In Newtonian physics, as far as I understand it, for a small object falling towards a massive object, the graphs of position, speed and acceleration vs time look like this:
(If the massive object is so massive, and the small object close enough, that the change in potential over the course of the fall is negligible, these curves approach a parabola, for distance over time, and a line for speed or acceleration over time)
What shapes would these curves have in general relativity, for an object falling towards a black hole, as seen from a stationary outside observer?
It is my understanding that for an outside observer, the object never reaches the event horizon of the black hole, since time dilation slows its speed to zero at the event horizon.
So over time the speed would first increase, but then decrease again to zero at the event horizon.
- the object is small enough that the gravitational pull it exerts on the black hole is negligible,
- the object is rigid enough and the black hole big enough that the object stays intact, despite tidal forces,
- the stationary observer has sensitive enough equipment to detect the position of the object despite red shift for most of the way (obviously not all the way to the event horizon),
what are the actual equations describing distance, speed and acceleration over time, that the observer would measure?
Is there some simple geometric shape that at least approximately describes these curves?