Black hole matter accumulation

[PeterDonis]

It's EH that is in the "infinite future" for external observer. Black hole interior is "beyond" infinite future. And because we define infinity as a limit where nothing is beyond that limit we get contradiction in terms.

Only if you switch meanings of the word "infinite" between the first and second sentences above. I have no problem with your first sentence, because I can map it easily to the actual math. Mathematically, the Schwarzschild "t" coordinate of the EH can be thought of as "plus infinity", so the region inside the horizon, which is to the future for any observer falling through the horizon, can be thought of as "beyond plus infinity". But the sense of "infinity" being used here is completely compatible, logically, with there being a region beyond infinity.

In your second sentence, you are using a different definition of the term "infinity", which explicitly rules out having a region "beyond" infinity. There are plenty of domains where that definition applies, but a black hole spacetime is not one of them. So there is no actual contradiction.

Therefore there is no black hole interior in Schwarzschild coordinates. And it is not because of chosen coordinate system but because of chosen simultaneity.

This is correct; in *Schwarzschild coordinates* (more precisely, in Schwarzschild *exterior* coordinates--you can construct Schwarzschild coordinates for the interior region too, but that is a separate coordinate chart, disconnected from the exterior one) there is no black hole interior, because the lines of simultaneity are constructed in such a way that the chart can only cover the exterior region.

It does not follow from this, however, that the interior region doesn't exist. It only follows that the Schwarzschild exterior chart doesn't cover the interior region. These are two different statements, and the second does not require or imply the first.

So my statement implies that simultaneity is not just convenience but rather physical fact.

Only when qualified as you do in the sentences I'm going to quote next. But the qualification is crucial, and it completely undercuts the claim you are trying to make.

Simultaneity is defined in such a way as to get isotropic speed of light. And there is only one "correct" way how to define simultaneity for any state of motion.

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.

But that's not all. It should be possible to convert consistently between reference frames that correspond to different states of motion.

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.

For example, anywhere in the exterior region, outside the EH, you can convert between Schwarzschild and Painleve coordinates easily. See, for example, the Wikipedia page on Painleve coordinates:

http://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

However, inside the horizon, where the exterior Schwarzschild coordinates don't cover, you can't convert between them and Painleve coordinates. You can, however, convert between *interior* Schwarzschild coordinates and Painleve coordinates. You can also convert between Painleve coordinates and other charts that cover the interior, such as ingoing Eddington-Finkelstein or Kruskal.

And I think there is complete chaos in GR regarding different coordinate charts and different (contradictory) simultaneity conventions they implement.

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.

 

[zonde]

Zonde,

please think carefully about some of the points I raised in my post #128 (with references). In particular, consider what is observed if a large globular cluster happened to undergo collapse to a supermassive black hole (there is no evidence of this happening, but there is no reason it could not). The key thing here is that for supermassive BH, you have the event horizon forming with very modest matter density. The stars are still many millions of miles apart when the event horizon forms. Key characteristics for an outside observer:

1) They see normal brightening as star density increases, at first.
2) The event horizon expands from the center out. First they see innermost stars slowing and reddening, the becoming invisible as too little light is emitted. This phenomenon expands out from the center until the whole cluster has effectively vanished from inside out. All that is left is a completely dark horizon, after finite time (rate of photon emission too low to observe; darker than any normal body in space, if all this occurred isolated, away from dust).

If you saw this, what would you conclude? It seems very hard to (for me) to see any possible interpretation than that the globular cluster still exists inside the event horizon, and is undergoing some history we cannot see (e.g. catastrophic collapse of some nature; probably not as neat as idealized solutions).

The other thing I point out in #128 is that the question of what is normally inside the horizon may be fully testable if there are exception to the cosmic censorship hypothesis. I give a number of references, including proposals for how this could be looked for.

The upshot, to me, is that accepting GR really does seem to require accepting the horizon interior as real, and this may actually be fully testable physics.

If you are in doubt about my description of what would be seen in the globular cluster collapse, part of this is discussed in MTW. The rest is discussed in some nice links George Jones can provide (but I can't lay hands on now). They discuss the formation and growth of the event horizon for collapsing dust, but the same would apply for the collapsing cluster. [The advantage of the cluster is that you can see inside it, at least ideally.]

Yes, I will consider your example and will post my answer later. With this thread growing so fast it's easy to loose track of discussion.

 

[zonde]

Only if you switch meanings of the word "infinite" between the first and second sentences above. I have no problem with your first sentence, because I can map it easily to the actual math. Mathematically, the Schwarzschild "t" coordinate of the EH can be thought of as "plus infinity", so the region inside the horizon, which is to the future for any observer falling through the horizon, can be thought of as "beyond plus infinity". But the sense of "infinity" being used here is completely compatible, logically, with there being a region beyond infinity.

What on Earth is this "sense of "infinity" being used here"
You just invent definitions on the fly that fit your needs?

In your second sentence, you are using a different definition of the term "infinity", which explicitly rules out having a region "beyond" infinity. There are plenty of domains where that definition applies, but a black hole spacetime is not one of them. So there is no actual contradiction.

What are these two definitions you are talking about? I know only one.

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.

Yes

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.

How we can test your statement?

 

[Q-reeus]

..That potential difference no longer shows up in the spatial parts of the metric, but if we compared the rate of time flow in the inner vacuum region to that at infinity, we would find it to be slower, by exactly the same factor as on the inner surface of the non-vacuum region...

