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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.

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closed as off-topic by WillO, Bill N, Jon Custer, Aaron Stevens, JMac Oct 25 '18 at 11:51

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    $\begingroup$ The moving observers are no longer 100 light-years from the star when the light reaches them. As such, we shouldn't expect the light to take 100 years to reach them. $\endgroup$ – probably_someone Oct 22 '18 at 14:19
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    $\begingroup$ "But they we're a hundred light years away when the event occurred" This assumes no relativity of simultaneity, right? "When the event occurred" is meaningless without specifying a reference frame, and all three observers are in different reference frames. And the distance of 100 light years is also relative too right? This question seems to suppose an absolute reference frame where the light is a specific distance away and takes a specific amount of time to travel that distance. $\endgroup$ – Aaron Stevens Oct 22 '18 at 15:09
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    $\begingroup$ If you do the work out you will find that even though the different observers measure difference distances and times, they will all see that the speed of light has the same value. $\endgroup$ – Aaron Stevens Oct 22 '18 at 15:10
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    $\begingroup$ Just because they measure light speed as constant, doesn't mean it is constant. Time dilation screws with our perceptions. $\endgroup$ – Zane Scheepers Oct 22 '18 at 17:32
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    $\begingroup$ @Zane Are you trying to understand the postulate that the speed of light is constant? Or are you thinking that this postulate is actually wrong? $\endgroup$ – Aaron Stevens Oct 23 '18 at 2:18
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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.

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  • $\begingroup$ Let's not assume that! Let's assume they are stationary, on the planet when the star, 100 ly away, goes nova. $\endgroup$ – Zane Scheepers Oct 22 '18 at 14:53
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    $\begingroup$ Then it is more complicated because they need to accelerate, but the result is essentially the same $\endgroup$ – Wolphram jonny Oct 22 '18 at 14:54
  • $\begingroup$ Yes. And one is travelling in the opposite direction, so slowing down time makes them see the explosion even later. $\endgroup$ – Zane Scheepers Oct 22 '18 at 14:56
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    $\begingroup$ Perhaps some explicit calculations will help the OP here for the non-accelerating case $\endgroup$ – Aaron Stevens Oct 22 '18 at 14:58
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    $\begingroup$ @Zane What do you mean "slowing down time"? In your question the observers are all moving relative to one another when they see the explosion. $\endgroup$ – PM 2Ring Oct 22 '18 at 15:00
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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.

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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.

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    $\begingroup$ Absolutely right. The invariance of c is a well established experimental fact. It isn’t true because SR says so, it’s the other way around: SR is correct because it agrees with this experimental fact (among many others) $\endgroup$ – Dale Oct 23 '18 at 3:13
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    $\begingroup$ @Dale your comment accurately explains the difference between mathematics and physics :) $\endgroup$ – Aaron Stevens Oct 23 '18 at 3:17
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    $\begingroup$ @AaronStevens this comment is where your mistake lies. 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. Time dilation affects our equipment the same way it affects our vision. The postulate states that we always measure the speed of light as constant. That much I agree with. But just because we measure it as constant, doesn't mean it is. That's my point. I'm not claiming we don't measure it as constant. $\endgroup$ – Zane Scheepers Oct 23 '18 at 9:56
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    $\begingroup$ @ZaneScheepers I didn't say it wouldn't effect our perception. What I mean is that it is not dependent on our perception. As for your other issue, I guess I just don't understand. Time dilation isn't some weird effect that screws everything up. It is an effect of the speed of light being constant in all inertial frames. So you can't say that time dilation makes the speed of light change, because then you are just being inconsistent with the theory. $\endgroup$ – Aaron Stevens Oct 23 '18 at 10:15
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    $\begingroup$ @AaronStevens no sir. It's the effect which causes us to always measure the speed of light as constant. Time dilation is the cause, Length contraction is the result. $\endgroup$ – Zane Scheepers Oct 23 '18 at 10:19
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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 $c$.

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  • $\begingroup$ So the closing velocity is not what the moving observer measures? $\endgroup$ – Zane Scheepers Oct 22 '18 at 19:51
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    $\begingroup$ Correct. If you want to know what the moving observer measures, you must calculate the effects of length contraction and time dilation, or in a more complete statement, the Lorentz transformation. $\endgroup$ – Andrew Steane Oct 22 '18 at 21:13
  • $\begingroup$ Well if it means that the observer perceives the light moving towards him at c, there should be no red shift. $\endgroup$ – Zane Scheepers Oct 22 '18 at 21:20
  • $\begingroup$ @ZaneScheepers Red shift is not caused by a change in observation of the speed of light. $\endgroup$ – Aaron Stevens Oct 23 '18 at 2:19
  • $\begingroup$ @AaronStevens it would, if we could observe the change in speed. How do I take this to chat? $\endgroup$ – Zane Scheepers Oct 23 '18 at 9:47
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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.

Special relativity employs Einstein synchrony convention for all inertial reference frames, which ASSUMES that one way speed of light is c.

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.

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  • $\begingroup$ Yes there is a way to measure the one way speed of light coming from a supernova. We can do it using voayager 1. If voyager were to capture the image of a supernova, and time stamp the image, then send the image to earth, we can compare it with when we see the supernova here on earth. We know exactly how far away voyager is and can triangulate how much sooner light hit voyager, before it reaches us. $\endgroup$ – Zane Scheepers Oct 25 '18 at 9:53
  • $\begingroup$ A lot of books and articles are devoted to this issue. The article in the Stanford Encyclopedia of Philosophy provides many links. No one has yet managed to measure the speed of light in one direction. At what time the voyager takes that picture? What its clock shows at this moment? There is a wonderful book by Max Jammer - "Conventionality of distant simultaneity", Researchers viewed hundreds of different setups of experiment without any success $\endgroup$ – Albert Oct 25 '18 at 10:04
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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 $

with:

$\gamma = 1/\sqrt(1-V^2/c^2)$

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.

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  • $\begingroup$ A light year is a distance. If they are all at the same location when the event occurs, the distance is the same for each of them. $\endgroup$ – Zane Scheepers Oct 22 '18 at 14:49
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    $\begingroup$ Distances are not invariant by the Lorentz transformation : en.wikipedia.org/wiki/Length_contraction $\endgroup$ – Bernard GODARD Oct 22 '18 at 15:44
  • $\begingroup$ That's great for calculations, but stars don't jump to new locations just because we are moving. They appear to change location due to time dilation. 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. $\endgroup$ – Zane Scheepers Oct 22 '18 at 17:29
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    $\begingroup$ Scheepers is mistaken. It does seem odd, but when you have understood relativity in full, then it will make sense to you. The "stars jumping around" here is a bit like what happens when you simply turn, i.e. rotate your coordinate system. Then the stars all swing around you, and seem to travel great distances around circular arcs in a blink of an eye, although nothing is really happening except your own choice of measuring system. In a similar way (similar, note, not identical), adopting a different state of motion implies a different set of measuring sticks and clocks, so distances change. $\endgroup$ – Andrew Steane Oct 22 '18 at 22:48
  • $\begingroup$ @AndrewSteane you're mistaken sir. I completely understand that the length contraction is a subjective perception resulting from time dilation. The observer still measures the speed of light as constant, subjectively. Objectively, however, the convergent speed between the light and the observer, has increased. $\endgroup$ – Zane Scheepers Oct 25 '18 at 9:44
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the concise answer: by moving the observers you changed the distance to the supernova, not the speed of light in your experiment.

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