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## Are "flowing space" models compatible with GR?

 Quote by zonde 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.

 Quote by PAllen [..] 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.

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

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 Quote by harrylin 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.
 Quote by harrylin 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.

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 Quote by PAllen 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?

 Quote by PAllen 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?

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 Quote by PAllen 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).
Well yes, my statement does not seem quite right when considering system of two bodies where we take into account gravity of both bodies. And certainly we are considering such situations.

About your second statement you might want to look at this wikipedia article describing classical two-body problem
and how exact solution is found for this two-body problem by using COM frame.
"Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between the masses changes with time."

This seems to contradict your statement that COM frame has no special role in calculations (I assume this was the meaning behind word "simpler").

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 Quote by zonde 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.
Einstein was philosophically attracted to Mach's ideas. Ideally, he felt there should be no physical significance to anything except relative motion. The idea of distinguishable accelerated motion in an empty universe was abhorrent to him. He hoped that general covariance and his GR program would show that inertial resistance to acceleration arose from motion relative the distant mass of the universe. SR's preference for inertial frames bothered him and was one of several major motivators for his development of GR. [Einstein later realized GR failed in his Machian objective, but still succeeded, in his view, of displacing or, at least weakening, any special position for inertial frames.]

Of course GR reduces SR sufficiently locally everywhere. That is built into the mathematical structure of pseudo-riemannian geometry in the same way local Euclidean geometry is built into Riemannian geometry.
 Quote by zonde 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.
No, this is not a correct statement of the way GR contains SR. The relationship is local not global. In general, a GR solution has no global coordinates that resemble Minkowski coordinates at all. In a limited sense you can say that for asymptotically flat spacetimes (which, by the way, does not include our universe), there are coordinates systems that approach Minkowski at infinity. However, not only are the 'too many' of them, they are not generally related by Lorentz transforms. This whole statement of your is pretty much a complete misunderstanding of the relationship between SR and GR.

As an aside, you should be aware that the GP coordinates used in the river model reduce Minkowski coordinates at infinity, just like SC ones do. In fact they reduce to exactly the same coordinates at infinity because they share the same center of symmetry and both make explicit the asymptotic flatness of SC geometry.
 Quote by zonde 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?
Yes. The only place for Lorentz transforms in GR is local (in the limit of a small region of spacetime where curvature= tidal gravity can be ignored).
 Quote by zonde What argumentation you can provide for this statement?
It is actually mathematically obvious. The metric for GP coordinates as given in the paper in post #1 of this thread is:

ds^2 = - dt^2 + (dr + βdt^2)^2 + r^2(dθ^2 + sin^2θd$\varphi$^2)

For a slice of constant t, you have dt=0. Then you have, for the spacial geometry of the slice:

ds^2 = dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2)

which is just the flat Euclidean metric in polar coordinates. Interesting, wouldn't you say, for coordinates that have no horizon coordinate singularity and go smoothly through the horizon to the singularity? The horizon itself is there as a physical phenomenon, but there is no coordinate singularity there, and no infinite coordinate time there. Those are artifacts of Schwarzschild coordinates.

Meanwhile, for t=constant slice for Scwharzschild coordinates, using e.g. the form given in:

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

you get:

ds^2 = (1/(1-R/r)) dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2)

(where I am using R for SC radius). This is non-Euclidean spacial geometry.

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 Quote by zonde Well yes, my statement does not seem quite right when considering system of two bodies where we take into account gravity of both bodies. And certainly we are considering such situations. About your second statement you might want to look at this wikipedia article describing classical two-body problem and how exact solution is found for this two-body problem by using COM frame. "Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between the masses changes with time." This seems to contradict your statement that COM frame has no special role in calculations (I assume this was the meaning behind word "simpler").
Yes, there is advantage to COM frame for two body in Newonian mechanics, but not for 3 body, so far as I know. For GR, there is also a some simplification in the numerical treatment (no exact treatment) of the two body problem in effective COM coordinates (these are called 'effective one body methods' in the literature); again, none for the 3 body problem. So I concede I overstated the case a little for the two body situation.

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 Quote by PAllen No, this is not a correct statement of the way GR contains SR. The relationship is local not global.
Yes yes local, not global. Appropriate limit is small enough region where we are comfortable with discarding difference between flat SR coordinate system and GR coordinate system.

