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:

enter image description here

(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.

Assuming that

  • 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?

  • $\begingroup$ The body would accelerate , but it cannot reach the speed c within the limit of the event horizon. It will fall into the event horizon at a finite speed, but what happens after that is not observable by an external observer as it will take light forever to reach the observer once beyond the event horizon. $\endgroup$
    – Lelouch
    Jul 3 '16 at 20:14
  • $\begingroup$ I don't think your description takes time dilation into account. As far as I understand, an object actually slows down when getting close to the event horizon, as measured by an outside observer. $\endgroup$
    – HugoRune
    Jul 3 '16 at 20:17
  • 2
    $\begingroup$ That's not even correct within the realm of Newtonian physics. What you have there is a the dynamics in a constant gravity field, whereas the correct description requires to use a 1/r potential. That's called the Kepler problem and its solutions are ellipses, hyperbolas and, in a limit case, a parabolic curve. $\endgroup$
    – CuriousOne
    Jul 3 '16 at 20:18
  • $\begingroup$ Fair enough, I simplified that example too much. I tried updating my question to clarify that these example shapes only approximately describe motion for the edge case. Still, they are merely meant to illustrate what sort of graph I am looking for in this question. $\endgroup$
    – HugoRune
    Jul 3 '16 at 20:55
  • $\begingroup$ What you are looking for does, strictly speaking, not exist. The motion of test particles in strong gravity is observer dependent. An infinity observer will see very different physics compared to an observer falling with the test body. Or, to put it differently, the observer independence of physics is the actual deception of nature in the non-relativistic weak gravity case. $\endgroup$
    – CuriousOne
    Jul 3 '16 at 21:13

I believe I understand what you are asking and the answer is that the observer does not see an accurate representation of what happens near the event horizon. In the first graph the line would curve up to infinity at the end because as the object fell into the hole the light would be slowed to a stop at the event horizon. As the object begins it's approach to the Black Hole at point A, there is not sufficient gravity to affect the speed of light emitted by the object. As the object gets very close to the Event Horizon and the gravity of the Black Hole has a considerable effect on the speed of light, the object appears to slow down from the perspective of the observer, but not relative to itself. Finally, immediately before the Event Horizon is reached the emitted light from the object is brought to an infinitely low speed and appears to the observer to stall.

enter image description here

An interesting follow-up question is would the object get dimmer as it approached the Event Horizon because the flux of photons was reduced?


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