At a black hole's event horizon, the escape velocity equals what?

At a black hole’s event horizon, the escape velocity equals the speed of light.

This is the familiar Newtonian shorthand for describing the boundary of a black hole. Escape velocity is the speed an object would need to move away permanently from a gravitational field. At the event horizon, that speed reaches light speed, so even light cannot escape to distant observers.

General relativity gives the more precise explanation: the event horizon is a boundary in spacetime beyond which every possible future-directed path leads inward. It is not a solid surface, and an astronaut crossing a sufficiently large black hole’s horizon might not notice anything locally unusual at that instant.

A common mix-up is to say that gravity suddenly becomes infinitely strong at the horizon. The horizon instead marks a causal boundary. For a non-rotating black hole, its radius is the Schwarzschild radius, calculated from the object’s mass. A distant observer sees approaching light become increasingly redshifted and the infalling object appear to slow, although the falling object crosses the horizon in finite proper time.