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From what I understand the expansion of the universe has no "center". If we're flying through space away from the "center of the big bang", there's basically no way to tell. Every two given points in space gets farther away from each other, and we can pick any point as center if we like.

I also understand that the speed of light is not relative to the speed of the source emitting the light. If I go on a train in $c/2$, turn on a flash light pointing forward, the light emitted from the flashlight will still travel at the speed of $c$.

Now here's my question: Why can't we set up a sphere with photodetectors with synchronized clocks on the inner walls, turn on a light in the center, record the exact time at which each photodetector detects the light, and compare the times to figure out if the sphere was traveling in some certain direction?

I mean if we turn on a lightbulb and light travels with the speed of $c$ in all directions at the same time, my intuition tells me that we should be able to figure out some form of "reference stand still".

(Tagging this with general relativity because I suspect that it's impossible to set up the experiment the way I like due to relativity.)

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Why can't we set up a sphere with photodetectors with synchronized clocks

To anyone moving relative to the sphere, the clocks aren't synchronized (relativity of simultaneity); the synchronization of spatially separated clocks is reference frame dependent and this is, in fact, the root of time dilation and length contraction.

Moreover, the process of (Einstein) synchronizing the clocks at rest with respect to the sphere guarantees that the one way speed of light measures $c$ in the sphere's reference frame.

In other words, as I've emphasized with bolding in the quote, by stipulating that the clocks are (Einstein) synchronized, you're stipulating that the photo-detectors will detect the flash of light from the center simultaneously according to the clocks.

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All the photodetectors will receive the flash at the same time.

Start by ignoring the expansion of spacetime: you've already established that the speed of light is the same for all observers, so the light at the centre sees the light moving away at $c$, and all the detectors see the light approaching at $c$. Since all the detectors are the same distance from the light source the travel time is the same for all of them.

Now what happens if we include the expansion of spacetime. The expansion means that if we pick any two points in space and wait, the distance between those two points will increase with time. Because the expansion is uniform the distance will increase at the same rate for any pair of points. That means the distance between the light and the photodetectors increases at the same rate for all the photodetectors, and therefore the travel time of the light from the source to the detectors is the same for all detectors.

I would guess you are still struggling to rid yourself of the notion that the expansion must be relative to some point, so unless the light just happens to be at that point the expansion will be uneven and will deform the original sphere of detectors into some other shape (an ellipsoid?). However this is not the case. The expansion is uniform in the sense that you can pick any two points anywhere and get the same expansion rate. I've expanded on this point in lots of answers. Rather than point you to a specific answer have a look at this search for lots of related discussions.

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  • $\begingroup$ I believe I understand what you're saying, but it's unclear to me whether you've answered the question. I'm not interested in the expansion of the sphere. The experiment with the sphere is just to figure out if I'm moving away from the other point, or if the other point is moving away from me (if either is the case of course). Here's a different formulation: Suppose we're on a train going on a straight line. We have no possibility to observe our surrounding so we don't know if we're going forward or backward... $\endgroup$
    – aioobe
    Mar 18, 2014 at 10:57
  • $\begingroup$ ...so we set up detectors at the front and at the back of the train, and then turn on a lightbulb on the middle of the train. Couldn't we then check which of the detectors receives the light first and from that judge if we're going forward or backward? $\endgroup$
    – aioobe
    Mar 18, 2014 at 10:58
  • $\begingroup$ @aioobe: no, that's the whole point. To all observers at rest with respect to the train light moves at the same speed of $c$. So for observers on the train the front and rear detector will receive the light at the same time. People standing on the train track will see the rear detector receive the light first, but that's because simultaneity is not invarient in relativity (special or general). Observers on the train (i.e. that's us in the expanding universe) see the light arrive simultaneously. $\endgroup$
    – John Rennie
    Mar 18, 2014 at 12:00
  • $\begingroup$ Aha... interesting. But my idea was that the detectors themselves had synchronized clocks. Will the rear and front detectors say that light was detected at the same time, regardless if the train moves forward or backward? If so, it feels like the photons moving in the same direction as the train have traveled faster (to "catch up" with the moving detector) than the photons that traveled against the direction of the train. $\endgroup$
    – aioobe
    Mar 18, 2014 at 12:09
  • $\begingroup$ @aioobe: Will the rear and front detectors say that light was detected at the same time, regardless if the train moves forward or backward? Yes. It seems strange, but all the weird effects in SR like time dilation and length contraction arise for exactly this reason. $\endgroup$
    – John Rennie
    Mar 18, 2014 at 12:11

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