Page:PrasadSpaceTime.djvu/6

fact, Lorentz built up a hypothesis the success of which lies precisely in the invariance of Optics for the group $G_{c}$. According to Lorentz every moving body should experience a contraction in the direction of motion in the ratio $1:{\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}$, where $v$ is the velocity of the body. This hypothesis sounds extremely fantastic. For, the contraction is not to be thought of as anything like the consequence of resistances in æther but purely as a present from above — as a concomitant of the circumstances of motion.

I will now show with our figure that the hypothesis of Lorentz is fully equivalent to the new conception of space and time by means of which it becomes much more understandable. For the sake of simplicity let us not reflect upon $y$ and $z$, and let us imagine to ourselves a spacially one-dimensional world. Then a strip perpendicular like the $t$-axis and a strip inclined to the $t$-axis are respectively the images of the course of a body at rest and of that of a body in uniform motion; each of these bodies retaining a constant spacial extension. If $OA'$ is parallel to the second strip, we may introduce $t'$ as the co-ordinate of time and $x'$ as the co-ordinate of space, and then the second body appears to be at rest and the first to be in uniform motion. We suppose now that the first body conceived to be at rest has the length $l$; in other words, the section $PP$ of the first strip on the $x$-axis is equal to $l\cdot OC'$, where $OC$ represents the unit of measure on the $x$-axis. Again we suppose that the second body conceived to be at rest has the same length $l$; in other words, the cross-section of the second strip parallel to the $x'$-axis, i.e., $Q'Q'$, is equal to $l\cdot OC'$. We have now these two bodies as images of two equal Lorentzian electrons, one at rest and the other in uniform motion. However, if we retain the original co-ordinates $x,t$, then the extension of the second electron is to be considered to be $QQ$, the section of the corresponding strip parallel to the axis of $x$. Now, it is evident that $QQ=l\cdot OD'$ since $Q'Q'=l\cdot OC'$. An easy calculation shows that, if $v$ represent the ${\tfrac {dx}{dt}}$ for the second strip,

$OD'=OC\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}},$

and, consequently, also

$PP:QQ=1:{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.$

And this is the sense of Lorentz's hypothesis of the contraction of electrons on account of their motion. On the other hand, if we look upon the second electron as at rest and thus adopt the system of reference $x',t'$,