Consider the spacetime around a static and spherically symmetric mass like a planet. From this setup we have a clear spatial origin for coordinates.
Outside the planet, the GR solution would be the Schwarzschild metric. Inside would depend on the details of the spherical object, but if we had the solution to this region as well, we could calculate a 'proper length' from a point at $r_1$ to the spatial origin with the line integral: $$R = \int_{0}^{r_1} \sqrt{g_{rr}}\ dr.$$
We could also take all the spacetime points at the given $r = r_1$ at time $t=0$, to get a spherical shell. I'm not quite sure what the integral for a "proper area" is, but I assume from the metric we could calculate the area of the spherical shell. If the spacetime was not curved, we'd expect: $$Area = 4 \pi R^2.$$ Since it is curved, there will be some "solid angle deficit" (or excess?). Maybe we could define it as: $$\Delta \text{solidangle}(R) = Area/R^2 - 4 \pi$$ I have a feeling this isn't the "best" way to define this, but the point is we can come up with some definition showing the difference relative to a sphere in flat spacetime.
My question is: Is there a better way to define this that is "more physical"? Is there some way we could notice (measure) some missing solid angle while staying on just one spherical surface?
What I'm ideally hoping for is something similar to the "angular deficit" around a cylinder mass ( Angular deficit ). A cylindrical surface well above the mass could still be mapped out with $z$ and $0 \leq \theta < 2\pi$, but we could perform experiments on this spatial surface to determine that in some physical sense, there is an "angular deficit", and it would be better to describe it with $\theta$ having a limit other than $2 \pi$. Maybe parallel transport (carry a gyroscope?), or some kind of phase difference for paths with winding number other than 0, were suggested in the other thread. I'm not sure if there is a good generalization to the "angular size" of a sphere.
