More volume with more mass?
Regarding to the Schwarzschild solution, is there more volume with more mass?
Lets take for example a space shuttle (spheric for simplicity) in an stationary orbit at the position $r_1$. The black hole around which it is orbitting has a mass $M$. Now, the mass is increased (not interesting, how). $M_2=2\cdot M_1$for example. The position $r_1$ is kept the same. Doesn't the volume of the spaceshuttle increase due to the mass increase and curvature increase?
According to a comment, the question might be similar to the question whether the volume of the space shuttle increases while moving nearer to the center of the black hole. The curvature there is stronger then outside.
As the metric is $ds^2 = -Bdt^2 + Adr^2 + \text{angular terms}$, and A is increasing while decreasing r, the volume should increase either. However, this seems to be dependent on the sign convention: If (+---) is used instead of (-+++), then the volume is decreasing, isn't it? I read somewhere, that <PERSON> used (+---) and now we are using (-+++) mostly and it is pure convention - but isn't the volume change important to classify?
Addendum: To make the question clearer I want to change it to the following thought experiment: Mankind finally managed to develop a material which completely isolates from gravity (please, don't ask me how). 483
Real hoverboards (see "Back to the future" 2) are used all-around the planet and, at last, the following experiment is done regularly in the physics undergrads course: They got an empty cube of this gravity-isolating material. In the middle of the cube, there is a (non-isolating) sphere which can be filled with a material of high gravitational mass. The volume of the cube is measured with and without the mass in the middle (probably by simply pressing water into it). Of course, the undergrads need to be very precise with the volume measurement, carefully keeping temperature the same and everything. Do they measure more volume inside the cube if the middle-sphere is filled with mass?
For clarification: volume is positive here, mass is also positive here. The material of the cube is magical. It just stays where it is and isolates gravity. (Hm, ok, I see: "It stays where it is" regarding inside the cube is different from "it stays where it is" regarding the room with the students. So, to define even this: It stays where it is regarding the room with the students) 651