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Oppenheimer-Snyder model of star collapse

  1. Nov 11, 2012 #1
    This comes from this thread https://www.physicsforums.com/showthread.php?t=647627&page=7 discussion in posts #103,#104,#107 and #108.
    The Oppenheimer-Snyder model was mentioned by PeterDonis as a more plausible model than the Schwarzschild spacetime, well this has an element of subjectivity, but one reason I don't share this view is because the only way to relax the highly idealized conditions required by the O-S model is to recurr to the Kruskal-Szekeres diagram for the Schwarzschild solution as is shown in MTW sec. 32.5 second paragraph. So how can one consider more plausible a model than the one it owes its plausibility to?

    Also I have a few things to clarify from this model.

    As I understand it the O-S model basically joins the exterior Schwarzschild to a contracting FRW spatially spherical solution, (a pressureless isotropic and homogeneous dust).
    I usually interpret the exterior Schwarzschild solution to refer to the Schwarzschild metric, outside the Schwarzschild radius, or region I in the K-S diagram and this leads me to a second dependency of the O-S model on the maximally extended Schwarzschild solution, since in order to sy that the Schwarzschild exterior includes region II and the event horizon one must obviously rely on the K-S diagram (that didn't exist in 1939) to begin with.
    I'm still not convinced that it is commonly understood that the region inside the Schwarzschild radius is also considered an exterior region, since then, what is the interior region?, the singularity by itself?

    More doubts about the model:
    Here is the abstract from the original paper from O-S
    "When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In the present paper we study the solutions of the gravitational field equations which describe this process. In I, general and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. In II, an analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. 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."

    My bold: the first sentence I bolded lists some of the conditions required for the model to hold, have they all been theoretically and empirically ruled out? If so how? And I mean by other ways other than the K-S spacetime mathematical solution, that is considered not plausible due to its implying white holes.

    And by gravitational radius are O-S referring to the Schwarzschild radius?
     
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  3. Nov 11, 2012 #2

    PeterDonis

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    I think you're misreading that paragraph. It only talks about relaxing one idealization, that of zero pressure inside the collapsing matter. It doesn't talk at all about relaxing spherical symmetry. The only reference to the K-S diagram is to show that, once the star has collapsed to R < 2M, no amount of pressure can stop it from collapsing, because inside the horizon, all timelike worldlines end in the singularity, not just geodesic ones. Pressure can make the worldlines of the infalling matter geodesic, but it can't make them not timelike.

    Also, note that the K-S diagram in MTW Figure 32.1b (the one that is referenced in the paragraph you refer to) is *not* a K-S diagram of the maximally extended spacetime. It's a K-S diagram of exactly the type of solution I described. The gray portion on the left is the region of spacetime occupied by the infalling matter; the white portion on the right (and below the singularity) is the vacuum region outside *and* inside the horizon (i.e., a portion of regions I and II of the maximally extended spacetime).

    The word "exterior" is used to mean two different things, which I agree is an unfortunate abuse of terminology. Sometimes it means "the vacuum region exterior to the horizon", and sometimes it means "the vacuum region exterior to the collapsing matter". In the O-S model the latter meaning is the one that's meant. As you can see from the K-S diagram that you referenced, the vacuum region includes portions of regions I *and* II of the maximally extended spacetime (see my comments above).

    No, of course not. The O-S model is a highly idealized model; nobody thinks otherwise. The problem with including all that other stuff is that nobody has an analytical solution that includes it. Numerical simulations, as referenced in MTW, still show the same qualitative behavior (formation of a horizon and collapse of the matter to form a singularity) when the other stuff is included.

    I'm not sure what you mean here. The O-S model uses a *portion* of the maximally extended Schwarzschild spacetime, which is what the "K-S spacetime mathematical solution" describes. There's no problem with doing that as long as you enforce the appropriate junction conditions at the boundary between the vacuum portion of the spacetime and the portion containing the collapsing matter.

    Yes.
     
