Are flowing space models compatible with GR?

[PAllen]

His inflow model does not lead to the same description by a far away observer as in standard GR (Einstein-Schwartzschild), but it does seem to reproduce the same phenomena so that it may be useful for computer simulations.

You have repeated this statement a few times. The only way I can understand this in a way that isn't trivially false is to emphasize description. So you are saying the description aka coordinate dependent interpretation is different. That is fine, as long as you realize that all physical observables are identical, as a mathematical fact.

Actually, as worded, I must call this false. GR does not specify coordinates, and includes all coordinate descriptions. If draw an equality between GR and coordinate dependent interpretations of SC coordinates, this is a false equality and don't bring Einstein into it.

 

[harrylin]

You have repeated this statement a few times. [..] So you are saying the description aka coordinate dependent interpretation is different. That is fine, as long as you realize that all physical observables are identical, as a mathematical fact.

Yes, as far as I verified, it predicts the same phenomena; that is exactly what I said.

 

[zonde]

The answer is of course that it uses a "cheap trick": if I understand it correctly then it has a discontinuity in the speed of light, right in the middle of a heavy body. And I'm not talking "black hole" here: I'm talking about the Earth or even a heavy piece of glass.

I can see how as a calculation tool this can produce correct predictions about observations. However as a physical model such a discontinuity is extremely ugly and it looks very unreasonable; there is to my knowledge no precedent for it in physics (except perhaps some abandoned theories that I don't know of). This appears to me as a strong argument against using such a toy model for the purpose of trying to understand what "really happens".

Yes I agree with you. This argument uncovers serious ugliness in river model.

And I will try to give my explanation for that argument (in the hope that it will make the ugliness more obvious).
River model kind of suggests that sink (or source) of the river is mass so that river terminates at massive particles. But this doesn't seem to work.
Imagine massive shell. River should terminate at the shell and inside shell you should have calm pond. But as I understand this disagrees with GR as we should experience time dilation inside empty massive shell.
Alternatively we can say that river continues inside massive shell and it has the same speed (!) everywhere inside the shell but then we have very strange point in the middle of the empty shell where rivers flowing in different directions meet.

 

[zonde]

They are physically different.

Thanks, this clears some confusion about your position.

I will write answer for the rest of your post later. I seems like it explains your point quite well so I want to think it over.

 

[PAllen]

Yes I agree with you. This argument uncovers serious ugliness in river model.

And I will try to give my explanation for that argument (in the hope that it will make the ugliness more obvious).
River model kind of suggests that sink (or source) of the river is mass so that river terminates at massive particles. But this doesn't seem to work.
Imagine massive shell. River should terminate at the shell and inside shell you should have calm pond. But as I understand this disagrees with GR as we should experience time dilation inside empty massive shell.
Alternatively we can say that river continues inside massive shell and it has the same speed (!) everywhere inside the shell but then we have very strange point in the middle of the empty shell where rivers flowing in different directions meet.

The river model really only works outside a massive body. It fully works only for BH/WH, which is all that is covered in Hamilton's paper.

Remember, it is just an interpretation of coordinate quantities in the Gullestrand-Panlieve coordinates (or, in a much more complicates way, the Doran coordinates for rotating BH). None of these coordinates/metrics extend inside a massive body; these are coordinates for a particular vacuum solutions, where the stress energy tensor is zero.

To use any metric like SC or GP for space time with a massive body, you use two metrics with junction conditions - one for the interior, one for the exterior. There are also solutions for attenuating fluid or dust - spherically symmetric non-vaccuum solutions where fluid or dust attenuates smoothly toward vacuum at infinity (where the solution becomes asymptotically Minkowski), but is nowhere an exact vacuum (the stress energy tensor is not zero except in the limit at infinity). If you use one of these fluid (dust is basically just pressure-less fluid) solutions, then the whole river model doesn't apply because you don't have the GP metric.

Right in post #2 and #3 of this thread it was pointed out that the river model has very limited applicability. It is just a tool for understanding two (mostly really one) specific metric forms (the generalization to Doran is rarely used). It is not a general theory, just a tool for understanding a special space time.

I think you are both making a big deal out of a special case visualization aid. If it rocks your boat, use it, if not don't. Within its stated domain, it is mathematically exact.

