Black hole matter accumulation

[Dale]

So I am stating that the same "infinite future" applies to set of observers that have parallel time dimension

OK, I think that the concept you are trying to describe here is a congruence, which is essentially a family of worldlines. For example, you could associate one worldline with each spatial location in Schwarzschild coordinates. This would give a family of timelike curves which could each represent a stationary observer relative to the central mass. Then this set of observers would all share the same "infinite future" region.

The existence of one congruence which share the same "infinite future" does not in any logical way forbid the existence of another set of congruences which share a different "infinite future". Your line of reasoning seems to be that there is a timelike congruence which ends up in the usual "infinite future" therefore all timelike congruences must end up in the same "infinite future". This is not sound logic.

i.e. it applies to global coordinate system as a whole.

Many spacetimes don't admit a single global coordinate system that covers the whole manifold. The Schwarzschild coordinates certainly don't cover the whole manifold.

 

[PeterDonis]

Well it would be nice to expand on that a bit. Best I could make of Einstein tensor is that it is divergenceless - so much for surmising about divergence as conceivable factor. Given the shell spherical symmetry and static state, can the operation of said tensor within shell wall be expressed entirely in terms of potential and gradients thereof, preferably in polar form? I think all but the T₀₀ term is operative as source on rhs, yes? So there should when it's all broken down, only be potential term and derivatives at work? So the specialness of matter region re tangent contraction should be explicable just in those terms. I think.

Some clarifications:

* The entire stress-energy tensor is the "source" on the RHS of the Einstein Field Equation (EFE). The LHS is the Einstein tensor, which is defined in terms of the Ricci tensor, which is a contraction of the Riemann curvature tensor.

* For the non-vacuum region in question, a reasonable stress-energy tensor would be a "perfect fluid" with non-zero pressure:

T_{00} = \rho

T_{11} = T_{22} = T_{33} = p

where \rho is the energy density and p is the pressure. The off-diagonal components are all zero. For the spacetime to be static and spherically symmetric, both \rho and p must be functions of the radial coordinate r only. Also, \rho should be positive everywhere in the non-vacuum region, but p will not be; a negative pressure is a tension, and in order for the non-vacuum region to be stable, there will have to be tension somewhere, to keep it from falling apart (a positive pressure everywhere would be possible only if the non-vacuum region went all the way down to the center, r = 0, without any hollow interior).

* The fact that the stress-energy tensor (and therefore the Einstein tensor) is divergenceless is one way of expressing local conservation of energy; the divergence of the stress-energy tensor, over some small piece of spacetime, is just the net energy going in or out of that piece of spacetime. The divergence being zero just means energy in equals energy out; i.e., energy is conserved. So you're right that this probably doesn't help much for this particular problem.

* You can still define a "potential" in the non-vacuum region, because you can do that for any spherically symmetric geometry. And you can still view the gradient of this potential as being the "acceleration due to gravity". I *think* that the effects on the metric, including its tangential part, in the non-vacuum region can be expressed in terms of the potential, but I'm not certain; the metric coefficients may be more complicated than that in the non-vacuum region. The only metric coefficient that I'm pretty well certain can be expressed entirely in terms of the potential is g_{00}, the timelike coefficient.

 

[Passionflower]

* For the non-vacuum region in question, a reasonable stress-energy tensor would be a "perfect fluid" with non-zero pressure:

T_{00} = \rho

T_{11} = T_{22} = T_{33} = p

where \rho is the energy density and p is the pressure.

Could you demonstrate that such a 'blob' could turn into a black hole?

 

[PAllen]

Could you demonstrate that such a 'blob' could turn into a black hole?

Q-reeus example has nothing to do with black holes. He is claiming Minkowski geometry inside a spherical shell and Schwarzschild outside somehow leads to a contradiction. This whole topic is really a hijacking of the original purpose of the thread - to discuss implications of the inability of a black hole interior (if it exists) to influence the outside.

 

[Passionflower]

Q-reeus example has nothing to do with black holes. He is claiming Minkowski geometry inside a spherical shell and Schwarzschild outside somehow leads to a contradiction. This whole topic is really a hijacking of the original purpose of the thread - to discuss implications of the inability of a black hole interior (if it exists) to influence the outside.

