Assume one initially has perfectly flat Minkowski vacuum. At some point in this Gedankenexperiment a point mass spontaneously appears at some location in Minkowski creating a Schwarzschild geometry. At some interval from the location of this point mass, how much proper time does it take for the space to appear Minkowski? Is there some sort of gradual change in the metric from Minkowski to Schwarzschild or does this change occur instantaneously?
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Generally, no. Suppose we have empty space $T_{\mu\nu} = 0$, in which case the field equations are,
$$R_{\mu\nu}-\frac12 g_{\mu\nu}R = 0$$
with solution $g_{\mu\nu} = \eta_{\mu\nu}$. Adding any kind of matter, we have a perturbation $\delta T_{\mu\nu}$, and we have that to first order $R_{\mu\nu} \sim \Delta_L h_{\mu\nu}$ where $h_{\mu\nu}$ is a perturbation.
How the curvature changes over time is then dictated by $h_{\mu\nu}$ and will depend on the problem. Since you inserted some new matter at some time, $\delta T_{\mu\nu}$ will be time dependent, and so we expect $h_{\mu\mu}$ to also be time dependent, not instantaneously changing the manifold.
JamalS
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