φ(v) φ(-v)
∴ φ(v) = φ(-v), ∴ φ(v) = 1.
§ 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
ξ2 + η2 + ζ2 = R2.
At time t = 0, the equation is expressed by means of (x, y, z, t,) as
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
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