Are flowing space models compatible with GR?

[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.

 

[harrylin]

In reverse order, by SC coordinates I mean the Schwarzschild form of metric [..]

Thanks for the clarification!

On the 1393 Einstein paper, I cannot read more of it because your link only allows reading one page. [..] The Oppenheimer-Snyder solution [..] Exactly what to make of this, and whether the result was in any way general, took time to work out. [..] 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"

Probably you need an institutional license for both papers (alternative is the library). And I already summarized the gist of both. Once more: their solution is that what for a non-infalling observer (a so-called "distant" observer) takes infinite time, corresponds to very little proper time for an infalling observer. Einstein concluded that "the Schwartzschild singularities don't exist in physical reality". I have no problem with those conclusions at all; the issue here came from our opinion that Hamilton's model disagrees with such Schwartzschild-based results, which I assume to be compatible with GR.
In this thread we are scrutinizing Hamilton's model, so I won't discuss more comments on the context of this thread in this thread.

 

[PAllen]

Once more: their solution is that what for a non-infalling observer (a so-called "distant" observer) takes infinite time, corresponds to very little proper time for an infalling observer. Einstein concluded that "the Schwartzschild singularities don't exist in physical reality". I have no problem with those conclusions at all; the issue here came from our opinion that Hamilton's model disagrees with such Schwartzschild-based results, which I assume to be compatible with GR.
In this thread we are scrutinizing Hamilton's model, so I won't discuss more comments on the context of this thread in this thread.

Ok, on this core issue, there is no disagreement between Hamilton's model and Schwarzschild conclusions:

- a key point is that your infinite time, for the umpteenth time, is coordinate time, as recognized right at the beginning of Einstein's 1939 paper. Coordinate time is not a physical observervable.

- Both SC metric and Hamilton river model agree that a free faller crosses horizon in finite physical time (time on their clock).

- Both SC metric and Hamilton river model agree that no light, signal, or causal influence can propagate from the horizon or inside to an external observer (while causal influences can freely propagate in the other direction - distant to interior).

It is impossible for them to disagree on any invariant, and all observables are invariants.

 

[Mentz114]

... and all observables are invariants.

We can measure components of tensors ( like x,y,z velocities and tidal forces) which are not invariant but frame dependent. Of course you could mean something different by 'observable'.

 

[PAllen]

We can measure components of tensors ( like x,y,z velocities and tidal forces) which are not invariant but frame dependent. Of course you could mean something different by 'observable'.

This was discussed at length in some threads by Ben Niehoff. A measurement of tensor components is a really a contraction of the tensor with the frame basis of specified world lines (of the instruments). It is thus invariant. In any coordinates or even no coordinates, once you have specified the instruments (thus physically chosen basis), you get the same results for the computation.

 

[Mentz114]

This was discussed at length in some threads by Ben Niehoff. A measurement of tensor components is a really a contraction of the tensor with the frame basis of specified world lines (of the instruments). It is thus invariant. In any coordinates or even no coordinates, once you have specified the instruments (thus physically chosen basis), you get the same results for the computation.

Basis dependent invariants. I can see what you mean. But the result still depends on the choice of instrument basis so it's not invariant in the sense that a scalar like charge (say) is.

Anyhow, thanks for the clarification of 'observable'.

 

[PAllen]

Basis dependent invariants. I can see what you mean. But the result still depends on the choice of instrument basis so it's not invariant in the sense that a scalar like charge (say) is.

Anyhow, thanks for the clarification of 'observable'.

Yes, not like a scalar field. However, a contraction produces a scalar invariant, and a physical measurement in GR is modeled as one or more contractions. This is really just the mathematical implementation of the required goal that changing coordinates you compute in must not change the result of any measurement. Also, that all observers see a given instrument reading the same way (No case of: I see it reading an B field of strength 3 along its axis; you see it reading no B-field. You may know that your instrument will read an E field and no B field, but you don't see my instrument reading any differently than I see it).

 

[Mentz114]

Yes, not like a scalar field. However, a contraction produces a scalar invariant, and a physical measurement in GR is modeled as one or more contractions. This is really just the mathematical implementation of the required goal that changing coordinates you compute in must not change the result of any measurement. Also, that all observers see a given instrument reading the same way (No case of: I see it reading an B field of strength 3 along its axis; you see it reading no E-field. You may know that your instrument will read an E field and no B field, but you don't see my instrument reading any differently than I see it).

