Black hole matter accumulation

[PeterDonis]

The problem here is that coordinates does not give clear picture about this geometry of simultaneity.

Schwarzschild coordinates don't, that's true; that's because their lines of simultaneity (the lines of constant Schwarzschild time t) all intersect at the horizon. But that is not true of other coordinate charts.

But I want to check that this geometry of simultaneity is consistent between two coordinate charts (Schwarzschild and Painleve). I have gut feeling that these two geometries are incompatible (they correspond to two physically different situations). And I either want to get rid of that doubt or confirm it.

It depends on what you mean by "physically different situations". Both sets of lines of simultaneity are in the same spacetime--the same global geometry--but they are two *different* sets of lines. The Schwarzschild lines of simultaneity only cover the exterior region of the global geometry. The Painleve lines of simultaneity cover both the exterior region and the interior region, and the horizon.

From that reasoning it seems that simultaneities corresponding to of black hole and white hole can not be jointly realized in single coordinate system (when we include region behind EH) and therefore they are mutually exclusive. But the only difference between black hole and white hole is this geometry of simultaneity.

The black hole and the white hole *can* both be "realized" in the same coordinate system: the maximally extended Kruskal coordinates. In those coordinates, the lines of simultaneity in the white hole region and those in the black hole region are "parallel"--they belong to the same global set of lines of simultaneity--but they are different subsets of the global set of lines of simultaneity, that don't overlap.

However, in an actual spacetime where a black hole is formed by the collapse of a massive body, the white hole region is not present; only the exterior, the "future interior" (the black hole region), and the non-vacuum region occupied by the collapsing massive object are present. As far as I know, the full spacetime geometry described by the maximally extended Kruskal chart, which includes the white hole region (and also a second "exterior" region), is theoretical only, and no one claims that any actual physical spacetime contains all the regions that are present in that theoretical spacetime.

Of course when we compare Schwarzschild and Painleve coordinate charts we should leave out black hole interior.

Yes, because the Schwarzschild chart (the exterior one) doesn't cover the black hole interior (or the horizon). But even in the exterior region the lines of simultaneity in the two charts are different sets of lines, "pointing" in different "directions".

Physical interactions is the thing that is important in physics. Round trip for light to EH is infinite and in Schwarzschild chart forward trip is equal to backward trip. So there can be no physical interactions between interior and exterior of Schwarzschild black hole.

Physical interactions don't require a "round trip" causal influence, just one-way is enough. Causal influences can still propagate into the interior from the exterior.

Sorry but your arguments about *geometry* are just hand waving.

If you really can't grasp the geometric way of describing the physics, there are other ways. They take a lot longer to talk about, which is why most physicists prefer the geometric description, at least for GR. But if you're getting hung up on the word "geometry", you're missing the point.

 

[Dale]

It does not make sense to speak about different "infinite futures" or the same "infinite future".

Sure it does. If you have any timelike congruence of curves simply parameterize each curve by the proper time, find the surface formed by all of the congruence curves at a given proper time, and then take the limit of that surface as the proper time goes to infinity.

If you get different results when you extrapolate function then they are different functions and not extrapolations with different "without-limits".

Again, that is only true if you have a 1D function. I.e. it is true for a single curve in a timelike congruence, but not for the whole congruence. If you have a function of multiple dimensions then you can easily have different infinite limits.

In any case, the Schwarzschild solution is valid for more than a single observer, so taking different infinite limits is reasonable. There is no reason to restrict ourself to a single observer or even a single timelike congruence.

 

[zonde]

Physical interactions don't require a "round trip" causal influence, just one-way is enough. Causal influences can still propagate into the interior from the exterior.

And now you even does not speak English as it seems. The part "inter-" in the word "interaction" stands for "two-way".

From wikipedia:
"Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect."

 

[zonde]

Sure it does. If you have any timelike congruence of curves simply parameterize each curve by the proper time, find the surface formed by all of the congruence curves at a given proper time, and then take the limit of that surface as the proper time goes to infinity.

Sure, now you gave definition and the term acquired sense.

Again, that is only true if you have a 1D function. I.e. it is true for a single curve in a timelike congruence, but not for the whole congruence. If you have a function of multiple dimensions then you can easily have different infinite limits.

You can't find directly limit at infinity for a function of multiple dimensions.
Intermediate step is to express it as single-dimensional function. And depending on how you do that you will have different single-dimensional functions.

In any case, the Schwarzschild solution is valid for more than a single observer, so taking different infinite limits is reasonable. There is no reason to restrict ourself to a single observer or even a single timelike congruence.

There are reasons to restrict ourselves to single timelike congruence. One reason is simplicity of calculations. We might benefit from timelike congruence that is spatially static in respect to gravitating object.

Other reason is consistency of global observations. Actually I think that consistency of global observations is the way how we can meaningfully speak about "correct" geometry of simultaneity.

 

[PeterDonis]

And now you even does not speak English as it seems. The part "inter-" in the word "interaction" stands for "two-way".

From wikipedia:
"Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect."

All right, if you don't like the word "interaction" in reference to one-way causal influences, I'll just say "causal influence", and I'll dispute your claim that two-way "physical interactions" are what are important in physics. I think that "causal influences" are what are important in physics, whether they are one-way or two-way. That was the point I was trying to make.