I know the formula for the event horizon is $$R_s = 2GM/c^2.$$ At this distance the escape velocity equals the speed of light so nothing can escape from a black hole from this distance or less. But escape from is not the same as leave. Take the earth as an analogy. The escape velocity at the surface is 11.2 $km/s$. Any object with this speed or less remains captured by the earth, ie it is doomed to fall back to earth. But such objects can and do leave the surface of the earth for a period of time depending on their velocity. Is the same true for a black hole? But if it is, what might stop such an object, temporarily outside the event horizon from colliding with a second object and being given enough velocity to escape the black hole?
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3 Answers
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What's the depth of a bottomless well? The radial parameter in the Schwarzschild solution isn't what you think: it's the circumference of a sphere around the black hole divided by $4\pi$. It's not the "distance from the black hole".
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No, you’re using Newtonian intuition on a strongly relativistic situation, here Newtonian gravity isn’t even a rough approximation. With a black hole, once you are inside the Schwarzschild radius, space time curves so strongly that every possible direction points towards the singularity.
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One way to link our intuitive perception of gravity to what happens at extreme situations as the event horizon of a black hole is the concept of geodesics.
The Newton differential equation for gravity, for a test mass where no other force is present, can be interpreted as a geodesic equation (see Newton Cartan theory).
So, the example of a body going up until a given height and getting back is a particular solution of this geodesic equation, under some boundary conditions (initial position and velocity).
What happens near the EH of a black hole is that this Newtonian approximation can not be used, and the geodesic differential equation comes from the Schwarzschild metric (for a non rotating BH). And there are no solutions going out of the BH, no matter the boundary condition of initial velocity, if the initial position is at the EH radius or less.
answered Jul 31, 2023 at 2:18