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Given are two cases of relative motion between an observer and a light source:

  1. the speed of light is always observed to be the same regardless whether the light source is approaching the observer or the observer is approaching the light source, and

  2. the speed of light is always observed to be the same regardless whether the light source is receding from the observer or the observer is receding from the light source.

Time dilation in Special Relativity resolves the first case in keeping light speed constant because the slowing of time (time dilation) keeps the same distance travelled per elapsed time on the observer's clock. But SR does not seem to resolve the second case because the observer's clock would have to speed up (not slow down) to maintain the same distance per elapsed time on the observer's clock. Please explain.

Dale
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  • What gets you into trouble here is that your reasoning is based on the assumption that the leading edge of the wave (the leading wavefront of the light ray) is one and the same for both of the two observers. It's not. What observer A, who is moving, currently identifies as the leading edge is in the future (scenario 1.), or in the past (scenario 2.) for observer B who is static. In observer B's notion of "now", what the observer A is looking at hasn't happened yet (1.), or has already happened (2.); the leading edge that B is seeing is not in the same part of spacetime as what A is looking at. – Filip Milovanović Feb 02 '23 at 19:08
  • Do not change a question after it has been answered in a way that invalidates the existing answers. Please post a new question as I described earlier – Dale Feb 08 '23 at 05:20
  • @Dale : Got it. Thanks. –  Feb 08 '23 at 06:01

2 Answers2

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You are neglecting the relativity of simultaneity and length contraction. All three are needed to explain the invariance of $c$.

Specifically, the usual derivation is to assume the invariance of $c$ along with the principle of relativity as postulates. These two postulates imply the Lorentz transform which implies all three of time dilation, length contraction, and relativity of simultaneity, and vice versa.

However, it is also possible to reverse the proof. You can start with time dilation, length contraction, and the relativity of simultaneity and derive the Lorentz transform which then implies the invariance of $c$ and the principle of relativity. You cannot do the derivation from time dilation alone, you need all three (the relativity of simultaneity is the one usually neglected by new students).

Once you have the Lorentz transforms, deriving the invariance of $c$ in both directions (your two cases) is straightforward:

Suppose that you have a flash of light at $(t,x)=(0,0)$ then the part of the flash going to the right is given by $x_+=c t$ and the part of the flash going to the left is given by $x_-=-c t$.

If you plug those expressions into the Lorentz transform and simplify then you get $x'_+=ct'$ and $x'_-=-ct'$. So using the full Lorentz transform which includes the relativity of simultaneity immediately shows that the speed of light does not depend on the direction in either frame.