PAllen, could you please give me the chapter and page number of MTW where you claim it is stated the formation of a black hole is in finite time?

 

[PAllen]

PAllen, could you please give me the chapter and page number of MTW where you claim it is stated the formation of a black hole is in finite time?

Section 33.1 is the clearest such discussion.

 

[noblackhole]

The black hole does not exist. It is a figment of an unscientific imagination. The alleged signatures of the black hole are an event horizon and an infinitely dense point mass singularity since the singularity is alleged to have mass but no volume. But a mass is not a point and a point is not a mass. Infinitely dense point masses are mathematical artifices; they are centres of mass, and hence have no physical existence. One can buy a bag full of marbles but one cannot buy a bag full of centres of mass of the marbles. Also, nobody has ever found an event horizon and nobody has ever found an infinitely dense point mass singularity and so nobody has ever found a black hole despite the claims of the astrophysical scientists that they have found them all over the place. All claims for the discovery of black holes are wishful thinking. It is also claimed frequently in the literature that the escape velocity of a black hole is the speed of light in vacuum. If that were true then light (photons) can escape from the black hole so that it can be seen by all observers and that material bodies can leave the black hole but not escape, merely travel out radially to a finite distance then fall back, as a ball thrown into the air on the Earth. On the other hand it is also claimed by the very same astrophysical scientists that nothing can even leave the black hole, including light so that no observers can see the black hole or material bodies leave it. This is a contradiction: merely a play on the words “escape velocity”. Furthermore, escape velocity is an implicit two body concept – one body escapes from another body. However, all black hole “solutions” pertain to a universe that contains only one mass, that of the black hole itself. There are no known solutions to Einstein’s field equations for two or more masses and no existence theorem by which it can even be asserted that the field equations contain latent solutions for two or more masses. The Principle of Superposition does not apply in General Relativity and so one cannot simply pile up masses in any given spacetime. An implicit two-body relation such as escape velocity cannot rightly appear in what is by definition a one body problem. For these reasons it is meaningless to talk of black holes existing in multitudes, interacting with one another or with other matter. Now the quantity r appearing in the so-called “Schwarzschild solution”, which I point out is not even Schwarzschild’s solution, has never been correctly identified by astrophysics. It is variously called the radius, the radial coordinate, the coordinate radius, the radial space coordinate, the radius of a 2-sphere, the area radius, the circumferential radius, and even a gauge choice that defines what r is. That this r is nevertheless always treated as the radius is manifest in the fact that a particular value of it always called the “Schwarzschild radius” of a black hole, namely the radius of the event horizon of the black hole – the radial distance from the infinitely dense point mass singularity to the event horizon of the black hole. All of these notions of what r is in the so-called “Schwarzschild solution” are incorrect because this r is not even a distance let alone a radius in the Schwarzschild manifold. It is easily proven that this r is in fact the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section of the related manifold, and so it is not even a distance let alone a radius in the Schwarzschild manifold. In addition, according to Einstein and his followers, his Principle of Equivalence and his Special Relativity must manifest in his gravitational field. Now both the Principle of Equivalence and Special Relativity are defined in terms of the presence of multiple arbitrarily large finite masses and photons and so neither can manifest in the spacetime of a black hole bearing in mind that all black hole “solutions” pertain to a universe that contains only one mass, that of the black hole itself. Moreover, Special Relativity forbids infinite densities and so forbids infinitely dense point mass singularities in General Relativity. It does not matter how it is alleged infinitely dense point mass singularities are formed in General Relativity because they cannot be reconciled with Special Relativity. Finally, the real Schwarzschild solution forbids black holes. One can easily verify this merely by reading Schwarzschild’s actual paper on the subject. Most astrophysical scientists have evidently never even read Schwarzschild’s actual memoir, given all the false claims that they attribute to him. What goes by the name of “Schwarzschild’s solution” in the literature is in fact a corruption of Schwarzschild’s actual solution, committed by David Hilbert in late 1916. It is from Hilbert’s corruption that the black hole was originally and incorrectly conjured.

 

[Drakkith]

Forgive me if I don't take your wall of text seriously noblackhole.

 

[Passionflower]

Section 33.1 is the clearest such discussion.

