Page:CarmichealPostulates.djvu/15

or

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

But $t_{1}$ and ${\overline {t}}$ are measures of the same interval of time, $t_{1}$ being in units belonging to $S_{2}$ and ${\overline {t}}$ being in units belonging to $S_{1}$. Hence to the observer on $S_{1}$ the ratio of his time unit to that of the system $S_{2}$ appears to be ${\sqrt {1-\beta ^{2}}}:1$. On the other hand, it may be shown in exactly the same way that to the observer on $S_{2}$ the ratio of his time unit to that of the system $S_{1}$ appears to be ${\sqrt {1-\beta ^{2}}}:1$. That is, the time units of the two systems are different and each observer comes to the same conclusion as to the relation which the unit of the other system bears to his own.

This important and striking result may be stated in the following theorem:

Theorem III. If two systems of reference $S_{1}$ and $S_{2}$ move with a relative velocity v and β is defined as the ratio of v to the velocity of light estimated in the manner indicated above, then to an observer on $S_{1}$ the time unit of $S_{3}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the time unit of $S_{2}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (MVLA).

Let us now bring into play our postulate $R'$. In theorem I. we have already seen that a logical consequence of M and $R'$ is that the velocity of light, as observed on a system of reference, is independent of the direction of motion of that system. Now, if c and ${\overline {c}}$ as estimated above differ at all, that difference can be due only to the direction of motion of $S_{1}$, as one sees readily from postulate $R'$ and the method of determining these quantities. Hence the statement which we made above as assumption A is a, logical consequence of postulates M and $R'$. Therefore we are led to the following corollary of the above theorem:

Corollary. Theorem III. may be stated as depending on (MVLR') instead of on (MVLA).

Let us now go a step further and employ postulate $R''$. From theorem I. and postulates $R'$ and $R''$ it follows that the observed velocity of light is a pure constant for all admissible methods of observation. If we make use of this fact the preceding result may be stated in the following simpler form:

Theorem IV. If two systems of reference $S_{1}$ and $S_{2}$ move with a relative velocity v and β is the ratio of v to the velocity of light, then to an observer on $S_{1}$ the time unit of $S_{1}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the time unit of $S_{2}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (MVLR).

Let us subject these remarkable results to a further analysis. Theorem III., its corollary and theorem IV. all agree in the extraordinary conclusions