Update: I have updated the the question, see below.
The radial excess due to the curvature of spacetime is a well-known prediction of General Relativity that is often mentioned in introductory texts, and then surprisingly (to me) never mentioned again.
Feynman's description in section II-42-3 of the Feynman lectures is this:
Suppose we were to measure the surface of the earth very carefully, and then dig a hole to the center and measure the radius. From the surface area we could calculate the predicted radius we would get from setting the area equal to $4\pi r^2$. When we compared the predicted radius with the actual radius, we would find that the actual radius exceeded the predicted radius by the amount given in Eq. (42.3). … it turns out the the earth has 1.5 millimeters more radius than it should have for its surface area.
To my knowledge there have been no experimental tests of this. This has led me to wonder whether it would be feasible to do so. The challenges I am aware of are:
- The large amounts of mass necessary to have a measurably meaningful effect (for a uniform density sphere of the mass of the earth the effect is predicted to be around 1.5mm). I take ‘measurably meaningful’ to mean within technically realisable limits and large enough that we are not solidly in the realm of quantum mechanics, i.e. the radial excess should be suitably larger than the de Broglie wavelength of the measured mass
- The difficulty in accounting for deviations from spherical symmetry
- The contribution from neighbouring massive bodies
The last one particularly is causing me some difficulty, as I don’t know how to quantify the effect of neighbouring bodies on the radial excess of a given object. The non-linearity of the Einstein Field Equations leads me to believe that this may be a non-trivial problem. Am I wrong? Is there a simple way to estimate the order of magnitude effect on the radial excess of a body from nearby masses?
My questions are then the above paragraph, and whether in general such a measurement is technically feasible, now or even in principle.
Edit: I have added this update to make explicit that the effect I am mentioning here is unique to General Relativity, unlike e.g. the bending of light by a gravitational source, which also occurs in purely Newtonian gravity (albeit with a different factor).
We can see this by noting that the proper distance between two points involves an integral with the metric tensor in the integrand (see e.g. the answer to this question How to calculate spatial distance in space-time?). In Newtonian gravity this would just be the Euclidean metric, and we would obtain the standard flat prediction for the radius.
It is this relativistic effect I am referring to, not the generic prediction of the bending of light. That is also why I mentioned my difficulty in understanding whether such an experiment would be feasible, since the presence of inhomogeneities would break the spherical symmetry, and thus make the Schwarzschild metric an unacceptable ansatz. Can we estimate the order of magnitude deviation from the simple result of Feynman if we have a more complicated metric?