- equivalence of mass and energy

- with an increase in the speed of the reference body, the rate of time on it slows down

- with an increase in the speed of the body, its linear size decreases

- as the speed of the body increases, its mass increases

- when the speed of a body approaches the speed of light, its linear size tends to zero, and the mass of the body tends to infinitely large

- invariance (invariance) of the space-time interval between events

- invariance of causal relationships

Correspondence of SRT and classical mechanics: their predictions coincide at low speeds (much lower than the speed of light).

The application of SRT to the description of mechanical processes in which the speeds of bodies are comparable to the speed of light is called relativistic mechanics .

The reason for SRT is the invariance of the speed of light

P.V.Putenikhin

[email protected]

Abstract 2

Derivation of SRT from the principle of constancy of the speed of light 2

Derivation of SRT from the principle of relativity 7

Analysis of SRT principles 11

Literature 14

annotation

Einstein laid two principles at the basis of SRT. However, in order to obtain the Lorentz transformations and all the relativistic consequences from them, only one principle (postulate) is sufficient - the invariance of the speed of light. This principle is the root cause of the Lorentz transformations, the only necessary and sufficient condition for their conclusion, as well as for the declaration of the principle of relativity and equality of all inertial frames of reference. Obtaining Lorentz transformations from the principle of relativity is possible, but with the obligatory consideration of the principle of constancy of the speed of light.

Derivation of SRT from the principle of constancy of the speed of light

All SRT conclusions - Lorentz transformations and relativistic relations are obtained as correct mathematical conclusions. Therefore, SRT is inherently a mathematical theory, it has all its features: the methodology of inference, the initial postulates. Although Einstein based SRT on two postulates (principles), we can say that SRT is actually based on a single postulate: on the invariance of the speed of light in all IFRs - the principle of constancy (invariance) of the speed of light. We will show this - we will derive the Lorentz transformations and the main consequences of them, using only one assumption for this: the speed of light " c" always the same, regardless of whether the ISO is moving or at rest. Otherwise, we can say that the speed of any photon is equal to the speed of light, wherever it is measured: in a moving or resting ISO. Exactly this general definition of the principle of constancy of the speed of light. It does not include references to the source of this photon and the state of motion of the source (or receiver), which are redundant. The statement about the limit of the speed of light is also derivative from the principle of constancy of the speed of light, its consequence: if the speed of light is constant in all ISOs, then it automatically becomes the fastest possible speed. Let's call this principle of the constancy of the speed of light the basis of the theory, and all the expressions obtained with its use - a consequence of this principle (postulate), consequences, conclusions of the theory.

To conclude, consider a platform of length L, which is crossed by a photon emitted by an unknown source and/or simply flying by. As is customary in SRT, we will consider two inertial frames of reference - stationary K and moving K. A photon for observers on the platform will fly through it in time t 0 = L / c. Let's keep the notation close to that adopted in SRT:

L" is the length of the platform in the inertial reference frame K";

L is the length of the platform in the inertial frame K;

t" is the time interval (time) during which the photon flies through the platform and returns back in the frame K";

t is the time interval (time) during which the photon flies through the platform and returns back in frame K.

An observer in the moving system K" considers it to be at rest and calculates that the photon will overcome the platform in time (round trip):

On the contrary, an external observer sees: in one case, the light catches up with the mirror at the opposite end of the platform, and in the other, it flies towards the target:

Fig.1 Flight of a photon from the point of view of an external observer. The clock of the external (stationary) observer will show the time t, and the clock on the platform (moving) will show the time t".

The figure shows that for an external observer, the time of the photon's movement along the moving platform back and forth will be:

Let's transform the equation:

The expression for the second fraction looks like the square of some quantity. Let us denote this value by k (obviously, this value is greater than one):

We got the readings of two clocks: moving with the platform - t "and stationary, past which the platform is moving - t. Obviously, these readings differ. To find out how the "time in flight" of a photon through a moving platform has changed when considering it in different ISOs, we calculate the ratio of these indications:

Hence, after reductions, we get:

(1)

The time t" is the time (time interval) of the photon's flight through the platform for an observer located on this platform, and L" is the length of the platform for this observer. It is obvious that the observer did not notice anything after the acceleration of the platform, nothing happened for him, he, generally speaking, could not know that the platform was moving. Therefore, these two values are the initial ones, not reduced, those that were known before the start of the experiment. And what are the values of t and L? The observer who sees the movement of the platform, we consider motionless. Therefore, he sees a platform of length L and the time t for which the photon flew through the platform back and forth. We know that the clock on the platform began to run slower, that is, the time t" elapsed on the platform is less than the time elapsed in the fixed frame of reference t. Similarly, we conclude: in a fixed frame, the length of the platform is seen shortened to the value L, against the original length L ". However, in accordance with the accepted postulate of the constancy of the speed of light, we must recognize that if the path for the light has changed, then the travel time of the photon has also changed. And it changed in the same direction as the length of the platform - it decreased, and exactly the same amount as the platform was reduced, because these three quantities are related by the formula: t 0 = L / s, that is:

(2)

Substituting (1) into (2), we get:

From where, after transformations, we find:

and finally:

Substitute the value of k and convert it to the usual form:

(3)

Thus, a rod having length L"in the inertial system where it is at rest, has length
in that inertial frame relative to which it moves with velocity v in the longitudinal direction. We substitute (3) into (2) and find the same expression for time:

(4)

Thus, the moving clock begins to lag behind, its course slows down in relation to
, although from the point of view of that inertial system that moves with the clock, absolutely no changes have occurred in the clock.