 

[harrylin]

I almost forgot the original issue that started this topic, and which I did not cite in my first post; the first suggestion concerning this model was that it is not just a special case visualisation aid. But that can wait for later. Please bear with me a little more, for there's another thing that puzzles me: qualitatively, falling space resembles very much what I would expect to emulate Newton's theory of light propagation. So I wonder: how does it manage to reproduce twice the Newtonian prediction of light bending?

 

[PAllen]

I almost forgot the original issue that started this topic, and which I did not cite in my first post; the first suggestion concerning this model was that it is not just a special case visualisation aid.

Not sure how. The paper itself incorporates a proof of what Dalespam said explicitly in post #3: that the river model cannot apply to any case more general than the Doran metric. It does this with an argument based on counting degrees of freedom.

But that can wait for later. Please bear with me a little more, for there's another thing that puzzles me: qualitatively, falling space resembles very much what I would expect to emulate Newton's theory of light propagation. So I wonder: how does it manage to reproduce twice the Newtonian prediction of light bending?

I see no relation to Newtonian anything. You need to make this thought more precise before I can deal with it.

Please note, geodesics are invariant features of geometry. River model is visual aid for GP coordinates of SC geometry. Geodesics are therefore identical to SC coordinates (except with points labeled with different numbers). Light bending is a question of finding null geodesics. They are the same no matter what the coordinates. This is why I can't attach any conceivable meaning to your question.

 

[zonde]

The river model really only works outside a massive body. It fully works only for BH/WH, which is all that is covered in Hamilton's paper.

I don't think that it works for WH and therefore I am not sure that it works for BH either.

Can we explore WH river model?

 

[PAllen]

I don't think that it works for WH and therefore I am not sure that it works for BH either.

Can we explore WH river model?

River model with β < 1 (negative square roots) is white hole. One is just time reflection of other. If one works, the other works.

Can you try to describe why you think it "doesn't work". Since it is just an interpretation of alternate coordinates for SC geometry, which is the exact vacuum solution for spherically symmetry, "not working" seems equivalent to claiming the most heavily used solution of GR is false.

 

[harrylin]

[..] I see no relation to Newtonian anything. You need to make this thought more precise before I can deal with it.

[..] Light bending is a question of finding null geodesics. They are the same no matter what the coordinates. This is why I can't attach any conceivable meaning to your question.

I don't understand what is unclear about my question as it relates to the first tested difference between GR and Newton's mechanics. I'll elaborate and I hope that someone who knows that model better than me can give the answer. While I don't understand some of the details, the river model seems to accelerate everything including light in a Newtonian fashion towards the mass. If so, then for any light ray starting at infinity at the speed of light, the prediction would be Newtonian. Therefore I asked how that model manages to reproduce the GR prediction of star light bending around the Sun, which is twice the Newtonian value.

 

[zonde]

River model with β < 1 (negative square roots) is white hole. One is just time reflection of other. If one works, the other works.

Can you try to describe why you think it "doesn't work". Since it is just an interpretation of alternate coordinates for SC geometry, which is the exact vacuum solution for spherically symmetry, "not working" seems equivalent to claiming the most heavily used solution of GR is false.

Test mass falling down on white hole can't get in and can't get away. It just doesn't make much sense as combined mass of white hole and test mass can get bigger so it can result in situation where falling mass and mass of white hole should have new event horizon that includes falling mass ... I guess.

About your argument that "it is just an interpretation of alternate coordinates for SC geometry" - I do not agree. And the point is that mapping infinity onto finite coordinate (or vice versa) is not mathematically correct. And it changes physical situation behind it.

 

[PAllen]

Test mass falling down on white hole can't get in and can't get away. It just doesn't make much sense as combined mass of white hole and test mass can get bigger so it can result in situation where falling mass and mass of white hole should have new event horizon that includes falling mass ... I guess.

This is not correct. A test mass outside can't get into a white hole but it can certainly get away. A test mass in the white hole interior has no choice but to leave the white hole.

As a consequence, a white hole can never grow (if not joined with a black hole solution, it just ends). There is no process by which it can form - it must be causelessly originate in the infinite past.

No one considers this physically reasonable, but it arises because GR no more incorporates thermodynamics than Newtonian physics. In Newtonian physics, anything you see running movie backwards is just as allowed as the forward version. GR has the same mathematical symmetry.

About your argument that "it is just an interpretation of alternate coordinates for SC geometry" - I do not agree. And the point is that mapping infinity onto finite coordinate (or vice versa) is not mathematically correct. And it changes physical situation behind it.