Peter - thanks for your lengthy explanation, but if I'm reading you right here, the above I cannot agree with. You are saying there is a fundamental breach between the temporal and isotropic spatial metric dilation/contraction factors in the interior MM region. Finite redshift but lengths as per infinity values. Sure that gives you flatness of sorts, but I believe all components should have the same 'redshift' - including length measure. I'm posting a separate thread on this matter, given there were no takers to my #203. Cheers.

 

[Q-reeus]

Originally Posted by Q-reeus:
"Actually, penta-questionmark, it very much matters whether the coordinates 'match up'. If as I maintain the 'mismatch' is a pathological feature of SM and thus the EFE's, EH's and BH's turn out to be literally non-entities so then just quit worrying about a non-issue, period."

I understand perfectly what you are saying. You have no way to respond to my issue, so would I please go away. Yes, I will. Continue with your mathematical analysis and find an answer that seems pleasing. But, just because you can prove by legitimate mathematical derivation that 1=0, it still is not so. In the end, no matter how perfect the derivation, one does not equal zero.

?; I fear there has been a misunderstanding here. Your reply could be taken a number of ways (first person, second person, switching between etc). So rather than assume to know exactly what you are driving at, let me just say I was in no way implying you should 'buzz off' or such - far from it. There was no implied directive to you in saying "...so then just quit worrying about a non-issue, period." Rephrased I meant "so in that event we can quit worrying, as the problem would be imaginary and not worth any more consideration."
Actually I sympathise entirely with your finding that the standard position doesn't add up, but feel the resolution is to shop elsewhere for a viable theory of gravity, and not to struggle on trying to correct that which is broke. I don't need to tell you that is very much a minority opinion, but if you want my 2-cents worth on another slightly different angle to all this, try following the discussion here:
https://www.physicsforums.com/showthread.php?t=508950 (my entries start at #3) Decide for yourself what adds up!

 

[Dale]

It's EH that is in the "infinite future" for external observer. Black hole interior is "beyond" infinite future. And because we define infinity as a limit where nothing is beyond that limit we get contradiction in terms.

This argument would hold if the universe were 1 dimensional, but for any number of dimensions greater than 1 it doesn't hold. Consider a 2 dimensional manifold like the surface of a road with a fork in it. You can have two different "infinite futures" without any contradiction.

 

[DrGreg]

It's EH that is in the "infinite future" for external observer. Black hole interior is "beyond" infinite future. And because we define infinity as a limit where nothing is beyond that limit we get contradiction in terms.

In this context, infinity is a limit where there are no t coordinates beyond that limit. A lack of coordinates, in a particular choice of coordinates, does not imply non-existence. It just means those coordinates don't cover the whole of spacetime.

 

[Passionflower]

please think carefully about some of the points I raised in my post #128 (with references). In particular, consider what is observed if a large globular cluster happened to undergo collapse to a supermassive black hole (there is no evidence of this happening, but there is no reason it could not). The key thing here is that for supermassive BH, you have the event horizon forming with very modest matter density. The stars are still many millions of miles apart when the event horizon forms. Key characteristics for an outside observer:

1) They see normal brightening as star density increases, at first.
2) The event horizon expands from the center out. First they see innermost stars slowing and reddening, the becoming invisible as too little light is emitted. This phenomenon expands out from the center until the whole cluster has effectively vanished from inside out. All that is left is a completely dark horizon, after finite time (rate of photon emission too low to observe; darker than any normal body in space, if all this occurred isolated, away from dust).

If you saw this, what would you conclude? It seems very hard to (for me) to see any possible interpretation than that the globular cluster still exists inside the event horizon, and is undergoing some history we cannot see (e.g. catastrophic collapse of some nature; probably not as neat as idealized solutions).

For an observer far away from all of this how long do you think it takes for the event horizon to fully form?

 

[PeterDonis]

What on Earth is this "sense of "infinity" being used here"
You just invent definitions on the fly that fit your needs?

What are these two definitions you are talking about? I know only one.

I was not making up a definition on the fly. I was trying to explain how the word "infinity" was being used in the phrase "the black hole interior is beyond infinity", and how that usage of the term does not satisfy the definition you gave. DaleSpam and DrGreg both gave good responses to help clarify what I was saying. The key point is that the term "infinity" is more general than you are claiming it is; the definition you gave is one special case of the general definition, and the property that there can't be any region "beyond infinity" applies only to your special case, not to the general definition.

How we can test your statement?

Read the rest of my posts about why it is reasonable to believe that the interior region exists, even if no causal influences can reach us in the exterior region from the interior. The alternative, as I've posted several times now, is to believe that physics at the horizon suddenly starts working completely differently, for no apparent reason.

 

[PeterDonis]

Peter - thanks for your lengthy explanation, but if I'm reading you right here, the above I cannot agree with. You are saying there is a fundamental breach between the temporal and isotropic spatial metric dilation/contraction factors in the interior MM region. Finite redshift but lengths as per infinity values.

No, the lengths are "contracted" too if you compare them with lengths at infinity. I'll respond in more detail in the other thread.

 

[PAllen]

For an observer far away from all of this how long do you think it takes for the event horizon to fully form?

Don't know exact formula, but MTW claims it is 'fairly fast'.

