The Lorentz transformations

Relativistic mechanics - mechanics, which studies the motion of bodies with velocities close to the speed of light.

On the basis of the special theory of relativity, let us analyze the concept of simultaneity of two events that occur in different inertial frames of reference. This is the law of Lorentz. Suppose that we are given a stationary XOY system and a system X1O1Y1 that moves relative to the XOY system with velocity V. We introduce the notation:

ХОУ = К, Х1О1У1 = К1.

We will assume that in two systems there are special installations with photocells, which are located at the points AC and A1C1. The distance between them will be the same. Exactly in the middle between A and C, A1 and C1 is, respectively, B and B1 in the band of placement of electric lamps. These bulbs are simultaneously lit at the moment when B and B1 are facing one another.

Suppose that at the initial instant of time the systems K and K1 are combined, but their instruments are shifted relative to each other. During the motion of K1 relative to K at a speed V at some instant of time, B and B1 will equalize. At this point in time, the bulbs that are in these points will light up. The observer, which is in the system K1, fixes the simultaneous appearance of light in Al and Cl. Similarly, the observer in the K system fixes the simultaneous appearance of light in A and C. At the same time, if the observer in the K system fixes the propagation of light in the K1 system, he will notice that the light that went out of B1 does not reach simultaneously A1 and C1 . This is due to the fact that the system K1 moves with velocity v relative to the system K.

This experience confirms that according to the clock of the observer in the K1 system, the events in A1 and C1 occur simultaneously, and according to the clock of the observer in the K system, such events turn out to be not simultaneous. That is, the time interval depends on the state of the frame of reference.

Thus, the results of the analysis show that the equality, which is accepted in classical mechanics, is considered invalid, namely: t = t1.

Taking into account the knowledge from the foundations of the special theory of relativity and as a result of carrying out and analyzing a multitude of experiments, Lorentz proposed equations (Lorentz transformations) that improve the classical Galilean transformations.

Suppose that in the system K there is a segment AB whose coordinates of the ends are A (x1, y1, z1), B (x2, y2, z2). It is known from the Lorentz transformation that the coordinates y1 and y2, as well as z1 and z2, vary with respect to the Galilean transformations. The coordinates x1 and x2, in turn, vary with respect to the Lorentz equations.

Then the length of the segment AB in the system K1 is directly proportional to the change in the segment A1B1 in the system K. Thus, a relativistic shortening of the length of the segment is observed due to an increase in the velocity.

From the Lorentz transformation, the conclusion is the following: when moving at a speed that is close to the speed of light, the so-called time delay (the twins paradox) occurs .

Let the time between two events in system K be defined as t = t2-t1, and in the system K1 the time between two events is defined as t = t22-t11. The time in the coordinate system with respect to which it is considered to be fixed is called the system's own time. If the proper time in the system K is greater than the proper time in the system K1, then we can say that the speed is not zero.

In the mobile system K, the time is slowed down, which is measured in a stationary system.

It is known from mechanics that if the bodies move relative to a certain coordinate system with a velocity V1, and such a system moves relative to a fixed coordinate system with a velocity V2, then the velocity of the bodies with respect to the fixed coordinate system is defined as V = V1 + V2.

This formula is not suitable for determining the velocity of bodies in relativistic mechanics. For such a mechanics, where the Lorentz transformations are used, the following formula holds:

V = (V1 + V2) / (1 + V1V2 / cc).