I am not going have that debate again. See the dozens of other threads here as well as innumerable sources on the web or in books. The premise of this discussion is that BH an WH behave as experts say they do; against this backdrop, how does the river model work?

As long as you don't accept the standard interpretation of SC geometry (wherein the interior exists, and objects can cross the horizon in finite time on their clocks), then we can't have a meaningful discussion of a visualization of this model.

 

[PAllen]

I don't understand what is unclear about my question as it relates to the first tested difference between GR and Newton's mechanics. I'll elaborate and I hope that someone who knows that model better than me can give the answer. While I don't understand some of the details, the river model seems to accelerate everything including light in a Newtonian fashion towards the mass. If so, then for any light ray starting at infinity at the speed of light, the prediction would be Newtonian. Therefore I asked how that model manages to reproduce the GR prediction of star light bending around the Sun, which is twice the Newtonian value.

Now I understand your question. Before it wasn't at all clear to me.

I'll give two answers.

The answer is that won't satisfy you is the one I've already given - the river model as a mathematical model (not the picture at the beginning of the paper) is nothing but re-casting the GP coordinates and associated metric in terms of some new variables. It is thus inherently equivalent in all predictions to SC geometry using any other coordinates. The quantitative model is not the picture of a swimming fish.

However, now that I understand your thinking I can give you a more concrete response as well. The basic issue that there is nothing Newtonian in the operation of the model except that the formula for β matches Newtonian escape velocity (the river flows in Galilean fashion - but not following any fluid laws - as if each piece were a mathematical point). The first difference is the time coordinate tff. This reflects (compared to SC coordinate time) the accumulated effect of time dilation over free fall from infinity. The second difference is that as an object or light moves (other than free fall from infinity - not possible for light) it undergoes a generalized Lorentz boost (including spatial rotation) from moment to moment by the difference in β (treated as an ingoing radial vector) from place to place within the river flow. I see nothing Newtonian about this. To actually make a light bending calculation in the river model quantitatively, it would be necessary to integrate over the appropriate infinitesimal Lorentz boosts as the light travels. In practice, this is very cumbersome. This is why no one (that I know of - including the author) does actual calculations directly in this model. However, it is proven that the result is the same by the derivation of the model.

 

[harrylin]

[..] now that I understand your thinking I can give you a more concrete response as well. The basic issue that there is nothing Newtonian in the operation of the model except that the formula for β matches Newtonian escape velocity (the river flows in Galilean fashion - but not following any fluid laws - as if each piece were a mathematical point). The first difference is the time coordinate tff. This reflects (compared to SC coordinate time) the accumulated effect of time dilation over free fall from infinity. The second difference is that as an object or light moves (other than free fall from infinity - not possible for light) it undergoes a generalized Lorentz boost (including spatial rotation) from moment to moment by the difference in β (treated as an ingoing radial vector) from place to place within the river flow. I see nothing Newtonian about this. To actually make a light bending calculation in the river model quantitatively, it would be necessary to integrate over the appropriate infinitesimal Lorentz boosts as the light travels. In practice, this is very cumbersome. This is why no one (that I know of - including the author) does actual calculations directly in this model. However, it is proven that the result is the same by the derivation of the model.

OK, I see! Hamilton was apparently inspired by Visser, who based his model on Newtonian acceleration; and one of them suggests that it is particular useful as calculation aid for computing... Perhaps it would not be cumbersome for an FEA such as Comsol.
(Visser: "The heuristic is based on Newtonian gravity, the notion of local inertial frames [the Einstein equivalence principle], plus the use of Galilean coordinate transformations".)

Thanks.

I'll comment later on the original issue that led to this discussion.

 

[zonde]

This is not correct. A test mass outside can't get into a white hole but it can certainly get away. A test mass in the white hole interior has no choice but to leave the white hole.

As a consequence, a white hole can never grow (if not joined with a black hole solution, it just ends). There is no process by which it can form - it must be causelessly originate in the infinite past.

I will say it differently.

There is no limit how much mass can accumulate near the event horizon of white hole.

I am not going have that debate again. See the dozens of other threads here as well as innumerable sources on the web or in books. The premise of this discussion is that BH an WH behave as experts say they do; against this backdrop, how does the river model work?

As long as you don't accept the standard interpretation of SC geometry (wherein the interior exists, and objects can cross the horizon in finite time on their clocks), then we can't have a meaningful discussion of a visualization of this model.