Oh I absolutely understand that everyone must agree on what a certain instrument reads.
But different observers instruments may disgree.

 

[PAllen]

Oh I absolutely understand that everyone must agree on what a certain instrument reads.
But different observers instruments may disgree.

Right. And each instrument's measurements may be formulated as contractions of a tensor field and vectors derived from the instrument world line(s), producing one or more scalars. This is what guarantees that different observers and coordinate systems may explain a given instrument's reading differently, but the result will never difffer.

That is, I may say: your instrument measures a B field because it is moving relative to the E field; you say: there is a B field that my instrument (at rest) measures. Mathemetically, the measurement in either frame is characterized by contraction of a tensor field and vectors derived from the instrument.

 

[Mentz114]

Right. And each instrument's measurements may be formulated as contractions of a tensor field and vectors derived from the instrument world line(s), producing one or more scalars. This is what guarantees that different observers and coordinate systems may explain a given instrument's reading differently, but the result will never difffer.

That is, I may say: your instrument measures a B field because it is moving relative to the E field; you say: there is a B field that my instrument (at rest) measures. Mathemetically, the measurement in either frame is characterized by contraction of a tensor field and vectors derived from the instrument.

Everything after the word 'right' is a repetition of what I've already agreed. I'm not trying to argue or disagree.

 

[harrylin]

Here are a few things that may need more elaboration. PAllen, sorry if I did not make the subject matter clear enough for you:

"In this model, space flows like a river through a flat background, while objects move through the river according to the rules of special relativity. " - Hamilton

We are here not discussing a precise transformation equation (which is necessarily only valid for a particular case) but a physical model, which pervect presumed to be a serious one.
Equations fail with a little change of situation, while a physical model accommodates itself to the changed situation. Zonde understood this subtle point of the topic, see my post #29 where I explained this, and his post #38.

A correct, functional physical model does not require tinkering with mapping, although one can make of course maps from it. Which brings me to the next point:

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.
[..]
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.

I already stated, without really explaining, that Hamilton puts the equivalence principle on its head. I have been giving examples of how Hamilton's model is inconsistent with any GR reference system - including free fall - from post #29 onward. I'll try to explain it better by building on my earlier illustration in post #56.

First, let's recall GR's equivalence principle; certainly the free fall case is valid and standard use of GR. And no Einstein free fall reference system will break either the law of inertia nor the law of local constancy of the velocity of light.

My illustration of yesterday does not sufficiently "catch" what is going on in Hamilton's model. It comes closer if we take a second Mercator projection, put that one its head, and stick it to the first one, with the North poles against each other. Now the description looks like something much more fancy, as the traveller from the equator to the North pole and on to the equator on the other side does not need to make bends anymore. Of course the resulting patchwork is still not a globe and neither is it really a Mercator projection anymore. And it still has a velocity discontinuity at the North pole, but some people might not notice.
A true Mercator projection is centred around the area of interest, and admits that it provides a distorted perspective except very close to the centre. It is similar with the equivalence principle, which knows no flowing space. I gave the example of a hole in the Earth, or in any heavy piece of matter: Hamilton's model is in several ways a fake Einstein free fall frame, and it breaks GR's laws of inertia and local speed of light. This is why I concluded that not only in spirit but also in action, the model does not explain GR; it is its antithesis. It is not compatible with GR in that sense, even though it gives for a number of cases the right answers.

 

[PAllen]

Here are a few things that may need more elaboration. PAllen, sorry if I did not make the subject matter clear enough for you:

"In this model, space flows like a river through a flat background, while objects move through the river according to the rules of special relativity. " - Hamilton

We are here not discussing a precise transformation equation (which is necessarily only valid for a particular case) but a physical model, which pervect presumed to be a serious one.
Equations fail with a little change of situation, while a physical model accommodates itself to the changed situation. Zonde understood this subtle point of the topic, see my post #29 where I explained this, and his post #38.

In this case, the whole model is special case. Hamilton not only admits this, but derives it in has paper. Every part of:

"In this model, space flows like a river through a flat background, while objects move through the river according to the rules of special relativity.

applies only to two special geometries. If you want to be pejorative, the whole river model is a trick for understanding perfectly ideal black holes (or regions of spacetime that match a portion of such - e.g. outside the earth, but not on or in the earth) - and for nothing else in GR. This is why few authors on relativity besides Hamilton (maybe Visser, on occasion?) bother with it.

