System of concentric shells: What general relativity predicts in such a situation?

Question

If there's a system of a point mass or point particle with mass $m$ and a structure such that - The structure is made of many concentric shells with certain thickness and the radius of those spheres are constantly changing but all remain concentric. These shells have a volume mass density $d$. And the total mass remains same.

The point mass is set into orbit around the structure. Newton's theory of gravity says that the point mass trace the same orbital path regardless of the changes in the structure (the spheres are concentric thus the resultant force due individual interactions with the point mass in the orbit will be the same as any between the point mass and a point particle placed at that centre with same total mass - Newton's shell theorem).

But what the General relativity will predict for such a system? What kind of changes into the orbit do one will observe?

Answer 1

The spacetime outside a spherically symmetric arrangement of mass is described by the Schwarzschild metric. This is a consequence of Birkhoff's theorem.

So the changes in the interior structure of your object make absolutely no difference to an object outside it. The orbit will be exactly the same as if the object was unchanging, or indeed if it was a black hole. The only way the orbit could be affected is the the object was undergoing changes that broke the spherical symmetry.

Answer 2

First ask yourself what Newtonian gravity would say.

Would a particle orbit around the center of a perfectly spherical star, or would the star and particle co orbit their common center of mass? Obviously the latter, but the former is a good approximation where you ignore the effect of the particle on the star.

So how about general relativity? Same deal. If you ignore the mass of the particle and its affect on the spherical shells then it doesn't matter how the shells are moving in and out (assuming the particle stays outside that outer shell). The metric from the shells alone doesn't change (Birkhoff's Theorem).

But the metric due jointly to the shells and the particle does change. And in fact the parts of the shell that are closer to the particle are affected more strongly by the particle. And so when the shell as a whole is closer to the altitude of the particle it is affected more strongly. So if the outermost shell and the particle get close, then ignoring the affect of the particle on the outer shell becomes a big issue.

So in reality there would be gravitational waves and the particle and the shells would eventually merge unless the particle somehow built up enough stress on the shells to cause them to explode, and they would have to be pretty sensitive to do that.

And in fact the parts of the shell that are closer are affected more strongly by the particle and when the shell is closer it is affected more strongly. - couldn't understand these lines. Yes, the metric jointly to the shells and the particle will change in such a case. But what kind of stress are you talking about(like tidal forces ?)

The effect of the particle on the shell will stress the shell because of tidal forces from the particle. Could be small if they stay far away, but could be large if they get close.