But please. Why do you have to refer to SC coordinates? Can't we discuss river model referring only to GP coordinates?

 

[Dale]

And the point is that mapping infinity onto finite coordinate (or vice versa) is not mathematically correct. And it changes physical situation behind it.

This is incorrect, but I agree with PAllen that it doesn't belong in this thread. I recommend that you study the second chapter of Carroll's GR lecture notes, available on arxiv.

 

[harrylin]

[..] the point is that mapping infinity onto finite coordinate (or vice versa) is not mathematically correct. And it changes physical situation behind it.
[..]

But please. Why do you have to refer to SC coordinates? Can't we discuss river model referring only to GP coordinates?

Yes please, it's good to know that it's not a real river model but let's stick to Hamilton's model and not discuss again other coordinate systems. I was going to elaborate on the conclusion that we arrived at concerning the mapping as not everything was said about it; but I had overlooked your comment about mapping infinity into finite coordinates. Do you perhaps mean that what is infinite Schwartzschild coordinate time transforms to finite proper clock time at the Schwartzschild radius? I see no problem with that and it's even the standard* GR solution. So I don't think that it's an issue here.

* Oppenheimer and Snyder, "On Continued Gravitational Contraction", Physical Review vol.56, 1939

 

[zonde]

Do you perhaps mean that what is infinite Schwartzschild coordinate time transforms to finite proper clock time at the Schwartzschild radius? I see no problem with that and it's even the standard* GR solution. So I don't think that it's an issue here.

This mapping is not a problem.
The problematic mapping is between SC and GP coordinates. I see the problem in the statement that two interpretations (coordinates) are equivalent - clock does not experience any proper time beyond some moment "tx" i.e. it stops versus clock experience some more proper time beyond moment "tx".

 

[PAllen]

This mapping is not a problem.
The problematic mapping is between SC and GP coordinates. I see the problem in the statement that two interpretations (coordinates) are equivalent - clock does not experience any proper time beyond some moment "tx" i.e. it stops versus clock experience some more proper time beyond moment "tx".

This thread was opened to discuss the river model of an black hole as conventionally interpreted - with horizon and interior; and whether the river model makes any different predictions from the any other interpretation of said BH. It is not appropriate to hijack this thread to raise, for the umpteenth time, that you reject the standard interpretation of BH's altogether.

 

[harrylin]

This mapping is not a problem.
The problematic mapping is between SC and GP coordinates. I see the problem in the statement that two interpretations (coordinates) are equivalent - clock does not experience any proper time beyond some moment "tx" i.e. it stops versus clock experience some more proper time beyond moment "tx".

Ah yes, that incompatibility was rather well demonstrated by the cited disagreeing statements in the black hole thread. That was also what led to this discussion, which was meant to find out if GR allows for such extremely opposing views as expressed by Hamilton vs. Oppenheimer and Schwartzschild. As I do not question the compatibility of Schwartzschild's model with GR, we here only discuss Hamilton's flowing space model (compare: http://www.jstor.org/stable/1968902).

I learned a lot from this interesting discussion and will give a wrap-up about Hamilton's model later today.

 

[harrylin]

Here's my wrap-up of this discussion topic as I now see it, with some elaborations - a lot in fact, as I had no time to write it all down until now.

This thread originated with the black hole thread, as truly incompatible opinions appear to result from GR. This is not just a matter of perspective; whereas according to Einstein¹, Oppenheimer² and modern followers a clock would stop ticking at the Schwartzschild radius if it could reach it, but it will never happen; according to Hamilton³ and a number of others, an object can (and will) fall through that radius. That is a contradiction of predicted events. It would be a poor theory if GR permits such a disagreement. As I don't doubt the standard solution (and Einstein was definitely "in" it, thus it's a mystery why PAllen wrote "and don't bring Einstein into it"), I started this thread for a more critical look at Hamilton's flowing space model.