Here the observant reader will notice a "contradiction" known as the "stroke paradox". It's contrived formal paradox, so to speak, the paradox of the letter, but not the spirit. In our case, we ourselves chose the notation of the times. How to designate the so-called " internal time ISO" is quite arbitrary.

Equations (3) and (4) clearly imply the limit of the speed of light "c" - no IFR can move with a speed v > c, since in this case radical expression becomes negative. Also, in the considered method for deriving the above equations, the principle of relativity is visible: we could carry out all the calculations by swapping the IFRs under consideration, and get exactly the same result.

Let us derive from the postulate (principle) proclaimed above the remaining consequences of the theory under consideration. To do this, we need to show explicitly two frames of reference K and K":

Fig.2 In a fixed inertial reference frame K, the clock has a coordinate x, and in a moving inertial reference frame K" after the time t - the coordinate x".

To the inertial reference frame K are tied coordinate axes XYZ, and to the moving system K" - coordinate axes X"Y"Z". In the figure, the Z and Z" axes are not shown. In initial moment time t=t"=0, the origins of the fixed frame K and the moving frame K" (position I) coincide. After the lapse of time t in the stationary frame K, the mobile frame K" moved away (position II), and the distance between the origins of the two reference frames became v t. Let's transform the coordinates of the stationary system K into the coordinates of the moving system K". It can be seen from the figure that the coordinate of the clock from the point of view of the system K" is equal to:

Where 0B" and 0A" are the lengths of the segments on the 0X axis from the point of view of the moving frame K" (taking into account their signs, since in the frame K" the clock moves in the negative direction). It is obvious that the lengths of these segments from the point of view of the mobile system K" are shortened with respect to their real size in the stationary state in frame K. Therefore, in order to calculate their lengths in the moving frame K", we must use the relation (3) obtained above for the segments:

respectively, the second segment:

We substitute these quantities in the original equation and get:

This equation shows which coordinate in the system K "will have a fixed clock with a coordinate x in a stationary system K after a time t of motion with a speed v. Consider what time the moving clock will show. We know that when moving, they lag behind the stationary ones. Apparently, the longer faster clock move, the more they lag behind. It is clear that in this case the clock moves away from the stationary ones by some distance. I wonder which one? To find out, consider the figure:

Fig.3 After the time t has elapsed, the moving clock will move to the point with the x coordinate and will show the time t", which will be less than the time t in the fixed reference frame K.

The moving system K" has moved from position I at time t=t"=0 to position II. At the same time, the clock shows the time t and t ", respectively, the coordinate of the moving clock from the point of view of the stationary system K is equal to x. We transform equation (4) as follows:

In the last expression of the composite equality, we make an obvious change v t = x:

(5)

Thus, after the lapse of time t, moving with speed v the clock will move away x and will show the time t", and we get everything classical equations Lorentz transformations (we add the last two for obvious reasons - movements only along the X axis):

; ; y" = y;z" = z.

The last and most mysterious of three famous The main consequences of the Lorentz transformations - the relativity of simultaneity will be derived in the traditional way. Let two events occur on the X axis in the inertial frame K at the points x 1 , x 2 at the same time t. We note the moments of these events t" 1, t" 2 in the system K". According to the obtained formula (5), we find:

,
.

We see that t" 1 is not equal to t" 2 , that is , two events that are simultaneous with respect to K turn out to be different in time with respect to K". This discrepancy in time is the greater, the farther apart from each other, from the point of view of the K system, the places where they occurred:

So, having obtained equations that exactly coincide with the equations of the Lorentz transformations in SRT, we have shown that the Lorentz transformations and the main consequences of them can be derived using the only thing guess: speed of light c" always the same, regardless of whether the ISO is moving or at rest. Therefore, this assumption, the postulate is the only a necessary and sufficient condition for the appearance of Lorentz transformations and all their consequences. Therefore, there are sufficient reasons to believe that the mathematics of the kinematic section of SRT is elementary mathematical problem for high school students of the form "A train left point A for point B ...".

Derivation of SRT from the principle of relativity

It was shown above that in order to derive all the Lorentz consequences of SRT, one (second) postulate is sufficient - about the constancy of the speed of light. But there is also an opposite approach: to obtain the same consequences, another (first) postulate is sufficient - the principle of relativity (equality of all IFRs). Moreover, it is argued that the principle of the constancy of the speed of light is generally superfluous. However, in the process of deriving SRT from the principle of relativity, a parameter inevitably appears that plays the same role in the Lorentz equations as the speed of light. That is, the principles of the constancy of the speed of light and relativity are still interconnected.

We will show this by using to a large extent the methodology of S. Stepanov. Let us write the resulting equations for the transformations of time and coordinates between two inertial frames of reference in following form:

x" = f(x, t, v), t" = g(x, t, v) (6)

The problem will be considered as a purely mathematical, idealized one. Therefore, we assume that these coordinate and time transformations are linear functions:

(7)

The coefficients k, m, n, p are functions depending on the relative speed of reference systems v .