 

[PeterDonis]

Q-reeus, after reading the other thread it looks like you got tangled up in issues which have nothing to do with what you wanted to find out. I think that's because the question you actually asked in that thread is not the one you are asking in this thread. So I think I'd better respond here.

In this thread, we talked about the "anisotropy" in the exterior Schwarzschild spacetime, in the sense that if we take two spherical "shells" at radius r and r + dr, their areas will be the "Euclidean" values of 4 pi times the radial coordinate r or r + dr, squared (since that's how the radial coordinate is defined), but the distance between the shells will be *larger* than the coordinate differential dr; there will be more distance (and volume) between the shells than Euclidean geometry would lead us to expect.

In the other thread, you asked about distortion of a small spherical object placed at some radius r in the exterior Schwarzschild spacetime. You seemed to imply in the other thread that the "anisotropy" I just described implies a physical distortion of a small spherical object placed at rest at a given radius. It does not, and that's why the other thread got so tangled up. When you say "distortion", you are implying (without meaning to, I think) that there are actual physical stresses in the small spherical object, caused solely by the coordinate "anisotropy" described above, and everyone jumped on that. But that can't be the case, because I can remove the "anisotropy" locally by a coordinate transformation, so it can't cause any actual observable physical effects, like stresses.

What I think you meant to ask is this: suppose I try to pack the space around a black hole with small spherical objects, each of very tiny radius, between radial coordinates r and r + dr. When I pack the objects tangentially, I will find that they pack as though distances are Euclidean: at radius r I can pack just enough objects to cover an area 4 pi r squared. But when I pack the objects radially, I will find I can pack *more* of them than the radial coordinate differential dr would suggest; there will be more volume to pack the little spherical objects in between radius r and r + dr than Euclidean geometry would indicate based on the difference in areas between those two radial coordinates. So how will this anisotropy in packing appear to an observer very far away, at a radius much larger than r?

This difference in "packing" is a real physical effect, but it does not physically distort the little spherical objects being packed at all. An observer sitting next to one of the little spherical objects would see that it looked and behaved exactly as it would in a flat, gravity-free spacetime, except for the fact that it is accelerated (because it is sitting at rest relative to the black hole, so it can't be in free fall), so there is a slight compressive stress along its radial dimension (but we can make that small enough to ignore by making the black hole large enough). But a faraway observer would see that, as we saw above, there would be more objects packed between radius r and r + dr than Euclidean geometry would allow based on the difference in areas. The faraway observer could interpret this as a distortion of the spheres--that radially, lengths are somehow "contracted", but not tangentially--but this is a matter of *interpretation*, not physics. The faraway observer could just as well interpret the observations as telling him that the geometry of space is not Euclidean.

Similarly, if you go back to my lengthy explanation, as one descends through the non-vacuum region from its outer to its inner surface, the number of objects that can be packed radially between r and r + dr gets "less larger" than the Euclidean value, until at the inner surface, it is exactly the Euclidean value again. So if we packed the entire interior vacuum region with little spheres, its volume would be exactly the Euclidean volume as a function of the r coordinate of the inner surface of the non-vacuum region, 4/3 pi r cubed. But suppose the non-vacuum region were transparent, and we had somehow packed all of it with little objects, as well as the exterior region. What would a faraway observer see? Suppose we stop packing at a radius r which is significantly larger than the outer radius of the non-vacuum region, but still much smaller than the radius where the faraway observer is. Then the faraway observer would see three regions: an exterior vacuum region where spheres grow more tightly packed radially, compared to tangentially, as you go inward; a non-vacuum region where spheres grow less tightly packed, radially, as you go inward, until the packing returns to its Euclidean value at the inner surface; and an interior vacuum region where the packing is Euclidean.

Again, all of this is physically valid and consistent; but the faraway observer can choose to *interpret* it in different ways. He could interpret it as a distortion of the little objects being packed; they grow more distorted radially as you descend in the exterior vacuum region, then less distorted again as you descend through the non-vacuum region, so that the interior vacuum region is free of distortion. Or he could interpret it as telling him that space in the exterior vacuum and non-vacuum regions is non-Euclidean. In either case, though, if the faraway observer wants to compare lengths in the interior vacuum region with lengths "at infinity", he has an additional factor to consider. The area of the inner surface of the non-vacuum region is 4 pi r_i squared (I guess I don't feel like typing itex tags today). But r is just a radial coordinate; there is no requirement that an increment dr of radius in the interior vacuum region corresponds to the same physical distance as an increment dr of radius "at infinity". To establish the actual correspondence, we have to look at the full metric at the inner surface of the non-vacuum region, and we find that it looks like this, as DrGreg posted in the other thread (hm, I'll have to type some tex after all):

$$d\tau^{2} = A^{2} dt^{2} - B^{2} \left[ dr^{2} + r^{2} \left( d\theta^{2} + sin^{2} \theta d\phi^{2} right) \right]$$

Notice the factor B^2 in front of the spatial part of the metric. That factor is due to the difference in "potential" between the inner surface of the non-vacuum region and the potential "at infinity". It also tells us the relationship between the radial coordinate r and actual physical distances in the interior vacuum region. But since the factor multiplies the entire spatial part of the metric, not just the radial part, the "distortion", if you want to call it that, is now isotropic; distances in *all* directions are different, as a function of r, than they are at infinity.

