Curvature of spacetime inside hollow sphere

[Tony Stark]

If mass curves spacetime in its vicinity, then consider the following case-
Take a heavy hollow lead sphere which has 2 smaller lead balls placed in it. The Outer Sphere will curve spacetime around itself and thus will have its own gravity, but what about the 2 balls placed in it? The spacetime already have been curved by the outer sphere, then how will the two smaller balls have their own gravity?

 

[A.T.]

The spacetime already have been curved by the outer sphere, then how will the two smaller balls have their own gravity?

Space time geometry inside the big shell with the smaller balls is different than without them.

 

[Nugatory]

Space time geometry inside the big shell with the smaller balls is different than without them.

Outside too, of course. (A.T. already knows this, I'm just not sure that OP does).

 

[Tony Stark]

Space time geometry inside the big shell with the smaller balls is different than without them.

The geometry of spacetime is taken invariant in General Relativity, so how can it be different..

 

[A.T.]

The geometry of spacetime is taken invariant in General Relativity

Where did you get that idea from?

 

[PeterDonis]

The geometry of spacetime is taken invariant in General Relativity

"Invariant" means "the same for all observers" (or in all frames of reference). It does not mean "the same for all possible distributions of matter and energy". If you change the distribution of matter and energy (for example, by putting two smaller lead balls inside a hollow lead sphere, instead of just having the hollow lead sphere with nothing inside it), you change the geometry of spacetime.