# Are flowing space models compatible with GR?

#### harrylin

Are "flowing space" models compatible with GR?

Recently papers have been published with new "flowing space" models for GR. In particular a "flowing river" model by Hamilton:
http://ajp.aapt.org/resource/1/ajpias/v76/i6/p519_s1? [Broken]
http://arxiv.org/abs/gr-qc/0411060

"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. "

I had the impression, years ago, that such models had been disproved because of giving predictions that do not match GR nor experiment; but perhaps this new one is different in a subtle way that escapes me.
In Einstein's GR, space, although of free choice, is taken as stationary reference, relative to which bodies are moving; it's a bit surprising for me if GR is equally compatible with a flowing space model in which a kind of ether flows like a river or waterfall relative to space.

Before starting a test example I'd like to be sure to understand it correctly:

- although Hamilton applies the model to black holes, it should work in general (such as near the Earth) if valid
- he pictures gravitation like an ether flow towards the mass
- an object at rest in the river is entirely unaffected by the flow

Is that correct?

This topic came up in another thread:
[..] Hamilton's river model's are [..] just a conceptual aid. They change not a single equation or rule for computing an observable.
That is for me the question!

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#### PAllen

Re: Are "flowing space" models compatible with GR?

As I read this paper it is just about providing a conceptual interpretation of GP coordinates for spherically symmetric spacetime, and the Doran metric for Kerr-Newman spacetime. Everything here is interpretation of a geometry via particular coordinates and associate metric; more specifically, what these coordinates say about the local measurements of a particular class of observers. Since the geometries are well known exact solutions of GR, the coordinates and associated metrics are well known, unless there are gross mathematical errors, I don't understand how one can even talk about 'different predictions' or 'right versus wrong'. There is only the question of helpful vs. non-helpful, which is personal choice.

#### Dale

Mentor
Re: Are "flowing space" models compatible with GR?

To further what PAllen said, my understanding is that this specific interpretation is limited to the coordinates mentioned above. It does not necessarily apply to other spacetimes in general. So I don't think that you can consider this model to be any kind of an alternative to GR, just a nice conceptual aid for certain coordinates.

#### harrylin

Re: Are "flowing space" models compatible with GR?

As I read this paper it is just about providing a conceptual interpretation of GP coordinates for spherically symmetric spacetime, and the Doran metric for Kerr-Newman spacetime. Everything here is interpretation of a geometry via particular coordinates and associate metric; more specifically, what these coordinates say about the local measurements of a particular class of observers. Since the geometries are well known exact solutions of GR, the coordinates and associated metrics are well known, unless there are gross mathematical errors, I don't understand how one can even talk about 'different predictions' or 'right versus wrong'. There is only the question of helpful vs. non-helpful, which is personal choice.
The issue is GR-compatible interpretation; does that mean YES to my three questions?

#### harrylin

Re: Are "flowing space" models compatible with GR?

To further what PAllen said, my understanding is that this specific interpretation is limited to the coordinates mentioned above. It does not necessarily apply to other spacetimes in general. So I don't think that you can consider this model to be any kind of an alternative to GR, just a nice conceptual aid for certain coordinates.
Then it is supposed, as I assumed, to work with the Earth's field - right?

#### Dale

Mentor
Re: Are "flowing space" models compatible with GR?

Yes, neglecting any small deviations from the ideal symmetry.

#### PAllen

Re: Are "flowing space" models compatible with GR?

The issue is GR-compatible interpretation; does that mean YES to my three questions?
I don't understand what you mean by GR-compatible interpretation. If equations and predictions are the same, what would be a non-GR compatible interpretation?

Yes, to the first of your questions. Any spacetime that can be sufficiently closely modeled by SC geometry or Kerr-Newman geometry can use this interpretation. Near earth qualifies, as does near sun.

I refuse to answer the second because I don't sufficiently understand what you mean by aether; Hamilton makes one aside reference to this in the text, and refers to one paper in the notes. Thus, I don't know the overalap between his and your concepts.

On the third question: yes for a sufficiently small object. The model, as I understand it, accounts for tidal forces by different river velocity in different places. So an object of any finite size will undergo tidal stresses, as required by GR.

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#### PeterDonis

Mentor
Re: Are "flowing space" models compatible with GR?

In Einstein's GR, space, although of free choice, is taken as stationary reference, relative to which bodies are moving
What (I think) you mean by "space" here is not what the river model means by "space" as in "space is flowing into the black hole." The "space" in the sense of "the spatial coordinates in the Painleve chart" is not flowing inward; a given radial coordinate r refers to the same 2-sphere (i.e., the same "distance from the hole") at all times. The "space" that is flowing inward is something different, a "space" constructed from the frame fields of Painleve observers.

- although Hamilton applies the model to black holes, it should work in general (such as near the Earth) if valid
Yes. It works for the exterior vacuum region around any static, spherically symmetric gravitating body. Of course, in the case of the Earth, the inward "river velocity" never gets anywhere close to the speed of light.

- he pictures gravitation like an ether flow towards the mass
Kinda sorta; the term "ether flow" may have lots of undesirable connotations.

- an object at rest in the river is entirely unaffected by the flow
Except in so far as the flow "carries" it inward.

That is for me the question!
The river model is an *interpretation* of GR, not a different theory. (More specifically, it's an interpretation to help you visualize the gravity of a static, spherically symmetric body, based on using Painleve coordinates to describe the field.) All of the equations are the same as for Painleve coordinates (since they *are* the equations of Painleve coordinates), and all the observables are the same as GR (since they *are* the observables predicted by GR).

#### pervect

Staff Emeritus
Re: Are "flowing space" models compatible with GR?

I'm not terribly fond of the "flowing river of space-time" model that Hamilton has. But - it is a better alternative than the time stops at the event horizon" model. Which I gather was his main intention. (I had an opportuity to talk to him over a wiki article some time back. As I recall he complained he had some trouble getting it published).

The main reason I'm not fond of the model is that there isn't any way to build a "space-flow-o-meter" to detect flowing space-time. So it's really more of a mental crutch or visual aid, not something you can measure. It's probaby a better visual aid than "stopping time" though.

#### Pierre007080

Re: Are "flowing space" models compatible with GR?

As a "popular physicist" I find the Flowing Space model extremely useful. It ties in with Einstein's insistence that the free falling frame is inertial. With the modern developments ascribing energy and various fields to "vacuum" I cannot understand the way everybody treads so lightly around the aether issue.
The concept that EVERYTHING (including space with it's structural fields, Higgs field, dark matter and dark energy) is being attracted by a massive body and all moving at the same velocity seems a fact that cannot be reasoned away and a better place to start understanding gravity than the complicated equations of GR that only a few can make sense of.
What are the simplistic issues that disproves this model? Be gentle Guys!

#### harrylin

Re: Are "flowing space" models compatible with GR?

I don't understand what you mean by GR-compatible interpretation. If equations and predictions are the same, what would be a non-GR compatible interpretation?

Yes, to the first of your questions. [..] Near earth qualifies, as does near sun.
[..] the second because I don't sufficiently understand what you mean [..]
On the third question: yes for a sufficiently small object. [..]
Thanks. My purpose with this thread is to understand how such a model can give GR predictions, by means of a simple example in a post later today. For the second question, PeterDonis gives some valuable feedback:
What (I think) you mean by "space" here is not what the river model means by "space" as in "space is flowing into the black hole." The "space" in the sense of "the spatial coordinates in the Painleve chart" is not flowing inward; a given radial coordinate r refers to the same 2-sphere (i.e., the same "distance from the hole") at all times. The "space" that is flowing inward is something different, a "space" constructed from the frame fields of Painleve observers. [..]
It works for the exterior vacuum region around any static, spherically symmetric gravitating body. Of course, in the case of the Earth, the inward "river velocity" never gets anywhere close to the speed of light. [..]
This may be the issue; however as you next formulate it, sounds exactly as I understood it.

To elaborate, I understood it as an imaginary inward flow relative to a static coordinate background, such that SR can be applied relative to that imaginary inflowing medium. According to an observer on Earth the total effect is then the combined effect of "inflow" plus SR effects relative to the "river".

#### harrylin

Re: Are "flowing space" models compatible with GR?

