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Consider a spaceship traveling (hypothetically) at the speed of light with respect to an inertial observer. What happen to a light clock in the spaceship according to time dilation time? Does it stop in the spaceship with respect to the inertial observer, so that the same thing happens to a light clock, so the light ray in the light clock stops? How can I imagine this? I'm totally confused.

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    $\begingroup$ There are no clocks, light clocks or otherwise, that have speed c relative to any inertial reference frame (IRF) according to special relativity (SR). As has been stated here many, many times, an entity with speed c in an IRF has speed c in all IRFs which is to say the entity is not at rest in any IRF. Stating that the spaceship is (hypothetically) at the speed of light is to state that (hypothetically) SR is wrong. What theory then shall we use to answer your question since you've discarded SR? $\endgroup$ – Alfred Centauri Jun 30 '17 at 2:29
  • $\begingroup$ the spaceship traveling at the speed of light with respect to anybody means division by zero regarding the period of ticking of the light clock. $\endgroup$ – robert bristow-johnson Jun 30 '17 at 6:10
  • $\begingroup$ @Alfred Centauri However at no point in the question does he mention anything from the pov of some "light speed inertial frame". He says "does it stop with respect to the inertial observer?" where that "observer" is an outside one looking at it. $\endgroup$ – The_Sympathizer Jun 30 '17 at 7:15
  • $\begingroup$ @mike4ty4, what is it? A light clock with speed c? It doesn't exist. Does the question what does something that doesn't exist look like to an inertial observer make sense? $\endgroup$ – Alfred Centauri Jun 30 '17 at 10:51
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A "ship" that could go at the speed of light would have to be itself made of light, or something similar to it (e.g. some other massless particle). So a "light clock" at the speed of light would just be three photons in succession, essentially (the other two serving as useless "mirrors"). This simply travels as a unit, the photon in the middle does not bounce against the others but keeps a constant distance to them as seen to the outside observer. You could thus say it records no time.

It is right as the other comments have said, you cannot transform to an inertial frame moving along with the object, but you can talk about what happens from the pov of the outside, and the above is just that, it records no time. What this goes to also suggest is that nothing which travels at the speed of light is capable of undergoing any internal change or evolution. And in fact, this is observed with particle physics as well: all massless particles are completely stable and impervious to decay.

I'd think this is a legit to say "time is frozen at 'c'" -- even though there is no inertial frame possible to move at the speed of light. After all, time dilation is not measured in the object's own frame, but to an outside observer. Time essentially measures the evolution of something, that is, the progression of change, if it is not evolving at all, it is not experiencing time. There is no need to assign a rest frame to make sense of that.

Furthermore, because of this, you cannot also ask what a "being" would "see" that moves at the speed of light because nothing can evolve, meaning no consciousness and so no information processing is possible. Were your body's particles somehow to "magically" loose mass and move at the speed of light, yet maintaining coherence in their shape, you would be dead, transformed into the most rigid corpse imaginable.

The above also shows that motion at the speed of light is qualitatively different from motion at other speeds, in particular, there are things that can happen (internal evolution) for motion at slower speeds that cannot happen at this speed, and furthermore, there is no rest frame for this speed. Indeed, there are actually 3 qualitatively different domains of speed in relativity theory: $v < c$, $v = c$, and $v > c$. It is impossible to accelerate something between these three domains, and what domain a particle inhabits depends on what its mass is: respectively positive, zero, and imaginary. Only the first two appear to be realized in the physical universe: the third leads to difficulties with causality and the causality of our universe looks to be rigid on top of not seeing anything that moves in that domain, so such a domain is strongly disfavored empirically. The "principle of relativity" which says there is no universal rest or motion only applies within a domain, or perhaps even better, only with the domain $v < c$ since that is the only one in which we can clearly make sense of an inertial frame. Things moving in the other domains (in our universe, just the domain $v = c$, apparently) are indeed in an absolute state of motion, again illustrating this qualitative difference.

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From the inertial observer's point of view, the light clock is stopped. The light ray moves parallel to the spaceship and never reaches either mirror. Because light always travels at the same speed, if it moves along with the spaceship (at the speed of light), then it can't have any velocity perpendicular to the spaceship, since that would result in a larger total velocity.

From the spaceship passenger's view, everything is normal, including the light clock. They will observe that time has stopped outside the spaceship.

The important thing to remember about relativity is that it describes how to take observations in one reference frame and predict what a second observer ina different reference will observe.

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    $\begingroup$ There is no spaceship passenger's view, i.e., no inertial reference frame, no rest frame for the passenger, the light clock, or the spaceship if, as the OP stipulates, the spacecraft has speed c in an IRF. $\endgroup$ – Alfred Centauri Jun 30 '17 at 2:32
  • $\begingroup$ @AlfredCentauri True, but sometimes sensible conclusions can be drawn from taking the limit as velocity goes to c. $\endgroup$ – Mark H Jun 30 '17 at 2:54
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    $\begingroup$ Sometimes sensible conclusions can be drawn but not in this case. The second paragraph isn't sensible at all because it's just plain wrong according to SR. If the spacecraft has speed $c$ in an IRF, it has speed $c$ in all IRFs (invariance of the speed $c$), i.e., there is no spaceship passenger's view where everything is normal including the light clock because the spaceship is not at rest in any IRF; to all inertial observers, the spaceship is moving with speed c. $\endgroup$ – Alfred Centauri Jun 30 '17 at 3:02

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