21
Physics, Astronomy & Cosmology / Re: Which bit of the Shell theorem is not working?
« on: 13/10/2023 19:41:50 »
Sorry for all the posts, but thought about this some more, and want to avoid stating things that sound unconditional.
I think that under Newtonian law, the sort of logic that you spelled out above is valid in an inertial (as opposed to expanding) frame, and that our test particle does indeed acceleration towards any arbitrary origin that you select. Hence, lacking any recession rate of that arbitrarily selected origin, all that homogeneous matter will collapse in due time. This is barring wrenches in the works like dark energy which isn't compatible with Newton's theories anyway.
So how does switching from Newton to GR affect that? Well, the shell theorem certainly has trouble under GR. Under GR, there is no possibility of a uniform gravitational field, and yet the shell theorem can show that an off-center spherical hollow in an otherwise solid homogeneous spherical mass will (like an infinite sheet of mass) produce a uniform gravitational field. Since GR doesn't allow this, the shell theorem doesn't work in general, and only approximates a solution for low mass densities.
GR of course also doesn't allow a uniform distribution of mass in an inertial frame, so any treatment using such a frame is wrong right out of the gate.
We can choose to put an origin somewhere and draw a sphere of radius r around that. Now, consider a particle of Hydrogen on the boundary (surface) of that sphere. By the shell theorem, we can see that the Hydrogen particle is attracted to the origin by gravity. We also see that we can ignore the attraction to anything outside that sphere of radius r.Back to the OP argument, since one can 'frame' my rebuttal in a way that refutes my rebuttal.
I think that under Newtonian law, the sort of logic that you spelled out above is valid in an inertial (as opposed to expanding) frame, and that our test particle does indeed acceleration towards any arbitrary origin that you select. Hence, lacking any recession rate of that arbitrarily selected origin, all that homogeneous matter will collapse in due time. This is barring wrenches in the works like dark energy which isn't compatible with Newton's theories anyway.
So how does switching from Newton to GR affect that? Well, the shell theorem certainly has trouble under GR. Under GR, there is no possibility of a uniform gravitational field, and yet the shell theorem can show that an off-center spherical hollow in an otherwise solid homogeneous spherical mass will (like an infinite sheet of mass) produce a uniform gravitational field. Since GR doesn't allow this, the shell theorem doesn't work in general, and only approximates a solution for low mass densities.
GR of course also doesn't allow a uniform distribution of mass in an inertial frame, so any treatment using such a frame is wrong right out of the gate.
The following users thanked this post: Eternal Student