# Are flowing space models compatible with GR?

PF Gold
P: 5,059
 Quote by harrylin Yes I insist: he presents it as a physical model in the frame of the astronomers. If the most basic properties of flow (such as continuity) do not apply to his "flowing space" model then it is a big misrepresentation - "not even wrong". Thanks for the clarification!
Fine. I personally have never used this model to solve problems or gain understanding of BH geometries. However, I would note that, as you quote, he says flowing space not flowing fluid. He never described fluid properties. He does talk about space properties (flat background; flat spatial slices). He also does describe in both words and equations how the river is to be used for computations or predictions. I agree there are parts that mislead to a fluid analogy: the word 'river'; pictures with water; descriptions of photons as swimming fish. To the extent that these suggest fluid properties for the river, they cause confusion.

If I were describing the method I would call it a flow of preferred frames having certain properties: the motion of one of the frames in relation to SC coord r coordinate can be obtained by Newtonian laws; the relation of one frame to another can be described in galilean terms; anything moving in one of these preferred frames follows SR locally, and experiences boosts from frame to frame. It is especially the first two parts of this that amount to coincidences for this this geometry.
PF Gold
P: 1,376
 Quote by PAllen 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.
So as I understand it invariants are invariant by definition so that if result of a function is different after coordinate transformation then we adjust the function so that the result stays the same.

So my statement should have been like this:
"And we have semi-invariants that are invariant only within certain class of coordinate systems (spacetime metric is semi-invariant under Lorentz tranformation)."

Is this right?
P: 3,187
 Quote by PeterDonis It joins region I to the collapsing FRW dust until the dust collapses to a small enough radius that an event horizon forms; after that it joins region II to the collapsing dust (until the dust collapses to r = 0 and the singularity forms). So both vacuum regions (I and II) are present in the complete model. [..] I believe numerical simulations have been done that relax the idealizations and still yield a spacetime that looks qualitatively similar [..]
The model that you refer to is not the model that Oppenheimer-Snyder introduced in 1939, for they obtained quite the contrary, for example: "it is impossible for a singularity to form in a finite time". I mentioned in the other thread a modern simulation that qualitatively agrees with their model.

It is the discussed disagreement between such Schwarzschild models (incl. Einstein's) with Hamilton's "flowing space" model (and others) that led to this thread about Hamilton's model.

This thread is already too long, and those other models are not the topic here; please start a new thread on other models if you like to discuss them more in depth.
P: 3,187
 Quote by PAllen Fine. I personally have never used this model to solve problems or gain understanding of BH geometries. However, I would note that, as you quote, he says flowing space not flowing fluid. He never described fluid properties. He does talk about space properties (flat background; flat spatial slices). He also does describe in both words and equations how the river is to be used for computations or predictions. I agree there are parts that mislead to a fluid analogy: the word 'river'; pictures with water; descriptions of photons as swimming fish. To the extent that these suggest fluid properties for the river, they cause confusion.
I was indeed just referring to "flowing space", not even demanding any liquid-like properties. A Lorentz boost is a transformation from a synchronised system "at rest" to a newly synchronised system that according to rest observers is "moving" in space. A body or system that is modelled as being at rest in space, even space that "itself flows in Galilean fashion through a flat Galilean background" cannot undergo a Lorentz boost. That is a total mix-up.
 If I were describing the method I would call it a flow of preferred frames having certain properties: the motion of one of the frames in relation to SC coord r coordinate can be obtained by Newtonian laws; the relation of one frame to another can be described in galilean terms; anything moving in one of these preferred frames follows SR locally, and experiences boosts from frame to frame. It is especially the first two parts of this that amount to coincidences for this this geometry.
OK. Thanks again!
Physics
PF Gold
P: 6,135
 Quote by harrylin The model that you refer to is not the model that Oppenheimer-Snyder introduced in 1939, for they obtained quite the contrary, for example: "it is impossible for a singularity to form in a finite time".
If you're going to quote, quote fully. They said the singularity does not form in a finite time according to a distant observer. They also found that the singularity does form in a finite time according to an observer who falls in with the collapsing matter. MTW section 32.4 and Box 32.1 go into this in some detail; the description I gave of the O-S model is taken from theirs, which specifically references the O-S 1939 paper.

