- #1
Markus Hanke
- 259
- 45
I am trying to understand the implications of the Birkhoff theorem as it applies to a cavity within a spherically symmetric, stationary thin shell. Is the region of space-time within the cavity described by the actual Minkowski metric diag{1,-1,-1,-1}, or is it some other metric with constant components ? We know that whatever it is, it must be constant at all points in the cavity.
I am finding this somewhat confusing, because the vacuum metric within the cavity must smoothly connect to the interior metric of the thin shell itself, which in turn must smoothly connect to the exterior Schwarzschild metric. The idea is that, globally, space-time for this scenario is everywhere smooth and differentiable, even at the boundaries of the shell. I might be wrong here, but that seems impossible if the metric in the cavity is strictly Minkowskian, or not ? At the same time it is clear that the Riemann curvature tensor must vanish everywhere inside the cavity.
Physically this boils down to the interesting question of gravitational time dilation - is a clock at rest within the cavity gravitationally time dilated as compared to an ideal, hypothetical clock at infinity ? If the vacuum metric within the cavity is Minkowskian, then the answer is clearly no; however, this seems to be in contradiction to Newtonian physics, where the gravitational potential within the cavity is everywhere constant, but not vanishing ( I am assuming the convention that the potential vanishes at infinity ). I understand that the value of the potential is arbitrary, but the question is whether it is the same inside the cavity as it is at infinity. It would seem that this cannot be the case.
Any comments will be much appreciated !
I am finding this somewhat confusing, because the vacuum metric within the cavity must smoothly connect to the interior metric of the thin shell itself, which in turn must smoothly connect to the exterior Schwarzschild metric. The idea is that, globally, space-time for this scenario is everywhere smooth and differentiable, even at the boundaries of the shell. I might be wrong here, but that seems impossible if the metric in the cavity is strictly Minkowskian, or not ? At the same time it is clear that the Riemann curvature tensor must vanish everywhere inside the cavity.
Physically this boils down to the interesting question of gravitational time dilation - is a clock at rest within the cavity gravitationally time dilated as compared to an ideal, hypothetical clock at infinity ? If the vacuum metric within the cavity is Minkowskian, then the answer is clearly no; however, this seems to be in contradiction to Newtonian physics, where the gravitational potential within the cavity is everywhere constant, but not vanishing ( I am assuming the convention that the potential vanishes at infinity ). I understand that the value of the potential is arbitrary, but the question is whether it is the same inside the cavity as it is at infinity. It would seem that this cannot be the case.
Any comments will be much appreciated !