This chapter talks about light getting fainter, but the fact is that a dust ball collapsing from a finite R value takes a finite time for an observer on the 'surface' to reach the EH but for a far away observer it takes an infinite amount of time. For closer observers it is faster very close to the horizon but at the horizon is infinite for any outside observer.

 

[PAllen]

This chapter talks about light getting fainter, but the fact is that a dust ball collapsing from a finite R value takes a finite time for an observer on the 'surface' to reach the EH but for a far away observer it takes an infinite amount of time. For closer observers it is faster very close to the horizon but at the horizon is infinite for any outside observer.

Consider the matter already (and always) inside the eventual event horizon. What do you think happens to it? Does it jump outside the horizon to be suspended there? Does it cease to exist? Consider the surface of last influence. You know beyond this point that anything falling through the horizon will find all the original matter already in the singularity. This surface is a well defined surface outside the horizon. MTW explicitly calls it the 'time of formation of the black hole'.

You can try read this in strained ways, but I think the clear intent here (and in later sections, and in earlier sections on black hole collapse, rather than eternal black holes) is that black holes form in finite time - the singularity inside and the horizon outside.

If I get a chance I'll try to track down the links George Jones gave in another thread that went into detail about how the actual event horizon forms from the inside out. That is not covered in MTW, so far as I know.

 

[Passionflower]

Consider the matter already (and always) inside the eventual event horizon.

How about one step at a time:

So you agree with me that for each outside observer the formation of a black hole happens in infinite time?

I am very interested by what you mean already inside the eventual event horizon as a dust ball is a ball not a shell.

More realistic would be a fluid ball, however who can do that?

 

[PAllen]

How about one step at a time:

So you agree with me that for each outside observer the formation of a black hole happens in infinite time?

No, I don't agree. I agree that an external observer never directly sees matter cross the event horizon. However, the outside observer knows the star is a ball of matter. When it goes dark, and has well defined event horizon radius, I find it nonsensical to conclude anything other than that most of the star's matter is inside the event horizon (it always was inside). Also, that when it goes dark, I call that an event horizon. Further, it seems clear that's what MTW authors mean.

I am very interested by what you mean already inside the eventual event horizon as a dust ball is a ball not a shell.

More realistic would be a fluid ball, however who can do that?

I don't understand how you don't understand. You have a ball of gas. No horizon, no catastrophic collapse yet. There is matter at its center. I collapses, its surface goes red, then dark. I call that the event horizon (as do everyone I have ever read on GR). Where is the matter that was always at the center of ball?

 

[Passionflower]

No, I don't agree. I agree that an external observer never directly sees matter cross the event horizon. However, the outside observer knows the star is a ball of matter. When it goes dark, and has well defined event horizon radius, I find it nonsensical to conclude anything other than that most of the star's matter is inside the event horizon (it always was inside). Also, that when it goes dark, I call that an event horizon. Further, it seems

I don't understand how you don't understand. You have a ball of gas. No horizon, no catastrophic collapse yet. There is matter at its center. I collapses, its surface goes red, then dark. I call that the event horizon (as do everyone I have ever read on GR). Where is the matter that was always at the center of ball?

What about that matter? I think you misunderstand.

It is the same with Earth there is matter everywhere.

We only have a black hole when all the matter is beyond the horizon!

We are talking about the formation of a black hole right?

 

[PAllen]

What about that matter? I think you misunderstand.

It is the same with Earth there is matter everywhere.

We only have a black hole when all the matter is beyond the horizon!

No, I disagree with that. Let's say the ball of gas is big enough so 1% in the center is enough to catastrophically collapse. The event horizon forms in the center, moves outward (it does not carry any matter with it), and the matter inside collapses to singularity, but not all at once. By the time the whole thing has gone dark, the singularity (or QG alternative) has existed for some time.

 

[Passionflower]

No, I disagree with that. Let's say the ball of gas is big enough so 1% in the center is enough to catastrophically collapse. The event horizon forms in the center, moves outward (it does not carry any matter with it), and the matter inside collapses to singularity, but not all at once. By the time the whole thing has gone dark, the singularity (or QG alternative) has existed for some time.

References?

So does that mean you are now agreeing that a collapsing dust ball from a given R value (or infinity) takes an infinite time to form an EH for all outside observers?

