Will using a digital mirror (or other mirror)that is traveling at the
speed of light toward a light beam double the speed of the refelcted
light?

Hello and thank you for your question.

Commenter kottekoe-ga is entirely correct in stating that the very
foundation of special relativity is that the speed of light in a
vacuum is always exactly the same.

In your question, you assume that the mirror is traveling at the speed
of light.  This might be a physical impossibility (because no matter
how little its mass, no less than an infinite amount of energy would
be needed to accelerate it to that speed), but the answer is no
different than if it were travelling at, say, 99.9999% of the speed of
light.  Anyway, the paper that I cite at the end of this answer
reaches the same conclusion when the mirror does travel at the speed
of light (as long as it isn't assumed to <exceed> that speed).

Physicists Albert Michelson and Edward Morley in their famous
experiment actually measured the speed of light in two directions - -
one direction where the speed of the earth moving through the universe
would seemingly boost the beam of light and another, perpendicular to
the first, where it would not.  The light beam was found to be going
at the same speed in both cases.
http://en.wikipedia.org/wiki/Speed_of_light#Relativity

Here is a more complete explanation of the phenomenon:
SPECIAL RELATIVITY: LORENTZ TRANSFORMATIONS
http://cmtw.harvard.edu/Courses/Phys16/l1_latex/l1_latex.html

And here is a more technical paper on the subject.  Note the author
requires only that the mirror velocity "never <exceeds> the speed of
light."
Reflection of Light From a Moving Mirror:Derivation of the
Relativistic Doppler Formula without Lorentz Transformations
http://www.phys.ufl.edu/~malik/education/refl.pdf

Search terms used:
"special relativity" fundamental mirror speed light
"special relativity" fundamental mirror speed light site:.edu

Thanks again for bringing us your question.  

Google Answers Researcher
Richard-ga

This is a simple answer to what I think is a complex question.  The
answer is as high level as science can explain.  I still think that
there is some experimentation that can be done here to prove that we
can increase the speed of light.  I am certain - I beleive in God -
certain that we are able to increase the speed of light in a universe
this size.

That is a question that remains unanswered and only theorized. A
similar question is, "If I am traveling at the speed of light and turn
on my headlights, what will happen?"

tutuzdada-ga

The answer to this is known as well as anything in physics. As
described in the other thread with the same question, the answer is an
unequivocal NO. The very foundation of special relativity is that the
speed of light in a vacuum is always exactly the same. The
implications of this are profound and well established experimentally
and theoretically.

There is a small problem in your assumption. As you increase the speed
of the mirror its mass will also increase. that means it'll take more
energy to increase the speed further. as the speed of the mirror
approaches the speed of light the amount of energy required will tend
to infinity.

This does not invalidate any of the detailed answer given by Richard
which is well considered and exactly on point; however, you might be
interested to learn that, at the very cutting edge of science, there
is actually debate going on about whether ?constants? such as the
speed of light, are actually constant.

June 05 Scientific American has a good lay discussion titled
?Inconstant Constants? which you may find interesting.

It is a fascinating topic and fun to think about, but please don?t
read too much into the article, these scientists are talking about
something very important to physicists but not something which has
much bearing on the universe we have to live in daily. They are
discussing far distant times, places, or other universes (M theory
predicts as many as 10 to the 500th power) but, of course, if physical
constants are different, you couldn?t be alive there since life as we
know it depends on the same physical laws which led Einstein to his
theories.

By the way, any questions about what will happen "if you are traveling
at the speed of light" aren't realy questions since, as far as we
know, in this universe you can't travel at the speed of light so
physicists don't really think about that sort of problem.