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A plane mirror image approaches the object at the same rate the object approaches the mirror. So the speed of approach is twice the speed at which the object approaches the mirror. If the object approaches the mirror at a speed greater than half the speed of light (c), does that mean that the speed of approach between object and image will exceed the speed of light?

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No, see the second box of this hyperphysics page –  Kyle Kanos Oct 8 '13 at 18:09
@KyleKanos is right, but note that for a curved mirror, it is possible for the image to move at speeds greater than $c$, and there's nothing wrong with that. The image isn't an object, it's just an imaginary point in space from which rays appear to have originated. –  Ben Crowell Oct 8 '13 at 19:26

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The real question to ask is, if this were true, could information be transmitted FTL? An easy way to answer these questions is usually by saying the light source is a photon gun that shoots a continuous stream of photons, and drawing out the path of the stream.

A similar question that can be answered the same way is the one where you have two "photon guns" oriented such that their streams are crossing, and then quickly (at a speed ~.9c) turning them so their streams are parallel. You might naively think that their intersection point flies away FTL.

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This doesn't directly address the question. The logic in the second paragraph also doesn't quite make sense -- rapidly changing the intersection point does not require motion at speeds like $.9c$. –  Ben Crowell Oct 8 '13 at 19:39

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