  4. Nov 11, 2012 #3
    I know that paragraph only refers to the pressureless idealization, I only brought it up to show the dependence of the O-S model on the posterior K-S extended solution, because you were claiming the O-S model was physically more plausible than the latter. IMO this is a meaningless statement given the commented dependence of one model on the other i.e. the O-S model at least originally when it was first published, seems to refer to the collapsing star before the BH singularity is formed, and subsequently this paper has been interpreted in the light of the progress made more than 20 years later by Kruskal and others.
     
  5. Nov 11, 2012 #4

    Dale

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    The Schwarzschild spacetime would require any black hole to pre-date the big bang, and the OS spacetime does not. So calling it "more plausible" seems reasonable to me.
     
  6. Nov 11, 2012 #5

    PeterDonis

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    I don't agree that the O-S model "depends" on the "extended K-S solution". What you are calling the "extended K-S solution" is just the maximal analytic extension of the Schwarzschild geometry. It's a mathematical object. There is no physical principle that I'm aware of that makes the maximal analytic extension of a manifold logically prior to just using a portion of that manifold in a physical solution.

    Another way of looking at this is to observe that the maximally extended Schwarzschild geometry, as described by K-S, requires the entire spacetime to be vacuum. This immediately makes the full geometry unsuitable for a model where matter is present, such as the O-S model. But since the EFE is local, there's no requirement that we use the *entire* maximally extended manifold; in fact, looking at it that way gets things backwards. We don't pick the portions of the different manifolds (regions I and II of extended Schwarzschild, plus collapsing FRW) first; we solve the EFE first, and then *discover* which portions of what manifolds arise when we develop the global solution.

    Can you give actual quotes from the original paper that support this view? My impression from reading the abstract (which appears in MTW) is that the original O-S model already includes all three regions I referred to (FRW region containing collapsing matter, vacuum region outside the horizon, and vacuum region inside the horizon) plus the singularity. Certainly that's how MTW describe the model, and they don't give any impression that their description was something "interpreted" later that wasn't present in the original O-S model.
     
  7. Nov 12, 2012 #6
    Let me try and explain what I mean by the O-S model relying on the extended Schwarzschild mathematical solution, when thinking of it as a BH model.
    The FRW dust plus Schwarzschild exterior model only describes the situation of a collapsing star (not charged and not rotating) from the moment the contraction of the star starts , up to the instant previous to the singularity being the only entity inside the Schwarzschild radius of the star (so there is no longer FRW isotropic dust, and therefore no more O-S model). Right at that point the mathematical model of the maximally extended Schwarzschild spacetime takes the place.
    Further as commented above if one wants to relax the requirement of the O-S model concerning the star fluid being pressureless, one does it (or at least it is done in MTW) alluding to the fact that the causal logic, that is, the expected consequence of the O-S collapse model is the extended Schwarzschild spacetime wich doesn't care about the initial conditions of the collapse since it is an eternal exact solution of the EFE.

    So saying that the O-S model is just a local version that needs not rely on the global spacetime solution misses the causality of the collapse model, and if one wants to add physical plausability to it like not demanding exactly zero star pressure one also needs the extended mathematical model to account for the final result of the collapse.

    So I'm still finding hard to separate the physical plausability (or lack of) of one model from the other.
    Also historycally, if one looks at the timeline of the 1939 paper citations, one can see it was basically ignored until the beginning of the sixties when the mathematical models by Kruskal et al. were published.
     
  8. Nov 12, 2012 #7

    Dale

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    Schwarzschild BH has existed forever, even before the big bang, OS BH has existed for a finite time. Existing for a finite time is more plausible than existing forever. What is so hard to understand about that?
     
  9. Nov 12, 2012 #8

    Dale

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    Here is a geometric analogy: If I have a car with a 3 m long interior and I go to a lumber yard and they have only 4 m long pieces of lumber, then I might ask them to cut off 1 m of length and I might even ask them to taper it nicely.

    Then, regardless of the fact that the cut lumber was based on the long lumber, and regardless of the fact that beyond the taper the cut lumber has the same shape as the long lumber, and regardless of the fact that historically lumber yards ignored the sizes of cars from the development of tapered cuts in 1939 until the beginning of the sixties, it is clearly more plausible that the cut lumber will fit in my car.
     
  10. Nov 12, 2012 #9

    PeterDonis

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    Yes, that's the initial condition, that the star is at rest with some finite radius.

