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Are flowing space models compatible with GR? 
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#91
Nov812, 11:31 PM

PF Gold
P: 1,376

About your second statement you might want to look at this wikipedia article describing classical twobody problem and how exact solution is found for this twobody problem by using COM frame. "Adding and subtracting these two equations decouples them into two onebody 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"). 


#92
Nov812, 11:32 PM

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PF Gold
P: 5,027

Of course GR reduces SR sufficiently locally everywhere. That is built into the mathematical structure of pseudoriemannian geometry in the same way local Euclidean geometry is built into Riemannian geometry. 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. ds^2 =  dt^2 + (dr + βdt^2)^2 + r^2(dθ^2 + sin^2θd[itex]\varphi[/itex]^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[itex]\varphi[/itex]^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/(1R/r)) dr^2 + r^2 (dθ^2 + sin^2θd[itex]\varphi[/itex]^2) (where I am using R for SC radius). This is nonEuclidean spacial geometry. 


#93
Nov812, 11:38 PM

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PF Gold
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#94
Nov812, 11:51 PM

PF Gold
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#95
Nov912, 11:54 AM

PF Gold
P: 1,376

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. 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: [tex]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))[/tex] 


#96
Nov912, 01:48 PM

P: 3,186

[/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):  http://casa.colorado.edu/~ajsh/schwp.html 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'. 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. 


#97
Nov912, 02:17 PM

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PF Gold
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#98
Nov912, 02:17 PM

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PF Gold
P: 5,027

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. 


#99
Nov1012, 07:50 AM

P: 3,186

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 freefall 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 r_{s}  Hamilton's model: "classical" increase, I think; towards double for approaching r_{s} I base this on the requirement that even a "flowing river" cannot accumulate light in transit. 


#100
Nov1012, 09:34 AM

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PF Gold
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 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). 


#101
Nov1012, 10:17 AM

PF Gold
P: 1,376

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? 


#102
Nov1012, 10:44 AM

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PF Gold
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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. The answers to many questions are bit different for the WH only versus WHBH 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. ] 


#103
Nov1012, 11:22 AM

P: 3,023

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)." 


#104
Nov1012, 11:40 AM

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PF Gold
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#105
Nov1012, 11:45 AM

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#106
Nov1012, 12:49 PM

P: 3,186

Thanks for the clarification! 


#107
Nov1012, 02:32 PM

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#108
Nov1012, 04:09 PM

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PF Gold
P: 6,079

(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 stellarmass 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. 


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