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I have looked online to no avail. There is two competing answers and I am curious to know which one is right.

Someone asked me this question. If you are traveling at the speed of light can you see your reflection in a mirror in front of you?

My answer to the question is no, I would figure that in order for that to happen the light reflecting off you that would appear in the mirror must travel faster than the speed of light to actually reach the mirror (which we all know is impossible).

He says the answer is yes, that it is all relative to the current frame of reference.

Can anyone validate the correct answer with possible references?

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Possible duplicate: physics.stackexchange.com/q/8163/2451 –  Qmechanic Aug 25 '12 at 16:24
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4 Answers

up vote 9 down vote accepted

This question cannot really be answered because you cannot travel at the speed of light. See Accelerating particles to the speed of light

If you were massless, you would always travel at the speed of light. However, in that case you would not perceive the passing of time. In relativity, the time that passes for an observer depends on the proper time. The proper time for a light-like trajectory is always zero, so photons themselves do not experience the passage of time.

If you travel very near to the speed of light - perhaps 99.9% light speed relative to Earth, you would still be able to view yourself normally in a mirror you carried with you. That is ensured by the principle of relativity, which states that all physical processes work the same way at any constant speed.

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So if I am traveling at 99.9999% the speed of light, I will see my reflection in a mirror exactly the same way as if I were traveling at say our current speed on Earth (assuming both speeds are constant)? –  John V. Aug 24 '11 at 15:25
@John V. Yes, that's right. –  Mark Eichenlaub Aug 24 '11 at 16:25
Thanks. Do you know any good articles I can read more about the last comment of your answer? –  John V. Aug 24 '11 at 16:37
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Your friend is correct that it depends on the reference frame (see previous answers). Both of these responses are correct, the only difference being that Mark assumes you and the mirror share a reference frame and are measuring your speed relative to another frame whereas Zassou assumes that you are measuring your speed relative to the mirror (and traveling toward it).

With regard to Zassou's answer, it is true that Lorentz contraction occurs, but, since you are the observer, it would appear as though mirror were "contracted like a pancake" and that it appears to experience time more slowly relative to you. From your point of view, everything would appear normal except for the absurdly fast approaching mirror. It is also worth noting that if you assume the opposite direction of travel your image would be red-shifted.

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You can not travel the speed of light, you can only get close to it and the closer you get the more relativistic effects you experience, but we can continue talking about the problem you posed in the context of approaching the speed of light.

From your perspective

As you approach the speed of light, the mirror actually gets closer from your perspective and the speed of light both approaching you from the mirror and leaving you moving toward the mirror is $c$. However, you still perceive the mirror moving toward you. So if the mirror is moving toward you at $0.99 c$ and the light from it is coming at you at $c$, then the light from the surface of the mirror only departs at a relative speed of $0.01 c$, which for the context of this discussion we'll say is not very fast.

From the mirror's perspective

The mirror is a part of an inertial reference frame that "observes" (which is really just a formalism for relativity, different than seeing with photons obviously) you moving toward it at $0.99 c$ and let's also note that you are length-contracted like a pancake and experience time more slowly according to the mirror. Light comes from the spaceship (I presume) at $c$ and moves toward the spaceship at $c$. The light that the spaceship emits only moves $0.01 c$ faster than the speed of the spaceship.

In both cases it is agreed that the light from the spaceship emitted at a time $t$ before collision with the mirror only hits the mirror a small amount of time before the spaceship itself slams into the mirror (specifically $0.01 c t$, and yes I know this isn't an objective time measure as I have used it here). In the limit of going exactly the speed of light, then of course, no light from the spaceship is able to reach the mirror before the collision. A reflection of the spaceship in-travel is impossible in this case.

So what do you see?

You see reflections in the mirror from before you started your speed-of-light trip. If we talk about going close to the speed of light, like $0.99 c$ then you see a highly blue-shifted version of yourself for a short period of time (relative to the duration of the trip) right before you crash into the mirror.

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What speed will your reflection appear to be going? Does length compression / time dilation apply to it, or can it move faster than c because it's not a physical thing (like the dot of a laser projected onto the moon can move faster than c) –  Random832 Aug 24 '11 at 18:54
@Random That's a very good question that I could make a complete separate answer for. From the "perspective" (which isn't what is "seen", but what is interpreted by rules of relativity) of the spaceship that moves at $0.99 c$ its reflection should be approaching at $1.98 c$, but what you said about this being "false" is correct. More interestingly, the appearance due to solely the light rays (although blue-shifted) would limit to infinity velocity as the spacecraft velocity approaches c. It only sees its reflection for a short time and the reflection travels the same distance in that time. –  AlanSE Aug 24 '11 at 19:14
Thanks - on further thinking about it, I managed to justify it to myself - if there's an actual ship behind the mirror, which exactly 'mirrors' (no pun intended) the first ship's actions (relative to the mirror, and simultaneously from the point of view of an observer near the mirror), I couldn't think of any consistent solution other than the reflection always lines up with the other ship. I do have some questions about blueshifting, that I will need to think through some more. –  Random832 Aug 24 '11 at 20:20
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At relativistic speeds, one does not simply add and substantial relative speeds. In other words, light reflecting off of a mirror traveling at 0.99c does not bounce back at (1-0.99)c, but bounces back at c. Worth reading up more in this.

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