At 05:47 PM 9/3/2008, Eugene Salamin wrote:

Test your physical intuition. We know that when light reflects from a stationary mirror

(1) The frequency of the reflected light equals the frequency of the incident light,

(2) The direction of the incident ray, the direction of the reflected ray, and the mirror normal are coplanar, and

(3) The angle of reflection equals the angle of incidence.

Suppose however, that the mirror is moving parallel to its surface. Do these three principles of reflection continue to hold?

Gene

The links to papers reporting experimental tests give only the abstracts. The Nature paper describes an experiment with frequency resolution of one part in 10^16. Unfortunately, the authors, in their excitement, forgot to include the conclusions in the abstract. Can someone tell us what were the measured results? Meanwhile, I will stick out my neck, and present my analysis of the problem. Let's solve the problem two ways, first by Lorentz transforming from the rest frame of the mirror, and second by a clever group theoretic trick. A plane wave of light is is described by a wave vector k which points in the direction of propagation and has length 2 pi/lambda, lambda being the wavelength. The angular frequency (= 2 pi frequency) is c |k| = ck. In relativity, this 3-vector is combined with angular frequency to form the 4-vector which gets Lorentz transformed. For simplicity assume units with c = 1. In the rest frame of the mirror, which is taken to be the xy-plane, the wave vectors can be resolved in direction cosines, and then the incident and reflected wave 4-vectors are ki = k (1, cos a, cos b, -cos c), kr = k (1, cos a, cos b, cos c). If we now transform to a frame moving with velocity v/c = tanh s along the x-axis, the Lorentz transformation matrix is block diagonal. It is the identity in the yz-plane, and in the tx plane is [[cosh s, sinh s], [sinh s, cosh s]]. In the frame in which the mirror is moving, we thus have ki'[t] = kr'[t] = k (cosh s + sinh s cos a) = k', ki'[x] = kr'[x] = k (sinh s + cosh s cos a) = k' cos a', ki'[y] = kr'[y] = k cos b = k' cos b', - ki'[z] = kr'[z] = k cos c = k' cos c'. Although there is a Doppler shift, from k to k', between the two frames, in the one moving frame, the frequencies of the incident and reflected light are the same. Also in the one moving frame, direction cosines of the two light rays are the same parallel to the mirror, and reversed perpendicular to the mirror, just as for a fixed mirror. Thus all three reflection principles hold for a mirror moving parallel to its surface. There is one caveat. The frequency in the rest frame of the mirror is different from that applied in the lab frame. If the material no longer behaves as a mirror in its rest frame at the shifted frequency, then it will not behave as a mirror in the lab frame. Now let's solve the reflection problem the clever way. The incident plane wave has a space-time variation given by (again with c = 1) exp(i (ki[x] x + ki[y] y + ki[z] z - ki t)). The physical arrangement is invariant under translation in the xy-plane. Hence the factors exp(i ki[x] x) and exp(i k[y] y), which constitute a one-dimensional representation of the translation group, should continue to describe the reflected wave. Likewise, the situation is invariant under translation in time, so the factor exp(- i ki t) should also persist. So kr[x] = ki[x], kr[y] = ki[y], kr = ki. We also must have k[x]^2 + k[y]^2 + k[z]^2 = k^2. So we can solve for kr[z]: k[z] = +/- ki[z]. The incident wave goes toward the mirror, the reflected wave away, so kr[z] = - ki[z]. This demonstrates the three stated principles of reflection. The proof remains valid whether the mirror is stationary or moving uniformly at constant velocity parallel to its surface. [What happens if light reflects from a rotating disk?] The principle of translation invariance implies that the wavelength is sufficiently long that it cannot sense the microcrystalline or atomic structure of the mirror material. The same idea can be used to derive Snell's law for refraction for a stationary medium of refractive index n. It is a consequence of Maxwell's equations that the transmitted wave vector kt has length n times longer than in vacuum. This is clear since the frequency stays the same (time translation invariance), the speed of light goes down by n, the wavelength goes down by n, and the wave vector goes up by n. For the transmitted wave vector, by xy translation invariance, kt[x] = ki[x], kt[y] = ki[y]. Thus kt[x]^2 + kt[y]^2 + kt[z]^2 = kt^2, ki[x]^2 + ki[y]^2 + kt[z]^2 = n^2 ki^2, ki^2 sin(theta[i])^2 + n^2 ki^2 cos(theta[t])^2 = n^2 ki^2, sin(theta[i])^2 = n^2 sin(theta[t])^2, sin(theta[i]) = n sin(theta[t]), Snell's law. There is a physical interpretation of the relation between translation invariance and the preservation of the exp(i k[x] x) factors. For a wave exp(i k.r), the momentum of a photon is p = hbar k. The preservation of exp(i k[x] x) is the preservation of the momentum component p[x]. A system that is invariant under translation in a certain direction cannot alter the component of photon momentum along that direction. Gene