Why does the light intensity increase as I approach a distance light source?

Analogy: assume that I have constant rain fall and I have a water bucket to collect this rain. If I am rest relative to the earth, I will catch a certain amount of rain. However, if I now move towards the rain, I will increase the amount of water I collect. Now I want to apply this idea to a source emitting photons instead of rain drops.

I image a distance light source (distance star) that is emitting a constant number of photons such that I can ignore the $1/r^2$ fall off of the intensity (that is, the intensity does not change very much over some appreciate distance). So instead of a water bucket, I now have a light collector that measures light intensity. Here is my question: Does the light intensity measured by my light collector that is moving towards the distant star increase or stay the same when compared to a light collector at rest realtive to the distant star?

My first thought was that the light intensity would increase because the light is blue shifted and higher energy photons will therefore produce a higher intensity. However, I believe that there might be another contribution due to length contraction. Since an observer moving towards a light source has to account for length contraction, does this mean that there is an increase in photon density? If so, this higher photon density in the frame of the light collector will also contribute to increasing the light intensity.

Can someone verify or correct my thinking.

In this question I showed that the power received from a light source moving away from the receiver at speed $\beta=v/c$ is $$P = \frac{P_0}{\gamma^2 (1+\beta)^2} = P_0 \frac{1-\beta}{1+\beta}$$ taking into account redshift, time dilation, and the effect of the changing travel distance for the photons.

Your question is essentially the same, except that you have the source and receiver approaching; in that case the sign of $\beta$ changes and you get a small increase in the received power.

• Thanks Rob, the link was real clear in answering my question. – Carlos Jun 27 '14 at 17:10

I don't think this is necessarily a relativity problem. The situation you describe is basically a plane wave (of either raindrops or photons), so the number of particles per unit area per unit time hitting a surface remains constant. In either case (rain or light), moving the surface towards the source will increase the number of particles you collect as you "sweep out" a volume.

Now, there's another wrinkle: raindrops have both mass and momentum, and you're not measuring the relative momentum as you move the bucket upwards (just the total mass). It gets worse when you use a "light detector," as exactly what is "detected" depends on the type of detector. A silicon photodiode, for example, detects a certain percentage of incident photons within its spectral sensitivity range. You get no information about the energy of each photon, just that it created a photoelectron. Perhaps you need a bolometer (essentially a blackbody which absorbs photons and converts to heat), which would give you an estimate of the total energy collected.

• Great comments. If you have the time, I would love to see how you can calculate this using EM waves. Rob's link above uses special relativity and it is real clear. Even though I didn't stated, I always assumed a bolometer-type detector was used to estimate the energy collected. – Carlos Jun 27 '14 at 15:46
• I'd be very interested to see whether a non-relativistic approach reproduces the small-$\beta$ result of $P/P_0 \approx 1+2\beta$. – rob Jun 27 '14 at 16:27
• @rob hard to say, since small$\beta$ implies $P = P_0$ . My guess is that unless I include the energy required to maintain constant detector speed (compensating for the absorbed photon momentum), there won't be any change. – Carl Witthoft Jun 27 '14 at 17:06
• I don't agree that the first-order effect is zero; your argument about "sweeping out a volume" gives $P/P_0 = 1+\beta$. Any sufficiently massive detector can have negligible speed change after absorbing a given amount of light. – rob Jun 27 '14 at 17:19