The first thing that struck me was that in spirit the model is not just the equivalence principle on its head, it is even the antithesis of what Einstein had in mind with GR. The equivalence principle of GR has that a gravitational field creates the same phenomena as acceleration relative to an inertial frame; and according to Einstein's GR one may even pretend that an inertial frame is in rest in a gravitational field. That permits according to the theory to define, not a "flowing" but a "stationary" space:

"motion "in space" .[..] “space,” of which, we must honestly acknowledge, we can not form the slightest conception, and we replace it by “motion relative to a practically rigid body of reference.”"⁴

Thus when Einstein admitted that GR implies some kind of an ether, it was certainly not of the "flowing space" kind:

"The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualities"⁵

Next we discussed what I called a "cheap trick": apparently Hamilton's model even has a discontinuity in the speed of light, right in the middle of a heavy body. As a physical model such a discontinuity is extremely ugly and it looks very unreasonable. Of course, the topic of this thread is slightly different, but also according to GR the following is a basic law of nature:

"A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line."⁶

To elaborate a little more and clarify that this has nothing to do with "vacuum" or not:
Suppose that the Earth has a tunnel right through, from one side to the other. Consider the kind of equation of motion that GR allows for a stone that falls through the centre of the Earth. And similarly, what "distant" descriptions of velocity as function of time does GR permit for a light ray passing through that hole. Right in the middle of the Earth, the space-time constants are "flat"; surely GR allows no infinitely rapid change in velocity at that point. That violates the law of inertia and the law of "local" constancy of the speed of light.

I thus came to the conclusion that even if Hamilton's model accurately matches predictions of currently verifiable observations, it does not correspond to the concepts of GR: it is the antithesis of Einstein's "stationary" space and as we understand Hamilton's model, it violates laws of nature that are fundamental to GR for common, "down to Earth" situations.

In summary, the "flowing river" models apparently fails the test of a light ray and a stone falling through a hole in any heavy body. It is then an unphysical mapping with deformation and discontinuity, similar to remapping this map:

http://www.1worldglobes.com/lg_image_windows/world_stage_lg.htm

to this map:

http://en.wikipedia.org/wiki/Mercator_projection

And that brings me to the post that initiated this topic:

[..] Space time is curved, like the surface of the Earth. You can make maps of it, like you can make maps of the Earth's surface. But they won't / can't be to scale except for small regions (frames). The metric describes how the particular part of the map is distorted. To oversimplify greatly, the closer the metric is to unity, the less the distortion.

Considering that Hamilton spends a good part of his time describing a journey into a black hole, (complete with visuals), do you really think it's an accurate reading of him to say that he supports your "time stops at the event horizon, so we don't have to worry about what comes after" idea?

(That was semi-rhetorica., I can say that I certainly don't, and I would be surprised if you did if you thought about it a bit more. Though I've been surprised in this manner before, alas.)

The conclusions from this discussion help me to elaborate on my preliminary answer there.

Someone who at a constant proper velocity moves towards the North pole, will on the Mercator projection get an increasing velocity and become stretched out. Then at the North pole there is a discontinuity that gives away the conformal mapping, as he supposedly makes an infinitely fast turnaround along the top of the map.

Hamilton mentions a "conformal factor" in his paper. And despite the fact that he admits that "According to the Schwarzschild metric, at the Schwarzschild radius rs, proper radial distance intervals become infinite, and proper time passes infinitely slowly", he also writes:
"[the river model] explains how an extended object will be stretched radially by the inward acceleration of the river"
- which is based on his apparently distorted map, and is the contrary of the interpretation of Einstein and Oppenheimer. I have the impression that Hamilton is, so to say, carried away by his own model.

For me this discussion was a big eye opener. Thanks again to everyone who gave feedback.

1 Oppenheimer and Snyder 1939, "On Continued Gravitational Contraction", Physical Review vol.56
2 Einstein 1939 http://www.jstor.org/stable/1968902
3 Hamilton 2008 http://arxiv.org/abs/gr-qc/0411060
4 Einstein 1916 https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory
5 Einstein 1920 https://en.wikisource.org/wiki/Ether_and_the_Theory_of_Relativity
6 Einstein 1922, The meaning of relativity

 

[PAllen]

This thread originated with the black hole thread, as truly incompatible opinions appear to result from GR. This is not just a matter of perspective; whereas according to Einstein¹, Oppenheimer² and modern followers a clock would stop ticking at the Schwartzschild radius if it could reach it, but it will never happen; according to Hamilton³ and a number of others, an object can (and will) fall through that radius. That is a contradiction of predicted events. It would be a poor theory if GR permits such a disagreement. As I don't doubt the standard solution (and Einstein was definitely "in" it, thus it's a mystery why PAllen wrote "and don't bring Einstein into it"), I started this thread for a more critical look at Hamilton's flowing space model.