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 Quote by PAllen Einstein was philosophically attracted to Mach's ideas. Ideally, he felt there should be no physical significance to anything except relative motion. The idea of distinguishable accelerated motion in an empty universe was abhorrent to him. He hoped that general covariance and his GR program would show that inertial resistance to acceleration arose from motion relative the distant mass of the universe. SR's preference for inertial frames bothered him and was one of several major motivators for his development of GR. [Einstein later realized GR failed in his Machian objective, but still succeeded, in his view, of displacing or, at least weakening, any special position for inertial frames.]
I'm not sure you are interpreting it correctly. There of course is physical significance to accelerated motion but as I understand it he felt that there should be no absolute reason for that physical significance. And with that I tend to agree.
As I imagine it physical significance is acquired from some rather physical field. Einsteins idea would be that properties of this field are determined by distant stars. Have to say I don't feel exactly the same way. As I see there should be something more besides distant stars (some other state of physical matter). But I am saying this just to point out how far I am ready to go along Einstein's viewpoint (as I understand it) and not to start discussion about my viewpoint.

 Quote by PAllen It is actually mathematically obvious. The metric for GP coordinates as given in the paper in post #1 of this thread is: ds^2 = - dt^2 + (dr + βdt^2)^2 + r^2(dθ^2 + sin^2θd$\varphi$^2) For a slice of constant t, you have dt=0. Then you have, for the spacial geometry of the slice: ds^2 = dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2) which is just the flat Euclidean metric in polar coordinates. Interesting, wouldn't you say, for coordinates that have no horizon coordinate singularity and go smoothly through the horizon to the singularity? The horizon itself is there as a physical phenomenon, but there is no coordinate singularity there, and no infinite coordinate time there. Those are artifacts of Schwarzschild coordinates. Meanwhile, for t=constant slice for Scwharzschild coordinates, using e.g. the form given in: http://en.wikipedia.org/wiki/Schwarzschild_metric you get: ds^2 = (1/(1-R/r)) dr^2 + r^2 (dθ^2 + sin^2θd$\varphi$^2) (where I am using R for SC radius). This is non-Euclidean spacial geometry.
You have minor error in GP metric - an extra ^2. It sould be like this:
ds^2 = - dt^2 + (dr + βdt)^2 + r^2(dθ^2 + sin^2θdφ^2)

Anyways, this is interesting argument and it requires adequate answer. Well, I don't have one.
But it does not solve the question but rather deepens it (at least from my perspective) as it can't address objections that where raised in this thread against river model (btw I am not sure if you didn't understood my argument in post #50 or you simply wanted to examine it more thoroughly for possible flaws).

Speaking about Schwarzschild coordinates for me it seems like it would be more meaningful to speak about isotropic coordinates. But my guess would be that they still would not make simultaneity slice Euclidean. And I would like to make sure that they are physically equivalent to Schwarzschild coordinates.

With isotropic coordinates I mean the one you get from this metric:
$$ds^2=\frac{(1-\frac{\mu}{2r})^2}{(1+\frac{\mu}{2r})^2}dt^2-(1+\frac{\mu}{2r})^4(dr^2+r^2(d\theta^2+sin^2 \theta \; d\varphi^2))$$

 Quote by PAllen 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[..].
I did not suggest anything else!
[/QUOTE] There is no such thing in GR as a 'distant perspective'.[/QUOTE]
Well, that is terminology that others use, see for example Ben's "distant observer" in his book chapter on GR:
- http://www.lightandmatter.com/html_b...ch27/ch27.html
and, not to forget, Hamilton (maybe he influenced me despite everything):

Of course, in GR there are simply different coordinate systems; in the literature and discussions colourful names are given that are more informative than S and S'.

 [..]
For a last time: please stop discussing and criticising here other things than Hamilton's river model.
 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. [..]
I illustrated that the same can be said (and is rightly said) for the Mercator projection. [EDIT:] Interestingly, that projection maps a big distance to zero distance; the mapping disagreement is just so to say inverse as with Hamilton's model. I don't think that either is "an interesting argument" that "requires adequate answer".