So the faraway observer, looking at the interior vacuum region packed with little spherical objects, would find that the relationship between the interior volume of the region and its surface area, at the inner surface of the non-vacuum region, was Euclidean; but he would see its actual, physical area and volume, as shown by the number of little spheres packed into it, to be "distorted" by the B factor, if he tried to use the radial r coordinate to establish a relationship between the space inside the interior region and the space at infinity. But an observer inside the interior vacuum region would see no "distortion" at all; the natural coordinates for him to use are "rescaled" from the above ones, using the factors A and B, so the metric takes its standard Minkowski form.

 

[atyy]

Possibly relevant: http://arxiv.org/abs/0903.0100

Key science questions
• What can gravitational wave astronomy tell us about new physics?
• How does quantum gravity manifest itself far below the Planck energy?
• Are the massive dark central objects in galaxies really Kerr black holes?
• Can naked singularities form?

 

[PeterDonis]

as one descends through the non-vacuum region from its outer to its inner surface, the number of objects that can be packed radially between r and r + dr gets "less larger" than the Euclidean value, until at the inner surface, it is exactly the Euclidean value again.

On re-reading, I realized there is another possible confusion here, because as I say later on, the metric in the interior vacuum region is still "length contracted" compared to the metric at infinity, in terms of the r coordinate, even though the relationship between area and radius is Euclidean. The faraway observer, looking at how the packing of the little objects changes as you descend through the non-vacuum region, could interpret what he sees this way: radial lengths continue to "contract" through this region, but now *tangential* lengths start to contract as well (they did not in the exterior vacuum region), and the tangential lengths contract *faster* than the radial lengths, so that when the inner surface of the non-vacuum region is reached, the packing of the little objects is now isotropic again; the relationship between radial and tangential packing is now Euclidean, but *all* lengths are now "contracted" compared to lengths at infinity.

 

[Q-reeus]

Q-reeus, after reading the other thread it looks like you got tangled up in issues which have nothing to do with what you wanted to find out. I think that's because the question you actually asked in that thread is not the one you are asking in this thread. So I think I'd better respond here.

In this thread, we talked about the "anisotropy" in the exterior Schwarzschild spacetime, in the sense that if we take two spherical "shells" at radius r and r + dr, their areas will be the "Euclidean" values of 4 pi times the radial coordinate r or r + dr, squared (since that's how the radial coordinate is defined), but the distance between the shells will be *larger* than the coordinate differential dr; there will be more distance (and volume) between the shells than Euclidean geometry would lead us to expect.

In the other thread, you asked about distortion of a small spherical object placed at some radius r in the exterior Schwarzschild spacetime. You seemed to imply in the other thread that the "anisotropy" I just described implies a physical distortion of a small spherical object placed at rest at a given radius. It does not, and that's why the other thread got so tangled up. When you say "distortion", you are implying (without meaning to, I think) that there are actual physical stresses in the small spherical object, caused solely by the coordinate "anisotropy" described above, and everyone jumped on that.

Yes and no. I have always clearly understood that the only locally induced and locally observed distortions are those due to mechanical strain under tidal (2nd derivative of potential) and/or 'g' (1st derivative of potential) forces. I made it real clear in that other thread these effects were to be discounted. All that was of interest was the 'pure' 0th order metric contractions/distortions as measured 'at infinity', and further made clear there these are not locally observable phenomena. But you are very right that all that somehow has gotten lost on folks who seem to think I meant something else. [edit: replaced 'metric' with potential' in parentheticals above - my slipup]

But that can't be the case, because I can remove the "anisotropy" locally by a coordinate transformation, so it can't cause any actual observable physical effects, like stresses.

Agreed, as per above.

What I think you meant to ask is this: suppose I try to pack the space around a black hole with small spherical objects, each of very tiny radius, between radial coordinates r and r + dr. When I pack the objects tangentially, I will find that they pack as though distances are Euclidean: at radius r I can pack just enough objects to cover an area 4 pi r squared. But when I pack the objects radially, I will find I can pack *more* of them than the radial coordinate differential dr would suggest; there will be more volume to pack the little spherical objects in between radius r and r + dr than Euclidean geometry would indicate based on the difference in areas between those two radial coordinates. So how will this anisotropy in packing appear to an observer very far away, at a radius much larger than r?

Spot on - somebody actually got it, at least to this stage of the issue.

This difference in "packing" is a real physical effect, but it does not physically distort the little spherical objects being packed at all. An observer sitting next to one of the little spherical objects would see that it looked and behaved exactly as it would in a flat, gravity-free spacetime, except for the fact that it is accelerated (because it is sitting at rest relative to the black hole, so it can't be in free fall), so there is a slight compressive stress along its radial dimension (but we can make that small enough to ignore by making the black hole large enough). But a faraway observer would see that, as we saw above, there would be more objects packed between radius r and r + dr than Euclidean geometry would allow based on the difference in areas. The faraway observer could interpret this as a distortion of the spheres--that radially, lengths are somehow "contracted", but not tangentially--but this is a matter of *interpretation*, not physics. The faraway observer could just as well interpret the observations as telling him that the geometry of space is not Euclidean.

So far, we agree entirely, except for the interpretation vs physics thing. If the distant observer finds a contracted radial value, that has to be owing to physics. It's just that the effect of that physics is only non-locally evident. But this is probably just pedantry, so on with this.