The "don't bring Einstein into it" reference of mine was any claim that Einstein believed one set of coordinates was better than another (he did not); and Einstein had no doubts that all observables must be coordinate invariant. Whatever Einstein believed about black holes, he would contend are true in all coordinate systems. I am aware Einstein proposed a number of arguments against BH's ever forming in the real world, and had interpretations of them, that are considered incorrect by almost all experts today - but never was the rationale that you can only use one preferred coordinate system. For that matter, Einstein flip flopped twice on the question of existence of gravitational waves.

Please note that every conclusion derivable from Kruskal coordinates can be derived in SC coordinates by taking limits from the two sides of the EH.

GR is not at fault for contradictory predictions. People are. In particular, Einstein was known for trying to adjust the way to interpret GR when it made predictions that violated his intuition.

If you polled modern relativity experts, you would find a very large majority that agree the classical GR insists that object do fall through the event horizon, and reach the singularity, in finite time on their clocks; and that this is not in any way contradictory with outside observers not being able to see this; and the SC coordinate time is a 'book keepers time' that has no physical meaning except for a special class of observers. Very different distribution of answers would arise if you asked how likely these things are in the real world rather than in classical GR.

 

[PAllen]

I am not sure you recognize that Hamilton's rive model is a specialized interpretation, of two special case GR geometries. Use of this model is not part of any general understanding of BH, EH, etc. The general modern consensus comes from studying such solutions in coordinate independent ways, and from the global methods developed by Hawking, Penrose, and others. It is not derived or understood by most using Hamilton's river model.

 

[harrylin]

[..] I am aware Einstein proposed a number of arguments against BH's ever forming in the real world, and had interpretations of them, that are considered incorrect by almost all experts today - but never was the rationale that you can only use one preferred coordinate system. [..]

Obviously not; who would think such a thing?

[..] that object do fall through the event horizon [..] is not in any way contradictory with outside observers not being able to see this [..]

Indeed, that was not an issue!

I am not sure you recognize that Hamilton's rive model is a specialized interpretation, of two special case GR geometries. Use of this model is not part of any general understanding of BH, EH, etc. The general modern consensus comes from studying such solutions in coordinate independent ways, and from the global methods developed by Hawking, Penrose, and others. It is not derived or understood by most using Hamilton's river model.

This thread was triggered by the above-mentioned post by perfect; and while it was just about one model, it was very instructive for me.

 

[PAllen]

The first thing that struck me was that in spirit the model is not just the equivalence principle on its head, it is even the antithesis of what Einstein had in mind with GR. The equivalence principle of GR has that a gravitational field creates the same phenomena as acceleration relative to an inertial frame; and according to Einstein's GR one may even pretend that an inertial frame is in rest in a gravitational field. That permits according to the theory to define, not a "flowing" but a "stationary" space:

There are two aspects to the principle of equivalence:

- that acceleration via applied force can be treated (almost) as a gravitational field
- that free fall can be treated (almost) as at rest.

The river model that you think is so anathema is simply defining a 'river' as a particular family of free falling frames. It is absolutely consistent with the second flavor of the principle of equivalence above.

 

[PAllen]

Next we discussed what I called a "cheap trick": apparently Hamilton's model even has a discontinuity in the speed of light, right in the middle of a heavy body. As a physical model such a discontinuity is extremely ugly and it looks very unreasonable. Of course, the topic of this thread is slightly different, but also according to GR the following is a basic law of nature:

"A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line."⁶

To elaborate a little more and clarify that this has nothing to do with "vacuum" or not:
Suppose that the Earth has a tunnel right through, from one side to the other. Consider the kind of equation of motion that GR allows for a stone that falls through the centre of the Earth. And similarly, what "distant" descriptions of velocity as function of time does GR permit for a light ray passing through that hole. Right in the middle of the Earth, the space-time constants are "flat"; surely GR allows no infinitely rapid change in velocity at that point. That violates the law of inertia and the law of "local" constancy of the speed of light.

I thus came to the conclusion that even if Hamilton's model accurately matches predictions of currently verifiable observations, it does not correspond to the concepts of GR: it is the antithesis of Einstein's "stationary" space and as we understand Hamilton's model, it violates laws of nature that are fundamental to GR for common, "down to Earth" situations.