 Quote by harrylin It is the discussed disagreement between such Schwarzschild models (incl. Einstein's) with Hamilton's "flowing space" model (and others) that led to this thread about Hamilton's model.
They don't disagree; they give exactly the same answers for all observables. They are just two different coordinate charts on the same spacetime geometry. That's been said repeatedly throughout this thread and nobody has refuted it.
P: 3,187
 Quote by PeterDonis If you're going to quote, quote fully. They said the singularity does not form in a finite time according to a distant observer. [..]
I quoted fully; the quote is from page 456, second column and your bold face words are not there. Presumably they implied distant observer, which is the issue here.
 They don't disagree
This thread was a spin-off of the thread mentioned in the first post, because some people here claimed that such accounts are no good, even criticising me for citing them. And we repeatedly mentioned their disagreement with Hamilton: "it takes, from the point of view of a distant observer, an infinite time for this asymptotic isolation to be established", while Hamilton has us see that according to us distant observers the falling observer will quickly fall through - at Newtonian fall speed in our Universal Time (for a black hole that is at rest wrt us)*.
And once more, their model is not the topic here. If you want to discuss it, please start it as a topic. I will not reply again about that other model here.

*"Free-fall coordinates reveal that the Schwarzschild geometry looks like ordinary flat space, with the distinctive feature that space itself is flowing radially inwards at the Newtonian escape velocity " [..] "Answer to the quiz question 9: The star does in fact collapse inside the horizon, even though an outside observer sees the star freeze at the horizon. The freezing can be regarded as a light travel time effect. As described here, space can be regarded as falling into the black hole, reaching the speed of light at the horizon, and exceeding the speed of light inside the horizon. [..] it just takes an infinite time for the information that [the star] has collapsed to get to the outside world. "
Physics
PF Gold
P: 6,135
 Quote by harrylin I quoted fully; the quote is from page 456, second column and your bold face words are not there.
You didn't give a reference for your quote, so I wasn't sure exactly where you were quoting from. I didn't mean to imply that my bold face words were a direct quote; you'll note that I didn't put them in quotation marks.

Here's a quote from the abstract of the 1939 O-S paper (referenced by George Jones in this thread):

 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; an external observer sees the star asymptotically shrinking to its gravitational radius.
That makes clear the distinction I was describing.

 Quote by harrylin Hamilton has us distant observers see the falling observer quickly fall through - at Newtonian fall speed in our Universal Time (for a black hole that is at rest wrt us).
You are misinterpreting what Hamilton says; he nowhere says that distant observers will "see" this. You are putting an interpretation on the flat background in Hamilton's model that Hamilton himself does not put on it. He makes clear that the flat background is not physically observable and doesn't correspond to anything physically observable. It's just an aid to visualization, one which evidently is not really helpful for you. As PAllen said, that's fine; just don't use his model.

 Quote by harrylin And once more, their model is not the topic here. If you want to discuss it, please start it as a topic. I will not reply again about that other model here.
Understood. TrickyDicky has started a separate thread:

