Suppose a pulse of light is sent from $A$ to $B$ in a rigid triangle $ABC$. While the pulse is moving from $A$ to $B$, what is it's velocity relative to $C$?

The special character of the photon among other particles make me hesitate to use the standard approach with polar coordinates $(r,\varphi)$ to find a velocity $\frac{dr}{dt}$ relative the point $C$.

I know this is a naive question, but I would be very pleased to see a non-naive answer.

Can the movement of a pulse of light between two stars be described with polar coordinates?

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    Are you asking for the linear or the angular speed? (If you're thinking that the light appears to vary in speed as it travels, you're actually talking about the angular speed.) – probably_someone Oct 15 at 11:21
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    @probably_someone: I didn't think of the angular speed. – Lehs Oct 15 at 11:23
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    It's just $c$. Were you expecting a different answer? – knzhou Oct 15 at 11:31
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    Of course, C can only calculate this speed, they can't determine it from direct observation, since the light is traveling from A to B, it's not traveling to C. – PM 2Ring Oct 15 at 12:53
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    The answer depends entirely on what you mean by "speed relative to C", and since you are the one who invented this (non-standard) term, only you can know that. – WillO Oct 15 at 13:59
up vote 19 down vote accepted

The speed of light is the same in every direction; specifically, $c=299,792,458$ m/s. This fact was proven by Michelson and Morley in 1887, when they measured the speed of light very precisely using the Michelson-Morley interferometer. They proved more than that, though: the speed of light is not only independent of direction, it is also independent of the speed at which the observer is moving. This was one of the main discoveries that gave Special Relativity its credibility in the early 20th century.

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    It might not align with OPs understanding of "relative to C" - this answers assumes OP is asking for the norm of the velocity vector. – WorldSEnder Oct 15 at 15:00
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    @WorldSEnder: I changed the question from 'speed' to 'velocity' after this answer. – Lehs Oct 15 at 15:12
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    They didn't prove it - they measured it. – Thorbjørn Ravn Andersen Oct 15 at 22:55
  • @WorldSEnder: No, the norm of the velocity vector is zero. – Ben Crowell Oct 15 at 23:39

In this case the speed would be same for anything sent from A to B, as seen from C. Since there is no relative velocity between the points A, B and C, the magnitude of the speed of any object going from A to B, as seen from A, B or C, or any other stationary point with respect to the triangle, would be the same.

For light, the magnitude of its speed would be the same even if the points where moving with respect to each other, the speed of light is the same for all frames of reference, independent of their relative position or velocity.

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    Is it bad to consider a vector from $C$ to the line $AB$ and expect velocity components? – Lehs Oct 15 at 13:40
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    This is the best answer, IMO. SR need not be invoked here. The same would be true if A threw a tennis ball at B. – Bridgeburners Oct 15 at 15:08

Maybe I would put something nuanced here. From STR we know that light velocity is constant and independent from the observer,and is equal C. But this is true for any observer, which means: "in observer's inertial frame of reference". But there is a question which we should ask: how it is measured?

You see it is far from being easy to measure speed of light in local areas, say several kilometres, but as we are asking such abstract questions, maybe we should focus on measurement aparatus, or experimental settings. Usually there's no way to measure speed of the light directly, but via measuring of time of some emission and then the time after absorption. After that we should calculate the speed. We between experimental system, and calculational results, a lot of models have to be used, from kinematics or optic and electrodynamics via light-matter interactions up to quantum mechanics in some cases.

As your system consists of triangle, and observer is in C whilst emission and absorption occurs in A and B respectively, we may ask: how it may influence our measurement and calculations. It may be far from obvious! There's a lot of details in between.

Of course if we assume that "all the rest" is known and under control, there is no need to drive into details. But real experimental settings rarely are so clear and fully controlled. So in this "triangle setting" may not be optimal for some measurements.

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    Sending an electron from $A$ to $B$ would neither be easy to measure from $C$, but there are theories about it's motion relative $C$. – Lehs Oct 15 at 15:07
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    There's a lot of this type experiments. I have just wanted to put some attention to detail of the measurement proces. For purely mathematical theories like STR we are usually omitting this details, saying observer is able to "measure" event, without specific details. But in physics there are always some details here. – kakaz Oct 15 at 15:19

Before even beginning to answer the question, there is one thing that needs to be cleared up about the question. The whole idea of "viewing light from the side" is more complex than you think it is.

If a rocket ship were traveling from A to B, and it were equipped with some kind of running light, you might be able to see the rocket ship from point C. But what you are actually seeing is light (photons, if you will) that were emitted from the running light, and happened to be aimed in the right direction to reach point C, where you are located.

But if a photon is on its way from A to B, it doesn't emit other photons that go off in all other directions, some of which end up at C. The only way a photon is going to emit photons towards C is if it interacts along the way. If the photon smashes into an atom, for instance, and the atom gains some energy that it releases as other photons, then you might be able to see that from C. But now the original photon is never going to reach point B, because it smashed into an atom on the way.

So we need a whole bundle of photons, released from point A towards point B, that interact in some way along the track between A and B. Each of these interactions will cause a different point of light, and these will reach you at different times and from different directions. From this you may be able to infer the speed of that bundle of photons going from A to B.

I'm pretty sure that the speed will turn out to c, the speed of light. Any change in distance caused by the angle of view will be compensated for by a change in time delay, so it will all come out in the wash. That's how I see it.

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