I don't think there's an energy -> mass -> energy transformation going on. Rather the crystal enters a different energy state while it 'has' the photon.
Photons have zero rest mass, but light has an energy of $h\nu$, where $\nu$ is the frequency, so this gets added on to the mass of our thing.
Oversimplified explanation: Photons are massless so they can go as fast as they want without needing kinetic energy, but even massless things can store energy (think of the massless springs you studied in high school physics). Energy is the same thing as mass, but we still refer to photons as massless, in the same way that we think of an (imaginary) massless spring as such; we don't like to think of potential energy as _being the object_, just somehow _there_.
This makes more sense if you study physics, in no small part due to the fact that we make it all rigorous (though not to the satisfaction of the mathematicians, I guess...).
> Does this mean that if you move an object away from Earth's gravitational influence it will increase in mass because it has more potential energy?
Gravitational potential energy is a little different. It's the energy that the object would have if it fell to earth, it doesn't actually exist until it starts falling. (correct this, i'm sure i'm wrong)
> What about if you heated something up, does it gain mass then?
Yes, but it's incredibly miniscule to the point that it's not worth looking at in most situations
> How about sound? Does the sound of my voice cause imperceptible ripples of increased mass as it vibrates everything that it touches?
This one I think does but it's going to be so small that you also won't notice it. What you would be able to notice though are the subtle changes in density at the peaks and troughs of the wave.
> Is a charged battery heavier than an empty one?
Maybe. This depends on the chemistry of the battery more than anything. Some of them will oxidize as it gives out energy and give off the oxygen when charging. While the energy will contribute slightly to the mass because of the chemcical bonds, the battery will change mass more because of the lost/gained atoms than any other cause.
> Is glass heavier with light travelling through it?
I really don't know how to answer this one. The photon never really becomes part of the glass as far as I'm aware, but some will be absorbed and raise the temperature so maybe?
Think about the opposite - nuclear fission. The energy released when nuke goes off comes from the difference of mass between original atoms and products of the split. This difference is the energy stored in strong nuclear force, binding protons and neutrons together.
I belive I kind of understand what you are saying.
I am confused as to whether or not the photons actually stop moving. Since they halt them by shutting off the transparency it seems that the photons would get trapped and end up bouncing around being absorbed slowly.
Is that why the effect is temporary? The light eventually gets absorbed and disperced though the crystal?
> Since light is effect by gravity, it must have a mass
Sorry, this is completely false. Light is affected by gravity because gravity is really a curvature of spacetime.
* I guess "curvature of spacetime" is perhaps not the most satisfying explanation of why light is redshifted as it moves out of a gravitational potential well (i.e. if you shoot a laser from earth into space). Nonetheless the answer is still "because of relativity".
Uh, E = mc^2 is not the full equation. That is, in fact, a special case for an object with no momentum in a given frame.
The actual equation is E^2 = (mc^2)^2 + (pc)^2, where the p is momentum. And, unsurprisingly, the old "p=mv" isn't the whole story, either. The (magnitude of the) momentum of a photon is given by h/λ where h is Planck's constant and λ, of course, is the wavelength.
As such, at no point is it necessary for light to have mass and it is believed that, in fact, photons are entirely and truly massless. Relativistic mass--which some take to mean photons actually gain mass--is a confusing and misleading concept that doesn't really have a whole lot to do with actual mass and doesn't really help explain anything.
"Effective mass"--which is more related to stopping photons, but not really related to the Einstein equation you gave--also doesn't really mean the photon gains mass but it does mean that interactions within the crystal give effects similar to if the photon had mass.
Keep in mind these descriptions are just interpretations of observations. Another intelligent civilization could explain the observations in a wholly different way. The observations show that light is affected by gravity (more precisely, by centers of gravitational attraction) the same as material objects are. We have alternate generally accepted ways of looking at this, such as the river model of space [1], where space is moving inward toward a center of gravitational attraction, carrying along in its "current" everything including light rays.
Correct. Photons are responding to the curvature in space-time, not directly to the gravitational field. Doing a second check, light has inertia-mass, but not rest-mass.