We will assume that at the initial moment of time t=t"= 0 system origins coincide x=x"= 0. The coordinate of the origin of the moving reference system is described by the equation x=vt . Substitute x"= 0 and x=vt into the first equation and get:

from where we find:

(8)

Now we substitute x= 0 and x"=vt into both equations and we get:

after simplification:

and then after substituting from the second equation into the first and taking into account (8) we get:

We insert the obtained relations into the initial equations (7):

Let us introduce the notation (substitutions):

The introduced parameters (substitutions) are functions of speed, but in the future, for brevity, we will write them without a sign of functionality - without brackets with an argument v. Taking into account these simplifications, the transformations between reference systems take the final form:

(9)

To determine the entered parameters γ and σ, based on the principle of relativity (the first postulate of SRT) - the equality of all inertial frames of reference, we consider three such arbitrary IFRs - K 1, K 2 and K 3. We establish that the K 2 system moves relative to K 1 with a speed v 1 , system K 3 - relative to K 2 with speed v 2 and system K 1 - relative to K 3 with a speed v 3 =-(v 1 +v 2):

Fig.4 Three frames of reference moving relative to each other.

Let's mark the coordinate x and time t digital indices corresponding to the numbers of the systems to which they belong, and write down the transformations for each of them:

Substitute x 2 and t 2 from the second system of equations to the third:

Let's open round brackets:

Let's take it out of brackets common factors:

and group common members:

The resulting equations should have (and have) the same form as the equations of system (9). This means that, as in the system of equations (9), in this system the coefficients for the first terms in the equations are the same coefficient:

After reduction and elementary transformations we get:

From this equality it follows that the following relationship have the same value for all frames of reference, regardless of the speed of their movement:

(10)

We denoted this ratio by the square of the value (constant) "c" - by the first letter of the word "const". Let us explain why it is necessary to equate the ratios exactly to the square. It follows from the second equation of system (9) that all obtained ratios have the dimension of the square of the velocity. To verify this, we analyze the dimensions of the values (the index "size" means that it is not the value that is considered, but the dimension of the values):

It is obvious that in brackets there are quantities with the dimension of time. It follows that the square of the dimension of the constant "c" is equal to the square the dimension of speed, and the value "c" itself has, respectively, the dimension of speed:

This means that all relations (10) are equal square some value "c".

Equations (9) must also be valid for inverse transformation when the frames of reference "swap". The relative velocity then changes its sign:

Let us substitute the values of primed quantities from the original system (9) into this equation:

and finally:

(11)

From relations (10) we find:

Substitute this value in (11) and get:

As a result of the transformations, we get:

(12)

Function γ(v ) is even. This is evident from the following considerations. If we turn the axes of two frames of reference by 180 o, then the speed will also change its sign. This is the same as if we were looking at these systems through a mirror (the rear view mirror of a car): the directions of the axes and movement will reverse. Therefore, the first equation of system (9) will look like:

Comparing these equations, we get:

Expanding the brackets:

and we get the parity sign of the function:

(13)

We substitute the obtained value (13) into (12) and find:

Now we find the value of the gamma function:

and substitute it into equations (9):

;
(14)

With these two equations, one can easily derive all other consequences of the Lorentz transformations, as shown in the previous section. Analysis of the principles of SRT So, we have derived the explicit form of the equations (6) for the transformation between two inertial frames of reference and obtained the Lorentz equations (14), in which we were forced enter some constant With , whose value we, strictly speaking, do not know. The meticulous reader, probably, has long been keeping in mind the thought: when, finally, and how the author of the article will declare this constant the speed of light. According to some authors, this question is not simple. For example, S.Stepanov considers (he has this constant α is reciprocal to our constant - c) that " functional form transformations between observers of two inertial frames of reference are completely determined up to a constant α . Finding her out values and sign- This is an experimental question. fundamental constant α could turn out to be zero, but in our World it is greater than zero. Faculty of Physics St. Petersburg State University S.N. Manida (his value g is also the reciprocal of our constant c): “introduces some constant value, the dimension of which is the inverse square of the velocity. This value is the same in all reference systems, and its numerical value cannot be inferred from any general principles. The experimental value of this quantity g=c -2 , where c - the speed of light in vacuum ". "We derived the ratios from the principle of relativity and obtained as a consequence the constancy of the speed c in all inertial frames of reference. It is important to note the fundamental difference this approach to the conclusion of the Lorentz transformations from the generally accepted. The constancy of the speed of light in all inertial frames of reference is an experimental fact established with a certain degree of accuracy. The above conclusion is not based on this fact, it only follows Existence speed that is the same in all inertial reference frames. ”At one of the forums on the Internet, an analysis of an article by Feigenbaum was published, devoted, in particular, to the derivation of SRT relations from the principle of relativity. It says: "To bring out" special theory relativity” (SRT), the postulate of the constancy of the speed of light is not needed. This means that it is possible that the speed of light is not constant (if it is less than the fundamental constant C). SRT formulas do not logically depend on the postulate of constancy of the speed of light. Feigenbaum writes that SRT could have been discovered back in the days of Galileo. All that is needed for this is the principle of equality of systems moving uniformly relative to each other (the principle of relativity of Galileo) and the isotropy of space. what are relativistic effects. The fundamental constant standing in relativistic formulas not necessarily equal to the speed of light. Only experience can determine its value. If the speed of light is less than this constant, then photons must have mass and, like any massive particles, experience gravitational attraction, which, perhaps, explains the phenomenon of beam bending near massive bodies. The above considerations are reasonable, however ... Be that as it may, the use of only the principle of relativity to derive SRT is inevitable compels us, requires against our will to introduce a certain constant, strongly reminiscent of the speed of light in the Lorentz transformations in the "standard" (Einsteinian) SRT. That is, the principle of relativity by itself is still insufficient to obtain relativistic effects. AT without fail he needs an assistant - a light-like constant. Let's try to assume that this constant is not the speed of light. But it has the dimension of speed and, therefore, it is the speed of something. But what? Let's see what properties it has. Einstein's SRT has a section in which he analyzes Maxwell's equations and concludes that they are invariant under Lorentz transformations. Einstein's Lorentz transformations are based both on the principle of relativity and on the postulate of the constancy of the speed of light. Therefore, if Maxwell's equations are invariant with respect to these transformations, then the principle of relativity in Einstein's interpretation is valid and valid. Then the question arises: if the principle of relativity is observed in the form of the invariance of Maxwell's equations with respect to Lorentz transformations, then how can they be simultaneously invariant with respect to other pseudo-Lorentz transformations, in which there is not the speed of light, but some other constant? How can you imagine that there are two different principles of relativity? One of them is the principle of relativity, which Einstein refers to when deriving the Lorentz equations containing the speed of light as an invariant. The second is the principle of relativity of Feigenbaum, Manid and Stepanov, from which the same Lorentz transformations are derived, but containing a certain constant similar to the speed of light, but not equal to it. In this case, only two conclusions are possible: either the Lorentz-Einstein equations do not correspond to the principle of relativity, or the found light-like constant is the speed of light. Further. From the basic Lorentz equation (14), we see that the speed of light is the maximum possible speed. No frame of reference can move with this or greater speed, since zero appears in the denominator or Square root from negative number:

But exactly the same equation appears when deriving transformations from the principle of relativity, but not with the speed of light, but with another similar constant. That is, in this case, no frame of reference can already move at a different speed, with a different maximum. It is obvious that this "other" speed cannot be less than the speed of light if it claims to be the maximum possible speed, since the speed of light has been reliably measured. Hence, it can only be greater than the speed of light (equality identifies them). Therefore, in this case, the speed of light is not the maximum possible speed. The well-established concepts of Lorentz invariance, light-like and time-like intervals, Hawking's light cone, Schwarzschild radius, etc. lose their meaning. But Einstein obtained the maximum possible speed using both the principle of the constancy of the speed of light and the principle of relativity. And again it turns out that the principle of relativity of Einstein and the principle of relativity of Stepanov - Manida - Feigenbaum are two different the principle of relativity, because they give different meanings maximum possible speed. Two different principles relativity for one theory is a complete absurdity. The derivation of the Lorentz equations on the basis of only one postulate of the constancy of the speed of light also contradicts the equations derived on the basis of the principle of relativity of the “second kind” (with the interpretations of Feigenbaumi and others). That is, these two principles - the constancy of the speed of light and the "new" relativity - turn out to be incompatible in this case. The constancy of the speed of light contradicts the principle of relativity (“second kind”). In other words, in the principle of relativity of the “second kind”, the speed of light is not an invariant, and reference systems become unequal, since the flow of physical processes in them depends on the speed of their movement: the speed of light can be added to the speed of the system.

All these absurd consequences are removed if we take the value of the constant, equal to speed Sveta. Then it inevitably follows: to derive all the consequences of SRT, Lorentz transformations, at least, it is impossible to do without the postulate of the constancy of the speed of light, and, as a maximum, for their derivation, only this postulate is generally necessary and sufficient - only it does not lead to rumors about the unclear constant. By itself, the postulate of the invariant of the speed of light includes the main element of the principle of relativity - the same flow physical phenomena dependent on the speed of light. And this, in accordance with the well-known opinion of Lorentz, is almost all natural phenomena. This principle of relativity appears in in a certain sense a consequence of the invariance of the speed of light, dependent on it, which, apparently, rejects the interpretation of the principle of relativity by Feigenbaum and his associates.

Considering the seriousness of the arguments of the cited authors, we can say that objectively they are the strongest refutation of Einstein's special theory of relativity, chopping, as they say, the theory to the very root, rejecting it in fact. fundamental level- theoretical, as opposed to the arguments of traditional alternatives, anti-SRT-in with their countless thought experiments. Einstein's two postulates are inseparable; one does not exist without the other. The principle of relativity gives rise to the principle of the constancy of the speed of light. The phrase is symmetrical for a reason: on the one hand, the use of the principle of relativity leads to the emergence of the principle of the constancy of the speed of light, and on the other hand, the use of the principle of the constancy of the speed of light means the proclamation and use of the principle of relativity. Who begets whom? Everyone - everyone! Indeed, the principle of relativity, as the principle of equality of all inertial frames of reference, proclaims that in all these frames there is one and the same maximum speed, the same velocity invariant, the same form of Maxwell's equations, and when deriving the Lorentz equations, it inevitably "generates" the same velocity constant for all systems, and this constant inevitably manifests itself as the speed of light. On the other hand, the principle of the constancy of the speed of light means nothing more than the equality of all systems in relation to this speed, which is at least part of the principle of relativity. The derivation of the Lorentz equations from the principle of the constancy of the speed of light gives them the same unambiguously the same form as in the derivation based on the principle of relativity. And this means that the principle of relativity is the same for both approaches, that there is only one principle of relativity - this is a principle that, as an integral part, contains the principle of constancy of the speed of light, equality, and is itself a direct consequence of the principle of constancy of the speed of light. Literature

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Lecture: Invariance of the modulus of the speed of light in vacuum. Einstein's principle of relativity

Galileo's principle of relativity

To understand what happens to bodies that move at high speeds, one should consider Galileo's principle of relativity in more detail.

