[Pg 47] is an invariant which refers to an infinitely short portion of the four-dimensional line which represents the motion of the material particle. The physical significance of the invariant may easily be given. If the time axis is chosen in such a way that it has the direction of the line differential which we are considering, or, in other words, if we reduce the material particle to rest, we shall then have ; this will therefore be measured by the light-seconds clock which is at the same place, and at rest relatively to the material particle. We therefore call the proper time of the material particle. As opposed to , is therefore an invariant, and is practically equivalent to for motions whose velocity is small compared to that of light. Hence we see that has, just as the , the character of a vector; we shall designate () as the four-dimensional vector (in brief, 4-vector) of velocity. Its components satisfy, by (38), the condition We see that this 4-vector, whose components in the ordinary notation are is the only 4-vector which can be formed from the velocity components of the material particle which are defined in three dimensions by [Pg 48] We therefore see that must be that 4-vector which is to be equated to the 4-vector of momentum and energy whose existence we have proved above. By equating the components, we obtain, in three-dimensional notation, We recognize, in fact, that these components of momentum agree with those of classical mechanics for velocities which are small compared to that of light. For large velocities the momentum increases more rapidly than linearly with the velocity, so as to become infinite on approaching the velocity of light. If we apply the last of equations (43) to a material particle at rest ( = 0), we see that the energy, of a, body at rest is equal to its mass. Had we chosen the second as our unit of time, we would have obtained Mass and energy are therefore essentially alike; they are only different expressions for the same thing. The mass of a body [Pg 49] is not a constant; it varies with changes in its energy.[11] We see from the last of equations (43) that becomes infinite when approaches 1, the velocity of light. If we develop in powers of , we obtain, The second term of this expansion corresponds to the kinetic energy of the material particle in classical mechanics. [11]The emission of energy in radioactive processes is evidently connected with the fact that the atomic weights are not integers. Attempts have been made to draw conclusions from this concerning the structure and stability of the atomic nuclei. Equations of Motion of Material Particles. From (43) we obtain, by differentiating by the time , and using the principle of momentum, in the notation of three-dimensional vectors, This equation, which was previously employed by H. A. Lorentz for the motion of electrons, has been proved to be true, with great accuracy, by experiments with -rays.