Davide- We're very pleased to help out.
1. You are correct regarding E. You are also correct regarding m if it is simply defined to be E/c
2, i.e. the inertial mass, the quantity that appears in
p=m
v. Some people prefer to use m to describe the rest mass, sqrt(E
2-p
2c
2)/c
2. For an isolated system, it too is conserved since both E and
p are conserved, so delta_m is zero either way. If someone decides to describe the system as the sum of a number of subsystems (e.g. elementary particles) then the sum of their separate energies is conserved so long as the interaction energies are not important. Then the sum of their inertial masses is conserved. The sum of their rest masses need not be conserved, since the sub-systems can exchange momentum.
2. yes
3. Yes, the rest mass changes when the non-kinetic part of the energy changes. Note that due to the conservation laws, that cannot happen due to any purely internal processes in an isolated system, but requires interactions with the outside. When people say that some nuclear reaction changes the rest mass, what they are really doing is taking the sum of the rest masses of some parts, not the rest mass of the total system. Those two quantities can be very different, as you can easily work out in two-particle examples.
As far as the gravitational potential energy goes, I hesitate to give too confident an answer since my knowledge of General Relativity is sketchy. However, to lowest order you do include the gravitational potential energy in the total energy and thus in the mass calculations. You can see where this leads to complications, since the gravitational energy is a source of the mass which enters into the gravitational fields, leading to the famous non-linearity of General Relativity. For our purposes here, consider the following situation: two stars in a highly elliptical mutual orbit. Some of the energy goes back and forth between kinetic and gravitational potential forms. If they contributed differently to the inertial mass, then momentum conservation would require that the entire binary speed up and slow down from the point of view of someone who says it has net momentum. From the center of mass point of view it wouldn't accelerate at all. That would make even approximate inertial frames impossible. So the gravitational potential energy counts as part of the total energy.
These were great questions, very clear.
Mike W.
(published on 04/23/2011)