jerromyjon said:
I'm not trying to compare manifolds or reference frames or anything that serious, I just want to have a basic conceptual understanding of "things that occur in nature"... I'm certainly not anywhere near the level to care how the velocity has to relate to something computable. Able to be calculated. Is there any other reason it matters?
Unfortunately, if you want even a basic understanding, you need to realize that velocity has to be measured relative to something. Furthermore, we need to check that your understanding of velocity is what we are talking about, otherwise we can't possibly communicate.
To measure a velocity, we need to define an observer who has zero velocity. Presumably this observer is hovering above a black hole by means of a powerful rocket. This is not a very complete description, but hopefully it's complete enough. If that's not what you had in mind, then the answer I'm going to give will perhaps not be right, but you'd need to explain what you mean by velocity in order for us to communicate.
I find that people in general find it somewhere between hard and impossible to explain what they mean, so what I try to do is to offer an explanation of what I mean, in hopes that they can compare it to what they mean. This sometimes works (and sometimes doesn't, if they don't follow the explanation of what I mean). Unfortunately, if they don't follow the explanation of what I mean, I see no way to proceed with the discussion.
How do we measure velocity? Well, we have two observers, and one of them (preverably both of them) carry around clocks and rods with them. Then one observer uses his clocks to measure the time interval it takes the other observer to pass by the length of his rod.
We find that velocity is reciprocal - it doesn't matter which one of the observers has the clocks and rods , we get the same answer either way.
So, let's get into the details. An observer "at rest" exists only outside the event horizon of a black hole. Said observer "at rest" needs to have a proper acceleration away from the black hole, or else they'll fall in. Something needs to hold them in place.
At the event horizon, the required proper acceleration is infinite - no material observer can "hold station" at the event horizon of a black hole. If we want the observer to measure the velocity, it needs to be a material observer (one that is not moving at light speed). There is a FAQ that explains why there is no reference frame of an observer moving at "c", the explanation is not terribly complex but some people get hung up on this point anyway. For the moment I'll just refer to the FAQ on this point rather than digress and break the thread of what I want to explain.
So, the observer "at rest" is presumed to be a material observer, who carries along some clocks and measuring rods, and he uses these clocks and measuring rods to measure the velocity of the infalling observer.
With this approach, the velocity of the infalling observer at the horizon can only be computed as a limit, by taking observers closer and closer to the event horizon. To make a long story short, that limit is "c", the speed of light. So when we speak informally, we say that an infalling observer crosses the event horizon at "c", though what we actually mean is this limiting process.
There is perhaps an easier way to do the same thought experiment that provides more insight. We still need two obserers, one moving, one "at rest", to measure the relative velocity between. However, we put the clocks and rods on the falling observer, and have the falling observer measure the relative velocity of the stationary observer, instead of the other way around. As we remarked earlier, this gives us the same answer. The two numbers turn out to be the same, measuring the velocity is a reciprocal process.
Doing things this way though clarifies what happens at the event horizon. There is no "observer" with clocks and rods located exactly at the event horizon, but we can imagine a light pulse moving out from the black hole that's "stuck" at the horizon. This doesn't qualify as an observer, because a light pulse can't have clocks (or measuring rods). To see why, you'd have to expand your interest to look at things you say you are not interested in, but turn out to be important to answer your question even though you think you're not interested.
With this approach, we can see that the relative velocity between the light pulse (located at the event horizon), and the infalling observer (who is a material observer with clocks and rods) must be equal to c, because the velocity of a light pulse relative to a material object is always "c". The trick here is the event horizon is not some sort of normal place. It can't be. Assumign that it is, and that there is some sort of "reference frame" that exists there leads to problems, the problems which you are probably encountered. The solution is reasonably simple - don't do that, don't assume there is a "reference frame" at the event horizon.
The math tells us the same thing, except we instead talk about the poor behavior of the Schwarzschild coordinates at the event horizon.
