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 A0 /8π, by the principle of relativity, as the energy of light in the moving system.

Thus A0 /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:

(x − lct)2 + (y − mct)2 + (z − nct)2 = R2 .

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

(βξ − lβξv/c)2 + (η − mβξv/c)2 + (ζ − nβξv/c)2 = R2 .

If S is the volume of the sphere, and S0 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 E0 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 A0002 (−c cos φ000 + 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.

§ 9. Transformation of the Maxwell-Hertz Equations

when Convection-Currents are Taken into Account

We start from the equations denotes 4π times the density of electricity, and (ux , uy , uz ) the velocity-vector of the charge. If we imagine the electric charges to be invariably coupled to small rigid bodies (ions, electrons), these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies.

Let these equations be valid in the system K, and transform them, with the assistance of the equations of transformation given in §§ 3 and 6, to the system k. We then obtain the equations

19 Since—as follows from the theorem of addition of velocities (§ 5)—the vector (uξ , uη , uζ ) is nothing else than the velocity of the electric charge, measured in the system k, we have the proof that, on the basis of our kinematical principles, the electrodynamic foundation of Lorentz’s theory of the electrodynamics of moving bodies is in agreement with the principle of relativity. In addition I may briefly remark that the following important law may easily be deduced from the developed equations: If an electrically charged body is in motion anywhere in space without altering its charge when regarded from a system of co-ordinates moving with the body, its charge also remains—when regarded from the “stationary” system K—constant.