- #1
michael879
- 696
- 7
I've been looking into this relatively simple problem out of curiosity and I have to say I've thoroughly confused myself, hopefully someone can provide some insight:
Imagine you have a massless (for simplicity) spherical shell with some overall charge on it that starts at a radius of "infinity". What happens to the intertial and gravitational masses as you slowly shrink the shell?
I eventually want to be able to answer this within the general relativistic framework, but to start I'm just focusing on the simpler classical aspects (i.e. Minkowski space-time, Newtonian gravity, spherical instead of cylindrical symmetry) to get a feel for it. The way I see it:
1) At first it has 0 mass (all kinds) and some constant charge Q.
2) Assume it is contracted at a constant velocity to avoid radiation
3) As it shrinks, it builds mass from the electrostatic energy being put into it
4) As the mass grows, a negative contribution to the mass from the gravitational energy is also put into it
5) At some point (either a finite radius or at 0 radius) you should reach an equilibrium where the gravitational and electrostatic forces cancel out and you're left with a stable spherical shell of charge Q and some mass M.
Now my confusion is about this equilibrium point, and whether or not its stable. Clearly it is unstable against perturbations in the +r direction. The -r perturbations are far less trivial though, and it seems to me that either:
a) It is a saddle point equilibrium and the shell once again begins to gain mass and collapse to a point
b) The mass starts to decrease and the shell is pushed back to equilibrium by the electrostatic pressure
c) The most sensical to me: The equilibrium point is at r=0, M is some function of Q, and there don't exist any perturbations in the -r direction
The second point of confusion is on the negative energy contribution from gravitational effects. If you were to set Q=0, and start with some non-zero mass it seems that if you compressed the shell enough it would end up with a negative mass? Clearly I'm doing something wrong, but I don't see what..
The third thing I'm confused about is where the mass is located in an originally massless spherical shell. Clearly the energy density is entirely outside of the shell, so it would seem to me that what I'm viewing as a shell with electromagnetic contributions to its own inertial mass is actually more like a massless ball with some mass density surrounding it (from the electrostatic field). If this were the case it would seem to me that an observer on the surface of the shell would feel no gravitational attraction towards it, as all of the mass is outside of the shell and spherically symmetric. So while it makes sense to say that the inertial mass of the ball increases as you compress it, is it incorrect to say the same about it's gravitational mass? Because if that's the case it is clearly in conflict with the equivalence principle (granted, I know this is a classical discussion atm but I would expect the equivalence principle to hold up at least approximately!)
Imagine you have a massless (for simplicity) spherical shell with some overall charge on it that starts at a radius of "infinity". What happens to the intertial and gravitational masses as you slowly shrink the shell?
I eventually want to be able to answer this within the general relativistic framework, but to start I'm just focusing on the simpler classical aspects (i.e. Minkowski space-time, Newtonian gravity, spherical instead of cylindrical symmetry) to get a feel for it. The way I see it:
1) At first it has 0 mass (all kinds) and some constant charge Q.
2) Assume it is contracted at a constant velocity to avoid radiation
3) As it shrinks, it builds mass from the electrostatic energy being put into it
4) As the mass grows, a negative contribution to the mass from the gravitational energy is also put into it
5) At some point (either a finite radius or at 0 radius) you should reach an equilibrium where the gravitational and electrostatic forces cancel out and you're left with a stable spherical shell of charge Q and some mass M.
Now my confusion is about this equilibrium point, and whether or not its stable. Clearly it is unstable against perturbations in the +r direction. The -r perturbations are far less trivial though, and it seems to me that either:
a) It is a saddle point equilibrium and the shell once again begins to gain mass and collapse to a point
b) The mass starts to decrease and the shell is pushed back to equilibrium by the electrostatic pressure
c) The most sensical to me: The equilibrium point is at r=0, M is some function of Q, and there don't exist any perturbations in the -r direction
The second point of confusion is on the negative energy contribution from gravitational effects. If you were to set Q=0, and start with some non-zero mass it seems that if you compressed the shell enough it would end up with a negative mass? Clearly I'm doing something wrong, but I don't see what..
The third thing I'm confused about is where the mass is located in an originally massless spherical shell. Clearly the energy density is entirely outside of the shell, so it would seem to me that what I'm viewing as a shell with electromagnetic contributions to its own inertial mass is actually more like a massless ball with some mass density surrounding it (from the electrostatic field). If this were the case it would seem to me that an observer on the surface of the shell would feel no gravitational attraction towards it, as all of the mass is outside of the shell and spherically symmetric. So while it makes sense to say that the inertial mass of the ball increases as you compress it, is it incorrect to say the same about it's gravitational mass? Because if that's the case it is clearly in conflict with the equivalence principle (granted, I know this is a classical discussion atm but I would expect the equivalence principle to hold up at least approximately!)