I think I have come across something that invalidates something one of my Physics professors said back in college, though. Someone check me if I'm wrong. We talked about how increasing the potential energy of an object will increase its mass. This seems perfectly reasonable, especially if one takes mass to be equivalent to rest energy. Thus, if we develop a super storage battery and dump a few trillion Joules of electrical energy into it, we would find the mass of the charged battery to be higher than that of the empty battery. As I recall, he also mentioned something else. Let us take a pair of identical massive balls, connected by a massless, friction free spring. They sell them at imaginary hardware stores all over. If we either compress or stretch the spring, the mass of the system should rise, since we have increased its potential energy. Let's assume we've stretched it. If we release the balls, they will begin moving towards each other, converting the potential energy of the system into kinetic energy for each of the masses. The kinetic energy of the system will not change, of course, because the center of mass doesn't move. Now, taking mass to be Lorentz invariant, which all the smart boys and girls are insisting is the case, the mass of each ball does not change. As the balls move toward the point where the spring is relaxed, the mass of the system should return to its original value. The balls then start compressing the spring, until they eventually come to a halt WRT each other and start moving apart again. In short, we've created ourselves a nice, little perpetual clock - patent pending. The interesting thing is we see the mass of the clock regularly increase and decrease at twice the fundamental frequency of the clock. Am I on track so far? So now let's increase the number of masses and spring into the trillions. In other words, let's talk about a gas. Well, sort of. In a gas, the molecules are not really very influenced very much by the other molecules except as a pair of them get very close together, at which point the pair undertake some extremely high accelerations. Most of the time is spent by most of the molecules moving in a fairly straight, if rather short, line at a more or less constant velocity. It seems to me, then, that dumping heat into a perfectly rigid container filled with a gas would not increase its mass nearly as much as we would expect, despite the fact we have increased its potential energy a huge amount. I submit this is because we haven't really increased its potential energy. It only seems that way at the macroscopic level. Rather, we have actually increased its kinetic energy, which doesn't increase its mass. The reason it doesn't look like we have increased its kinetic energy is all the individual momentum vectors sun to zero (well, almost zero - we'll ignore Brownian motion, here). Have I run amok, here? It also seems to me the same would hold for a solid or a liquid, just not quite as strongly, since the molecules in non-gaseous object are much closer together, thus spending much more of their time accelerating. In any case, I think my prof was wrong.