Blackhole Escape Velocity

Good4you

I've heard it said that any object that wants to escape another object's gravitational pull must travel at the object's escape velocity. I assume this is incorrect, and would only apply to something with an initial velocity at the escape velocity (like a bullet from a gun would need to travel at the Earth's escape velocity to not fall back to earth.) Is it correct that if it gradually exerts force like a space elevator then something could travel much slower than the escape velocity?

1) So as a typical space shuttle or rocket does not expend all of its fuel at launch, but releases it gradually; does that mean that the space shuttle travel's much slower than the Earth's escape velocity.

2) If a black hole is defined as an object with an escape velocity equal to the speed of light, why could something gradually exerting force (like a space elevator) not "climb" its way out of the black hole's gravitational pull?

JesseM

Yes, powered flight allows you to escape Earth's pull without reaching escape velocity.
Some popular articles on black holes may define them this way but it isn't really accurate. The reason you can't escape a black hole doesn't really have any perfectly good analogy in terms of Newtonian forces and escape velocities, you really need the theory of general relativity to understand it. Are you familiar with the idea of a light cone in relativity? Basically if you pick an event, the "future light cone" of that event is all the points in spacetime that could receive a signal from the original event which travels at the speed of light or slower; since no signal can travel faster than light in relativity, no point in spacetime outside the future light cone can be influenced by the event in any way. In general relativity light cones can be "tilted" by the curvature of spacetime, and it works out that for an event at or inside the event horizon of a black hole, the entire future light cone is tilted to lie inside the event horizon, so that no signal from the event can ever reach the region outside the horizon. Here's a diagram showing the future light cones of various events on the path of a particle falling into a black hole (its event horizon represented by a cylinder, the vertical dimension of the diagram standing for the time dimension), from the textbook Gravitation by Misner/Thorne/Wheeler:



This page has some similar diagrams at the bottom, one showing more clearly how for an event exactly on the horizon, the light cone has tilted over enough so it becomes impossible for anything in the future light cone to be outside the horizon: