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I have seen time dilation explained more or less in this way:

If you are in a rocket, racing a photon, and your rocket is almost at the speed of light, technically you would see the photon moving away at way less than the speed of light. But that doesn't happen, because the speed of light is always the same. To solve this problem, your time would dilate, so you would still see the photon moving away at speed of light, and you would experience time slower.

I think this case makes sense. But what if the photon was moving in the opposite way? What if instead of moving away from the rocket, it was actually coming towards it from far away? Technically, the pilot would see the photon moving at more than the speed of light (the sum of speed of light and the velocity of the rocket).

I suppose that is also impossible, as the speed of light is always constant. But if the pilot's time were to dilate in this case (be experienced slower), wouldn't he perceive the photon even faster than before (even faster than speed of light and speed of rocket combined)?

How do I approach this problem? Would time dilate or contract in this case?

Qmechanic
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  • While it might not be an exact duplicate, my answer here should hopefully address your question at least in some part. – Philip Dec 23 '20 at 13:10
  • @Philip thanks! I am not looking necessarily for a definitive answer, but could you point out here where my logic is failing? How would you approach the problem? –  Dec 23 '20 at 13:30
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    I believe the answer I linked does exactly that. Time dilation is a special case of the Lorentz transformations, which holds when certain conditions are satisfied. Thinking about it as a "reason" for the speed of light to be constant is not correct, and certainly causes more confusion than it clears up. Both time dilation and the relativistic velocity addition formulae are results of the Lorentz Transformations. – Philip Dec 23 '20 at 13:34
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    Because of Relativity of simultaneity. https://en.wikipedia.org/wiki/Relativity_of_simultaneity – R. Emery Dec 23 '20 at 13:53
  • I'm not sure where you got that definition of time dilatation, but it seems extremely unclear to me. Maybe in context it's better? Where did you find that? – Brick Dec 23 '20 at 14:18
  • @Brick I found it here and comments therein. I think it makes sense, please let me know your opinion of it. Thanks. –  Dec 23 '20 at 14:29
  • This part is obviously false: "technically you would see the photon moving away at way less than the speed of light". I think you got yourself confused by reading Reddit instead of a credible book. – Brick Dec 23 '20 at 14:36

2 Answers2

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Here's a spacetime diagram on rotated graph paper
that suggests how any inertial observer arrives
at the same value of the speed of light.

RRGP-robphy-speedOfLight

The velocity of the worldline of light (along the light cone)
can be gotten by considering a vector along the light cone.
The velocity is the slope, the ratio of its spatial-component to its temporal-component.

From the diagram, all inertial observers obtain the same speed for light, for both forward-directed and backward-directed light signals.

What is displayed here geometrically on this spacetime diagram
can be expressed in other ways, such as the equation provided by @Dale 's answer.

From the OP's comment

but I don’t understand how the pilot’s perception of time would adjust to compensate for the “higher velocity” light would assume. How do you approach this problem? – Roberto Valente

The pilot's perception of time (i.e. her light-clock ticks along her worldline [her timeline]) is accompanied by the pilot's perception of space (i.e. her light-clock ticks along her sense-of-space [her spaceline]). This is a visualization of the Lorentz transformation that was by stated by @Dale in his reply to the OP's comment.

robphy
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  • I actually like your answer better than mine. +1 – Dale Dec 23 '20 at 18:20
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    @Dale Thanks. I'm just trying to encourage "spacetime thinking", especially to support (often numerous) standard explanations and formulas. – robphy Dec 23 '20 at 18:35
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Technically, the pilot would see the photon moving at more than the speed of light (the sum of speed of light and the velocity of the rocket).

In the pilot’s frame the light would be coming towards him at exactly c. To obtain this you need to use the correct relativistic velocity addition formula. In Newtonian mechanics the relative speed is just $v’= v+u$ but in relativity it is $$v’=\frac{v+u}{1+vu/c^2}$$

As you can see, for $v=\pm c$ this gives $v’=\pm c$ regardless of $u$. It doesn’t matter if the light is moving towards or away from the observer. Either way it moves at c in any inertial frame.

Dale
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  • Thanks. I believe your answer is correct, but I don’t understand how the pilot’s perception of time would adjust to compensate for the “higher velocity” light would assume. How do you approach this problem? –  Dec 23 '20 at 13:53
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    It is not sufficient to consider merely time dilation. You must use the full Lorentz transform. The velocity addition formula comes from the Lorentz transform, not just time dilation. – Dale Dec 23 '20 at 14:18