In special relativity, all motion is relative. But in the presence of black hole, all motion is with respect to black hole. The curvature of spacetime depends on how far we are away from the black hole. If there are three observer, two of them moving close to the black hole and third observer at Infinity at rest with respect to the black hole. Then schwarzschild metric only describes the what happens with respect to the stationary observer because black hole is only static in his frame. If the remaining two observer are moving around the black hole then black hole isn't stationary right? So how would they describe their coordinate and what would metric tensor look like in their frame? I have studied Gullstrand Painleve coordinate, but they are obtained via transformation of schwarzschild metric where we only transform the time coordinate from inertial observer at Infinity to proper time of freely falling observer. The $(r,\theta,\phi)$ stay the same, but they are reinterpreted as measuring proper distance on constant proper time slice. If I come back to special relativity where changing frame of reference, meant transforming all the coordinate not just one of them.
When we talk about length contraction and time dilation in special relativity. We are comparing two different observer and their measurements of length and time.
For example, if I am trying to measure distance, we don't just substitute $dt=0$ in both frames of references due to relativity of simultanity. Derivation to the argument is in this post
However when we are doing the same in general relativity, we compare two observers but at rest with $dt=0$ or $dr=0$.
$$dr' = \frac{dr}{\sqrt{1-\frac{2GM}{r}}}$$
$$d\tau = \sqrt{\left(1-\frac{2GM}{r}\right)}dt$$
What happens if the observers are moving relative to one another in the presence of black hole? This feels like it brings back the specific reference frame back into the picture. Here we are concerned with observers and their motion relative to black hole. Even the origin of this coordinate system lies at the center of black hole.
The 4 velocity of stationary observer is: $$U^{\mu} = \left(\frac{1}{\sqrt{1-\frac{2GM}{r}}},0,0,0)\right)$$ and a freely falling observer, where we measure $r$ from the center of black hole. $$U^{\mu} = \left(\frac{1}{1-\frac{2GM}{r}}, -\sqrt{\frac{2GM}{r}},0,0\right)$$ In Gullstrand Painleve coordinate, the same velocity transforms into: $$U^{\mu} = \left(1, -\sqrt{\frac{2GM}{r}},0,0\right)$$ and stationary schwarzschild observer at finite radial distance $r$ is $$U^{\mu} = \left(\frac{1}{\sqrt{1-\frac{2GM}{r}}},0,0,0)\right)$$
How can we use the relative velocities of these observer with respect to black hole to determine relative velocity with respect to them. I understand that velocities at different point can not be transformed from one observer to another observer perspective but, each particle or observer moving on spacetime gives rise to a vector field and we can transform them given a coordinate transformation between two observers. The observers must have a transition function between their chart which would allow one to do so. I understand that no coordinate is globally inertial and I don't want them to be. I wanna know coordinate system of freely falling observer where the time measured is proper time and distances are proper distance. My question is how do we figure that out and their possible relationship with tetrad transformation of observer.
