On The Electrodynamics Of Moving Bodies Part:9

§ 8. Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors

Since A²/8π equals the energy of light per unit of volume, we have to regard A'²/8π, by the principle of relativity, as the energy of light in the moving system. Thus A'²/A² would be the ratio of the “measured in motion” to the “measured at rest” energy of a given light complex, if the volume of a light complex were the same, whether measured in K or in k. But this is not the case. If l, m, n are the direction-cosines of the wave-normals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:—

We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k

The spherical surface—viewed in the moving system—is an ellipsoidal surface, the equation for which, at the time τ = 0, is

If S is the volume of the sphere, and S' that of this ellipsoid, then by a simple calculation

Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and E' when measured in the moving system, we obtain

and this formula, when φ = 0, simplifies into

It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.

Now let the co-ordinate plane ξ = 0 be a perfectly reflecting surface, at which the plane waves considered in § 7 are reflected. We seek for the pressure of light exerted on the reflecting surface, and for the direction, frequency, and intensity of the light after reflexion.

Let the incidental light be defined by the quantities A, cos φ, ν(referred to system K). Viewed from k the corresponding quantities are

For the reflected light, referring the process to system k, we obtain

Finally, by transforming back to the stationary system K, we obtain for the reflected light

The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently A²(c cos φ−v)/8π. The energy leaving the unit of surface of the mirror in the unit of time is A'''²(−c cos φ''' +v)/8π.

The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the product Pv, where P is the pressure of light, we obtain

In agreement with experiment and with other theories, we obtain to a first approximation

All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.

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