 

[PAllen]

My illustration of yesterday does not sufficiently "catch" what is going on in Hamilton's model. It comes closer if we take a second Mercator projection, put that one its head, and stick it to the first one, with the North poles against each other. Now the description looks like something much more fancy, as the traveller from the equator to the North pole and on to the equator on the other side does not need to make bends anymore. Of course the resulting patchwork is still not a globe and neither is it really a Mercator projection anymore. And it still has a velocity discontinuity at the North pole, but some people might not notice.
A true Mercator projection is centred around the area of interest, and admits that it provides a distorted perspective except very close to the centre. It is similar with the equivalence principle, which knows no flowing space. I gave the example of a hole in the Earth, or in any heavy piece of matter: Hamilton's model is in several ways a fake Einstein free fall frame, and it breaks GR's laws of inertia and local speed of light. This is why I concluded that not only in spirit but also in action, the model does not explain GR; it is its antithesis. It is not compatible with GR in that sense, even though it gives for a number of cases the right answers.

Most of what you say genuinely makes no sense to me, specifically, what it has to do with the river model.

The river model does not have any violation of local speed of light or inertia that I can see. Light moves through the river at all times with local speed of c relative to the co-moving observer at that point. The river itself, which is a non-observable abstraction, moves at speed > c relative to infinity; but > c coordinate speeds are actually quite common in GR.

I definitely don't see where inertia is violated since SR applies in comoving frame; and each comoving frame is a 'free fall from infinity' frame.

The real break down of the river model is that it wants to describe the motion of the river against a flat background, governed by Newtonian laws (then apply SR at a local frame at each point in the river, and boosts based on the difference in river velocity between points). This conceit can only be made to match GR for the very special geometries described in the paper. [As admitted and proven in Hamilton's paper.]

 

[zonde]

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.

I am not going to respond to all your statements in this post as it will lead us away from topic at hand. So just one thing now.

When we speak about gravity of gravitating body we always view it in the rest frame of gravitating body.

Do you agree?

 

[PAllen]

I am not going to respond to all your statements in this post as it will lead us away from topic at hand. So just one thing now.

When we speak about gravity of gravitating body we always view it in the rest frame of gravitating body.

Do you agree?

No, not at all. What if there are two or three similar gravitating bodies in mutual motion? Then, in Newtonian physics, you might view it in the COM frame, but it wouldn't by any simpler than any inertial frame (in the Newtonian sense). In GR, for such a scenario, there is no such thing as COM frame. You pick any reasonable coordinates (not frame, since the problem isn't local, and frames are strictly local in GR).

It seems you cannot let go of the idea that there really are preferred frames and preferred coordinates. That is the antitheses of relativity.

 

[zonde]

It seems you cannot let go of the idea that there really are preferred frames and preferred coordinates.

Yes indeed.
In SR they are frames where speed of light is isotropic (and has particular value).

That is the antitheses of relativity.

From Einstein's 1905y SR paper:
"... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate ..."
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good."

I think you are a bit confused.

 

[PAllen]

Yes indeed.
In SR they are frames where speed of light is isotropic (and has particular value).From Einstein's 1905y SR paper:
"... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate ..."
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good."

I think you are a bit confused.

That is SR. I agree for SR. The issue is GR. Einstein considered these features of SR a defect. In GR there are no global inertial frames at all. In Einstein's view, general covariance in GR removed these defects (irrespective of whether it was a 'theory filter' - which in 1917, he conceded to Kretschmann that it was not).

Further, in GR, SC coordinates are as far as you can imagine from being an 'inertial frame'. As I mentioned, lines of constant r have proper acceleration approaching infinity near the EH. It is actually GP coordinates that are built around a family of inertial frames. One sign of this is that 3-spaces of constant GP time are flat, Euclidean space; 3-spaces of constant SC time are non-Euclidean.

 

[harrylin]

[..]
The river model does not have any violation of local speed of light or inertia that I can see. Light moves through the river at all times with local speed of c relative to the co-moving observer at that point. [..]
I definitely don't see where inertia is violated since SR applies in comoving frame; and each comoving frame is a 'free fall from infinity' frame.

