What is the median shape of the surface of the Earth

Factual Questions
1

I have been obtaining amusement from perusing (not entirely passively) some discussions amongst Flat Earth Conspiracy Theorists and rational persons, and a question occurred to me.

According to General Relativity, the Earth moves in a straight line (Newton’s First Law), but in the context of the local spacetime gradient, that line is a geodesic that looks approximately like an ellipse in Euclidean space (with small distortions due to gravitational influences less dominant than the Sun).

My question is, can the curve of the Earth’s gravity well be measured (directly or indirectly) to compare the median shape of the surface to the geodesic at that altitude? Is the surface of the Earth in fact flat in the GR context? Instinct would suggest that what would define the circular geodesic for the surface is actually below the surface, meaning that the surface of the Earth is not flat but slightly or somewhat concave.

2

I’m not sure where to start with this, because I think there are some misconceptions implicit in this question, but I’m not sure precisely where they are.

First off, the geodesics that are relevant in General Relativity aren’t the shortest paths through three-dimensional space; they’re the shortest distances through four-dimensional spacetime. And if we’re including time as a dimension, then the “surface of the Earth” is a three-dimensional structure, not two-dimensional.

Second, assuming that we’re only talking about spatial dimensions, here, and the bit about GR was just a digression (so the surface of the Earth is two-dimensional), the geodesics we’re discussing would be, by definition, along the surface of the Earth. And since the Earth isn’t perfectly spherical, the geodesics won’t be perfectly circular.

Third, if we do approximate the Earth as a sphere, so that, on that sphere, the geodesics are circles, then it would presumably make most sense to pick the sphere that most closely approximates the surface in some way (RMS deviation from the surface, or same volume, or something), in which case that approximating sphere would be above the actual surface in some places (mostly higher latitudes) and below the actual surface in others (mostly lower latitudes).

Fourth, are we discussing the solid Earth itself, or its gravity well? If its gravity well, the usual way of discussing it is via equipotential surfaces, surfaces along which there isn’t any “uphill” nor “downhill”, and something confined to the surface, set at rest on any point of it, would stay at that point. There are an infinite number of such surfaces, nested within each other; the one that most closely approximates sea level is usually called the “geoid” (you’d expect sea level to be exactly on an equipotential surface, but due to currents and such, there can be minor irregularities).

3

Yes, I was trying to find out whether the surface of the Earth is “flat” in the context of GR spacetime distortion. This would be a 3-dimensional assessment: is the surface effectively 2D planar in 3D space. But, then you throw in the part about 4D spacetime and bollix up my premise.
       I guess considering the time dimension would require inclusion of the inertial reference frame, kind of but not quite like mixing in SR. If the centripetal acceleration imposed by gravity matches the centrifugal inertia of rotation, perhaps one could regard it as truly flat – except, being a sphere, the centrifugal inertia (which is already way too low at its max) is a downward curve from equator to pole.
       Hence, I have to conclude that the surface of the Earth is concave, like a bowl, probably with a catenary profile with respect to lattitude.