Is it possible to have perfect white or mirrors?

I think it's more that the color of a surface is more complicated than the simple model you are using will allow for.

But you will always get some absorbtion of some wavelengths.
To get something very white - look at TiO₂ and ZnO spectra.

 

Visible light, while only a small part of the electromagnetic spectrum, is still quite a range so I don't see how you could EVER have a material that could reflect all the frequencies perfectly. I think I'm saying the same thing Simon said.

As for a perfect mirror ... good question. I THINK I remember reading that there is some reason it's impossible for any mirror to be perfect, but I don't recall what the rationale for that statement was, and don't know if it's true.

 

iirc: classically you can get perfect reflection for one wavelength by interference methods - and, then, there is total internal reflection if we call the inner surface a "mirror".

On the scale of photons - all bets are off: there is no such thing as a "surface" the way we are used to thinking of it. See the Feynman QED descriptions of reflection ... you can increase the total reflection from a surface by removing more than half of it.

 

You could argue that an otherwise perfect mirror would have to recoil under the momentum of the light that it is reflecting and that this would red-shift the reflected light.

 

Of course not- you also can't fabricate a perfectly flat surface or a frictionless surface. Some real materials (for example, Spectralon and InfraGold) provide >99% reflectivity over a reasonably broad wavelength range:

http://www.pro-lite.co.uk/File/spectralon_material.php

http://www.labsphere.com/products/reflectance-materials-and-coatings/infragold.aspx [Broken]

 

Can a black hole send a beam of light back to its source if the beam comes in at the right angle? Without changing the light?

 

Sure, but from an infinitely narrow crossection, so any real light beam will get at least somewhat scattered.

 

You specified reflectivity at visible wavelengths. However, your question about thermodynamics implies broader conditions. So I think you were asking if it was possible for a mirror that reflects all the energy from electromagnetic waves completely. I don’t think there are any theoretical limits for such reflectivity restricted to the visible wavelengths. However, I would need to examine the Kramers-Kronig relationship more closely to determine if what you say is true restricted to the visible wavelengths.
The existence of a mirror that reflects 100% of the electromagnetic wave energy from its surface does not contradict any law in thermodynamics. Or at least I haven’t seen any such theorem which says a mirror that completely reflects violates thermodynamics.
However, the existence of such a mirror probably contradicts the Kramers-Kronig relations. The Kramers-Kronig relations relate the spectrum of the index of refraction to the spectrum of the absorption.
According to the Kramers-Kronig relationships, the reflectivity is high when the absorptivity is strong. The converse is not always true. However, there is always a strong absorption somewhere in the frequency spectrum when there is a high reflectivity somewhere in the reflectivity spectrum. You can’t have a high reflectivity without a strong absorption, somewhere.
The indices of refraction of two media at a surface between them determine the reflectivity. In order to reflect 100% over any spectral range, the index of refraction of one of the media has to be infinite. When you apply the Kramers-Kronig relations to such high index of refraction, one would get rather high oscillations in absorption at high frequencies. A high index of refraction over a spectral range implies that the material has a high absorptivity toward the higher frequencies of this range.
I suspect that you are intuiting this type of relationship. So you should look into the Kramers-Kronig relations to see if you are right.
The Kramers-Kronig relations have a physical basis in a property called “macroscopic causality”. However, here are links to introduce you to Kramers-Kronig relations.


http://en.wikipedia.org/wiki/Kramers–Kronig_relations
“The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. These relations are often used to calculate the real part from the imaginary part (or vice versa) of response functions in physical systems because causality implies the analyticity condition is satisfied, and conversely, analyticity implies causality of the corresponding physical system.[1] The relation is named in honor of Ralph Kronig[2] and Hendrik Anthony Kramers.”

http://media.wiley.com/product_data/excerpt/26/04700219/0470021926.pdf
“thus. the reflectance is substantial while absorption is strong.”

http://en.wikipedia.org/wiki/Refractive_index
“The real and imaginary parts of the complex refractive index are related through the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.”

 

That is correct ... that is what Snell's Law predicts.
However: Snell's Law is only obeyed on average.