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There is a standard diagram for deriving time dilation in special relativity. So say there's a ship in deep space with observer O' in inertial reference frame S' moving to the right with respect to an observer O in inertial reference frame S. In frame S', the path of light in his light clock is vertical whereas in frame S it follows a triangular path. I am not questioning that, so let's not go through that derivation here.

What I am questioning is how observer O explains the non-vertical path of light in his reference frame, on an intuitive level, given that the light source is pointing up for him also. For instance, if there were a ball thrown up at in S', observer O would explain its path in S by saying that before being tossed the ball had momentum to the right by virtue of being on the ship and it retains this momentum when only a vertical force is applied to toss it up. Something along those lines, to explain the light path.

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Imagine the light-source is a cylinder pointing vertically upwards in frame S'. A photon of our light beam travels up the cylinder and out into space. For observer O' it appears to travel upwards, but as you state, for O it is at an angle.

This doesn't actually require special relativity. The same effect would happen if the spaceship was a train and the light-source was a gun (i.e., non relativistic speeds). The bullet would appear to travel straight up according to O' but at an angle according to O. The angle would be accounted for by the fact that the gun was in motion and the bullet was travelling to the right at the same time according to O.

Special relativity comes in because in the train example, O measures the bullet speed by the length of the diagonal of the vertical and horizontal velocities (so it appears greater than O' sees it) whereas with light both observers measure the light beam travelling at $C$. Since the $C$ is the same for O and O', other elements of the calculation have to change, but this is then just the time dilation question.

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  • $\begingroup$ Interesting. So you're saying that in the S frame the light source is not pointing straight up. Which effect or effects - length contraction, time dilation, relativity of simultaneity - is responsible for this? Dimensions perpendicular to the direction of relative motion are not affected and lengths in S' are contracted as measured in S. How would that account for a longer tilted cylinder? $\endgroup$
    – Not_Einstein
    Commented Aug 10, 2020 at 1:10
  • $\begingroup$ I am not sure how I would characterize it. Which direction is directly "up" is a property of the reference frame. There are a lot of these paradoxes. See Ladder Paradox and at the end there are links to many others. I think you'll find one that is close to your example. $\endgroup$
    – rghome
    Commented Aug 10, 2020 at 12:50
  • $\begingroup$ I did some more searching and found this similar question from three years ago (The special theory of relativity angle contradiction): physics.stackexchange.com/questions/313375/… I think @Hans de Vries answer is especially interesting (although I didn't yet follow all of it). Apparently there is no tilt of the light source but the relativity of simultaneity comes in to play in redirecting the wavefront in the S frame. $\endgroup$
    – Not_Einstein
    Commented Aug 10, 2020 at 20:34
  • $\begingroup$ You are right - I misunderstood what you were getting at. I have updated the answer, which I don't believe requires relativity. But the other answers give more detail than mine. $\endgroup$
    – rghome
    Commented Aug 10, 2020 at 21:58

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