This I chalk up to your misunderstanding the points made in #2 and #3 of this thread. The river model only applies to two specific geometries. It can be applied outside the Earth (to excellent approximation), but its validity ceases as soon as you reach the earth. It doesn't matter whether you drill through the Earth or not - the fact that Earth is there means that below its surface you immediately deviate from the geometry the river model applies to. If, instead of the earth, you had a black hole with event horizon, then the river model would continue to apply. The Earth has no event horizon or singularity at all.

 

[PAllen]

Harrylin,

You appear to quote Hamilton as follow:

"According to the Schwarzschild metric, at the Schwarzschild radius rs, proper radial distance intervals become infinite, and proper time passes infinitely slowly"

In the paper by Hamilton beginning this thread, I can find nothing resembling this quote.

On another note, the 1939 paper by Einstein you reference is very careful to say:

"Further, it is easy to show that both light rays and material particles take an infinitely long time (measured in "coordinate time") in order to reach ..."

Note Einstein is careful to highlight the coordinate, not physical nature of this observation.

Please be aware that investigations into the nature of the EH were just beginning at this time, and full understanding of the nature of the horizon and the purely coordinate singularity there did not settle down until well into the 1960s.

You would, I hope, admit that many things were discovered about pure classical EM after the death of Maxwell.

 

[zonde]

The "don't bring Einstein into it" reference of mine was any claim that Einstein believed one set of coordinates was better than another (he did not);

Certainly he did. For particular body preferred coordinates are it's rest frame coordinates. And there is only one such a coordinate system for every body.

So what it has to do with river model. And the thing is that river model changes simultaneity convention. The same goes for GP coordinates.

Let's compare this with simpler model in SR. Let's say we have a point in space (line in spacetime). We have a body moving toward this point (we view the situation in point's restframe). Now we remap coordinate system of point's restframe so that simulataneity on the ray starting at the point and going through the body toward infinity is according to moving body's restframe. And we extend this remapping in spherically symmetric way around the point.
Now the question is - would we expect any pathologies in this new coordinate system? And would we try to say that laws of physics hold in this particular coordinate system?
There are certain similarities between this coordinate system and GP coordinates. So this way we can analyze if we allow in GR some freedom that we would not allow in SR.

 

[PAllen]

Certainly he did. For particular body preferred coordinates are it's rest frame coordinates. And there is only one such a coordinate system for every body.

So what it has to do with river model. And the thing is that river model changes simultaneity convention. The same goes for GP coordinates.

Let's compare this with simpler model in SR. Let's say we have a point in space (line in spacetime). We have a body moving toward this point (we view the situation in point's restframe). Now we remap coordinate system of point's restframe so that simulataneity on the ray starting at the point and going through the body toward infinity is according to moving body's restframe. And we extend this remapping in spherically symmetric way around the point.
Now the question is - would we expect any pathologies in this new coordinate system? And would we try to say that laws of physics hold in this particular coordinate system?
There are certain similarities between this coordinate system and GP coordinates. So this way we can analyze if we allow in GR some freedom that we would not allow in SR.

In GR, Einstein felt its biggest contribution was general covariance, which despite controversy about how much it means, meant that all coordinate systems are equal. Einstein viewed the preference for inertial frames in SR a fundamental weakness of the theory. There is no other possible interpretation of Einstein's writing on this.

The reference I made about "don't bring Einstein into it" was a preference for SC coordinates. I stand by the view that Einstein would have considered such a preference an abomination.

In GR, there is no such thing as a global frame even for an inertial body. For a hovering body in SC geometry, there isn't even an inertial local frame, because such a body is not inertial. However, if you want to consider local inertial frames, there is an unambiguous answer that is coordinate independent (because local frames are just a matter of the local basis on a world line) - an inertial frame crosses the event horizon in finite time in that frame, and continues to the singularity. This was proved by Robertson in the early 1940s.

Your view of coordinate system features is quite wrong. GP coordinates represent a collection of free fall frames which is the GR analog of rest frames. SC coordinates represent frames of non-inertial observers, with proper acceleration approaching infinite for near horizon.

 

[harrylin]

[..] You appear to quote Hamilton as follow:

"According to the Schwarzschild metric, at the Schwarzschild radius rs, proper radial distance intervals become infinite, and proper time passes infinitely slowly"

In the paper by Hamilton beginning this thread, I can find nothing resembling this quote. [..]

Thanks for spotting that! The discussion with pervect was about http://casa.colorado.edu/~ajsh/schwp.html, and when citing his post I forgot to do MULTIQUOTE and to add "on his web page" before that citation (and now it's too late to edit).

