Page:CarmichealPostulates.djvu/14

Now, from the assumed relations among the systems S, $S_{1}$ and $S_{2}$ and from the homogeneity of space it follows that the two observations which we have supposed to be made must lead to the same estimate for the velocity of light. This is readily seen from the fact that the observations were made in such a way that the effect due to either the absolute value or the direction of the motion of the systems $S_{1}$ and $S_{2}$ is the same in the two cases. In other words, if we denote by $L_{1}$ and $L_{2}$ the quantities measured on $S_{1}$ and $S_{2}$ respectively, then the relation of $L_{1}$ to $S_{1}$ is precisely the same as that of $L_{2}$ to $S_{2}$; and hence the numerical results are identical, as one sees from the definition of systems of reference. Therefore we have $c_{1}=c$.

Let us now suppose that the observer at A is watching the experiment at B, To him it appears that B is moving with a velocity v; we shall assume that the apparent motion is in the direction indicated by the arrow. To the observer at B it appears that the ray of light traverses BD from B to D and returns along the same line to B. To the observer at A it appears that the ray traverses the line BEF, F being the point which B has reached by the time that the ray has returned to the observer at this point. If EG is perpendicular to $l_{2}$ and $d_{1}$ is the length of EF as measured by means of units belonging to $S_{1}$, then, evidently, GF (when measured in the same units) is $\beta d_{1}$ where $\beta =v/{\overline {c}}$ and ${\overline {c}}$ is the (apparent) velocity of light as estimated in this case by the observer at A. From the right triangle EFG it follows at once that we have

$d_{1}={\frac {d}{\sqrt {1-\beta ^{2}}}}$

Now, if ${\overline {t}}$ is the time which is required, according to the observer at A, for the light to traverse the path BEF, then we have

${\frac {2d_{1}}{\overline {t}}}={\frac {2d}{{\overline {t}}{\sqrt {1-\beta ^{2}}}}}={\overline {c}}$

So far in our argument in this section we have employed only those of our postulates which are generally accepted by both the friends and the foes of relativity. Now we come to the place where the men of the two camps must part company.

Let us introduce for the moment the following additional hypothesis.

Assumption A. The two estimates c and ${\overline {c}}$ of the velocity of light obtained as above by the observer at A are equal.

Now we have shown that c is equal to $c_{1}$, Hence we may equate the values of $c_{1}$ and ${\overline {c}}$ given above; thus we have

${\frac {2d_{1}}{t_{1}}}={\frac {2d}{{\overline {t}}{\sqrt {1-\beta ^{2}}}}}$