, if the second system is moving relatively to the first. Only the invariant velocity preserves its value in all systems, no matter with what velocity they be moving relatively to one another. The value of this invariant velocity enters as a characteristic constant into the equations of transformation. Hence, if we wish to find those transformation relations that hold physically, we must find out the singular velocity that plays this fundamental part. To determine it is the task of the experimental physicist. If he sets up the hypothesis that a finite velocity can never be such an invariant, the general equations of transformation degenerate into the transformation-relationships of the principle of relativity of Galilei and Newton. (This hypothesis was made, albeit unconsciously, in Newtonian mechanics.) It had to be discarded after the results of the Michelson-Morley and Fizeau's experiment had justified the view that the velocity of light
plays the part of an invariant velocity. Then the general equations of transformation degenerate into those of the "special" principle of relativity of Lorentz and Einstein.
This remodelling of the co-ordinate-transformations of the principle of relativity led to discoveries of fundamental importance, as, for example, to the surprising fact that the conception of the "simultaneity" of events at different points of space, the conception on which all time-measurements are based, has only a relative meaning, that is, that two events that are simultaneous for one observer will not, in general, be simultaneous for another. [2] This deprived time-values of the absolute character which had previously been a great point of distinction between them and space co-ordinates. So much has been written in recent years about this question that we need not treat it in detail here.
[2]The assertion, "At a particular point of the earth the sun rises at 5 o'clock 10'6"," denotes that "the rising of the sun at a particular point of the earth is simultaneous with the arrival of the hands of the clock at the position 5 o'clock 10'6" at that point of the earth." In short, the determination of the point of time for the occurrence of an event is the determination of the simultaneity of happening of two events, of which one is the arrival of the hands of a clock at a definite position at the point of observation. The comparison of the points of time at which one and the same event occurs, as noted by several observers situated at different points, requires a convention concerning the times noted at the different points. The analysis of the necessary conventions led Einstein to the fundamental discovery that the conception "simultaneous" is only "relative inasmuch as the relation of time-measurements to one another in systems that are moving relatively to one another is dependent on their state of motion. This was the starting-point for the arguments that led to the enunciation of the "special principle of relativity."
The new form of the equations of transformation by no means exhausts the whole effect of the principle of relativity upon classical mechanics. The change which it brought about in the conception of mass was almost still more marked.
Newtonian mechanics attributes to every body a certain inertial mass, as a property that is in no wise influenced by the physical conditions to which the body is subject. Consequently, the Principle of the Conservation of Mass also appears in classical mechanics as independent from the Principle of the Conservation of Energy. The special principle of relativity shed an entirely new light on these circumstances when it led to the discovery that energy also manifests inertial mass, and it hereby fused together the two laws of conservation, that of mass and that of energy, to a single principle. The following circumstance moves us to adopt this new view of the conception of mass.
The equations of motion of Newtonian mechanics do not preserve their form when new co-ordinates have been introduced with the help of the Lorentz-Einstein transformations. Consequently, the fundamental equation of mechanics had to be remodelled accordingly. It was then found that Newton's Second Law of Motion: force = mass x accel. cannot be retained, and that the expression for the kinetic energy of a body may no longer be furnished by the simple expression
, which involves the mass and the velocity. Both these results are consequences of the change which we found necessary to make in our view of the nature of the mass of matter. The new principle of relativity and the equations of electrodynamics led, rather, to the fundamentally new discovery that inertial mass is a property of every kind of energy, and that a point-mass, in emitting or absorbing energy, decreases or increases, respectively, in inertial mass, as is shown in [Note 5] for a simple case. The new kinematics thereby disposes of the simple relation between the kinetic energy of a body and its velocity relatively to the system of reference. The simplicity of the expression for the kinetic energy in Newtonian mechanics rendered possible the revolution of the energy of a body into that (kinetic) of its motion and of the internal energy of the body, which is independent of the former. Let us consider, for example, a vessel containing material particles, no matter of what kind, in motion. If we resolve the velocity of each particle into two components, namely, into the velocity, common to all, of the centre of gravity and the accidental velocity of a particle relative to the centre of gravity of the system, then, according to the formulæ of classical mechanics, the kinetic energy divides up into two parts: one that contains exclusively the velocity of the centre of gravity and that represents the usual expression for the kinetic energy of the whole system (mass of the vessel plus the mass of the particles), and a second component that involves only the inner velocities of the system. This category of internal energy is no longer possible so long as the expression for the kinetic energy contains the velocity not merely as a quadratic factor; so we are led to the view that the internal energy of the body comes into expression in the energy due to its progressive motion, and, indeed, as an increase in the inertial mass of the body.