Proper time vs null path

Question

I've heard three seemingly contradictory facts, so apparently I'm not understanding at least one of them correctly.

"Proper time along a path is the time elapsed for a clock that travels along that path."

"Proper time along a null path is zero."

"There are no preferred inertial reference frames, so if I travel at the speed of light, my own clock that I take with me will tick like normal for me."

So how can it tick like normal but then not tick at all? Which of these three statements is incorrect?

The third statement is a physical impossibility, so you should ignore it. — Marco Ocram, Nov 09 '23 at 14:00
This may help - How does a photon experience space and time? — mmesser314, Nov 09 '23 at 14:11
In relativistic theory (rt), we call the proper time $\tau$ of an object the time measured in “the” frame of reference of this object, that is to say in a frame of reference where it is at rest, but it does not exist this 'rest' reference for a photon, in addition, there is no massive particle which reaches the speed of light in rt. — The Tiler, Nov 09 '23 at 15:16

score 0 · Answer 1 · answered Nov 09 '23 at 14:55

Starting from one reference frame, you can think about what happens in another reference frame via a Lorentz transformation. This is often referred to as a boost, because the two frames have different velocities. There is no inertial reference frame with $v=c$, because there is no Lorentz transformation to connect that frame to any other. The Lorentz transformation depends on the term $\gamma = (1 - v^2/c^2)^{-1/2}$, which is undefined at $v=c$.

This is related to the fact that nothing can accelerate and reach the speed of light. Your third statement is conditioned on if I travel at the speed of light, but this is impossible. Nothing with mass can travel at the speed of light, so whatever comes after that conditional doesn't matter.

Proper time vs null path

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