On Poynting Vector and Photon Velocity Directions

G.R.Dixon, 5/28/2006

The direction of radiant energy flux can be expressed as the Poynting vector’s direction and/or as photon velocity direction. For example, suppose that in inertial frame K a plane wave propagates in the negative y-direction. Let its field vectors in the xz-plane be

, (1a)

. (1b)

The Poynting vector at the origin is then

. (1c)

This power flux can also be expressed as a flow of photons, each with velocity

. (2)

From the perspective of frame K’, moving at speed u relative to K, the general field transformations indicate that

, (3a)

, (3b)

, (3c)

, (3d)

. (3e)

If qS is the acute angle that S’ makes with the y’-axis, then

. (4)

In K’, vy (Eq. 2) transforms to

, (5a)

. (5b)

And of course

. (6)

Defining qv’ to be the acute angle that v’ makes with the y’-axis, we find that

. (7)

Thus

, (8)

quite as qS = qv = 0.

Let a point light source be at rest on the y-axis of K. D, the distance to the source, is adequately great that in the vicinity of the origin the waves are practically plane. At time t’ = t = 0 the source will also be found to be at y’ = D on the y’-axis. However, since qS’ = qv’ is nonzero, this is not where an observer at rest at the origin of K’ will perceive the source to be. Fig. 1 illustrates.

Figure 1

Perceived Source Position, K’ Observer

This disagreement regarding where the light source is measured to be and where it is perceived to be lies at the heart of stellar aberration. The perceived distance to the source (at t’ = 0) is

. (9)

Among other things, Eq. 9 implies that the light perceived by the K’ observer will be somewhat dimmer than that perceived by the K observer. The K’ observer will also perceive a frequency down shift, owing to time dilation, and a frequency up shift owing to the Doppler effect. Any net frequency shift will imply a disagreement regarding the energy of individual photons.

As previously discussed, the perceived offset (from the y’-axis at time t’ = 0) is not a general field effect. For if the source has a net charge, then the K’ observer will find its divergent electric field to point along the y’-axis at time t’ = 0. It is only the radiant energy flow, in the radiant part of the net field, which results in the perceived offset.

To the extent electric interactions in divergent fields are modeled as exchanges of "virtual" photons, this result may imply a fundamental difference between real light photons and virtual photons.