Page:The principle of relativity (1920).djvu/48

 § 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks.

Let us consider a rigid sphere (i.e., one having a spherical figure when tested in the stationary system) of radius R which is at rest relative to the system (K), and whose centre coincides with the origin of K then the equation of the surface of this sphere, which is moving with a velocity v relative to K, is

ξ^2 + η^2 + ζ^2 = R^2

At time t = 0, the equation is expressed by means of (x, y, z, t,) as

x^2/([sqrt](1 - v^2/c^2))^2 + y^2 + z^2 = R^2.

A rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition—when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes

R [sqrt](1 - v^2/c^2), R, R.

Therefore the y and z dimensions of the sphere (therefore of any figure also) do not appear to be modified by the motion, but the x dimension is shortened in the ratio 1: [sqrt](1 - v^2/c^2); the shortening is the larger, the larger is v. For v = c, all moving bodies, when considered from a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become