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A particle of light is traveling towards earth at the same moment an apple falls from a tree.

The photon crosses paths with the path of the apple, but barely misses the apple, and instead hits the ground at the same moment the apple hits the ground.

From the perspective of the photon, time has not passed, and yet during that moment its path with the apple has crossed.

If, according to the photon, time is singular, and the place in space is the same for the photon as the apple at one point, why do they miss each other?

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There is no frame of the photon. Any conclusion that you derive from such a frame could be terribly wrong. You have solved the problem in an appropriate frame, so you have solved it: in no frame the two objects touch.

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  • $\begingroup$ Why is there no frame for the photon? Would it matter if we were talking about a photon or an object going very nearly the speed of light? $\endgroup$
    – J.Rockefeller
    Mar 8, 2017 at 0:05
  • $\begingroup$ There is no reference frame for a photon for various reasons. I will state the simplest. 1. One of the postulates of relativity could be phrased as "Things moving at the speed of light in one frame move at the speed of light in every frame". If you were trying to "follow" a photon, you would be seeing it traveling at $c$, for any velocity that you could have. 2. Look at the Lorentz boost matrix, that in SR one uses to switch frames. The $\gamma$ factors are $\gamma=(1-v^2/c^2)^{-1/2}$. If $v\to c$, this becomes singular, so the boost is not well defined. $\endgroup$
    – Salvatore Baldino
    Mar 8, 2017 at 10:11
  • $\begingroup$ I actually needed two comments. This trick works exactly at $c$. If a body a fraction of the velocity of light, any fraction up to (but not including) 1, you could define a frame where the body is stationary. In the universe, we have two type of particles: massive and massless. Massive particles (like the electron) can have any velocity $<c$, and you can always define a frame where those particles are stationary. Massless particles (like the photon) move always at $c$, in every frame, and there is no stationary frame. $\endgroup$
    – Salvatore Baldino
    Mar 8, 2017 at 10:16
  • $\begingroup$ This is great, thank you Salvatore! The idea that massless particles always travel at c and never have a stationary frame is really helpful. Thanks! $\endgroup$
    – J.Rockefeller
    Mar 10, 2017 at 0:01

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