 

[PAllen]

Anyway, you asked where MTW said what I said (that the black hole forms in finite time). In 33.1 they say:

"one can think of the surface of last influence as the birthpoint of the black hole".

This surface exists far outside the event horizon. So go argue with them, I'm done for now.

 

[PAllen]

References?

So does that mean you are now agreeing that a collapsing dust ball from a given R value (or infinity) takes an infinite time to form an EH for all outside observers?

No I don't agree. I distinguish visually seeing matter cross an event horizon from existence of the event horizon. I believe my usage is standard. There is an ongoing astronomy project to 'image the event horizon' at the center of the milkyway. They expect success.

 

[Passionflower]

Anyway, you asked where MTW said what I said (that the black hole forms in finite time). In 33.1 they say:

"one can think of the surface of last influence as the birthpoint of the black hole".

This surface exists far outside the event horizon. So go argue with them, I'm done for now.

I can write down the integrals for the proper time on the surface of the dust ball from a given r0 value and the proper time of an observer an infinite distance away and also any stationary observer. The result is that the proper time for the outside observers for a black hole to fully form (the collapsing dust ball form) is infinite.

 

[PAllen]

I can write down the integrals for the proper time on the surface of the dust ball from a given r0 value and the proper time of an observer an infinite distance away and also any stationary observer. The result is that the proper time for the outside observers for a black hole to fully form (the collapsing dust ball form) is infinite.

The event horizon is what prevents you from seeing matter cross it. The fact that you see object redshifting infinite amount getting blacker than CMB background is because the event horizon already exists. And at the point you cross the surface of last influence, it is reasonable (and MTW do so) to conclude the singularity inside the event horizon has fully formed.

 

[Passionflower]

By the way MTW says he same as I do. On page 847:

Hence, to the distant astronomer, the collapsing star appears to slow down as it approaches its gravitational radius: light from the star becomes more and more red-shifted. Clocks on the star appear to run more and more slowly. It takes an infinite time for the star to reach its gravitational radius; and as seen by the distant astronomer the star never gets beyond there.

 

[PAllen]

By the way MTW says he same as I do. On page 847:

Hence, to the distant astronomer, the collapsing star appears to slow down as it approaches its gravitational radius: light from the star becomes more and more red-shifted. Clocks on the star appear to run more and more slowly. It takes an infinite time for the star to reach its gravitational radius; and as seen by the distant astronomer the star never gets beyond there.

Emphasize the word 'seen'. Why is this seen? Because of an event horizon. What is the state of the matter inside the star? Any application of GR says the matter has collapsed to a singularity.

 

[PAllen]

Consider a time when a collapsing mass has gone dark. It has an apparent radius. From its history, we know there is matter inside this radius. The idea that the manifold ends at or just below the visible surface is untenable.

Consider the definition of an event horizon. The surface from which no light can ever reach infinity. At this time (for an outside observer, when the star has gone completely dark), it is clear that light epsilon inside the dark surface can never reach infinity. Thus the event horizon already exists.

Once the outside observer passes the surface of last influence, unless a different law than GR is used, the matter inside the event horizon will collapse quickly to a singularity.

 

[zonde]

The qualification, in bold, is crucial, as I said. For any given state of motion, there is a "correct" way to define simultaneity that respects the Einstein clock synchronization convention (which is what "isotropic speed of light" refers to). However, that only applies to that particular state of motion. A different state of motion can have a different "correct" simultaneity convention.

For example: if an observer is hovering at a constant radial coordinate above a black hole's horizon, there is a "correct" simultaneity convention for him, which is the one used in the Schwarzschild interior coordinates. However, if an observer is freely falling towards the hole, there is a different "correct" simultaneity convention for him, which is the one used in Painleve coordinates.

What this means is that the "lines of simultaneity" for Schwarzschild coordinates are *different lines* than the ones for Painleve coordinates. "Different lines" is an invariant, coordinate-free statement; the two sets of lines of simultaneity are different geometric objects. And it's perfectly possible for different sets of lines to cover different regions of spacetime; in this case, one set happens to reach into a region of spacetime that the other set does not. See below.

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

For now I have only the things that I explained in my post #237. 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.