+ this clarification ended up on the next page

 

[TrickyDicky]

No one considers this physically reasonable, but it arises because GR no more incorporates thermodynamics than Newtonian physics. In Newtonian physics, anything you see running movie backwards is just as allowed as the forward version. GR has the same mathematical symmetry.

what do you mean by "no more incorporates thermodynamics than Newtonian physics"?

AFAIK this is not correct, see the FAQ about this on the cosmology subthread. GR doesn't have the same time symmetry as Newtonian physics unless you are referring to the static solutions only, but you shouldn't attribute this to GR in general.

 

[harrylin]

Some comments invite for more elaboration of my wrap-up but - that is for later. Now I just have time to address a few side issues brought up by PAllen that would lead us away from the topic if we don't keep it brief:

[..] the 1939 paper by Einstein you reference is very careful to say:

"Further, it is easy to show that both light rays and material particles take an infinitely long time (measured in "coordinate time") in order to reach ..."

Note Einstein is careful to highlight the coordinate, not physical nature of this observation.
[..]
You would, I hope, admit that many things were discovered about pure classical EM after the death of Maxwell.

If you had read the whole paper then you would know that it pretends the contrary of what you suggest. As I have no issues with that, this thread discusses and criticizes Hamilton's model. If you have doubt about the compatibility of Einstein's and Oppenheimer's analyses with GR, please don't hesitate to start a topic on that.

Concerning Maxwell, I am not aware of any serious misapplication of his own theory by him. If you do, it could be interesting (but please, not in this thread!)

[..] The reference I made about "don't bring Einstein into it" was a preference for SC coordinates. I stand by the view that Einstein would have considered such a preference an abomination.

I guess that with "SC coordinates" you meant spherical coordinates. If so, I'm mystified by your comment; surely we all agree that such coordinates only are preferred for mathematical convenience, in order to easily make precise calculations. That is why everyone including Hamilton uses them; it's irrelevant for the topic at hand.

 

[PAllen]

what do you mean by "no more incorporates thermodynamics than Newtonian physics"?

AFAIK this is not correct, see the FAQ about this on the cosmology subthread. GR doesn't have the same time symmetry as Newtonian physics unless you are referring to the static solutions only, but you shouldn't attribute this to GR in general.

Disagree. In classical GR, without adding anything about BH entropy (which is quantum), any solution time reversed is also a solution. This is trivial to show.

 

[PAllen]

...
If you had read the whole paper then you would know that it pretends the contrary of what you suggest. As I have no issues with that, this thread discusses and criticizes Hamilton's model. If you have doubt about the compatibility of Einstein's and Oppenheimer's analyses with GR, please don't hesitate to start a topic on that.

...

I guess that with "SC coordinates" you meant spherical coordinates. If so, I'm mystified by your comment; surely we all agree that such coordinates only are preferred for mathematical convenience, in order to easily make precise calculations. That is why everyone including Hamilton uses them; it's irrelevant for the topic at hand.

In reverse order, by SC coordinates I mean the Schwarzschild form of metric as opposed to:

- GP coordinates
- Lemaitre Coordinates
- Kruskal coordinaes
- Eddington-Finkelstein coordinates
- etc. etc.

All of these are spherical in the sense of having a radial type of coordinate and a theta,phi part of the metric. All of these describe the same geometry, and are connected by coordinate transformation.

On the 1393 Einstein paper, I cannot read more of it because your link only allows reading one page. From that page, Einstein is only arguing about the physical plausibility of the formation of a BH, not the interpretation of SC geometry as a solution of the equations. Further, let's note he died 20 years before completion of the singularity theorems. He was not one to reject mathematical proofs.

The Oppenheimer-Snyder solution does have an EH and a singularity. It also has the feature that evidence for this never reaches a distant observer. Exactly what to make of this, and whether the result was in any way general, took time to work out. However, every major feature of the modern view of black holes in classical GR was present and computable in this solution. For example, right from their abstract: "The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day"

 

[TrickyDicky]

Disagree. In classical GR, without adding anything about BH entropy (which is quantum), any solution time reversed is also a solution. This is trivial to show.

Sure, if this was what you were trying to convey by the post I quoted from you, then we agree. But I didn't understand that.
To me the time symmetry is specified by the existence of a time-like Killing vector. You can find it in GR static solutions or in Newtonian mechanics, but it is not a general feature of all GR solutions.