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Pencilvester

I have been autodidactically working on understanding the mathematical concepts that general relativity is based on, so I've never had anyone to ask questions to (until it occurred to me to find a forum like this). I started with linear algebra and vector calculus, and worked my way up to a basic understanding of tensor analysis, topology, manifold theory, and differential geometry, so if you use math in your answer be aware that I am not completely proficient in these latter subjects.

Anyway, I was watching the Leonard Susskind lectures on GR from the Stanford youtube channel, and in the one where he talks about the Schwartzschild metric and what it means for objects falling into black holes, he is adamant that the apparent slowing down (in velocity, not necessarily time dilation) of objects near the event horizon and only crossing said horizon after an infinite amount of time is merely a peculiarity of the choice of coordinates. His reason seems to be that it takes a finite proper time for the object to cross the event horizon, therefore, from the perspective of the object, nothing strange happens near the event horizon.

While I completely agree that it

*would*take a very finite amount of proper time to cross the horizon, it seems to me that the object would always be destroyed before it could ever actually cross. Let's suppose that you're standing at a comfortable distance "above" the black hole with a rubber ball and a flashlight. Drop the rubber ball so that its trajectory will go straight to the black hole (only motion through t (time) and r (radius from center of black hole)) and as it begins to fall, start strobing the flashlight in its direction (and never stop). Eventually, I will call the amount of time between pulses "dt" for obvious reasons, but for now dt will retain its normal meaning of a small, arbitrary change in coordinate time. So if I've understood correctly, the Schwartzschild metric looks like this:

dτ^2 = (1 - s/r)dt^2 - (1/(1 - s/r))dr^2, where "s" is the radius of the event horizon (really 2GM and using units so that c=1). I've omitted the dΩ term for simplicity. Working this metric around as the Legrangian to get information about the freely falling ball introduces total engergy, "E". I can then manipulate and solve for dr/dt, and then linearly approximate dr by multiplying both sides by dt, then plug this approximation back in to the metric, which yields dτ = m*dt(1 - s/r)/E. It's easy enough to see that as r approaches s from the outside, dτ approaches 0. Now if we interpret dt as the length of time between light pulses, we can see that as the ball approaches the event horizon, the length of time between light pulses that the ball experiences, dτ, approaches 0 and therefore the frequency of pulses approaches infinity. I might not know a ton about how light interacts with matter, but I would expect that this would deliver a good deal of energy to the ball in a very short amount of (proper) time and just utterly destroy it, or at least turn it into a melange of tiny high energy particles and radiation, or perhaps just radiation.

But then there's the matter of light itself, according to the Schwartzschild metric, only crossing the event horizon after an infinite amount of coordinate time as well, but light doesn't experience the passage of proper time (dτ for a light ray is always 0), so a statement like "it takes a finite amount of proper time for a light ray to cross the event horizon" would be meaningless.

So unless I'm missing something (and that's why I'm posting this, to see if anyone can tell me what I'm missing), it seems to me that everything that falls "into" a black hole just gets slowly eaten away by incoming photons before it ever crosses the horizon (a process that would indeed take a very short amount of proper time from the perspective of the infalling object). So then what is the purpose of discussing the interior of a black hole? If all of the frames of reference outside of a black hole all agree that nothing, not even light, crosses the event horizon until all of time has passed, and the black hole evaporates before then, why can't we just consider a black hole to just be a dense shell made up of radiation (and matter being slowly blasted into radiation)?

Someone might respond by saying, "If all of the matter and energy is on the outside of the event horizon of a black hole, then wouldn't it no longer be a black hole? Wouldn't we see them?" Firstly, unless I misunderstand Steven Hawking, or unless he's wrong, black holes do, in a sense, emit radiation, so they aren't completely invisible. Secondly, any light rays that find their way out from near the event horizon to an observer a safe distance away from the black hole would be incredibly redshifted, right? They would be so low energy, it would be difficult to detect them using normal instruments(?). I haven't really explored redshift or blueshift too thoroughly, so I'll have to rely on someone with more experience than I to confirm or refute that.

Which finally brings me back to my reason for posting this: I have only had books, my curiosity, and a few Leonard Susskind videos to teach me about GR, so if there's anyone reading this that could give me insight as to what I'm missing here, it would be much appreciated.