GR86 said:
there must be a reason you said above the event horizon, not at or below it.
Yes. The reason is that, as I said, the rope has a finite breaking strength, and the force that can be exerted at its top end is finite. That means that, at the point on the rope where it breaks, there will be a finite proper acceleration at the instant of breakage. That requires the break to happen above the horizon; there is
no point at or below the horizon where the proper acceleration would be finite with an unbroken rope. See further comments on that below.
GR86 said:
We know gravity bends space
No. We know gravity
is spacetime curvature. More precisely,
tidal gravity is spacetime curvature.
GR86 said:
objects in free fall will fall in a straight line but through curved space.
No, objects in free fall will travel along worldlines that are geodesics of curved
spacetime.
I am making this key distinction because it is not trivial. It makes a
huge difference in the predictions that your model makes.
GR86 said:
a rope being pulled is not free falling.
That's true. But it doesn't mean spacetime curvature doesn't affect the rope. Spacetime curvature doesn't just affect geodesic (free-falling) worldlines. It affects all worldlines.
GR86 said:
space and time are not separable, so as the gravity of the black hole distorts space it also distorts time.
No. The black hole's spacetime geometry
is spacetime curvature. It's not that "gravity distorts spacetime". "Gravity" is not a cause. Gravity
is spacetime curvature. They're the same thing.
GR86 said:
are you saying the rope breaks due to the sheer pulling of gravity, or bc of the differences of "time" from outside the horizon vs inside?
Neither. The rope breaks when the tension in the rope exceeds its finite breaking strength. The tension in the rope is ultimately caused by the spacetime curvature at and near the horizon, combined with the pull being exerted on the top end of the rope that prevents that end from freely falling. But the effect of spacetime curvature can't be usefully described as "pulling of gravity" or "differences of time".
GR86 said:
This brings me back to a question i asked recently, how much is time distorted at an event horizon vs here on earth.
I already answered this question: it's meaningless. The concept of "time dilation" has no meaning at or below the event horizon.
GR86 said:
A slight tug on the rope from a distance, would be a massive yank on the rope within the horizon, due to time variance correct?
Not quite. There is a germ of truth here, but it has to be stated carefully.
Consider first a
static rope, entirely outside the horizon, and not falling or rising, just hovering, suspended from its upper end. If the upper end is very far away from the horizon, and the lower end is very close, then yes, the tension in the rope, or more precisely the upward proper acceleration at a given point on the rope, will be much, much larger at the lower end than at the upper end. But this is not due to "time variance"; at least, that's not a useful way of looking at it. A better way to look at it would be to say that force "redshifts" as you gain altitude just as the frequency of light signals does.
However, note that I just said "entirely outside the horizon". It is
impossible for even an infinitesimal piece of the rope to be static (hovering at constant altitude) at or inside the horizon; it
must be falling inward. In other words, there is
no force that you can apply to the rope, no matter how large, at or inside the horizon that will make it hover, even for an instant.
To put this another way: if you look at the force that has to be applied to an infinitesimal piece of the rope to make it hover, as a function of altitude (more precisely, of radial coordinate), that force increases without bound ("goes to infinity" in the more usual phrasing, but that phrasing is really not quite correct) as you approach the altitude of the horizon. But, as I've said, the breaking strength of the rope is finite: so there will be
some altitude above the horizon at which the force that is required to make the rope hover exceeds the rope's breaking strength. At that point, the rope
has to break. And that point will always be above the horizon. It doesn't matter that the force applied at the top of the rope can be smaller (a lot smaller, if the top of the rope is at a high altitude), and gets increased as you go down the rope. The point is that the force has to exceed the rope's breaking strength at some point above the horizon.