Dale
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  • But these don't explain the constancy of c: the constancy of c is an experimental fact, used as a postulate in the development of relativity. – John Doty Feb 02 '23 at 18:28
  • Agreed. That is why I wrote the third sentence of the answer. I just didn’t see any reason to pick on the OP about that any further – Dale Feb 02 '23 at 18:49
  • "The invariance of c" does not attribute the postulate to experiment. Physics is not math: postulates don't come from nowhere. – John Doty Feb 02 '23 at 18:52
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    Where else do you think that postulates would come from besides experiment? – Dale Feb 02 '23 at 18:56
  • The problem is that students don't learn where the postulates come from in physics. We implicitly teach the misconception that it's just math. Your answer reinforces that misconception. – John Doty Feb 02 '23 at 19:00
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    Then write your own answer to address that. My answer is correct and I don’t feel the need to revise it to cater to your personal preference – Dale Feb 02 '23 at 19:02
  • Your answer is mathematically correct. This is a physics forum. – John Doty Feb 02 '23 at 19:05
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    The answer is also physically correct. It is just not conforming to your personal preferences. Again, if you want such an answer then write it – Dale Feb 02 '23 at 19:09
  • Where's the physics in it? I see only mathematics. – John Doty Feb 02 '23 at 19:12
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    Then look more carefully – Dale Feb 02 '23 at 19:15
  • @Dale: I'm not sure what you mean when you stated "Or rather, the invariance of c along with the principle of relativity implies all three of time dilation, length contraction, and relativity of simultaneity." This statement (the operative word, "along with") suggests that the invariance of c cannot be entirely explaind by the effects of SR. If this is what you mean, then what else contributes to the invariance of c in both scenarios I had mentioned besides SR (i.e., besides all three SR effects which are due to tilting of the time axis for the "moving" frame)? –  Feb 02 '23 at 20:54
  • @Dale: You included the Lorenz Transform equations in your last paragraph. Are you suggesting that said equations incorporate not only SR effects but also some other non SR related cause or even that the invariance of c IS the other non SR related cause? If you mean the former, then what is this non SR related cause? If you mean the latter, then we're back to my original question - why is light speed invariant. –  Feb 02 '23 at 20:55
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    @Rob SR is the Lorentz transforms (LT). Usually we derive the LT from the invariance of c (IOC) and the principle of relativity (POR). This is what my “along with” comment was intended to convey, you need both IOC + POR. Once you have derived the LT then you can further derive length contraction (LC), time dilation (TD), and the relativity of simultaneity (RS). That is the usual approach. You want to go backward, which is fine. To go backward you need to use LC + TD + RS to derive the LT. Then you can derive POR and IOC from the LT. TD is insufficient by itself. All 3 are needed, LC, TD, RS – Dale Feb 02 '23 at 21:22
  • @Dale: But my question was (using the terms you used), what is IOC "derived" from. Again, why is the speed of light constant? Why is it constant in any inertial frame at all, let alone in all inertial frames? It would seem that the first case cited in my original question might be answered if IOC can be derived from LC, TD and RS instead of vice versa which you stated. But the second case I cited still baffels me. Please reread my original question if you need to. –  Feb 03 '23 at 06:32
  • @Rob The LT can be derived from LC, TD, and RS. I then showed that from the LT both of your cases are resolved. Your two cases are light going in opposite directions. That is why I showed both directions. I have edited substantially to clarify the answer – Dale Feb 03 '23 at 12:12
  • @Dale: Mathematics is a necessary tool to prove the accuracy of hypotheses. But proven hypotheses are ultimately appreciated in terms of human observation through the senses, i.e., translation of the math to everyday perceptual experiences because said translation allows for a more intuitive, meaningful grasp of the concept at hand. So, please explain your answer in the same verbal (non mathematical) context which I am about to present the problem (below). –  Feb 07 '23 at 05:03
  • @Dale: At the outset, my question does not concern a relationship between a given "stationary" frame and the corresponding "moving" frame. Instead, it concerns only a "stationary" frame. Therefore, what precise factors come into play to make the "stationary" observer record the same speed of light whether the light is approaching them or receding from them? –  Feb 07 '23 at 05:04
  • @Dale: If the recorded speed (distance per unit time) is invariant for this "stationary" observer, then as the light approaches, the space between it and the observer would have to be stretching by just enough to offset the decreasing distance in order to maintain the recorded light speed ("c"). And, as the light is receding, the space between it and the observer would have to be shrinking by just enough to offset the increasing distance in order to, once again, maintain the recorded light speed ("c"). –  Feb 07 '23 at 05:05
  • @Dale: But said stretching and shrinking of space would not apply for the "stationary" frame because length contraction of space in SR would only apply to the "moving" frame (which in this case would be the light itself as opposed to a "moving" mass). Time dilation and relativity of simultaniety wouldn't apply here because, again, we are concerned with only the elapsed time recorded on the "stationary" observer's clock. –  Feb 07 '23 at 05:07
  • @Dale: The following is, I think, a plausible explanation for this seeming conundrum. Unlike a moving mass which has already gained its momentum before being subjected to a force which accelerated it to its subsequent greater speed (the latter speed which, by analogy, we would be measuring), light is not a mass subject to Newtonian mechanics but is, instead, energy created at the very moment it is emitted from the moving mass. The light had no history of motion before it was created. So, its speed is independent of the speed of the moving mass. –  Feb 07 '23 at 05:09
  • @Dale: So, we are always measuring the speed of light, not a "moving" mass from which the light was emitted. All that said, my argument might explain why light speed is constant whether it is approaching or receding from a "stationary" observer, but why is this constant speed 299,792,458 meters per second? Why not any other speed? Your thoughts? –  Feb 07 '23 at 05:10
  • @Rob this is way too much for comments. You should open a new question with these details – Dale Feb 07 '23 at 05:38
  • @Dale: The moderator would then close my comment, stating that I had already asked this question. Furthermore, the moderator could have moved this to the "chat" forum. In any event, your reply was not one I was expecting. –  Feb 07 '23 at 15:58
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    @Rob usually if you link your new question directly to this one and clearly articulate the differences between the new one and this one then it will not be closed as a duplicate. Usually it will only get closed if you try to hide the fact that you asked a related question or if you don't explain the differences. Anyway, it is simply way too much for me to answer in comments, sorry – Dale Feb 07 '23 at 16:58
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The key to understanding SR is to understand the relativity of simultaneity, which is the cause of effects such as time dilation and length contraction.

Suppose you walk East past a stationary person at a metre per second, and just as you pass, that person flashes a light that heads off East and West. After a second in the frame of the stationary person, the light will be 229,792,458 metres away in both directions, but in your frame you will have walked a metre towards the East, so when the light is equidistant from the stationary person it is 229,792,457 metres ahead of you and 229,792,459 metres behind you, ie it is nearer to you in the direction you are walking and further from you in the opposite direction. Given that the speed of light, for you, is the same in both directions, in your frame the time associated with the position of the light ahead of you must be just under a second from the time of the flash, while the time associated with the position of the light behind you must be just over a second from the time of the flash.

More generally, what happens is that when you are moving relative to someone else, your time axis in spacetime is tilted relative to theirs. That means a plane of constant time for them is a sloping slice through time for you, and vice versa. If you ponder on that for long enough, you should come to realise that it explains time dilation, and it also explains why time dilation is symmetrical in a particular way.

Marco Ocram
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