So, let's imagine that we are on a ship whose cabin has no windows or any other openings through which one could look at the ship's surroundings. Question: can we determine whether the ship is moving uniformly or stationary? In this cabin, we can consider the same processes as if we were on Earth. We can consider the movement of the body along inclined plane, the movement of a body that falls or any kind of movement. But all of them will proceed in the same way as if they were taking place outside the ship on land.

Thus, we can conclude that if you are stationary or are in a system that moves uniformly, all physical processes proceed in the same way. And, therefore, it is impossible to determine how the ship behaves while in the cabin.

Thus, all systems moving uniformly, or at rest, are inertial.

According to Galileo's principle of relativity, all processes proceed in the same way in all IFRs.

Velocity invariance

Consider two IFRs, one of which is stationary and the other moves uniformly.

At the initial moment of time, the origin of coordinates of both systems coincides. After the movement starts, the time count starts. To determine the coordinates of a body in a moving reference system relative to a fixed one, you should use the formula:

Note that since the movement occurs along one axis, the change in the coordinate is noticeable only relative to it, all other parameters remain unchanged.

Using Galileo's relativity, one can determine the position of a moving system relative to one that is not moving.

And now let's imagine that a particle is still moving in this moving system. Let the speed of a given particle relative to the stationary system u, and relative to the moving system u 1 . Now we will look at how these two speeds are related.

We know that speed is the first derivative of a coordinate, so let's find the derivatives of the previous three equations:

Generalizing the three equations, we get:

This formula has long been familiar to us as the law of addition of velocities.

Einstein's principle of relativity

We said earlier that it is impossible to determine in which ISO we are moving or not, from the point of view of mechanics. But that we should try to do this from the point of view of other branches of physics.

It turns out that the laws of other branches of physics are not subject to Galilean relativity, this was proved by Maxwell. The scientist proved that the speed of light in vacuum is constant value, no matter how fast and how the system in which the experiments take place moves.

Imagine a situation in which you are moving on a high-speed ship at a speed 5*10 7 m/s. On the bow of this ship is a light bulb, the light of which propagates at a speed known to us. 3*10 8 m/s. This means that according to Galileo's principle of relativity, its speed relative to you reaches 3.5*10 8 m/s. But, as already mentioned, the speed of light cannot take on a value greater than the limit.

In addition to some changes regarding the addition of velocities, Lorentz noticed that bodies moving at speeds close to the speed of light noticeably shrink in size.

) and is the embodiment of the Lorentz invariance of electrodynamics. More generally, we can say that the maximum speed of propagation of the interaction (signal), called the speed of light, must be the same in all inertial frames of reference.

This statement is very unusual for our daily experience. We understand that speeds (and distances) change as we move from a system at rest to one in motion, while intuitively believing that time is absolute. However, the principle of the invariance of the speed of light and the absoluteness of time are incompatible. If the maximum possible speed is invariant, then time passes differently for observers moving relative to each other. In addition, events that are simultaneous in one frame of reference will not be simultaneous in another.

The invariance of the speed of light in the laboratory at rest relative to the surface of the Earth is firmly established experimentally. Of interest is the search for possible small deviations from this law.

Notes

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RELATIVITY- Theories of relativity form an essential part of the theoretical basis of modern physics. There are two main theories: private (special) and general. Both were created by A. Einstein, private in 1905, general in 1915. In modern physics private… … Collier Encyclopedia

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Lorentz transformation

Lorentz transformation- Lorentz transformations in physics, in particular in the special theory of relativity (STR), are the transformations that the space-time coordinates (x, y, z, t) of each event undergo when moving from one inertial frame ... ... Wikipedia

History of the theory of relativity- A prerequisite for the creation of the theory of relativity was the development of electrodynamics in the 19th century. The result of the generalization and theoretical understanding of experimental facts and patterns in the fields of electricity and magnetism were the equations ... ... Wikipedia

Lorentz transformations- Lorentz transformations are linear (or affine) transformations of a vector (respectively, affine) pseudo-Euclidean space that preserves lengths or, equivalently, scalar product vectors. Lorentz transformations ... ... Wikipedia

List of scientific publications by Albert Einstein- Albert Einstein (1879-1955) was a well-known theoretical physics, who is best known as the developer of general and special relativity. He also contributed huge contribution in the development of statistical mechanics, especially ... ... Wikipedia

THEORY- (1) system scientific ideas and principles that generalize practical experience, reflecting the objective natural laws and regulations that form (see) or a section of any science, as well as a set of rules in the field of any knowledge million ... ... Great Polytechnic Encyclopedia

Einstein's principle of relativity and the Lorentz transformation

One of the most important physical constants is the speed of light in vacuum c, that is, the speed of propagation of electromagnetic waves in space free of matter. This speed does not depend on the frequency of electromagnetic waves, and its current value is c = 299,792,458 m/s.

In the vast majority of cases, this value can be taken equal to c = 3 108 m/s with sufficient accuracy - the error is less than 0.001.

And it is precisely “three hundred thousand kilometers per second” for the speed of light that most of us remember for our entire lives. Recall that 300,000 km is, in order of magnitude, the distance from the Earth to the Moon (more precisely, 380,000 km).