I'm not terribly fond of the "flowing river of space-time" model that Hamilton has. But - it is a better alternative than the time stops at the event horizon" model. Which I gather was his main intention. [..]
It's exactly that issue that triggered this thread; for me it is evident that the two models are first of all incompatible as interpretation goes, and I don't get how they can give the same predictions.
Einstein's model gives that, as interpreted from a distant system, the resonance frequency of an object in free fall near a black hole goes to zero. As a matter of fact, Einstein's GR corresponds to a modified Lorentz ether (as long as one stays away from extremes such as black holes).
In contrast, Hamilton's "flowing river" model gives that, as interpreted from a distant system, the resonance frequency of an object in free fall near a black hole is completely unaffected (correct?). If the river model is like I think, then it can't work and the experts are stupid for not having noticed (or more nicely put: I would be much smarter than them). Much more likely is that I misunderstand it, and I want to know what.
As a "popular physicist" I find the Flowing Space model extremely useful. It ties in with Einstein's insistence that the free falling frame is inertial. [..] The concept that EVERYTHING (including space with it's structural fields, Higgs field, dark matter and dark energy) is being attracted by a massive body and all moving at the same velocity seems a fact that cannot be reasoned away and a better place to start understanding gravity than the complicated equations of GR that only a few can make sense of.
What are the simplistic issues that disproves this model? Be gentle Guys!
What Einstein argued is not at all like a free falling space if I understand what Hamilton means with that. It's a bit subtle; for example Hamilton's "river" can exceed the speed of light near a black hole in our galaxy. But I think that it's time to grab the bull by the horn and discuss a simple example (next post; I now came up with an extremely simple test case, but later today as I must work now).

PS. For a really simple model that doesn't match GR exactly (but it should match all experiments so far), see my new thread here: https://www.physicsforums.com/showthread.php?t=647616

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#### PAllen

Re: Are "flowing space" models compatible with GR?

It's exactly that issue that triggered this thread; for me it is evident that the two models are first of all incompatible as interpretation goes, and I don't get how they can give the same predictions.
Einstein's model gives that, as interpreted from a distant system, the resonance frequency of an object in free fall near a black hole goes to zero. As a matter of fact, Einstein's GR corresponds to a modified Lorentz ether (at least far away from black holes).
In contrast, Hamilton's "flowing river" model gives that, as interpreted from a distant system, the resonance frequency of an object in free fall near a black hole is completely unaffected (correct?). If the river model is like I think, then it can't work and the experts are stupid for not having noticed. Much more likely is that I misunderstand it, and I want to know what.
Somehow, you need to be more precise about what incompatibility you see (perhaps in your upcoming post). As I see it, we have the same solution under discussion (the SC geometry). For this, we have multiple coordinates we can place on it (like rectilinear and polar on a flat plane). For one of these coordinates (GP coordinates), you can use an analogy to describe the experience of specific class of local frames as flowing in a river, and describe other local frames in relation to these (via SR). There is only one set of physical laws involved: the EFE globally; SR locally. I remain unable to conceive of conflict.

Maybe the key is here: "Einstein's model gives that, as interpreted from a distant system, the resonance frequency of an object in free fall near a black hole goes to zero. As a matter of fact, Einstein's GR corresponds to a modified Lorentz ether (at least far away from black holes)"

This is wrong. Or, at best, it takes approximate treatment of one solution as the whole theory. Only in your mind have I ever seen Einstein so maligned as to confuse an approximate treatment of a special case of his own theory as the whole thing.

What you say about distant regions is an approximation. Useful, perhaps, but not = to the theory as a whole.

What you say about resonant frequency is nonsense. Even for earth, we don't talk about the resonant frequency of hydrogen in a valley being different from the resonant frequency on a mountain top. We say they are locally unchanged. However, hydrogen emissions, decay rates etc. for an object in the valley observed from a mountain are reduced. This is of the same character as Doppler, and, in fact, the mathematical basis of all Doppler in GR is the same [I have described the math to you in another thread; I will repeat here if you request]. Treated in an exact manner, there are not two (or three) types of Doppler, but only one. Separating them is purely a computational convenience for special cases. Special case does not equal general case.

To sum:

1) GR, in no form, or coordinates, says the resonant frequency of hydrogen changes as it approaches an EH. The axiom of 'locally SR physics' is built into the mathematical framework of GR (that the tangent plane at every point has Minkowski metric).