Similarly, if you go back to my lengthy explanation, as one descends through the non-vacuum region from its outer to its inner surface, the number of objects that can be packed radially between r and r + dr gets "less larger" than the Euclidean value, until at the inner surface, it is exactly the Euclidean value again. So if we packed the entire interior vacuum region with little spheres, its volume would be exactly the Euclidean volume as a function of the r coordinate of the inner surface of the non-vacuum region, 4/3 pi r cubed. But suppose the non-vacuum region were transparent, and we had somehow packed all of it with little objects, as well as the exterior region. What would a faraway observer see? Suppose we stop packing at a radius r which is significantly larger than the outer radius of the non-vacuum region, but still much smaller than the radius where the faraway observer is. Then the faraway observer would see three regions: an exterior vacuum region where spheres grow more tightly packed radially, compared to tangentially, as you go inward; a non-vacuum region where spheres grow less tightly packed, radially, as you go inward, until the packing returns to its Euclidean value at the inner surface; and an interior vacuum region where the packing is Euclidean.

There is something lurking in the woodpile here, but more on that below.

Again, all of this is physically valid and consistent; but the faraway observer can choose to *interpret* it in different ways. He could interpret it as a distortion of the little objects being packed; they grow more distorted radially as you descend in the exterior vacuum region, then less distorted again as you descend through the non-vacuum region, so that the interior vacuum region is free of distortion. Or he could interpret it as telling him that space in the exterior vacuum and non-vacuum regions is non-Euclidean. In either case, though, if the faraway observer wants to compare lengths in the interior vacuum region with lengths "at infinity", he has an additional factor to consider. The area of the inner surface of the non-vacuum region is 4 pi r_i squared (I guess I don't feel like typing itex tags today). But r is just a radial coordinate; there is no requirement that an increment dr of radius in the interior vacuum region corresponds to the same physical distance as an increment dr of radius "at infinity". To establish the actual correspondence, we have to look at the full metric at the inner surface of the non-vacuum region, and we find that it looks like this, as DrGreg posted in the other thread (hm, I'll have to type some tex after all):

d\tau^{2} = A^{2} dt^{2} - B^{2} \left[ dr^{2} + r^{2} \left( d\theta^{2} + sin^{2} \theta d\phi^{2}
right) \right]Notice the factor B^2 in front of the spatial part of the metric. That factor is due to the difference in "potential" between the inner surface of the non-vacuum region and the potential "at infinity". It also tells us the relationship between the radial coordinate r and actual physical distances in the interior vacuum region. But since the factor multiplies the entire spatial part of the metric, not just the radial part, the "distortion", if you want to call it that, is now isotropic; distances in *all* directions are different, as a function of r, than they are at infinity.

So the faraway observer, looking at the interior vacuum region packed with little spherical objects, would find that the relationship between the interior volume of the region and its surface area, at the inner surface of the non-vacuum region, was Euclidean; but he would see its actual, physical area and volume, as shown by the number of little spheres packed into it, to be "distorted" by the B factor, if he tried to use the radial r coordinate to establish a relationship between the space inside the interior region and the space at infinity. But an observer inside the interior vacuum region would see no "distortion" at all; the natural coordinates for him to use are "rescaled" from the above ones, using the factors A and B, so the metric takes its standard Minkowski form.

Well this is basically what I expect should happen, but there is a problem that needs to be faced square on here. Specifically, an unintentional 'sneaking something under the table' issue re the 'absolute' value of the tangent metric during the descent through the shell wall process. As I have said elsewhere, explicitly for SC's (and trivially for MC's), there is no potential dependence for the tangent length components - anywhere. Directly implying they always maintains there gravity-free values - i.e. are invariant quantities. That becomes an issue re any happy union. It directly implies the radial component has to return to the potential-free value, otherwise interior region (r < r_a) isotropy is impossible. My view is that there is a de facto violation of the potential independence of tangent measure as a consequence of the mathematical fitting procedure. In other words, maths is forcing an unphysical union. It stems from a choice of those adjustable A and B parameters that provide for the exterior to interior matching I would suggest. You will have to convince me that tangent measure can somehow be contracted in the MM region, yet with no potential dependence to allow it! Good luck with that Peter, but many thanks for your efforts to date!

 

[Q-reeus]

On re-reading, I realized there is another possible confusion here, because as I say later on, the metric in the interior vacuum region is still "length contracted" compared to the metric at infinity, in terms of the r coordinate, even though the relationship between area and radius is Euclidean. The faraway observer, looking at how the packing of the little objects changes as you descend through the non-vacuum region, could interpret what he sees this way: radial lengths continue to "contract" through this region, but now *tangential* lengths start to contract as well (they did not in the exterior vacuum region), and the tangential lengths contract *faster* than the radial lengths, so that when the inner surface of the non-vacuum region is reached, the packing of the little objects is now isotropic again; the relationship between radial and tangential packing is now Euclidean, but *all* lengths are now "contracted" compared to lengths at infinity.

Ok well just caught this after posting on your previous long response. Sorry, but you will have to convince me tangent contraction is physically plausable given what I said there. Late for bed again.:zzz:

 

[PeterDonis]

You will have to convince me that tangent measure can somehow be contracted in the MM region, yet with no potential dependence to allow it!