Thus, the radio signal from the Earth reaches the Moon in a little more than one second.

The assumption that light travels not with infinite, but with a finite speed, was expressed many centuries before people could prove it experimentally. This was first done in the 17th century when astronomical observations strange "irregularities" in the motion of Jupiter's moon Io could only be explained on the basis of the assumption of final speed propagation of light (by the way, this first attempt to determine the speed of light gave an underestimate of ~ 214,300 km/s).

Up to late XIX centuries, the speed of light has interested researchers, mainly from the point of view of understanding nature electromagnetic radiation- it was not clear to physicists then whether they could electromagnetic waves propagate in a vacuum, or they propagate in a special space-filling substance - ether. However, the result of the study of this problem was a discovery that turned all the ideas about space and time that existed until then. In 1881, as a result of the famous experiments of the American scientist Albert Michelson,

installed amazing fact - the value of the speed of light does not depend on which frame of reference it is determined with respect to!

This experimental fact contradicts Galileo's law of addition of velocities, which we considered in the previous chapter and which seems obvious and is confirmed by our everyday observations. But light does not obey this seemingly natural rule of speed addition - relative to all observers, no matter how they move, light propagates at the same speed c = 299,793 km/s. And that the spread of light is movement electromagnetic field, not particles,

consisting of atoms does not play a role here. When deriving the law of addition of velocities (9.2), the nature of the moving object did not matter.

And although it is impossible to find anything similar in the experience and knowledge we have accumulated earlier, nevertheless, we must recognize this experimental fact, remembering that it is experience that is the decisive criterion of truth. Recall that we encountered a similar situation at the very beginning of the course, when we discussed the properties of space. Then we noted that to imagine the curvature three-dimensional space it is impossible for us - three-dimensional beings. But we realized that the fact of the "presence or absence" of curvature can be established empirically: measuring, for example, the sum of the angles of a triangle.

What changes need to be made to our understanding of the properties of space and time? And how, in the light of these facts, should we treat Galileo's transformations? Is it possible to change them so that they still do not contradict common sense when applied to the habitual movements of the bodies around us and at the same time did not contradict the fact of the constancy of the speed of light in all frames of reference?

The fundamental solution to these issues belongs to Albert Einstein, who created at the beginning of the 20th century. special theory of relativity (SRT), which connected the unusual nature of the propagation of light with fundamental properties space and time, manifested during movements at speeds comparable to the speed of light. In modern physical literature it is more often called simply relativistic mechanics.

Einstein later built general theory relativity (GR), where the connection between the properties of space and time and gravitational interactions is studied.

The SRT is based on two postulates, which bear the name Einstein's principle of relativity and the principle of constancy of the speed of light.

Einstein's principle of relativity is a generalization of Galileo's principle of relativity, discussed in the previous chapter, to all without exception (and not just mechanical) phenomena of nature. According to this principle, all laws of nature are the same in all inertial frames of reference. Einstein's principle of relativity can be formulated as follows: all equations expressing the laws of nature are invariant with respect to transformations of coordinates and time from one inertial frame of reference to another. (Recall that the invariance

equations is called the invariance of their form when the coordinates and time of one reference system are replaced in them by the coordinates and time of another). It is clear that, in accordance with the Einstein principle of relativity, no experiments at all can establish whether “our” reference frame is moving at a constant speed or it is stationary, more precisely, there is no difference between these states. Galileo postulated this impossibility in principle only for mechanical experiments.

The principle of constancy (more precisely, invariance) of the speed of light states that the speed of light in vacuum is the same for all inertial frames of reference. As we will soon see, it follows that c is the maximum of all possible physical speeds.

Both postulates are a reflection of experimental facts: the speed of light does not depend on the movement of the source or receiver; it also does not depend on the motion of the frame of reference in which experiments are carried out to measure it. In the principle of relativity, this is reflected in the recognition of the fact that not only mechanical, but also electromagnetic (light propagation) phenomena obey in all inertial frames of reference

the same laws.

From the statements formulated above follows the series important findings concerning the properties of space and time. First of all, new rules for the transition from one inertial frame of reference to another follow from them, within the framework of which the “obvious” Galilean transformations are only some special case, realized only when moving with velocities much less than c. To determine these new rules, consider light propagating from a point source located at the origin of a fixed reference frame K (Fig. 10.1 a).

The propagation of light can be represented as the propagation of a light front having the shape spherical surface in a frame of reference relative to which the light source is stationary. But according to Einstein's principle of relativity, the light front must also be spherical when it is observed in a frame of reference that is in uniform and rectilinear motion regarding the source.

Rice. 10.1 Light propagating from a point source located at the origin of a fixed frame of reference A light front must also be spherical when it is observed in a frame of reference that is in uniform and rectilinear motion relative to the source.

From this condition, we will now determine what the rules for the transformation of coordinates and time should be when moving from one inertial frame to another.

If the light source is at the origin of the frame of reference K, then for the light emitted at the moment t = 0, the equation for a spherical light front has the form

x 2 + y 2 + z 2 = (ct) 2 (10.1)

This equation describes a spherical surface whose radius R = ct

increases with time at a rate of s.

Let us denote the coordinates and time measured by the observer in the moving reference frame K "by letters with strokes: x", y", z", t". time, the origin of coordinates of the K1 system coincides with the position of the light source in the K system. Let, for definiteness, the K system move in the + x direction with a constant speed V relative to the K system (Fig. 10.1 b).

