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I want to show that it is impossible to construct a timelike curve between two points on the event horizon of a black hole. This should be an obvious fact, for example, by considering any particular model of a black hole, one can go to coordinates that extend over the horizon and by observing the deformation of the light cones as one approaches the horizon, one would notice that they become degenerate, which indicates that the horizon is a null hypersurface.
However, I want to prove this without assuming from the start that the horizon is a null hypersurface, but to simply use the formal definition:
The event horizon is the boundary of a region from which it is impossible to reach null infinity by any causal path.
Ideally, this impossibility could come out of this definition directly, however I don't see yet how it is impossible to have a timelike path that stays on the horizon(by this definition then it wouldn't be able to reach null infinity since it stays on the horizon).
I have a feeling this shouldn't be hard to see, but for some reason my mind got stuck on it, so any help would be appreciated.
However, I want to prove this without assuming from the start that the horizon is a null hypersurface, but to simply use the formal definition:
The event horizon is the boundary of a region from which it is impossible to reach null infinity by any causal path.
Ideally, this impossibility could come out of this definition directly, however I don't see yet how it is impossible to have a timelike path that stays on the horizon(by this definition then it wouldn't be able to reach null infinity since it stays on the horizon).
I have a feeling this shouldn't be hard to see, but for some reason my mind got stuck on it, so any help would be appreciated.