Now, I really think that the river model has been sufficiently discussed, with this thread nearing 100 posts. For different reasons several of us here including myself do not like it very much and recognise that it doesn't work as a general physical model for GR. Anything else is for another thread.

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 Quote by zonde Speaking about Schwarzschild coordinates for me it seems like it would be more meaningful to speak about isotropic coordinates.
These work fine as long as you're only interested in the region at or outside the horizon. Isotropic coordinates don't cover the region inside the horizon. (They actually double cover the region outside the horizon: 0 < r < mu/2 and mu/2 < r < infinity cover the same region.)

 Quote by zonde But my guess would be that they still would not make simultaneity slice Euclidean.
You're correct, they don't, because of the extra factor in front of the spatial part of the metric, which depends on r.

 Quote by zonde And I would like to make sure that they are physically equivalent to Schwarzschild coordinates.
They are; why would you think they weren't?

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 Quote by zonde You have minor error in GP metric - an extra ^2. It sould be like this: ds^2 = - dt^2 + (dr + βdt)^2 + r^2(dθ^2 + sin^2θdφ^2)
 Quote by zonde Anyways, this is interesting argument and it requires adequate answer. Well, I don't have one. But it does not solve the question but rather deepens it (at least from my perspective) as it can't address objections that where raised in this thread against river model (btw I am not sure if you didn't understood my argument in post #50 or you simply wanted to
I don't think there is any argument to answer there. What I get from checking back is a repetition of an argument you've made many times that I view as trivially false and has been refuted dozens of times on threads here, and would be a waste to revisit. Unless I am mistaken, the reference argument is that there is such a thing as an 'illegitimate' coordinate transform that changes physics. This is mathematically equivalent to claiming there is something wrong with claiming limit x->∞ (x/x+1) = 1. I take it as an argument that the foundations of differential geometry are wrong. I am not, ever, interested in debating that.

If this is not your argument, I don't know what you are referring to. If it is the white hole argument, I don't see the point. White holes cannot form by any process (mathematical fact; must 'exist in the past' for no reason); "no one" believes they exist in our universe. So debating their exact properties is not very interesting. Matter can get from inside to outside a white hole, and from near the WH horizon to further away. It can get closer to a WH, taking infinite time (fighting the river) to reach the horizon. However, any idea you have about pile up is wrong - the River model, as with any use of SC geometry, when talking about matter moving we are talking about 'test bodies' - bodies of vanishingly small mass. If you are talking about significant mass, the geometry ceases to be SC geometry at all (you do not have static exterior geometry), and all analysis with the SC metric is incorrect (whether interpreted the river model, directly with GP coordinates, or with SC coordinates). So again, I saw no real argument there to respond to.
 Quote by zonde . Speaking about Schwarzschild coordinates for me it seems like it would be more meaningful to speak about isotropic coordinates. But my guess would be that they still would not make simultaneity slice Euclidean. And I would like to make sure that they are physically equivalent to Schwarzschild coordinates. With isotropic coordinates I mean the one you get from this metric: $$ds^2=\frac{(1-\frac{\mu}{2r})^2}{(1+\frac{\mu}{2r})^2}dt^2-(1+\frac{\mu}{2r})^4(dr^2+r^2(d\theta^2+sin^2 \theta \; d\varphi^2))$$
They would not make the slice Euclidean (however, the spacial slice is conformally flat, for these). They also have another issue: they only cover the exterior geometry. Compared to the regular SC coordinates, they only cover r ≥ Schwarzschild radius.

 To my own surprise, thanks to a parallel discussion, I may after all have have found a simple and enlightening way to distinguish between predictions by Hamilton's model and GR. Applicable to both the solar system and a black hole. A voyager spacecraft is allowed to free-fall towards the Sun or even a black hole. Looking in forward direction to the stars it sends intensity and Doppler shift information back home. - GR: more than "classical" increase; even towards infinity for approaching rs - Hamilton's model: "classical" increase, I think; towards double for approaching rs I base this on the requirement that even a "flowing river" cannot accumulate light in transit.

