This is a question for people who know about relativity, not relativity-hating crackpots. Suppose that in box “A” you have two spheres sitting in contact with each other. In box “B” you have two spheres that are identical to those in box A, but in this box they are sitting some distance apart, giving them gravitational potential energy. Will the apparent mass of the two boxes be the same, or will box B appear to have more mass than box A due to its higher potential energy?
That's a very good puzzle; suppose we let some time pass so that the masses in box B move due to gravity and come together; now some work is done; some energy is expended. But the masses in box B are not necessarily the same as those in box A. They may be more massive because of they absorbed the energy of contact. I would hazard a guess and say that box B system is more massive.
I'm not sure, but I think the idea of gravitational potential energy is problematic in general relativity. What you need to look at is the energy-momentum tensor.
So far as I know, the energy-momentum tensor is not affected by gravitational potential energy. Am I wrong about that? So it would seem to me that according to relativity the apparent mass of box B would actually increase over time as the spheres accelerate toward each other, then (if they collide perfectly elastically) abruptly go back down when they come together. See my problem?
Look at it another way. Apply force to the spheres in box A to move them to the position of those of box B. Now box A has to contain more energy; work was required to move the spheres in box A apart. Still guessing box B is more massive; but only guessing. Edit: Continuing with the scenerio, allow the balls in box A to come together and freely bounce apart forming a system in which they continously move together and apart. Seems more massive.
energy is mass, but only the energy which is frame independant. (for example kinetic energy is not mass and produces no gravitational field.) Thermal enregy is mass as it is the same for all reference frames (I think, but not sure as if you add up all the individual particles's KE I am not sure the total is frame independent) Likewise the gravitational potential energy of the two separated balls is positive in all frames but I am not sure that it is frame independant. None the less I am almost sure there is more (mass and gravity) with the balls separated. Some one who can do it correctly with GR needs to tell us.
The four-momentum of a particle is a four-vector, so it is frame invariant. The energy is the timelike component of the four-momentum, so it is not frame invariant. But the mass is the norm of the four-momentum, so it is frame invariant. The four-momentum of a system is the vector sum of the four-momenta of the particles, so that is frame invariant as is its norm which is the mass of the system. However, the mass of the system is generally greater than the sum of the masses of the particles. This is basically for the same reason that the length of one side of a triangle is shorter than the sum of the lengths of the two other sides. -Dale
