How does density affect gravity? Say we have two masses, mass A and mass B. These two masses are identical in every dimension. The only difference is the density. Do they not curve the same amount of space-time, and if not, why?
 A: Let's assume our two masses are spherical and not rotating, and they have the same mass. In that case Birkhoff's theorem tells us the geometry outside the masses is the same in both cases i.e. the Schwarzschild metric. So if you are some distance $r$ away, where $r$ is greater than the radius of either object, then the curvature is exactly the same. You would not be able to tell the difference between the two objects from their gravitational fields.
However if one object is very dense while the either is far less dense, e.g. one is a solid sphere and the other a spherical shell, then you could get much closer to the denser object before meeting its surface. This means the spacetime curvature would be greater at the surface of the solid object than at the surface of the shell.

A: I know this is overly simplified, but here goes.
As I understand it, if the sun were to collapse to the size of a basket ball, the orbit of the earth would not change. The amount of mass determines the size of the spacetime depression.  Replace the sun with an actual basket ball, and we are now soaring through the cosmos.  So unless I am missing something ( I know I will be corrected if I. am)  The  denser object will have.a greater gravitational field.
