If 3 observers are on a planet which 100 light years from a star, and the star goes supernova, if one observer moves towards the star and one moves in the opposite direction, each observer will see the explosion at a different time. The observer who stays on the planet, sees the explosion a hundred years after it occurs. The person moving towards the star, sees it in less than a hundred years, while the person moving in the opposite direction, will see it more than 100 years after it occured. Doesn't this prove that the speed of light, relative to a moving observer, is not invariant.
closed as off-topic by WillO, Bill N, Jon Custer, Aaron Stevens, JMac Oct 25 '18 at 11:51
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The motion of the oberver does not affect the speed but if the distance changes then the light will have to move that additional distance, and thus the time cannot be the same: time=distance/velocity, same velocity but different distance gives different time.
However, things are more complicated than that due to length contraction. Let us assume that the two moving observers are already moving at a constant speed at the time of the explosion (as determined by the observer at rest on the planet) and at that time they are all at the surface of the planet and the three observers synchronize their clocks at that time. The two moving observers will experience lenght contraction, that is, to them the distance between the planet and the star is not 100 light years, but less. If they move fast enough, both will see the exposion "before" the person on the planet. By before meaning that they will measure less time in their clocks when the light reaches them.
Some comments by the OP in the question and other answers:
You're making stars jump around and exist in different locations at the same time, just to preserve the postulate that the speed of light is invariant in any reference frame.
Just because they measure light speed as constant, doesn't mean it is constant. Time dilation screws with our perceptions.$^*$
And in the question:
Doesn't this prove that the speed of light, relative to a moving observer, is not invariant.
It seems like you are doubting the postulate that the speed of light is the same for all inertial reference frames, which is why you are unsatisfied with the current answers that assumes this postulate is valid. And this is a valid issue to have. Any calculations using SR in your specific example could only show internal consistency with the theory. So how can we convince ourselves that this postulate is valid so that the other answers here are not just self-conistencies that do not explain our reality?
As always, we should look at experiments. Of course, there is the famous Michelson-Morley experiment which first gave validity to this idea that there is not a universal reference frame. This, combined with Maxwell's equations, gives motivation to think that the speed of light must be the same in all inertial reference frames. Many, many more experiments have also been shown to be consistent with this postulate (http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html)
Additionally we hold this postulate to be true because of how the theory that arises from the postulates does very well at explaining reality. QED, which combines quantum mechanics and SR, is the most well tested theory to date (I think this is true, correct me if I'm wrong). Your GPS would not work without correcting for SR (and GR as well). The point is that the postulates don't just make a self- consistent theory; they make a theory that explains reality very well, which is what physics is all about. Therefore, we believe the postulates that the theory is built on to be true due to experimental evidence of the postulates and the success of predictions following from those postulates.
Therefore, for your current specific example, any conclusion that is drawn that shows the speed of light is different for the various inertial reference frames must have an error either in how you understand the problem and its set up, or in how you are applying and understanding SR itself. I will direct you to the other answers here in how to treat your specific example.
$^*$ Something to realize is that time dilation has absolutely nothing to do with the finite time period it takes for light to travel from someplace to our eyes. It has nothing to do with our perception.
Questions like this are best answered by drawing the trajectories on a spacetime diagram. It all works out very nicely. You might find it helpful to note that there is a difference between "relative velocity" and "closing velocity". Relative velocity between A and B is the velocity which B has when observed in the rest frame of A. This never exceeds $c$, and always equals $c$ for light waves in vacuum. The closing velocity of A and B, as oberved in any given frame, is the difference of their velocities relative to that frame. The latter can take values up to $2c$. For example, the observer on the planet measures that some rocket is moving towards the star at speed $v$, and the star light is moving towards him on the planet at speed $c$. He may then correctly deduce that the rate at which the distance between the rocket and the supernova flash of light is decreasing, as measured in the planet frame, is $c+v$. This quantity is not the speed which the observer on the rocket measures, however. That observer finds that the flash approaches him at speed $c$.
Apparently you speak about one - way speed of light from the supernova towards the observer and whether this velocity is actually equal to c. I believe that it is important to understand some details.
Please note, that the one way speed of light of light cannot be measured even in principle. So as to measure the one way speed of light you need two spatially separated and synchronized clock. But, to synchronize these clocks you need to know one way speed of light.
It is well known that in the moving frame two - way speed of light is equal to c because of effects of time dilation and length contraction. However, there are no experiment, which allows to measure the one way speed of light of light from supernova towards the observer.
Thus, due to Einstein synchronization in Special Relativity "stationary on the planet" and "relatively moving" observers will make different conclusions about time of this event - explosion of supernova.
Synchronized clocks of their reference frames, that are adjacent to the "event" will show different time BECAUSE of the same synchronization procedure (Einstein synchronization) which assumes, that one - way speed of light is c. Sure, if you will measure one - way speed of light with Einstein - synchronized clocks, you will always measure, that one - way speed of light is equal exactly c.
Thus, they may say that explosion happened at different moments. However, they may also use non-standard clock synchronization(Reichenbach's), then they may also say, that event happened at the same moment and one - way speed of light relatively to that observer who moves towards the supernova was greater than c.
You are reasoning in the frame of the observers and while the speed of light is the same in that frame, the distance to the supernova is different in the frame of the moving observer so they are not all "100 light years away" at the time (on the observer clock which is different for the 3 observers) the light is emitted by the supernova.
Let us assume that observer 1 is at the origin of the "rest" frame and the supernova is along the X axis at coordinate $x_1$.
At t = 0 (clock of observer 1), t' = 0 (clock of observer 2), observer 2 is coinciding with observer 1 but moving towards the supernova with constant velocity V.
The supernova explodes at t=0.
Using the Lorentz transformation, we can compute the explosion time in observer 2 clock:
$t'_1 = -\gamma V x_1 / c^2$
(while for observer 1 the explosion is simultaneous with the passage of observer 2, for observer 2 it happened before) and its distance to observer 2 in its local reference system
$x'_1 = \gamma x_1 $
$\gamma = 1/\sqrt(1-V^2/c^2)$
For small velocity $\gamma$ is close to 1, then:
$t'_1 = - V x_1 / c^2$
In the newtonian approximation the observer moving towards the source will see the event happen $x_1/c - x_1/(c+V) \approx V x_1/c^2$ before. Same as what we just derived in the approximation $\gamma = 1$. Here the travel time is the same because we neglected length contraction assuming $\gamma = 1$ and of course the speed of light is the same, but the explosion happened "before" for observer 2.
the concise answer: by moving the observers you changed the distance to the supernova, not the speed of light in your experiment.