No--what I apparently have to convince you of is that, in a non-vacuum region, the tangential measure *does* depend on the potential. Obviously it has to, for the rest of what I said to work. To see why it does, consider why it *doesn't* in the exterior vacuum region: it's because the "direction of gravity" is purely radial--the only "gravity" that is "pulling" on you at a given point is the inward radial "gravity" of the distant massive object. In a non-vacuum region, that's no longer true; there is non-zero stress-energy surrounding any given point in the non-vacuum region, on all sides, so there is "gravity" pulling on any given point on all sides, not just a purely radial "acceleration" due to the distant gravity source as there is in the exterior vacuum region. So the effects of gravity in the non-vacuum region change *all* of the spatial components of the metric, not just the radial one.

I should note that I've oversimplified considerably in the above, because I'm just trying to give a quick heuristic explanation of how it can be possible for the tangential measure to be dependent on the "potential" in a non-vacuum region. So I should probably clarify that I am *not* saying the "acceleration due to gravity" in the non-vacuum region no longer points in the radial direction; it is still radial, just as it would be if you descended into the Earth's interior (idealizing the Earth as a sphere). All I am saying is that, in a non-vacuum region, "gravity" is more complicated than it is in a vacuum region, so the direction of the "acceleration due to gravity", which is the direction of the spatial gradient of the "potential", doesn't tell you as much as it does in a vacuum region.

 

[PeterDonis]

If the distant observer finds a contracted radial value, that has to be owing to physics. It's just that the effect of that physics is only non-locally evident.

What I meant by "interpretation" is that describing the physics as a "contracted radial value" depends on adopting a particular radial coordinate. Adopting a different one (such as the one used in isotropic coordinates for Schwarzschild spacetime) results in not having a "contracted radial value" in the sense of the radial coordinate having a different relationship to the physical distance measure than the tangential coordinates. But the actual physical "anisotropy", the fact that there is more volume inside a sphere of a given area than a Euclidean geometry calculation would lead one to expect, is still there, and will show up if we try to calculate actual physical areas and volumes using the isotropic coordinates. The physical area won't be a simple function of the radial coordinate any more, but it can still be calculated.

 

[DrGreg]

The tangential part of a spherically symmetric metric takes the form$$r^2(d\theta^2 + \sin^2 \theta \, \, d\phi^2)$$Note that there is an "r" in there, so if the radial coordinate r "undergoes contraction" (however you interpret that phrase), tangential distance gets contracted by the same factor, even though θ and $\phi$ don't change.

 

[Passionflower]

Don't know exact formula, but MTW claims it is 'fairly fast'.

Would you mind giving me the chapter and page number?

 

[Dale]

Yes and no. I have always clearly understood that the only locally induced and locally observed distortions are those due to mechanical strain under tidal (2nd derivative of potential) and/or 'g' (1st derivative of potential) forces. I made it real clear in that other thread these effects were to be discounted. All that was of interest was the 'pure' 0th order metric contractions/distortions as measured 'at infinity', and further made clear there these are not locally observable phenomena.

0 order effects of the metric are due to the choice of coordinates. For example consider the standard Minkowski metric.
$$ds^2=-dt^2+dx^2+dy^2+dz^2$$
And a transformation to anisotropic coordinates, $t=T$, $x=2X$, $y=Y$, $z=Z$:
$$ds^2=-dT^2+4dX^2+dY^2+dZ^2$$

They are mere artifacts of the coordinates and say nothing about the physics.

 

[PeterDonis]

Note that there is an "r" in there, so if the radial coordinate r "undergoes contraction" (however you interpret that phrase), tangential distance gets contracted by the same factor, even though $\theta$ and $\phi$ don't change.

Yes, but it can still be true that the volume between two spheres at r and r + dr is larger than the areas of the spheres themselves would lead one to expect, if the geometry of space were Euclidean. For the geometry in the interior vacuum region in Q-reeus' scenario to be Minkowski (even with coordinates scaled differently compared to those "at infinity"), this "mismatch", between the area of a sphere at r and the volume enclosed between it and another sphere at r + dr, has to go away by the time the inner surface of the non-vacuum region is reached. That means the tangential part of the metric has to change, in the non-vacuum region, *in addition* to the change it undergoes because r is getting smaller. (Which it can, because the stress-energy tensor in the non-vacuum region does not vanish.)

A while back I remember coming across a paper online that actually derived an expression for the line element in this type of scenario, in each of the three "regions" (exterior vacuum, "shell" non-vacuum, and interior vacuum), in terms of the r coordinate as I've defined it. But I haven't been able to find it again.

 

[George Jones]

A while back I remember coming across a paper online that actually derived an expression for the line element in this type of scenario, in each of the three "regions" (exterior vacuum, "shell" non-vacuum, and interior vacuum), in terms of the r coordinate as I've defined it. But I haven't been able to find it again.

Related stuff is treated in section 3.9 (and some stuff before this might be useful for looking at the metrics) from Eric Poisson's notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

which evolved into the excellent book, A Relativist's Toolkit: The Mathematics of black hole Mechanics.

I don't think that I'll have a chance to have a serious look at this for a few days.