I gave the map example but forgot to mention that of course no local observer on a Mercator projection notices anything strange, perhaps not even when crossing the patchwork boundary between two projections. Similarly I see nothing noticeable with the local Hamilton frame from the local frame's perspective. The defects are noticed in the mapping from the other frame ("the distant perspective") in Hamilton's model. GR demands that both descriptions obey the laws of nature that I mentioned earlier.

The real break down of the river model is that it wants to describe the motion of the river against a flat background, governed by Newtonian laws (then apply SR at a local frame at each point in the river, and boosts based on the difference in river velocity between points). This conceit can only be made to match GR for the very special geometries described in the paper. [As admitted and proven in Hamilton's paper.]

That's also a good point! Patchwork like that may be expected to have multiple failures.

 

[harrylin]

While I think that we have covered the topic well enough by now, there are a few loose ends:

[..] When we speak about gravity of gravitating body we always view it in the rest frame of gravitating body.

Do you agree?

I disagree, but of course everyone will agree that such a rest frame is a valid, relevant and often very convenient perspective.

Yes indeed.
In SR they are frames where speed of light is isotropic (and has particular value).

Similarly, in GR the speed of light in a small enough region is constant.

From Einstein's 1905y SR paper:
"... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate ..."
"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good." [..]

Of course, he meant it then to indicate a class of reference systems; and he extended that class with GR such that any form of motion became "relative" (the modern point of view is not unanimously in favour of that, see the physics FAQ, "Twin paradox"; but that is not in question in this thread). In Einstein's GR the slightly modified laws of mechanics - including the unmodified law of inertia - are valid "locally" in accelerating and non-accelerating reference systems.

 

[PAllen]

I gave the map example but forgot to mention that of course no local observer on a Mercator projection notices anything strange, perhaps not even when crossing the patchwork boundary between two projections. Similarly I see nothing noticeable with the local Hamilton frame from the local frame's perspective. The defects are noticed in the mapping from the other frame ("the distant perspective") in Hamilton's model. GR demands that both descriptions obey the laws of nature that I mentioned earlier.

No, that's not right. GR make no such demands, and cannot. Coordinate speed of light varies from c (either greater or lesser) in almost all coordinates in GR, including SC coordinates (in fact, I think it can be proven that in general spacetimes it is impossible to establish coordinates where the coordinate speed of light is c everywhere, in all directions). There is no such thing in GR as a 'distant perspective'. The belief that coordinate quantities SC coordinates represent physical characteristics of a 'distant perspective' is a classic error of giving physical meaning to coordinate quantities. In any coordinates, you get predictions about what any observer measures or sees about distant events by computing invariants as I have explained in other posts here. Any and all coordinates are tools to that end. You prefer one coordinate system over another primarily because it makes some class of calculations easier. In the case of SC geometry, each of the popular coordinates makes different cases easier to calculate or visualize.

That's also a good point! Patchwork like that may be expected to have multiple failures.

It is more of design limit than a failure. If I say: for x > 0, f(x)=x^2 is a bijection, do you respond: Well that fails if x allowed to be any real number? That's silly because it contradicts the hypothesis. Hamilton's paper and the river model are making the statement:

For SC or Kerr-Newman geometry (and no others), you can use this model to make exact predictions and visualize behavior.

It is then equally silly to respond: well, for other spacetimes you can't. You can argue (and this is my general opinion), that its limited applicability make it not a particularly useful method. Hamilton may argue (correctly) that most situations in astronomy are characterized by large regions where the river model applies to a good approximation, and it is easy to know when to not apply it. It's not a right/wrong decision.

 

[zonde]

That is SR. I agree for SR. The issue is GR. Einstein considered these features of SR a defect. In GR there are no global inertial frames at all. In Einstein's view, general covariance in GR removed these defects (irrespective of whether it was a 'theory filter' - which in 1917, he conceded to Kretschmann that it was not).

Where did you get that Einstein considered SR defective?
SR certainly is not defective. It has limited applicability. And GR is meant to overcome this limit. But that means one important thing - GR should reduce to SR at appropriate limits.
And that in turn means that if two different coordinate systems when reduced (under appropriate limits) to SR give coordinate systems that are not related by Lorentz transform then they describe two different physical situations.

So we can try to compare coordinate systems in that sense to find out if they are equal.

Would you still say that this somehow goes against Einstein's view?

One sign of this is that 3-spaces of constant GP time are flat, Euclidean space; 3-spaces of constant SC time are non-Euclidean.

What argumentation you can provide for this statement?