Page:CarmichaelPhilo.djvu/7

$t_{v}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {1}{2}}}}$ $l_{v}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}$

IV. The Notion of Length.

In the preceding section we saw that two observers A and B on relatively moving systems of reference $S_{1}$ and $S_{2}$ respectively are in disagreement as to units of length along a line l parallel to their line of relative motion. This disagreement is of a very peculiar character. To A it appears that $B$'s units are longer than his own. On the other hand, it seems to B that his units are shorter than $A$'s. In the two cases the apparent ratio is the same; more precisely, the unit which appears to either observer to be the shorter seems to him to have the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that which appears to him to be the longer. Although they are thus in disagreement there is yet a certain symmetry in the way in which their opinions diverge.

Let us suppose that these two observers now undertake to bring themselves into a closer agreement in measurements of length along the line l. Suppose that B agrees arbitrarily to shorten his unit so that it will appear to A that the units of A and B are of the same length. Then, so far as A is concerned, all difficulty has disappeared. How is B affected by this change? We see that the difficulty which he experienced is not disposed of; on the other hand it is greater than before. Already, it seemed to him that his unit was shorter than $A$'s. Now, since he has shortened his unit, the divergence appears to him to be increased. Moreover, the symmetry which we found in the former case is now absent.

Furthermore, if any other changes in the units of A and B are made we shall always find difficulties as great as or greater than those which we encountered in the initial case. There is no other conclusion than this: We are face to face with an essential difficulty — one that is not to be removed by any mere artifice. What account of it shall we render to ourselves?

This much is already obvious: The length of an object is not an absolute something; it depends upon the measurer in an essential way.

We have just spoken of the length of an object, a material object. One can hardly refrain from raising the question of the abstract notion of length as apart from any material thing having length. But this problem has new difficulties of its own. All lengths of which I have experience are lengths of material objects or lengths between material