I was musing today about black holes, and came across what seems to be a contradictory fact about black holes. If any matter were to actually fall into a black hole, it seems like it would need to gain infinite energy.
This is seen by considering an object close to the event horizon of a black hole. If we consider escape velocity for a particle at the event horizon, it has an escape velocity of the speed of light. The energy required to achieve this escape velocity would therefore be infinite. As we approach the event horizon, it seems like we should be able, then, to define an arbitrarily small distance across which the particle gains an arbitrary amount of energy.
As the object approaches arbitrarily close to the event horizon, it seems like its relativistic mass would therefore grow arbitrarily high, resulting in a corresponding increase in the gravity of the black hole, which seems impossible.
If we extrapolate back to the formation of the black hole, it then seems like it should be impossible for any particles to fall into the black hole, even as it forms. Instead, mass should be distributed in a probability cloud whose density asymptotically approaches zero at the event horizon of the black hole. New particles could then enter this cloud of incredibly dense (but not black hole dense) matter, where they would be captured rather than falling past the event horizon, within which there would be a vacuum.
If this were true, there would be no black holes, only hollow spheres of radiation and matter which externally look like black holes, since their gravity would be great enough that particles entering the sphere would exit at an incredibly slow rate. Interestingly, such spheres wouldn't suffer from the apparent information paradox that black holes do.
That or I'm missing an obvious explanation for why that's all wrong, which seems more likely because I'm neither a physicist nor an astronomer. What's the critical flaw in this argument?