So, the question of whether the event horizon can be reached or not, is geometrically equivalent to asking whether there are geodesics of spacetime that connect a shell at some distance to the event horizon, while remaining of finite geometric length. This question is a geometric one, and

*independent* of what coordinate system is used.

*Without reference to a specific coordinate system*, the proper distance between shells in Schwarzschild spacetime (note that this isn’t a reference to any coordinate system, just the name of this solution to the field equations) is, in terms of the metric, given by

This is not yet the length of any worldline, but simply geometric distance on the spacetime manifold. Your claim now was that this distance is infinite, so I would like you to show us how

*you* mathematically arrived at this conclusion. I am not interested in why you

*think* it should be infinite, I am interested only in a formal derivation, based on this particular solution to the field equations. So basically, if could evaluate the integral for us, and show how it diverges, that would be great. Once I see your mathematical derivation, I can then address the specific places where the actual problems occur, and we’ll take it from there. As you can (hopefully) see, the evaluated expression I gave earlier for this integral is quite finite, so you must have come up with a different expression, while mine is somehow wrong. I’d just like you to explicitly show this.

Can I please politely request that you restrict your response to addressing this one specific point, i.e. the mathematical evaluation of the above expression.