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 Quote by harrylin To my own surprise, thanks to a parallel discussion, I may after all have have found a simple and enlightening way to distinguish between predictions by Hamilton's model and GR. Applicable to both the solar system and a black hole. A voyager spacecraft is allowed to free-fall towards the Sun or even a black hole. Looking in forward direction to the stars it sends intensity and Doppler shift information back home. - GR: more than "classical" increase; even towards infinity for approaching rs - Hamilton's model: "classical" increase, I think; towards double for approaching rs I base this on the requirement that even a "flowing river" cannot accumulate light in transit.
Well, this is false. It is covered in the paper qualitatively in bullet 7 on p.16 of the paper. Quantitatively, there are two observations to be made:

- the demonstrated mathematical equivalence between river model and GP coordinates is
fully sufficient to establish the result is the same.

- You persist, when discussing quantitative predictions, on making up verbal interpretations of your own and not using the actual equations circa p.4 of the paper. Light must continuously change frames going 'up' the river, at each frame change being SR boosted by the change in β between the frames. Nothing 'classical' about this. It is, in fact, fairly well known that all GR redshifts (kinematic, 'gravitational', and cosmologic) can be modeled as incremental SR boosts in such a manner (I first came across a proof of this in J.L. Synge's 1960 book on GR). The ability to treat all GR redshift as kinematical (if desired) is actually far more general than the river model (completely general, in fact).

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 Quote by PAllen Unless I am mistaken, the reference argument is that there is such a thing as an 'illegitimate' coordinate transform that changes physics. This is mathematically equivalent to claiming there is something wrong with claiming limit x->∞ (x/x+1) = 1.
I don't follow you so I can't respond to that.

So I will try to go into more details about coordinate transformations. If we look at coordinate transformations we have different things. We have coordinate invariants that stay the same under any coordinate transformation. We have coordinate dependent quantities that change under transformation (but of course it's description that changes and not physical reality, unless we mess up our transformation - then we describe different physical reality). And we have invariants that are such only within certain class of coordinate system (spacetime distance is invariant under Lorentz tranformation).

Does this seems ok?

 Quote by PAllen If it is the white hole argument, I don't see the point. White holes cannot form by any process (mathematical fact; must 'exist in the past' for no reason); "no one" believes they exist in our universe. So debating their exact properties is not very interesting.
As a theoretical exercise it might be quite interesting. Please tell me how do you think - is mass attracted toward white hole (we speak about exterior of WH) or not?

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 Quote by zonde So I will try to go into more details about coordinate transformations. If we look at coordinate transformations we have different things. We have coordinate invariants that stay the same under any coordinate transformation. We have coordinate dependent quantities that change under transformation (but of course it's description that changes and not physical reality, unless we mess up our transformation - then we describe different physical reality). And we have invariants that are such only within certain class of coordinate system (spacetime distance is invariant under Lorentz tranformation). Does this seems ok?
Not quite. Invariants are invariant. There is no 'invariant under only some transforms'. Spacetime distance is invariant in both SR and GR for all coordinate transforms. A coordinate transform modifies the metric according to a defined rule. Using the new metric expression (it is really the same geometric object), proper distance and all invariants are the same. That's why they are called invariants.

What you can say about Lorentz transforms and SR is that they leave the metric in constant diagonal form. Other coordinate transforms make the metric more complicated. But invariants are always invariant. And all physical observations (in both SR and GR) are defined in terms of invariants.
 Quote by zonde As a theoretical exercise it might be quite interesting. Please tell me how do you think - is mass attracted toward white hole (we speak about exterior of WH) or not?
Do you want to talk about the maximal SC geometry (past eternal white hole joined to future eternal black hole), or a universe with just a white hole and no black hole (such an object must cease to exist - it is past eternal, not future eternal; its singularity must cease, and then its event horizon ceases - this is what defines it as white).

The answers to many questions are bit different for the WH only versus WH-BH universe. (To clarify: a WH only universe is the time reversal of a collapse to BH; it is like a big bang).

[edit: wait, for the purpose of this thread I guess the thing to discuss is obviously the river model of a white hole = white hole GP coordinates = 1/2 the complete SC geometry, in the same way the GP black hole is only 1/2 the complete geometry.

I'm not so familiar with the GP description of WH. I'll need a little time to work out some thngs, and post more when I have it done.
]

 Tags flowing space model, general relativity, river model