sha1000 said:
1) And why there is a need to precise these assumptions and approximations for this particular GR equation of the proper acceleration?
I mean, these assumptions must be made even for the classical Newtonian equation.
You asked if the equation you quoted was correct for a hovering object in GR. The answer is no in general, and in fact it isn't always possible to define "hovering", because if the geometry is time varying it can be impossible to find a sensible definition of "staying in the same place" - let alone calculate the acceleration necessary to hover. But in the case of the Schwarzschild geometry you can do, and the correct formula is the one you gave.
sha1000 said:
2) So there is no need to make the weak field assumption?
The formula is exact for the specific case of the vacuum around an uncharged non-rotating spherically symmetric body if all other masses (yourself included) are negligible. In other cases it's an approximation, or wildly wrong, or "not even wrong", depending on how badly you violate those assumptions.
sha1000 said:
3) Can we derive the Force equation (as we know it) from the proper acceleration?
Sure. But there are caveats here too. If you are measuring the rocket thrust needed to hover, measured locally, then you just multiply by the mass. If you are suspending the mass from a string then the force needed at the top of the string is different.
sha1000 said:
I have difficulty to understand why we can't give a simple definition of the distance R in GR.
Because the geometry is non-Euclidean and the distance across a circle isn't necessarily its circumference divided by ##2\pi##. In the case of a black hole the geometry is so far from Euclidean that there isn't a center of the circle - ##r=0## turns out to have more in common with a moment in time than a place in space, and there's no way to pass through it, even hypothetically. So we have to define a "radius" that isn't actually a distance in terms of the circumference of a circle (or the area of a sphere) at that radius.
I don't think all of this was clear in the early days of GR. Perhaps if it were we would define the Schwarzschild circumference as ##4\pi GM/c^2## and not use "Schwarzschild radius" at all.
sha1000 said:
We can measure the real distance between the center of the distant galaxy and any star in the system. Am I wrong?
It depends what you mean by "real distance", what measurement method you use and, in some cases, how distant. Measuring a distance to the surface of the Sun, or even its center, is simple enough in principle. But that distance will not be the distance round our orbit divided by ##2\pi## (even if our orbit were circular). The difference is tiny, but it's there.
The distance to another star in our galaxy or a nearby one gets a bit trickier to define because they aren't stationary. If you bounce a radar pulse off one (which is possible in principle) you'd get a different answer from what you'd get if you used a really long ruler, because the two methods respond differently to the changing geometry. You can calculate what either method will tell you and the results will differ by far less than typical measurement uncertainties, but it rather undermines the concept of a "real distance" to a star. It gets even more fun when talking about distances to distant galaxies, since there's the metric expansion of the universe to contend with - rulers don't necessarily work and radar doesn't either if you go far enough (the echo never returns, even in principle unless the universe is closed).
The fundamental truth is that a lot of "obvious" stuff about how you can measure things and standard relationships like ##C=2\pi r## rely on assuming Euclidean geometry in a static spacetime, and that is only an approximate model of reality.
We are fairly well into "you need to study GR properly to understand this" territory here. Can I ask why you asked your original question? There isn't a general answer to it (more precisely, the general answer is ##U^\mu\nabla_\mu U^\nu##, but that probably isn't helpful), so understanding what you are trying to do might enable us to give specific advice.