Page:SahaElectrodynamics.djvu/14

§ 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

$\xi ^{2}+\eta ^{2}+\zeta ^{2}=R^{2}$

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

${\frac {x^{2}}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{2}}}+y^{2}+z^{2}=R^{2}$.

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

$R{\sqrt {1-{\frac {v^{2}}{c^{2}}}}},R,R$.

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 $1:{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$ ; 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