2) There is no upper bound to Doppler factor between source and target, and in a variety of ways you can get infinite Doppler, and horizons: a uniformly accelerating rocket will have a horizon with infinite Doppler. A distant observer will see infinite Doppler for an object approaching an EH. There is a great similarity between these two cases. In both cases, that one observer (accelerating rocket; observer away from BH) sees a horizon has no bearing on what a different observer sees (in both cases, the 'horizon observer' observer has no specific awareness of the horizon seen by the other, and sees no sudden change in light frequency as they cross the horizon; this fact can be derived in all coordinates, even SC coordinates using limiting arguments).

3) There is no difference in character between mountain to valley Doppler on earth versus approach to an EH. There is only difference in degree. We don't require that that valley dweller consider time to be truly slower for them.

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#### PAllen

Re: Are "flowing space" models compatible with GR?

What Einstein argued is not at all like a free falling space if I understand what Hamilton means with that. It's a bit subtle; for example Hamilton's "river" can exceed the speed of light near a black hole in our galaxy. But I think that it's time to grab the bull by the horn and discuss a simple example (next post; I now came up with an extremely simple test case, but later today as I must work now).
The "river" exceeding the speed of light is analogous to an expanding universe solution with > c recession velocity between co-moving observers. In no way is it inconsistent with GR as Einstein understood it.

Again, it seems you have a personal theory (not just interpretation) that you derive from over-interpreting selected Einstein approximate computations for special cases; and you seem to believe this is the 'real GR' per Einstein.

#### PAllen

Re: Are "flowing space" models compatible with GR?

Let me be even clearer on what seems a key point:

Gravitational time dilation is a computed, coordinate feature that can be defined only in near static spacetime regions. It if very useful computationally, but is not a physical observable at all, ever.

What is an observable, and never requires the concept gravitational time dilation to compute, is Doppler. The Doppler factor in GR as in SR affects both signal frequency (and therefore, any other distantly observed clock rate as well) and wavelength.

Note that these statements apply almost verbatim to frame dependent time dilation versus Doppler in SR.

#### harrylin

Re: Are "flowing space" models compatible with GR?

Sorry, I made it too simple: originally I had in mind something like Gravity probe A, but now simplified it to a high tower with clocks.

However, with a high tower it is clear to me that such a river flow model should work: If I correctly understand it, in Hamilton's model the ground clock will have more SR time dilation than the top clock (correct?).

It should be an example that is more like gravity probe A but still simple enough for a forum discussion. I'll be back!

#### PAllen

Re: Are "flowing space" models compatible with GR?

Sorry, I made it too simple: originally I had in mind something like Gravity probe A, but now simplified it to a high tower with clocks.

However, with a high tower it is clear to me that such a river flow model should work: If I correctly understand it, in Hamilton's model the ground clock will have more SR time dilation than the top clock (correct?).

It should be an example that is more like gravity probe A but still simple enough for a forum discussion. I'll be back!
Correct.

If you are planning on bringing in rotation effects, you will need to understand the Doran metric part of Hamilton's article. I have not read this part through, myself yet. The 'river' gets complicated for the rotating case.

Also, recall Dalespam pointed out in #3: this is the limit of Hamilton's interpretation. He has not proposed any way to apply it to a more general scenario than an ideal rotating, massive gravitational source.

[edit: wait, it was probe B that was about rotational effects. Probe A was just a precise, scaled up, version of the tower scenario (using rocket and maser). I see it as exactly the same case. ]

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#### zonde

Gold Member
Re: Are "flowing space" models compatible with GR?

This is wrong. Or, at best, it takes approximate treatment of one solution as the whole theory. Only in your mind have I ever seen Einstein so maligned as to confuse an approximate treatment of a special case of his own theory as the whole thing.
Well you see, in SR any inertial frame is valid and it does not come into conflict with any other inertial frame. We can relate this with global one-to-one mapping.

This does hold for different coordinate maps for black holes.

And yet another thing is that these BH coordinate maps describe highly symmetric eternal objects i.e. their past is symmetric with their future. This is very special case. And basically not interesting as we believe that all BH have formed in finite past.
How many of these BH coordinate charts will fail miserably when applied to BH with finite past?

#### zonde

Gold Member
Re: Are "flowing space" models compatible with GR?

As I see the river model I don't see how it can work.