As we have already said, according to Einstein's second postulate, for an observer in a "primed" frame, the light front must also be spherical, that is, the equation of the light front in a moving frame must have the form

x "2 + y" 2 + z "2 \u003d c 2 t" 2 (10.2)

moreover, the value of the speed of light c here is the same as in the reference frame K. Thus, the transformations of coordinates and time from one of our reference frames to another must have such a property that, for example, after replacing with the help of these transformations in (10.2) " primed" quantities to "not primed" we must again obtain the equation of a spherical front (10.1).

It is easy to see that the Galilean transformations (9.3) do not satisfy this requirement. Recall that these transformations relate coordinates and time in two different systems reference by the following ratios:

x" = x - Vt, y" = y, z" = z, t" = t. (10.3)

If we substitute (10.3) into (10.2), we get

x 2 - 2xVt + V 2 t 2 + y 2 + z 2 \u003d c 2 t 2, (10.4)

which, of course, does not agree with equation (10.1). What should be the new transformations? First, since all systems are equal, the transition from some system to any other must be described by the same formulas (with its own value V), and the double application of transformations with the replacement of +V at the second step by

V should take us back to original system. Only transformations that are linear in x and t can have this property. It is useless to test for this relationship like

x" \u003d x l / 2 t 1/2, x" \u003d sin x

or the like.

Secondly, for V/c -> 0 these transformations must go over into Galilean transformations, the validity of which for low velocities cannot be questioned.

It is clear from equation (10.4) that we cannot leave the transformation t" = t unchanged if we want to destroy the unwanted terms -2xVt + V 2 t 2 in this equation, because in order to destroy them, it is necessary to add something to t .

Let's try transforming the view first:

x" = x-Vt, y" = y, z"= z, t" = t + bx, (10.5)

where b is a constant whose value must be determined. Then equation (10.2) takes the form

x 2 - 2Vxt + V 2 t 2 + y 2 + z 2 \u003d c 2 t 2 + 2c 2 bxt + c 2 b 2 x 2. (10.6)

Note that the terms on the left and right parts equalities containing the product xt cancel each other out if we accept

b \u003d -V / c 2, or t "= t-Vx / c 2. (10.7)

With this value of b, equation (10.6) can be rewritten as follows:

x 2 (1 - V 2 / s 2) + y 2 + z 2 \u003d c 2 t 2 (l - V 2 / s 2) . (10.8)

This is closer to equation (10.1), but there is still an undesirable factor 1 - (V 2 /c 2), by which x 2 and t 2 are multiplied.

We can also eliminate this factor if we finally write down the transformation of coordinates and time in the following form:

These are the famous Lorentz transformations, named after the Dutch theoretical physicist Hendrik Lorentz, who in 1904 derived formulas (10.9) and thus prepared the transition to the theory of relativity.

It is easy to check that when (10.9) is substituted into equation (10.2), the Lorentz transformations, as it should be, transform this equation into the equation of a spherical surface (10.1) in a fixed coordinate system. It is also easy to verify that when

V/c -> 0 the Lorentz transformations go over into the Galilean transformations (9.2).

10.2. Consequences from the Lorentz transformations. Length contraction and time dilation

From the Lorentz transformations, a number of consequences that are unusual from the point of view of Newtonian mechanics follow.

Length of bodies in different reference systems. Consider a rod located along the x-axis and resting relative to the reference frame K "(Fig. 10.2). Its length in this system is equal to l 0 = x" 2 - x "1 where x" 1 and x "2 are not changing with time t "coordinates of the bar ends. Relative to the system K, the rod moves together with the primed system with a speed v. To determine its length in this system, it is necessary to note

Rice. 10.2 reference systems K, K ". Relative to the system K, the rod moves together with the primed system at a speed v

coordinates of the ends of the rod x 1 and x 2 at the same time t 1 = t 2 = t. The difference between these coordinates l \u003d x 2 - x 1 will give the length of the rod measured in the K system. To find the relationship between l 0 and l, one should take that of the Lorentz transformation formulas that contains x", x and t, that is, the first of the formulas (10.9) According to this formula,

where we get

or finally

Thus, the length of the rod l, measured in the frame relative to which it moves, is less than the "own" length l 0 measured in the frame relative to which the rod is at rest. The transverse dimensions of the rod in both systems are the same. So, for a stationary observer, the dimensions of moving bodies in the direction of their movement are reduced, and the more, the more speed movement.

Duration of processes in different reference systems. Let at some point, which is motionless with respect to the moving system K", there occurs

some process lasting time At 0 = t" 2 - t" 1 . This may be the work of some device or mechanism, the oscillation of the pendulum of a clock, some change in the properties of the body, and so on. The beginning of the process corresponds in this system to the coordinate x "= a and time t" 1, to the end - the same coordinate x "2 \u003d x" 1 \u003d a and time t "2 Relative to the system K, the point at which the process occurs moves .According to formulas (10.9),

the beginning and end of the process in the system K correspond to the time points

where we get

Entering the notation t 2 - t 1 = At, we finally get:

In this formula, ∆t 0 is the duration of the process, measured by the clock in a moving frame of reference, where the body with which the process occurs is at rest. The interval At is measured by the clock of the system, relative to which the body is moving at a speed v. Otherwise, we can say that ∆t is determined by a clock that moves relative to the body with a speed v. As follows from (10.11), the time interval ∆t 0, measured by the clock, which is motionless relative to the body, turns out to be less than the time interval At, due to

measured by a clock moving relative to the body.

