If you change the position of an object with mass, at rest in both positions in relation to another object with mass, so that there is more potential energy in the system of the two masses,
does that change the rest mass of either or both objects?
I say:
By E=mc^2, the system has gained mass = (Kinetic Energy to move the objects)/c^2 = (the difference in potential between the first and second positions)/c^2.
I say that the individual rest masses of the two objects doesn’t change if the objects are at rest with respect to one another except moved farther apart.
Since the rest mass of either object of mass hasn’t changed, the field had to take up the mass.
My co-worker says:
the object you moved has gained mass in the amount of the K.E./c^2 relative its starting position once it again comes to rest.
You are correct. The famous equation “E = mc[sup]2[/sup]” is in fact a special case, the case where there is no potential energy. The full relativistic energy equation is:
E = mc[sup]2[/sup] = ( Mc[sup]2[/sup] + p[sup]2[/sup]c[sup]4[/sup] )[sup]1/2[/sup]
Where M is the proper rest mass and p is the momentum. This says nothing about potential energy. To encompass that, you need:
Total E = ( Mc[sup]2[/sup] + p[sup]2[/sup]c[sup]4[/sup] )[sup]1/2[/sup] + V
Yes that’s correct. The system has gained energy that cannot be transformed away, and energy that can’t be transformed away equals mass.
Not correct. Only the system has gained energy not the individual objects that comprise the system.
Move the object far enough away so that the gravitational effect from the other object is negligible. Does your friend think that if he was standing on the object he would weigh more?
As Ring and Spatial Rift 47 said, the locally-measurable mass of the object doesn’t change with its potential energy. You can see this by the Equivalence and Relativity principles, like this:
Suppose I have two heavy objects, of masses M and 100M, and also a test mass m. First I place my test mass at a distance 10R from the 100M mass, moving the mass M far away. The test mass has a potential energy -G(100M)m/(10R)=-10GMm/R, and a weight G(100M)m/(10R)[sup]2[/sup]=GMm/R[sup]2[/sup]. Next I place the test mass at a distance R from the M mass, moving the 100M mass far away. Now the test mass has a potential energy -GMm/R, but it still has a weight GMm/R[sup]2[/sup].
By the Equivalence Principle, both cases are locally indistinguishable from a test mass in flat space, measured in a frame accelerating at GM/R[sup]2[/sup]. Now by the Relativity Principle, any local experiment I perform on this test mass must give the same results in both cases. So the (locally-measurable) mass of my test mass must be the same for these two cases, even though the test mass is at higher potential energy in the second case than the first.