 

[atyy]

PAllen can tell us if this is right http://arxiv.org/abs/gr-qc/0211061

 

[zonde]

Zonde,

please think carefully about some of the points I raised in my post #128 (with references). In particular, consider what is observed if a large globular cluster happened to undergo collapse to a supermassive black hole (there is no evidence of this happening, but there is no reason it could not). The key thing here is that for supermassive BH, you have the event horizon forming with very modest matter density. The stars are still many millions of miles apart when the event horizon forms.

I have tried to consider something in this direction before. The problem for me is that I have no clear picture about interior of gravitating body.

You see the problem is because black hole does not form at once. It starts as a small black hole at the center of gravitating body and then grows larger.
But what are conditions for formation of this small black hole?
Say we can compare the same massive body in two different situations. in one case it is in empty universe and in the other case it is inside large cloud of dust. How gravity around the body changes in those two situations?

EDIT: There other things as well. One is question about initial conditions (how we can arrive at situation we are considering). Other is changes caused by increase of density - heating up and consequent expansion and cooling of individual stars.
But it's hard to discuss all questions at once.

 

[Passionflower]

Yes, but it can still be true that the volume between two spheres at r and r + dr is larger than the areas of the spheres themselves would lead one to expect, if the geometry of space were Euclidean.

The two graphs below show resp. the difference and ratio between the Schwarzschild volume between two shells and the Euclidean volume.

[PLAIN]http://img192.imageshack.us/img192/5892/volumex.png
[PLAIN]http://img850.imageshack.us/img850/6859/volumeratio.png

 

[zonde]

It's EH that is in the "infinite future" for external observer. Black hole interior is "beyond" infinite future. And because we define infinity as a limit where nothing is beyond that limit we get contradiction in terms.

This argument would hold if the universe were 1 dimensional, but for any number of dimensions greater than 1 it doesn't hold. Consider a 2 dimensional manifold like the surface of a road with a fork in it. You can have two different "infinite futures" without any contradiction.

You can't have two different "infinite futures". What you can do is you can construct two different dimensions and claim that they both can be considered as going in direction of "future" under different viewpoints. But under single viewpoint there will be only one "infinite future".

The other way how you can have "different infinities" is by redefining function under consideration. This is closer to different simultaneities as it seems to me. So I will describe this in more detail.

Lets say that we have function f(x)=g(x)+h(x) and it approaches infinity as x->x0. So how we can define g(x) and h(x)?
There are three possibilities:
1) g(x) and h(x) both approach infinity as x->x0.
2) g(x) approaches infinity as x->x0 but h(x) approaches some finite value h0.
3) h(x) approaches infinity as x->x0 but g(x) approaches some finite value g0.
Obviously we are not redefining infinity across 1), 2) and 3) but functions g(x) and h(x).

And this example is useful for another purpose as well. Let's say that f(x) is round-trip time for signal moving at c toward point that is at radius x from center of black hole but g(x) and h(x) are time of forward trip and backward trip respectively.
Then we can identify case 1) as frozen star
case 2) as white hole
and case 3) as black hole

While we can extrapolate h(x) in case 2) and g(x) in case 3) even after x has reached x0 physically meaningful is only f(x) that gives round-trip time.
And that is cornerstone of relativity. (That might be the reason why Einstein was not taking black holes seriously.)

 

[Dale]

You can't have two different "infinite futures". What you can do is you can construct two different dimensions and claim that they both can be considered as going in direction of "future" under different viewpoints. But under single viewpoint there will be only one "infinite future".

I am not certain that I understand what you are saying here. Let me rephrase it the way I would say it: A spacetime with multiple dimensions may have different regions considered to be the infinite future, but the worldline of a single observer will only have one infinite future.

If this is what you meant then I agree, otherwise could you clarify your meaning?

 

[Q-reeus]

0 order effects of the metric are due to the choice of coordinates. For example consider the standard Minkowski metric.
$$ds^2=-dt^2+dx^2+dy^2+dz^2$$
And a transformation to anisotropic coordinates, $t=T$, $x=2X$, $y=Y$, $z=Z$:
$$ds^2=-dT^2+4dX^2+dY^2+dZ^2$$

They are mere artifacts of the coordinates and say nothing about the physics.

I will now concede after looking again over the way ISC is formulated at http://en.wikipedia.org/wiki/Schwar...c.29_formulations_of_the_Schwarzschild_metric , with the r₁ there having a changed meaning to the standard r, that ISC's are indeed just an alternate expression and not different physics to what standard SC's predict. Happy? Given how things have developed here, will let the other thread die with dignity and answer your #16 there here:

"What is the differential comparison technique you have in mind? Perhaps something like sending a sending a light pulse from one side of the sphere to the other, sending a light pulse out to infinity at the start and a second pulse out to infinity at the end, and measuring the time between receiving the two pulses at infinity?"

Infinity takes a long time, but you more or less get that there are means to eliminate 'difficult' factors if present. Really in my example it's no more than the before/after thing - measure f, D, before shell present, and then the same with shell. Simple (in principle). But this is not really the issue.

 

[Q-reeus]

No--what I apparently have to convince you of is that, in a non-vacuum region, the tangential measure *does* depend on the potential. Obviously it has to, for the rest of what I said to work. To see why it does, consider why it *doesn't* in the exterior vacuum region: it's because the "direction of gravity" is purely radial--the only "gravity" that is "pulling" on you at a given point is the inward radial "gravity" of the distant massive object. In a non-vacuum region, that's no longer true; there is non-zero stress-energy surrounding any given point in the non-vacuum region, on all sides, so there is "gravity" pulling on any given point on all sides, not just a purely radial "acceleration" due to the distant gravity source as there is in the exterior vacuum region. So the effects of gravity in the non-vacuum region change *all* of the spatial components of the metric, not just the radial one...

