A space ship has maximum proper acceleration of $a_0$. How close can it fall freely towards a black hole before it can no longer accelerate away?

Question

Edit: To clarify, all motion is radial only.

Classically, the answer to this is pretty obvious. You just find the distance from the black hole at which the gravitational acceleration matches $a_0$ and then determine that once the spaceship gets closer to the black hole than this radius, it cannot escape the gravitational pull. Unfortunately though the Universe isn't classical.

But that doesn't change the fact that we feel gravitational acceleration in our non-inertial frames of reference. Now, this question has made me realise that most textbooks and courses on general relativity tend to deal with things following geodesics, so we never actually worry about problems like this.

In this case, we have a spaceship along a non-geodesic world line that is in a curved spacetime. Is there any limit (aside from the Schwarzschild radius) on how close it can get to the black hole before it can safely accelerate away? How do we account for the gravitational force on the spaceship in GR?

Answer 1

Assuming you are talking about radial motion the limit is when the proper acceleration for a stationary observer is equal to $a_0$. This is calculated in twistor59's answer to What is the weight equation through general relativity? The proper acceleration at a distance $r$ is given by:

$$ a = \frac{GM}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}} $$

Note that this is similar to the Newtonian result but modified by the extra factor of $1/\sqrt{1-\frac{2GM}{c^2r}}$. Equate this to the maximum acceleration of your spaceship, $a_0$, and solve for $r$ to get the smallest distance from which the spaceship can escape.

Note that the proper acceleration goes to infinity at the event horizon i.e. when $r = 2GM/c^2$. At the event horizon it is impossible to escape no matter how powerful an engine you have.