P: 3,187
 Quote by PeterDonis [..] You are misinterpreting what Hamilton says [..]
Sorry I do not, instead I used sloppy phrasing - corrected now!
Good - will see it a few days from now.
Physics
PF Gold
P: 6,135
 Quote by harrylin Sorry I do not, instead I used sloppy phrasing - corrected now!
I don't see anything in what you have actually quoted from Hamilton's web pages that translates to "according to us distant observers the falling observer will quickly fall through - at Newtonian fall speed". He talks about "space flowing radially inwards at the Newtonian escape velocity", but that doesn't say anything about what happens "according to a distant observer". He also says explicitly that "an outside observer sees the star freeze at the horizon".
PF Gold
P: 1,376
Quote by PeterDonis
 Quote by PAllen 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.
Just flip the sign of the dt dr term in the line element, so the "escape velocity" vector points outward instead of inward. I.e., the "river" is flowing outward at every point at the "escape velocity" instead of inward. For example, the river is flowing *outward* at the speed of light (relative to the flat background) at the horizon, and flows outward more and more slowly as you go further and further out (to a limit of zero outward velocity at infinity).
PeterDonis,
When we flip the sign in GP metric it does not change anything about infalling test mass as seen by outside observer, right?
So mass is still attracted (speaking in Newton terms) toward WH, right?
Physics
PF Gold
P: 6,135
 Quote by zonde When we flip the sign in GP metric it does not change anything about infalling test mass as seen by outside observer, right? So mass is still attracted (speaking in Newton terms) toward WH, right?
Yes, this is correct; a WH's gravity is attractive, just like a BH's gravity. However, it shows up in a different way. In the "river model" of a BH, space flows inward towards the BH, faster and faster as you get closer and closer. In the corresponding "river model" of a WH, space flows *outward* away from the WH, but slower and slower as you get farther and farther away. So objects that are ejected from the WH will decelerate as they rise, showing that the WH's gravity is attractive.
PF Gold
P: 1,376
 Quote by PeterDonis So objects that are ejected from the WH will decelerate as they rise, showing that the WH's gravity is attractive.
You carefully speak about objects ejected from WH.

Now I am very interested about object moving toward WH. It is going against the accelerating river and yet as it seems to me it is still accelerated toward WH as seen by outside observer so it double accelerates against the river flow.

What do you say?
Physics
PF Gold
P: 6,135
 Quote by zonde Now I am very interested about object moving toward WH.
There can be objects moving toward the WH, but they can never reach its horizon, because the horizon is moving inward at the speed of light from their point of view. (From the "river model" point of view, at the WH horizon space is flowing outward at the speed of light, so ingoing light can just manage to stay at the horizon; anything slower than light can't quite do so and will move outward.)

 Quote by zonde It is going against the accelerating river and yet as it seems to me it is still accelerated toward WH as seen by outside observer so it double accelerates against the river flow.
No, it's you that's double counting accelerations. In the river model, all of the "acceleration" caused by the hole (black or white, depending on which version of the river model you are looking at) is accounted for by the change in the "river flow" of space itself with radius. In the WH version of the river model, the WH's gravity is fully accounted for by the slowing down of the outward river flow as you get further from the hole. There's no extra "acceleration" beyond that.

Also bear in mind that the portion of both river models (BH and WH) that is outside the horizon is the *same* region of spacetime, just described by two different coordinate charts (ingoing Painleve for the BH river model, outgoing Painleve for the WH river model) which are adapted to two different families of observers. The observers who are "flowing with the river" in one model are not the same as the ones who are "flowing with the river" in the other model; another way of putting this is that the region of spacetime outside the horizon can be described by two *different* "rivers", the outgoing one and the ingoing one. But you have to pick one; you can't incorporate both into the same description, since that would be "double counting" the gravity of the hole(s).

The above may not be a very good description of what's going on; feel free to ask further questions if it's not clear.
PF Gold
P: 1,376
 Quote by PeterDonis No, it's you that's double counting accelerations.
It's sometimes such a challenge to talk with you PeterDonis. You can turn on it's head such a simple thing that I am at loss how to explain your mistake.

It's not hard to start believing that it's deliberate misinterpretation of my words.
You certainly noticed that I was talking about going inwards as you are describing that situation in first part of your response with: "There can be objects moving toward the WH ..."

But in second part of your response you are giving argument concerned with going outwards: "... the slowing down of the outward river flow as you get further from the hole ...".

PeterDonis are you trolling?
Physics
PF Gold
P: 6,135
 Quote by zonde It's sometimes such a challenge to talk with you PeterDonis. You can turn on it's head such a simple thing that I am at loss how to explain your mistake.
Perhaps I'm misunderstanding the question you're asking. Let me step back for a bit and try to describe things without phrasing it as an answer to a specific question.

