Page:LorentzContraction1921.djvu/2

second $\cos ^{2}\delta =0$, and $\mu =1$, the change will be the same as would be produced by a lengthening of $L_{1}$ in the ratio of 1 to $1+v^{2}/2c^{2}$. As no displacement of the fringes has been observed, we are led to the well-known hypothesis of a contraction of moving bodies in the direction of translation, in the ratio of 1 to $1-v^{2}/2c^{2}$.

We could now try to extend the above considerations to cases in which $v/c$, though always below 1, is no longer a small fraction. This would require somewhat lengthy calculations, into which, however, we need not enter here, because we know by the theory of relativity that the true value of the coefficient of contraction is ${\sqrt {1-v^{2}/2c^{2}}}$. I may remark here that there can be no question about the reality of this change of length. Suppose that, in studying the phenomena, we use a system of rectangular co-ordinates $x_{1},x_{2},x_{3}$, and a time $t$, and that in this system the velocity of light is $c$ in all directions. Further, let there be two rods, I. and II., exactly equal to each other, and both placed in the direction of $x_{1}$, I. at rest in the system of co-ordinates, and II. moving in the direction of its length with a velocity $v$. Then, certainly, if the length of a rod is measured by the differences of the values which the coordinate $x_{1}$ has at the two ends at one and the same instant $t$, II. will be shorter than I., just as it would be if it were kept at a lower temperature. I need scarcely add that if, by the ordinary transformation of the theory of relativity, we pass to new co-ordinates $x_{1}',x_{2}',x_{3}',t'$ in such a manner that in this system the rod II. is at rest, and if now we measure the lengths by the difference between the values of $x_{1}'$ which correspond to a definite value of $t'$, I. will be found to be the shorter of the two.

The question arises as to how far the dimensions of a solid body will be changed when its parts have unequal velocities, when, for example, it has a rotation about a fixed axis. It is clear that in such a case the different parts of the body will, by their interaction, hinder each other in their tendency to contract to the amount determined by ${\sqrt {1-v^{2}/c^{2}}}$. The problem can be solved by the ordinary theory of elasticity, provided only that this theory be first adapted to the principle of relativity. Indeed, we can still use Hamilton's principle: —

$\delta \int _{t_{1}}^{t_{2}}dt\int (T-U)dS=0$

(3)

($dS$, element of volume; $T$, kinetic, and $U$, potential, energy, both per unit of volume), if, by some slight modifications, the integral is made to be independent of the particular choice of co-ordinates. That this can be done, even in the general theory of relativity (theory of gravitation), is due to the possibility of expressing the length of a line-element in the four-dimensional space $x_{1},x_{2},x_{3},x_{4}\ \left(x_{4}=t\right)$ in "natural units" — i.e. in such a manner that the number obtained for it is the same whatever be the co-ordinates chosen — and of measuring angles in a similar way. As is well known, the length $ds$ of a line-element is given by the formula : —

$ds^{2}=\sum (ab)g_{ab}dx_{a}dx_{b}$

(4)

where the ten quantities $g_{ab}\ \left(g_{ab}=g_{ba}\right)$ are the gravitation potentials, and the angle $\delta$ between two elements is determined by

$\cos \delta ds\ d's=\sum (ab)g_{ab}dx_{a}d'x_{b}$

(5)

In the sums, each of the indices $a$ and $b$ is to be given the values 1, 2, 3, 4. When the value 4 is excluded, as will be the case in some of the following formulae, a Greek letter will be used for the index.

We can also find an invariant value $l$ for the distance between two material particles $P$ and $P'$ infinitely near each other. To this effect we have to consider the word-lines $L$ and $L'$ of these particles in the space $x_{1},x_{2},x_{3},x_{4}$. Let $Q$ be the point of $L$ corresponding to the chosen time $x_{4}$, and $Q'$ a point of $L'$ such that $QQ'$ is at right angles to $L$. Then the length of $QQ'$, determined by means of (4), will be the value required. Similarly, if $P''$ is a third particle, infinitely near $P$ and $P'$, and $Q''$ the point of its word-line so situated that $QQ''$ is perpendicular to $L$, the angle $P'PP''$ will be taken to be the angle between the elements $QQ'$ and $QQ''$ determined according to (5).

As to the co-ordinates $x_{1},x_{2},x_{3},x_{4}$, it may be recalled that, in a field free from gravitation, they may be chosen in such a manner ($x_{1},x_{2},x_{3}$ being at right angles to each other) that the velocity of light has the constant magnitude $c$; the potentials $g_{ab}$ will in this case have the values

$g_{11}=g_{22}=g_{33}=-1,\ g_{44}=c^{2},\ g_{ab}=0\,$ for $a\neq b\,$

These may be called the normal values of the potentials, and a system of co-ordinates for which they hold a normal system.

Let us now consider a solid body $M$, and let us first conceive it to be placed in a normal system of co-ordinates ($S_{0}$), and to be at rest in that system, free from all external forces. The body may then be said to be in its natural state, and its particles may be distinguished from each other by their co-ordinates $\xi ,\eta ,\zeta$ with respect to three rectangular axes fixed in the body. In all that follows, these quantities will be constant, and so will be the mass $\rho d\xi d\eta d\zeta$ of an element, $\rho$ being the density in the natural state.

We shall now suppose the body to be placed in a system of co-ordinates $x_{1},x_{2},x_{3},x_{4}\ (S)$, not necessarily normal, and to have some kind of motion in that system. It is this motion, in which $x_{1},x_{2},x_{3}$ will be definite functions of $\xi ,\eta ,\zeta$, and $x_{4}$, which we want to determine by means of Hamilton's principle properly modified.

In order to get the new U, I shall introduce the dilatations $\xi _{\xi },\eta _{\eta },\zeta$, and shearing strains $\xi _{\eta },\eta _{\zeta },\zeta$, with respect to the axes $\xi ,\eta ,\zeta$. These quantities are defined as follows: —

Let $P$, $P'$ be the particles $\xi ,\eta ,\zeta$, and $\xi +d\xi ,\eta ,\zeta$, and let $l$ be their distance in the state considered

no. 2677, vol. 106]