§ 8. Transformation of the Energy of the Rays of Light. Theory of the Radiation-pressure on a perfect mirror.
Since A²/8π is equal to the energy of light per unit volume, we have to regard A²/8π as the energy of light in the moving system. A′²/A² would therefore denote the ratio between the energies of a definite light-complex “measured when moving” and “measured when stationary,” the volumes of the light-complex measured in K and k being equal. Yet this is not the case. If l, m, n are the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface
(x - clt)² + (y - cmt)² + (z - cnt)² = R²,
which expands with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses, when regarded from the system k, i.e., the energy of the light-complex relative to the system k.
Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time τ = 0, the equation:—
If S = volume of the sphere, S′ = volume of this ellipsoid, then a simple calculation shows that:
If E denotes the quantity of light energy measured in the stationary system, E′ the quantity measured in the moving system, which are enclosed by the surfaces mentioned above, then