The maximally extended Schwarzschild spacetime has a total of *four* regions. The best way to globally visualize this spacetime and its four regions is using a Kruskal diagram, as seen for example here:

http://en.wikipedia.org/wiki/Kruskal...es_coordinates

Region I is the "normal" part of spacetime we're used to, the exterior region that's outside any horizons. Region II is the interior of the black hole. Region III is a *second* exterior region; and region IV is the interior of the white hole.

The standard "river model" covers regions I and II; that is, it views space as flowing inwards towards the black hole. However, note that in this model, there is *no* white hole. More precisely, the white hole portion of the maximally extended spacetime, region IV, is not covered by the standard river model; so it makes no sense within that model to talk about objects falling towards the white hole. Anything that falls inward will eventually fall into the *black* hole, region II.

There is also a second possible "river model", which is obtained by using outgoing Painleve coordinates instead of ingoing Painleve coordinates. This second "river model" covers regions IV and I; that is, it views space as flowing outwards from the white hole. In this model, we can talk about objects moving towards the white hole; but they can't possibly reach the white hole because its horizon is moving inwards at the speed of light.

Now if we look at the full extended spacetime, as shown on the Kruskal diagram, we can see that an observer in region I can move inward, at speeds approaching the speed of light; this corresponds to moving on a worldline that is tilted to the left at an angle approaching 45 degrees. Such an observer, if he were way down in the lower right corner of the diagram, might want to think of himself as moving towards the white hole. However, he will never reach the white hole; he will never reach region IV. Instead, he will eventually cross the black hole horizon and enter region II.

Also, if we look at the full extended spacetime, we can see that there are timelike worldlines that leave region IV, enter region I, and then leave region I and enter region II. Some of these worldlines will be geodesics, i.e., the worldlines of freely falling objects. (The Wikipedia page doesn't show any of these worldlines, but some of the figures in MTW do.) We can use either one of the two "river models" to describe what happens to objects that follow these worldlines:

- The standard river model will view the object as rising away from the black hole (like a ball thrown upwards), coming to rest, then falling back in and entering the black hole; but this model can't show where the object ultimately came from, because it ultimately came from the white hole, and the white hole isn't covered by the standard river model.

- The second river model will view the object as coming out of the white hole, rising upwards, coming to rest, then falling back down; but this model can't show where the object ultimately goes to, because it ultimately goes into the black hole, and the black hole isn't covered by the second river model.

But note that in *both* cases, the object starts by moving upward, then comes to rest, then falls back down; this shows that gravity is attractive throughout the spacetime. There is no region where anything is "repelled" by either the white hole or the black hole. Furthermore, the change in the object's motion, since it is freely falling, is entirely due to the change in river velocity along its trajectory; this is true regardless of which river model you use to describe its motion. This is why I said there is no "additional" acceleration, over and above that produced by the river.

(Remember that even though the second river model has the river flowing outwards, its velocity decreases as you go outwards. An object that comes to rest at a finite height is moving at *less* than the Newtonian "escape velocity", so it is moving *inward* relative to the river.)

Does this help any?

[Edit: I should probably also add that there are other worldlines in the maximally extended spacetime that are also relevant:

- There is a set of worldlines that starts from spatial infinity in the infinite past, and falls inward at exactly the Newtonian "escape velocity". This set of worldlines covers regions I and II, and these worldlines are used to construct the frame field of ingoing Painleve observers, which underlies the standard river model.

- There is a set of worldlines that starts at the white hole singularity and moves outward at exactly the Newtonian "escape velocity", eventually reaching spatial infinity in the infinite future. This set of worldlines covers regions IV and I, and these worldlines are used to construct the frame field of outgoing Painleve observers, which underlies the second river model.

It's a good exercise to work through how observers following these worldlines would describe the objects following the worldlines I described above, the ones that rise upward, come to rest, and then fall back in. This may help to reduce some of the confusion.]
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