# Are flowing space models compatible with GR?

PF Gold
P: 1,376
 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").
PF Gold
P: 5,060
 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.
PF Gold
P: 5,060
 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.
PF Gold
P: 1,376
 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.
PF Gold
P: 1,376
 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:
$$ds^2=\frac{(1-\frac{\mu}{2r})^2}{(1+\frac{\mu}{2r})^2}dt^2-(1+\frac{\mu}{2r})^4(dr^2+r^2(d\theta^2+sin^2 \theta \; d\varphi^2))$$
P: 3,187
 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.
Physics
PF Gold
P: 6,170
 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?
PF Gold
P: 5,060
 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.
 P: 3,187 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.
PF Gold
P: 5,060
 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).
PF Gold
P: 1,376
 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?
PF Gold
P: 5,060
 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.
]
P: 3,043
 Quote by PAllen 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.
" 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)."
Physics
PF Gold
P: 6,170
 Quote by TrickyDicky 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.

 Quote by TrickyDicky " 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.
Physics
PF Gold
P: 6,170
 Quote by PAllen 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).

 Quote by PAllen [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).

 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).
P: 3,187
 Quote by PAllen 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!
P: 3,043
 Quote by PeterDonis 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?

 Quote by PeterDonis 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.
Physics
PF Gold
P: 6,170
 Quote by TrickyDicky 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.

 Quote by TrickyDicky 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.

 Quote by TrickyDicky 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).

 Quote by TrickyDicky 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.

 Related Discussions Computers 14 General Physics 5 Astronomy & Astrophysics 13 Advanced Physics Homework 0 General Physics 3