Would <PERSON>'s law in the solar system be independent from a presumed different fundamental gravitational law?
A thought experiment:
Let's assume the gravitational field of a black hole without any neighbors would be stronger than the Schwarzschild metric in the Newtonian limit.
Now, we let a solar system (sun and planets, much smaller mass than the black hole) orbit around this black hole at $r>> R_s$, so that the tidal forces are nearly zero.
My question is, although the fundamental gravity law is "set" to be stronger, would the planets around the sun experience <PERSON>'s law?
The metric in the solar system can be calculated with the static, sperically symmetric solution of the field equations, namely $ds^2=-Bdt^2+Adr^2+\text{angular terms}$ using the Newtonian limit (asymptotically flat spacetime), lim A(r) = 1, because due to the free fall (orbit) around the black hole, there are no forces, no curvature experienced by the solar system as it is moving along the curvature in free fall. That would presumably lead to the derivation of the Schwarzschild metric.
The question therefore is, whether this solar system does experience an asymptotically flat space because of the free fall around the BH and is this asymptotically flatness of space independent from the underlying gravity law of the black hole?
Remark: Tidal forces are theoretically possible inside this solar system, because for example a planet experiences another force of gravity if it is behind the sun (in greater distance) in relation to the central black hole, than if it is in front of the sun (in nearer distance) in relation to the black hole. 397
However, these tidal forces are very small at galactic distances (see for example https://physics.stackexchange.com/questions/702248/are-there-tidal-forces-between-the-solar-system-and-the-galactic-centre), even if the gravitational field produced by the black hole were stronger. The difference measured within the solar system between "black hole gravity is stronger than Newtonian" and "black hole gravity is Newtonian" is even much smaller.
This is just a thought experiment. It comes from the idea whether it is possible to create an alternative model of gravity by abandoning the weak field assumption. 43