First to talk about river model we map GR curved spacetime to flat spacetime and introduce some medium who's properties cover up for effects described by curvature of spacetime.
So we have that some property (let's assume it is density) determines coordinate speed of light.
And as we talk about river model we have flux of that medium. But in order to have constant density we have to have dynamic equilibrium and that means that we have the same flux as we go closer and closer to gravitating object. And flux per surface unit increases as inverse square law.

So we have coordinate speed of light changing as a function c=k*(1/r1/2) while flux changes as k*(1/r2).
And I just don't really see how to connect these two function in physically meaningful way.

Please note that I don't say it's impossible. I just don't see a way.
Actually it feels like one possibility should still be explored. When we map curved spacetime to flat one we take coordinate r from GR coordinate charts and carry it over to flat coordinates i.e. we take projection of curved spacetime to flat one. But maybe we can use conformal map from curved spacetime to flat one? And that even seems mathematically more justified.

#### PAllen

Re: Are "flowing space" models compatible with GR?

Well you see, in SR any inertial frame is valid and it does not come into conflict with any other inertial frame. We can relate this with global one-to-one mapping.

This does hold for different coordinate maps for black holes.

And yet another thing is that these BH coordinate maps describe highly symmetric eternal objects i.e. their past is symmetric with their future. This is very special case. And basically not interesting as we believe that all BH have formed in finite past.
How many of these BH coordinate charts will fail miserably when applied to BH with finite past?
On your last question, none of them.

On your observations, I genuinely don't understand your point. What I was arguing against was apparent claim that free faller and distant observer represented two disconnected 'realities' rather than two equally valid observational points of view for the same overall universe. The SR analogy clearly favors the latter point of view.

#### PAllen

Re: Are "flowing space" models compatible with GR?

As I see the river model I don't see how it can work.

First to talk about river model we map GR curved spacetime to flat spacetime and introduce some medium who's properties cover up for effects described by curvature of spacetime.
So we have that some property (let's assume it is density) determines coordinate speed of light.
And as we talk about river model we have flux of that medium. But in order to have constant density we have to have dynamic equilibrium and that means that we have the same flux as we go closer and closer to gravitating object. And flux per surface unit increases as inverse square law.

So we have coordinate speed of light changing as a function c=k*(1/r1/2) while flux changes as k*(1/r2).
And I just don't really see how to connect these two function in physically meaningful way.

Please note that I don't say it's impossible. I just don't see a way.
Actually it feels like one possibility should still be explored. When we map curved spacetime to flat one we take coordinate r from GR coordinate charts and carry it over to flat coordinates i.e. we take projection of curved spacetime to flat one. But maybe we can use conformal map from curved spacetime to flat one? And that even seems mathematically more justified.
I don't understand how what you say above relate to the river model. The river model says the speed of light for frame carried with the flow is always c. Behavior for a local frame moving relative to a 'carried' frame is given my SR formulas.

The apparent slow down of light emitted near horizon as perceived by a distant observer comes from this light 'fighting the river flow' to get the the distant observer.

Can you explain what you mean about flux? I see nothing about this in Hamilton's paper.

Further, I note that this model really cannot be wrong because it is just geometry using GP coordinates; any observable comes out the same as SC coordinates by pure mathematical construction. Please don't bring into this thread your rejection coordinate independence of GR observables.

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#### harrylin

Re: Are "flowing space" models compatible with GR?

As I see the river model I don't see how it can work.[..]
Please note that I don't say it's impossible. I just don't see a way.
Actually it feels like one possibility should still be explored. When we map curved spacetime to flat one we take coordinate r from GR coordinate charts and carry it over to flat coordinates i.e. we take projection of curved spacetime to flat one. But maybe we can use conformal map from curved spacetime to flat one? And that even seems mathematically more justified.
I almost fully agree, but for a different reason which I will explain with a simple thought experiment. Einstein's GR is compatible with a modified Lorentz ether model just as he explained; with such a model one can use for experiments on Earth the ECI frame and SR, plus GR corrections for height. That is exactly the kind of mapping that Gravity probe A used and also what GPS uses for its satellites; it is not something that "should still be explored". Thus it was paradoxical for me that a kind of flowing ether model could give the same results. Of course, for example a voltage source can be replaced by a current source with the same effects, so I searched if I could come up with an example thought experiment where the effect is different according to my understanding; and I now came up with such a test case.

