Could You Escape an Event Horizon by increasing a Black Hole's velocity?

Question

I have an idea for trying to escape the black hole's event horizon that might make basically no sense and I want it double checked. Basically, I do know that you can't escape the event horizon unless you can move faster than light, but there is no way to move faster than light... But what if you fake it?

The basic idea is that, if you move the black hole in one direction and turn your ships thrusters on in the opposite, you could get to the point where the ship is moving away from the center of the black hole at faster than light speeds even though both objects are moving at FTL speeds. Basically, you are in a zone where you need to move at 1.01C to escape, you turn your ship on to move at 0.6C and you somehow (probably with outside help) get the black hole's center to move at exactly .41C in the exact opposite direction allowing you to escape. This seems like a relatively simple method that would theoretically allow a particle or a tiny fraction of mass or light to escape a black hole... but it seems so obvious that if it actually worked it would be common knowledge already... So why doesn't this work?

Answer 1

That is not how velocity works. Even without getting into general relativity, if you move away from a black hole at velocity $v_1$ and at the same time the black hole moves away at velocity $v_2$ (from some observer that remains at rest between them) then the relative velocity between you and the black hole isn't $v_1+v_2$ it is $$\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}$$

And this value is always less then the speed of light.

If you do want to get into the General Relativity, then you are even more doomed, since inside a black hole, every direction is "down". It's not just that you cannot get enough speed to go up, it is that literally up doesn't exist. Going faster will only make you go down faster, and so the singularity is inescapable.