Page:CarmichaelPhilo.djvu/6

Let us consider the units of length in $S_{1}$ and $S_{2}$. If A and B make measurements of length in the direction MN their results are in perfect agreement in the theory of relativity as in the classical mechanics; but the state of matters is very different when measurement is made in a line parallel to the line of relative motion of $S_{1}$ and $S_{2}$, as one sees from the following theorem:

Let l denote a line parallel to the line of relative motion of $S_{1}$ and $S_{2}$. Then to an ob server A on $S_{1}$ the unit of length of $S_{1}$ along l appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the unit of length of $S_{2}$ along l appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$.

Thus it appears that when A and B are measuring length in a line parallel to their line of relative motion they are in hopeless disagreement. What this result signifies we shall attempt to explain in section IV.

From the results stated above it may be shown that:

The velocity of light is a maximum which the velocity of a material body may approach but can never reach.

If one brings into consideration the mass of a moving body another result, essentially equivalent to that just given, may also be obtained; this may be stated as follows (see further discussion in section VIII. below) :

No finite force is sufficient to give a material particle a velocity as great as that of light.

In classical mechanics it is customary to assume that the mass of a given body is constant and that it is independent of the direction in which the mass is measured. But if the principle of relativity is accepted it follows that neither of these conclusions is valid. It turns out that the mass of a body depends on its velocity and also on the direction, relative to the line of motion of the body, along which that mass is measured. For convenience in distinguishing these measurements of mass we shall speak of the "transverse mass" as that with which we must reckon when we consider motion in a line perpendicular to the line of relative motion of the systems $S_{1}$ and $S_{2}$; when the motion is parallel to this line we shall speak of the "longitudinal mass" of the body.

The general conclusions of the theory of relativity concerning mass may now be stated as follows:

Let $m_{0}$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S, denote by $t_{v}$ its transverse mass, that is, its mass in a line perpendicular to its line of motion. Similarly, denote by $l_{v}$ its longitudinal mass. Then we have