Are flowing space models compatible with GR?

[zonde]

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?

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?

 

[PAllen]

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.

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.
]

 

[TrickyDicky]

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.

Right, but it is also a mathematical fact that the Schwarzschild spacetime with its white holes is the most physically plausible mathematical model we have of a black hole, the other three(R-N, Kerr, K-N) have much worse problems. So if one goes by the GR solutions (I mean their existence is the main reason we are willing to believe in black holes in the first place, since the astrophysical evidence might be compatible with other explanations), one should "believe" in white holes as much as one believes in black ones.
Anyway according to the wikipedia page on black holes:
" A much anticipated feature of a theory of quantum gravity is that it will not feature singularities or event horizons (and thus no black holes)."

 

[PeterDonis]

Right, but it is also a mathematical fact that the Schwarzschild spacetime with its white holes is the most physically plausible mathematical model we have of a black hole

No, it isn't, precisely because of the presence of the white hole. The most physically plausible model we have of a black hole spacetime, if we are restricting ourselves to simple models with analytical solutions, is the Oppenheimer-Snyder model, which joins a portion of regions I and II of the Schwarzschild spacetime (vacuum exterior and vacuum black hole interior) to a portion of a collapsing FRW spacetime.

" A much anticipated feature of a theory of quantum gravity is that it will not feature singularities or event horizons (and thus no black holes)."

Not sure which Wikipedia page you are referring to; the one I get when I google on "black holes" says at one point that quantum gravity is expected to feature black holes without singularities (i.e., event horizons but no singularities). That's my understanding of the current state of play (but I am not very familiar with the current state of play). Links to recent review articles would be helpful.

 

[PeterDonis]

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).

By a "WH only universe" I assume you mean the time reverse of the collapsing Oppenheimer-Snyder model? I.e., an expanding FRW region joined to a portion of regions IV and I of the maximally extended Schwarzschild spacetime (the white hole and the exterior.) If so, then yes, it is like a big bang (the expanding FRW portion), but with extra regions (the portions of regions IV and I).

[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.

Actually, the exterior vacuum region in both models (WH and BH) is the same region (region I of the maximally extended spacetime). The difference is that the WH model covers regions IV and I (with the past horizon in between), and the BH model covers regions I and II (with the future horizon in between).

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).

 

[harrylin]

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, [..] incremental SR boosts [..]

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!

 

[TrickyDicky]

No, it isn't, precisely because of the presence of the white hole. The most physically plausible model we have of a black hole spacetime, if we are restricting ourselves to simple models with analytical solutions, is the Oppenheimer-Snyder model, which joins a portion of regions I and II of the Schwarzschild spacetime (vacuum exterior and vacuum black hole interior) to a portion of a collapsing FRW spacetime.

I thought that model joined region I to the collapsing FRW dust, rather a model of collapse than of a stablished BH, but I'm not really sure, the model dates back to 1939, do you have any good current source about that model?

Not sure which Wikipedia page you are referring to; the one I get when I google on "black holes" says at one point that quantum gravity is expected to feature black holes without singularities (i.e., event horizons but no singularities). That's my understanding of the current state of play (but I am not very familiar with the current state of play). Links to recent review articles would be helpful.

http://en.wikipedia.org/wiki/Black_hole section 4.5 last paragraph it links to a couple of arxiv papers.

 

[PeterDonis]

I thought that model joined region I to the collapsing FRW dust

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.

rather a model of collapse than of a stablished BH

It's a model of both. Once the collapse is complete, the spacetime is Schwarzschild all the way to future infinity; but a WH doesn't magically appear in the past when the collapse is complete. The collapsing portion of the model *is* the past of the Schwarzschild portion.

but I'm not really sure, the model dates back to 1939, do you have any good current source about that model?

It's discussed in MTW, and (I believe) in Wald. AFAIK it's still a perfectly good model, just a very highly idealized one (perfect spherical symmetry everywhere and zero pressure in the collapsing FRW region). I believe numerical simulations have been done that relax the idealizations and still yield a spacetime that looks qualitatively similar (i.e., still a collapsing non-vacuum region surrounded by vacuum regions outside and inside a horizon).

http://en.wikipedia.org/wiki/Black_hole section 4.5 last paragraph it links to a couple of arxiv papers.

Thanks, I'll take a look. A couple of brief comments just looking at this section of the Wiki article:

(1) I agree that the existence of an upper mass limit for a neutron star, or any similar gravitationally bound static object, is critical in the prediction that stellar-mass BHs are common in our universe. I also agree that the numerical value of that limit depends on detailed knowledge of the possible states of very dense matter, which we don't currently have a good handle on. However, the *existence* of some such limit in the general range of 1.5 to 3 times the mass of the Sun, AFAIK, is pretty much a foregone conclusion, because a wide variety of possible equations of state have been modeled and all of them lead to *some* limit in that general range.

Also, there is a theorem due, I believe, to Einstein that says that there cannot be *any* static equilibrium for a gravitationally bound object with radius less than 9/8 of the Schwarzschild radius. A typical neutron star radius is already fairly close to that, and as the star gets heavier the radius gets smaller; so there doesn't seem to me to be a lot of room for exotic bound states (such as quark stars, etc.) that are much heavier than known neutron stars (all of which are, I believe, around 1.5 solar masses). (The 9/8 limit arises because pressure contributes to the Ricci tensor, so as the radius of a static equilibrium state approaches the 9/8 limit, the central pressure required to maintain equilibrium goes to infinity, since increased pressure also increases the inward force the pressure has to resist, in a positive feedback loop.)

(2) Regarding quantum corrections, I agree there are plenty of reasons to suspect that quantum corrections will remove the r = 0 BH singularity. I don't see the same sorts of reasons leading to a removal of the event horizon. On a quick skim of the arxiv paper that is referenced at that point in the Wiki article (footnote 115), I think the Wiki author was misinterpreting the term "singularities" in the arxiv paper to refer to the EH instead of (or in addition to) the r = 0 singularity. But I'll read through the paper in more detail when I get a chance, it's possible that there's a more complex picture there.

I should emphasize that (1) and (2) above are just my personal take on it; we still have a lot to learn about this area of physics.

 

[PAllen]

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.

 

[zonde]

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?

 

[harrylin]

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.

 

[harrylin]

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 modeled 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!

 

[PeterDonis]

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.

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.

 

[harrylin]

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 [STRIKE]us see [/STRIKE]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. "
- http://casa.colorado.edu/~ajsh/schwp.html; http://casa.colorado.edu/~ajsh/collapse.html#collapsed

 

[PeterDonis]

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.

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.

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:

https://www.physicsforums.com/showthread.php?t=651362

 

[harrylin]

[..] You are misinterpreting what Hamilton says [..]

Sorry I do not, instead I used sloppy phrasing - corrected now!

Understood. TrickyDicky has started a separate thread:

https://www.physicsforums.com/showthread.php?t=651362

Good - will see it a few days from now.

 

[PeterDonis]

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".

 

[zonde]

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,
what would be your answer to the question I asked PAllen?
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?

 

[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?

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.

 

[zonde]

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?

 

[PeterDonis]

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.)

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.

 

[zonde]

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?

 

[PeterDonis]

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–Szekeres_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.]

 

[micromass]

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