Relativistic mechanics is a mechanics that studies the motion of bodies with velocities close to the speed of light.
Based on the special theory of relativity, we analyze the concept of simultaneity of two events that occur in different inertial reference frames. This is the law of Lorentz. Let a fixed HOU system and a system X1O1U1 be given, which moves relative to the HOU system with speed V. We introduce the notation
HOU = K, X1O1U1 = K1.
We assume that in two systems there are special installations with photocells that 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 are, respectively, B and B1 in the basement of the placement of electric lamps. Such bulbs are simultaneously lit at the moment when B and B1 are one opposite the other.
Suppose that at the initial time, the systems K and K1 are aligned, but their devices are offset from each other. During the movement of K1 relative to K at a speed V at some point in time, B and B1 are equal. At this point in time, the bulbs that are at these points will light up. The observer, who is in the K1 system, captures the simultaneous appearance of light in A1 and C1. Similarly, an observer in system K captures the simultaneous occurrence of light in A and C. Moreover, if an observer in system K registers the propagation of light in system K1, he will notice that the light that has exited B1 does not reach A1 and C1 at the same time. . This is due to the fact that the system K1 moves with speed V relative to the system K.
This experience confirms that according to the observer’s clock in system K1, events in A1 and C1 occur simultaneously, and according to the observer’s clock in system K, such events will not be simultaneous. That is, the period of time depends on the state of the reference system.
Thus, the results of the analysis show that the equality, which is accepted in classical mechanics, is considered invalid, namely: t = t1.
Given the knowledge from the foundations of the special theory of relativity and as a result of conducting and analyzing many experiments, Lorentz proposed equations (Lorentz transformations) that improve the classical Galilean transformations.
Let the system K contain 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, change with respect to the Galilean transformations. The x1 and x2 coordinates, 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, there is a relativistic reduction in the length of the segment due to an increase in speed.
From the Lorentz transformation, we conclude the following: when moving at a speed that is close to the speed of light, the so-called time dilation (the paradox of twins) occurs .
Let in the system K the time between two events is defined as follows: t = t2-t1, and in the system K1 the time between two events is determined as follows: t = t22-t11. 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 equal to zero.
In the moving system K, a slowdown occurs, which is measured in a fixed system.
It is known from mechanics that if bodies move relative to a certain coordinate system with a speed V1, and such a system moves relative to a fixed coordinate system with a speed V2, then the speed of bodies relative to a fixed coordinate system is determined as follows: V = V1 + V2.
This formula is not suitable for determining the velocity of bodies in relativistic mechanics. For such mechanics, where Lorentz transformations are used, the following formula is valid:
V = (V1 + V2) / (1 + V1V2 / cc).