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This is the updated, more precise question--is this a paradox?:

Suppose a rocket traveling close to the velocity of light which emits a single photon from its midpoint at point A, illustrated below. The rocket is equipped with a single detector drawn in green at the front of the rocket. The velocity of light is independent of the velocity of the source, and thus an earthbound observer will note the photon's spherically-symmetric probabilistic wavefront expanding in the form of of the larger red circle C. An observer on the rocket will note the photon's spherically-symmetric probabilistic wavefront expanding in the form of of the smaller black circle D.

Let us run this single-photon experiment numerous times. Because the detector illustrated in green occupies a larger portion of the smaller Circle D, the observer on the spaceship will see the photon detected more often by the detector than will the earthbound observer. Because the detector illustrated in green occupies a smaller portion of circle C, the earthbound observer will see the photon detected less often at the detector than the rocket's observer.

enter image description here

One could imagine surrounding both Circle C and Circle D with similar detectors along the entire circumference. One could perform the single-photon experiment numerous times on numerous trips, using only the detectors on Circle C or only the detectors on Circle D.

On average, the earthbound observer will see the photon hit the illustrated green detector less often than will the observer on the rocket.

Can both the observer on earth and the rocket be right? IS not there a paradox here?

Suppose two flashes of light of equal intensity. If one measures one further away from the other, it will appear dimmer. And so it is that the passenger on Einstein's train will see the lightning flash behind them to be dimmer than the one in front of him.

In Einstein's book on relativity, he writes,

We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig 1.

Einstein Figure 1

People travelling in this train will with a vantage view the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises :

Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative.

When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A arrow B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M1 be the mid-point of the distance A arrow B on the travelling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves towards the right in the diagram with the velocity v of the train. If an observer sitting in the position M1 in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result:

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time ; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

And so it is that the passenger on Einstein's train will see the lightning flash behind them to be dimmer than the one in front of him. Is this not true?

Indeed if the passenger is traveling very close to c, the flash from behind them will appear to be very, very dim, as the intensity of light falls of as $r^2$. Is this not true?

Let us replace the two lightning strikes with light bulbs which the stationary observer standing at M observes to flash at the exact same time, just like the lightning strikes did. Will not the observer on the train conclude that the lightbulb behind them is dimmer than the one in front of them?

(@knzhou answers "Yes" below in the comments.)

Suppose then we consider a traveler on a spaceship with two light bulbs at either end, replacing the lightning strikes.

enter image description here

The space ship is traveling at .9 c relative to the earth. Will the traveler not see the flash from the light bulb behind him to be dimmer?

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closed as unclear what you're asking by knzhou, CuriousOne, user36790, honeste_vivere, Gert Jul 9 '16 at 2:39

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Yes, redshift occurs? What is the question? $\endgroup$ – knzhou Jul 8 '16 at 3:57
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    $\begingroup$ @knzhou also, light intensity falls of as r^2. do you admit that light intensity falls off as r^2? do you admit this is different from your redshift? Will not the passenger thus see a dimmer signal due to the fact they are further away and light intensity falls off as r^2? That is my question, as stated above. $\endgroup$ – Astrophysics Math Jul 8 '16 at 3:58
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    $\begingroup$ This depends on the kind of light. For example, if we model the lightning strike as a cylindrical source, it's actually a $1/r$ falloff. But generally, yes. $\endgroup$ – knzhou Jul 8 '16 at 3:59
  • $\begingroup$ Yes @knzhou, but the intensity still falls off. $\endgroup$ – Astrophysics Math Jul 8 '16 at 4:00
  • $\begingroup$ Yes. $\hspace{0mm}$ $\endgroup$ – knzhou Jul 8 '16 at 4:01
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A quick reprise of the situation:

Train

This is the view in the embankment rest frame at $t=0$. We'll take the train to have a length $2d$ and we'll choose the origins so the middle of the train is at the origin in both frames. To find out where the flashes occur in the train frame we use the Lorentz transformations and the results are:

$$ F^\prime_{\,1}(t,x) = \left(\gamma\frac{vd}{c^2}, -\gamma d\right) $$

$$ F^\prime_{\,2}(t,x) = \left(-\gamma\frac{vd}{c^2}, \gamma d\right) $$

So according to the passenger on the train the distance to both flashes is the same, but flash $2$ happens before flash $1$ so the passenger sees the light from $F_2$ before he sees the light from $F_1$.

The thought experiment is really intended to show breakdown of simultaneity i.e. that in the embankment frame the flashes are simutaneous while in the train frame they are not. However we can extend the experiment to consider intensity as well.

In the train frame the flashes occur at an equal distance, so the light from them travels an equal distance and therefore the $1/r^2$ factor is the same for both.

In the embankment frame the light travels different distances because the light from $F_1$ travels a longer distance to reach the passenger than the light from $F_2$ does, so the $1/r^2$ factor is different for the two flashes. How do we explain the difference?

The solution is simply that for the passenger on the train the light is Doppler shifted because the flashes are moving relative to him. Remember that Doppler shift changes intensity as well as frequency. Although the passenger sees the light from both flashes travelling the same distance, he sees $F_1$ to be red shifted and less intense while $F_2$ is blue shifted and more intense.

Now we consider the situation where the you replace the lightning flashes with light bulbs that are stationary with respect to the train. Now the passenger sees both flashes as equal brightness. How do we explain what the embankment observer sees?

And again the solution is just the Doppler shift. Now the embankment observer sees $F_1$ to be blue shifted and $F_2$ to be read shifted i.e. $F_1$ is brighter than $F_2$. So even though the light from $F_1$ has to travel farther, it's brighter to start with.

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  • $\begingroup$ Thanks for the excellent explanation @JohnRennie ! I have updated the question above with a related, more precise thought experiment. Would love to hear your thoughts! $\endgroup$ – Astrophysics Math Jul 9 '16 at 4:47

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