Are flowing space models compatible with GR?

PF Gold
P: 4,087
 Quote by PAllen ... 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'.
PF Gold
P: 5,059
 Quote by Mentz114 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.
PF Gold
P: 4,087
 Quote by PAllen 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'.
PF Gold
P: 5,059
 Quote by Mentz114 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).
PF Gold
P: 4,087
 Quote by PAllen 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.
PF Gold
P: 5,059
 Quote by Mentz114 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.
PF Gold
P: 4,087
 Quote by PAllen 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.
P: 3,187
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:
 Quote by PAllen I am not sure you recognize that Hamilton's rive model is a specialized interpretation, of two special case GR geometries. Use of this model is not part of any general understanding of BH, EH, etc. The general modern consensus comes from studying such solutions in coordinate independent ways, and from the global methods developed by Hawking, Penrose, and others. It is not derived or understood by most using Hamilton's river model. [..] 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.
PF Gold
P: 5,059
 Quote by 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.
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.
PF Gold
P: 5,059
 Quote by harrylin 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.]
PF Gold
P: 1,376
 Quote by PAllen 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?
PF Gold
P: 5,059
 Quote by zonde 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.
PF Gold
P: 1,376
 Quote by PAllen 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).

 Quote by PAllen 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.
PF Gold
P: 5,059
 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.
P: 3,187
 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.
P: 3,187
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.
PF Gold
P: 5,059
 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.
PF Gold
P: 1,376
 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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