Nice try, but sadly still not buying that as is. Still opting that the nice transition is purely mathematical artifact of a force-fit coordinate scheme. Here's the crux of the problem imo. We agree tangent spatials by SC's are invariant everywhere exterior to r_b. There is in that region the potential (1-r_s/r)^1/2, plus it's spatial derivatives to all orders, just as there is within the matter region of shell wall. Only essential difference I see is the relative size distribution of potential and derivatives. This gets down then to the *fundamental character* of the relation between potential and various metric components. It makes perfectly good sense that there is a fairly abrupt transition in the 'g' field from maximum at r_b, to zero at r <= r_a, and likewise for tidal terms - they explicitly are potential derivative *in nature* and must cease in the equi-potential interior. Where is there any analogous physical basis, in the tangent metric component case, for *total* exterior indifference to potential *and all it's derivatives*, yet a relatively steep dependence just within the matter region? 'Pulling in all directions' (I realize this is just you 'dumbing it down' for my benefit) just won't near cut it as explanation, as I have outlined above.

What 'essence' or geometric 'object' can be entirely absent >= r_b, yet there strongly for r_b>r>r_a, so as to explain it? And what's more it has to be shown to be cumulative in effect, and not a mere 'blip' that leaves no trace on exit past r<r_a, so to speak. Tall order indeed! Sole uniquely present identity I can think of might be divergence, but that seems most unlikely a solution, and in itself creates another issue. Namely, if divergence is truly absent exterior to r_b, this gives the lie to those claiming that in GR 'gravity truly gravitates'. What say you sir?
[One final comment: in #222 you mentioned agreement between yourself and DrGreg's finding in https://www.physicsforums.com/showpost.php?p=3559845&postcount=10, but I read him there as saying interior length are as at infinity, once the metric is applied. A misunderstanding?]

 

[PeterDonis]

What 'essence' or geometric 'object' can be entirely absent >= r_b, yet there strongly for r_b>r>r_a, so as to explain it? And what's more it has to be shown to be cumulative in effect, and not a mere 'blip' that leaves no trace on exit past r<r_a, so to speak.

Um, a non-zero stress-energy tensor, which means a non-zero Einstein tensor, which is the primary geometric object in the Einstein Field Equation? That is what does the work in the non-vacuum region, r_b>r>r_a. For r<r_a, the vacuum Einstein Field Equation is enough to ensure that the "potential" is constant at its value at r_a.

One final comment: in #222 you mentioned agreement between yourself and DrGreg's finding in https://www.physicsforums.com/showpost.php?p=3559845&postcount=10, but I read him there as saying interior length are as at infinity, once the metric is applied. A misunderstanding?

No, just a difference in terminology. What he means by "once the metric is applied" is "from the viewpoint of an observer in the interior vacuum region". Such an observer can't tell that he is not in the flat spacetime region at infinity by purely local measurements; locally the two regions look the same. Only by global observations can the two regions be distinguished.

 

[zonde]

In this context, infinity is a limit where there are no t coordinates beyond that limit. A lack of coordinates, in a particular choice of coordinates, does not imply non-existence. It just means those coordinates don't cover the whole of spacetime.

Infinity is imaginary (not real) limit where there are no real values beyond this limit no matter what choices you make.
Look if you say that function asymptotically approaches value a as it's argument approaches infinity then it means a is the limit no matter what you do with the argument.

Maybe there is some confusion with my argument that I can still clear up.
I can explain my argument in two steps rather than one:
1) in Schwarzschild metric interior of black hole is completely disconnected from exterior because there is no future beyond infinite future and there is no past before infinite past (where you could hope to connect interior with exterior).
2) there can be any number of spacetime patches that are completely disconnected from our spacetime. There can be even any number of universes that are completely disconnected from our universe. As they do not affect our reality in any way it can be stated that they are not real or alternatively they do not exist.

 

[DrGreg]

As they do not affect our reality in any way it can be stated that they are not real or alternatively they do not exist.

Even in flat spacetime, the "inside" of a Rindler horizon does not "affect the reality" of a Rindler observer (and it is "beyond" T=∞ in Rindler coordinates), so does the "inside" of a Rindler horizon exist?

(See post #75, the "inside" is the blue region, the observer is the black line, the red and green lines specify Rindler coordinates.)

 

[PeterDonis]

1) in Schwarzschild metric interior of black hole is completely disconnected from exterior because there is no future beyond infinite future and there is no past before infinite past (where you could hope to connect interior with exterior).

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.

 

[Dale]

Infinity takes a long time, but you more or less get that there are means to eliminate 'difficult' factors if present. Really in my example it's no more than the before/after thing - measure f, D, before shell present, and then the same with shell.

Yes, I get the idea, but different measurement techniques will give different answers which is why a complete description of the measurement technique becomes important. Once you have done that then the result of that measurement technique is guaranteed to be coordinate independent.

But this is not really the issue.

OK, so I am not sure what you still think the issue really is. You now understand that the anisotropy was merely an artifact of the coordinates, and I thought that the dissapearance of that anisotropy was what was bothering you.