Please be a little patient as I have a life with a job and I want to do better than most posters who come with such questions by making it as simple as possible (without making it too simple, which I did yesterday) and plugging in numbers with (IMHO) correct calculations.

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#### zonde

Gold Member
Re: Are "flowing space" models compatible with GR?

I don't understand how what you say above relate to the river model. The river model says the speed of light for frame carried with the flow is always c. Behavior for a local frame moving relative to a 'carried' frame is given my SR formulas.

The apparent slow down of light emitted near horizon as perceived by a distant observer comes from this light 'fighting the river flow' to get the the distant observer.

Can you explain what you mean about flux? I see nothing about this in Hamilton's paper.
Yes, the things about density where out of place. Sorry for my error.
But let me fix my error. Let's say we want to find out at at what speed the river is flowing. As time dilation is related to speed of river flow we can write time dilation as function of that speed using SR:
$$d=\sqrt{1-\frac{v^2}{c^2}}$$
But GR tells us what is time dilation as function of distance from gravitating body:
$$d=\sqrt{1-\frac{r_0}{r}}$$
So we can write speed of the river flow as function of distance from gravitating body:
$$\sqrt{1-\frac{v^2}{c^2}}=\sqrt{1-\frac{r_0}{r}}$$
$$\frac{v^2}{c^2}=\frac{r_0}{r}$$
$$v=c\;\sqrt{\frac{r_0}{r}}$$
But because the surface that the river is flowing trough is reduced as we approach gravitating body it should be speeding up. And that increase should follow inverse square law as speed of the river flow should be inversely proportional to the surface it is flowing trough.
$$v=k\;\frac{1}{r^2}$$

Two functions are obviously different as one contains r-1/2 but the other one r-2. Hence it doesn't work.

#### PAllen

Re: Are "flowing space" models compatible with GR?

Yes, the things about density where out of place. Sorry for my error.
But let me fix my error. Let's say we want to find out at at what speed the river is flowing. As time dilation is related to speed of river flow we can write time dilation as function of that speed using SR:
$$d=\sqrt{1-\frac{v^2}{c^2}}$$
But GR tells us what is time dilation as function of distance from gravitating body:
$$d=\sqrt{1-\frac{r_0}{r}}$$
So we can write speed of the river flow as function of distance from gravitating body:
$$\sqrt{1-\frac{v^2}{c^2}}=\sqrt{1-\frac{r_0}{r}}$$
$$\frac{v^2}{c^2}=\frac{r_0}{r}$$
$$v=c\;\sqrt{\frac{r_0}{r}}$$
But because the surface that the river is flowing trough is reduced as we approach gravitating body it should be speeding up. And that increase should follow inverse square law as speed of the river flow should be inversely proportional to the surface it is flowing trough.
$$v=k\;\frac{1}{r^2}$$

Two functions are obviously different as one contains r-1/2 but the other one r-2. Hence it doesn't work.
This is nonsense. The river is not a material fluid following fluid flow laws. It is a flow of imaginary (figurative sense, not √ -1) space, and its laws are as derived in the paper. In this regard, your first calculation is correct, and is consistent with equation (2) of the paper. Your second is something you made up that is wrong. You have refuted a nonsensical straw man.

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#### PAllen

Re: Are "flowing space" models compatible with GR?

Einstein's GR is compatible with a modified Lorentz ether model just as he explained; with such a model one can use for experiments on Earth the ECI frame and SR, plus GR corrections for height. That is exactly the kind of mapping that Gravity probe A used and also what GPS uses for its satellites; it is not something that "should still be explored".
I am curious to see this claim of Einstein in context. I suspect I would interpret it quite differently from you. Do have a reference?

The rest of the context you are discussing is not GR as a whole, but one specific simple solution that is maximally Newtonian (e.g. can be modeled with an effective potential depending only on position). Over and over you seem to draw an equality between this one solution, and comments Einstein may have made in respect to this one solution, as if they were the full content of the theory.

As to the specifics of interpreting this solution via the river model, this solution is exactly the one it was first designed for (then extended, with some difficulty, to the rotating perfect BH case). I don't understand how you can read the paper and not see the exact mathematical equivalence. All you do is transform between GP coordinates and SC coordinates.

Do you really believe using different coordinates to compute observables (which are all defined as invariants) can produce a different result?