An object cannot travel faster than the speed of light.
But given a video of an object travelling at $0.5c$, could one speed the video up so that the object appeared to be travelling at $1c$, $2c$ or more?
Or would something prevent this?
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$\begingroup$ Nothing would prevent this, but then the video wouldn't reflect something that would actually happen. $\endgroup$– knzhouCommented Jul 10, 2016 at 17:46
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3$\begingroup$ You could play the video backwards, or upside-down, or through a mirror, and those would still show possible physical situations. But there's no symmetry associated with speeding it up. $\endgroup$– knzhouCommented Jul 10, 2016 at 17:46
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$\begingroup$ Avoiding the "speeding the video up" issue, you can think that an object is approaching you at superluminal velocity because of an optical illusion. Jets from a quasar are one example: en.m.wikipedia.org/wiki/Superluminal_motion $\endgroup$– user108787Commented Jul 10, 2016 at 19:47
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2$\begingroup$ It depends on what you mean by "appear". An object that is traveling towards you does actually "appear" to be traveling faster than the speed of light. Let's say a nuclear rocket traveling at a speed of 0.8c takes off from Alpha Centauri some 4 light years away and you can see the enormous engine burn at takeoff here on earth. The light of the launch event took 4 years to get here. The rocket takes 5 years to get here, so the "apparent" travel time is only one year and the rocket seems to be traveling at 4c. $\endgroup$– CuriousOneCommented Jul 10, 2016 at 22:07
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1$\begingroup$ This seems related to questions like "can the edge of a shadow travel faster than the speed of light," which I'm sure have been covered before somewhere on this site. Edit: for example, physics.stackexchange.com/questions/48328/… $\endgroup$– sumelicCommented Jul 10, 2016 at 23:30
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2 Answers
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No. There is nothing to prevent such faster than light appearances. The rule is simple: No actual thing (information) can travel at a speed greater than the speed of light.
When the considered particle appears to travel at a speed greater than the speed of light in your video, there is a non-local distribution of information - set up a priori. This distribution exhibits itself as something travelling at a speed greater than that of light. But no actual particle/information travels at a speed greater than the speed of light in this entire scenario.
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Nothing can prevent you from speeding up the video, but there are ways to tell if you are looking at a sped-up video or at a real video of an object traveling at a speed close to or equal to $c$ (or greater than $c$, if we can assign any meaning to this concept).
Think about length contraction: if you are looking at a spaceship traveling at a speed $v$, from your frame of reference its length in the direction of motion will appear to be
$$L = \frac {L_0} \gamma = L_0 \sqrt{1-v^2/c^2}$$
where $L_0$ is the length of the spaceship in its rest frame.
If $v \in [0,c]$, $\gamma >1$, so the spaceship will appear shorter to you. If $v=c$, we would have $L=0$: the spaceship will look like a segment from your frame of reference (the contraction only happens in the direction of motion, so it would appear like a segment and not like a point).
This shortening effect will not be present in a sped-up video, so you would be able to tell if the video is really showing a spaceship moving at a speed close to $c$ or just a sped-up version of the same video.
If $v>c$, the formula above breaks down because $\gamma$ becomes imaginary, but you would still be able to tell a sped-up video from a real one...also because you know that no physical object can travel at a speed $v>c$.