Note that for the relativistic factors (Lorentz factors) of a reference frame moving with a speed V and/or a particle moving with a speed v, the designations

G \u003d 1 / √ (1 - V 2 / s 2)

and correspondingly

γ \u003d 1 / √ (1 - v 2 / s 2).

If this does not lead to confusion, the notation γ is used for both quantities.

Considering the flow of the process from the system X, we can define ∆t as its duration, measured by a stationary clock, and ∆t 0 - as the duration, measured by a clock moving at a speed v. According to (10.11),

∆t0< ∆t

so it can be said that moving clocks run slower , than a resting clock (meaning, of course, that in everything except the speed of movement, the clocks are completely identical).

The time ∆t 0 counted by the clock moving together with the body is called the “own time” of this body. As seen from (10.11), own time always less than the time counted by the clock moving relative to the body.

The effect of time dilation is symmetrical with respect to both clocks under consideration: for both observers from different frames of reference, the clock of the observer moving relative to him will go slower. Time dilation is an objective consequence of the Lorentz transformations, which, in turn, are a consequence of the constancy of the speed of light in all frames of reference. It is necessary to emphasize the fact that relativistic effects are by no means speculative. To date, SRT has been experimentally confirmed with very good accuracy. Of course, as V/c -> 0 formulas (10.10), (10.11) transform to the trivial

nonrelativistic limit. To observe nontrivial effects, it is necessary to study objects with V ~ s.

Phenomena observed in the study of elementary particles can serve as examples. One of the most visual experiences, confirming the relationship (10.11), is the observation in the composition of cosmic rays of one of the types of elementary particles called muons. These particles are unstable - they spontaneously decay into others. elementary particles. Muon lifetime measured under conditions when they

motionless (or moving at low speed) is approximately 2 10 -6 s. It seemed

If, even moving almost at the speed of light, muons can travel from the moment of their birth to the moment of decay, only a path equal to approximately 3 10 8 m/s) (2 10 -6 s) = 600 m. in cosmic rays in the upper layers of the atmosphere at an altitude of 20-30 km, still manage to in large numbers reach earth's surface. This is explained by the fact that 2 * 10 -6 s is the muon's own lifetime, that is, the time measured by the clock, which would "move along with

him." The time counted by the clock of an experimenter connected with the Earth's surface turns out to be much longer due to the fact that the speed of muons is close to the speed of light. Therefore, it is not surprising that the experimenter observes a muon range much greater than 600 m. It is interesting to consider this effect from the point of view of an observer "moving along with the muon." For it, the distance flying to the Earth's surface is reduced to 600 m in accordance with formula (10.10), so that the muon has time to

to fly it in 2 10 -6 s, i.e., in "its own lifetime".

The most impressive consequence of the Lorentz transformations is relativity of simultaneity of spaced events . If two events A and B occurred simultaneously at one point in space, then in any coordinate system t A =t B . Specific values, for example, t A and t "A may be different, but in each system the equality t" A \u003d t "B will remain valid. If, however, at t A \u003d t B it turns out that

x A ≠ x in, then in any other system, as it obviously follows from the Lorentz transformations, t A ≠t B .

Why did this circumstance go unnoticed before Einstein? Before Einstein, the notion of the existence of absolute space and absolute time was preserved explicitly or implicitly. But if there is no absolute frame of reference, there is no absolute simultaneity. Not only does absolute space disappear, but absolute time, which, according to Newton, flows "always in the same way, regardless of anything external." The SRT time depends on the frame of reference. Depends on the reference system and the time interval between two events, and the distance between two points. In the mechanics of Galileo-Newton, the coordinates of points depend on the reference system, but the distance between points A and B

(x A - x B) 2 + (y A - y c) 2 + (z A - z B) 2 \u003d l 2

does not depend on the system. In SRT mechanics, this quantity ceases to be an invariant. The interval between events becomes independent of the reference system, determined by the relation

s 2 AB \u003d c 2 (t A - t B) 2 - (x A - x B) 2 + (y A - y c) 2 + (z A - z B) 2.

Time becomes on a par with spatial coordinates, or, as G. Minkowski said, “space itself and time itself plunge into the river of oblivion, and only a kind of their union remains to live.” This is especially evident if, following Minkowski, one chooses not t, as such, but ict as the fourth coordinate. Then the interval will be written in symmetrical shape:

However, one should not perceive the four-dimensional space of Minkowski as a simple analogue of our three-dimensional world. Yet the fourth coordinate preserves the most important difference from the other three - unidirectionality, which, in particular, determines

causal relationships. Traveling back in time, as it was, remains impossible.

In view of the fact that, according to Lorentz, in contrast to Galileo, time is transformed, in addition to coordinates, the law of addition of velocities noticeably changes. If in the frame K the body moves with a speed v, which has components along the coordinate axes v x v y v z and the frame K "moves with a speed V along the x axis, for the components of the body velocity in the frame K" we obtain

Taking into account the fact that

Although the y" and z" coordinates are equal to y and z, respectively, the velocity components

along these axes in different systems are different, since the rates of time flow differ.

Does not appear unexpected fact that if v x is equal in absolute value to the speed of light - c, then this value will not change upon transition to any other frame of reference. After all, it is the invariance of the speed of light that is the criterion for the validity of the Lorentz transformations.

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