Yes, certainly. But the conversion only has to be possible in a region of spacetime that is covered by both frames ("coordinate charts" would be a better term). If one chart does not cover a region (such as the black hole interior), then there is no requirement that other charts have to be able to convert to or from it in that region.

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

There are certainly different charts, with different simultaneity conventions. However, you have not shown that any of them are contradictory. All you have shown is that the different charts cover different regions of the spacetime, and that the "coordinate lines" on the different charts, such as the lines of simultaneity, are different geometric objects. None of this is in any way contradictory.

Interiors of black hole and white hole are contradictory and that is only because of different geometry of simultaneity.

 

[zonde]

This is not correct. The *geometry* of the interior is not disconnected from the geometry of the exterior. They are connected, as can be easily seen by analyzing covariant or invariant objects like geodesics, curvature tensors, etc.

It is true that the interior Schwarzschild *coordinate patch* is disconnected from the exterior Schwarzschild coordinate patch; that is what is meant by statements about the "infinite future" and whether anything is "beyond" it. But that statement does not support your argument, because it only applies to a particular coordinate system; it is not a statement about the underlying geometry, which is what is important for the physics.

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.

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

 

[zonde]

But do you think that "inside" of a Rindler horizon exists for a Minkowski observer? If yes, then your interpretation of "existence" is observer dependent?

Well, yes observer dependent "existence" does not sound good.
I would like to say that existence of "real" things is observer independent.

But I suppose that in reality we have other things that we can consider to make our conclusions more likely correct. For example we might decide that one set of observers gives more contrived global picture than the other set of observers.

The Rindler time coordinate approaches ∞ as you approach the Rindler horizon, and coordinates can't go beyond ∞, so you are right. (The Rindler time coordinate equals the Rindler observer's proper time along his own worldline, and locally represents Einstein-simultaneity for any other observer at rest relative to the Rindler observer.)

Not sure that you understood my question. But I checked that unclear point myself and it turned out that my doubts where false.

 

[Dale]

Issue now, following #241, is to nail down just what property/operation of ET actually yields tangential contraction.

The tangential pressure components of the stress energy tensor. There is no tangential pressure in the vacuum region, there is in the matter region.

Here is a arxiv paper you may like. It uses an analytical model for the shells, so it is not the usual "step function" you would normally consider, but it describes things like the radial and transverse pressures:
http://arxiv.org/abs/0911.4822

 

[zonde]

OK, I think that the concept you are trying to describe here is a congruence, which is essentially a family of worldlines. For example, you could associate one worldline with each spatial location in Schwarzschild coordinates. This would give a family of timelike curves which could each represent a stationary observer relative to the central mass. Then this set of observers would all share the same "infinite future" region.

The existence of one congruence which share the same "infinite future" does not in any logical way forbid the existence of another set of congruences which share a different "infinite future". Your line of reasoning seems to be that there is a timelike congruence which ends up in the usual "infinite future" therefore all timelike congruences must end up in the same "infinite future". This is not sound logic.

I'm not sure I can make myself clear by I will try one more time.

It does not make sense to speak about different "infinite futures" or the same "infinite future".
Infinity is just an abstract idea to help one say how function or series or whatever behaves when it is extrapolated without limit.

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

And if we look back with what it started then it is pointless to say that some function will acquire different properties if we extrapolate it beyond "without-limit".

 

[Q-reeus]

Q-reeus example has nothing to do with black holes. He is claiming Minkowski geometry inside a spherical shell and Schwarzschild outside somehow leads to a contradiction. This whole topic is really a hijacking of the original purpose of the thread - to discuss implications of the inability of a black hole interior (if it exists) to influence the outside.

Agreed that's what I'm on about, but I strongly disagree it has nothing to do with BH's, as my opening salvo in #138 outlined. But I accept it could be taken as hijacking, and will therefore start my own thread just on that issue. Feel free to participate there. It will be titled "How does GR handle metric transition for a spherical mass shell?"

 

[Q-reeus]

Some clarifications:...

Peter - thanks again for useful input, but I'm taking PAllen's hint and vacating this thread. Hope we can continue this discussion in the new one. Cheers

 

[PeterDonis]

Peter - thanks again for useful input, but I'm taking PAllen's hint and vacating this thread. Hope we can continue this discussion in the new one. Cheers

No problem, looking forward to it.

 

[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.