    No, this is not correct. The "model" includes the entire spacetime to the future of the start of the star's contraction. You don't "switch models" when the collapse forms the singularity. For one thing, "when the collapse forms the singularity" depends on how you choose your spatial slices. In the Penrose diagram of the O-S model, the singularity is as far in the future as it gets--spacelike slices that are cut arbitrarily close to the singularity also come arbitrarily close to future infinity. See the diagrams on Hamilton's web page:

    http://casa.colorado.edu/~ajsh/collapse.html

    Even if we leave out the issues about "right at that point", this is not correct; if this were true, then a white hole would magically appear in the past instead of the collapsing star. The O-S model is a model of the *entire* spacetime to the future of the start of the star's collapse. The spacetime to the past of that point is not included in the model, but it certainly is not a white hole; the simplest assumption (which of course is not realistic) would be that the star simply sat there statically for an infinite time into the past. A more realistic model would include the rest of the universe, all originating from the Big Bang. In no case would we have a white hole or a maximally extended Schwarzschild spacetime; that spacetime, as I've said before, assumes that there is vacuum everywhere, and there can't be vacuum everywhere with a collapsing star present.

    First of all, the "expected consequence" still depends on exact spherical symmetry; in any real case there is not exact spherical symmetry, so Birkhoff's theorem doesn't apply and we can't say that the vacuum region will be Schwarzschild.

    Second, you're misunderstanding what I said about the EFE being local. See below.

    I didn't say the *model* was local, I said the *EFE* was local. That means that when I am putting together a global model for a spacetime, I don't have to use only one single solution; i.e., I don't have to use the entirety of one particular spacetime (one particular mathematical geometry). I can stitch together pieces of different geometries, as long as I satisfy the appropriate junction conditions when I do the stitching.

    Here's a simpler example to illustrate what's going on. A 2-sphere is a particular mathematical geometry. So is a cylinder that extends infinitely far in the direction along its axis. Each one can be described very simply in terms of coordinates on it. But I can also form a third geometry by taking half of the sphere and stitching it together with half of the cylinder; as long as I do the stitching right (I have to match up the radius of the 2-sphere with the radius of the cylinder, and orient the junction so the tangent vectors of the two surfaces match up at the boundary), meeting the appropriate "junction conditions", the resulting surface will be continuous and differentiable (I'm hand-waving on terminology a bit here, hopefully you can see what I mean), and will therefore be just as much of a legitimate mathematical geometry as the sphere and the cylinder. I won't be able to describe it quite as simply using coordinates on it, but it is still a perfectly good geometry, and it is perfectly self-contained; nothing in my description of it will have to take into account the "existence" of the other half of the sphere or the other half of the cylinder.

    Similarly, to form the O-S model, I take the maximally extended Schwarzschild spacetime and "cut" it along the boundary where the surface of the collapsing star is going to be, and use only the portion to the future of that boundary. I then stitch that portion together with a collapsing FRW spacetime, making sure that things match up along the boundary. (And, if I want to have a complete solution, I also stitch in something to the past of the initial spacelike surface where the star's collapse begins, so my complete model includes the entire past history of the star and its vacuum exterior. That will still only include a further portion of region I of the extended Schwarzschild spacetime, i.e., I will still be "cutting" that spacetime and only using a portion of it in my model.) The final solution therefore only contains a piece of the extended Schwarzschild spacetime, a piece comprised of a portion of region I and a portion of region II. The rest of the maximally extended spacetime is simply not there in the model, just as half of the 2-sphere and half of the cylinder were simply not there in the object I made by stitching a half-sphere and half-cylinder together.

    I don't understand what you're saying here. Causality just means the local light cone structure is continuous throughout the spacetime. As long as the junction conditions are satisfied, this holds when I stitch together pieces of different geometries.

    That may be so (I don't know enough about the citation history to know, perhaps you have a link?). So what?
     
    Last edited: Nov 12, 2012
  11. Nov 12, 2012 #10
    I'm not saying you have to "swith models" necessarily, just highlighting the domain of appliccation of each model.
    Penrose diagrams didn't even exist for more than 20 years after the paper that describes the O-S model.
    The O-S model predicts that "the total time of collapse for an observer comoving with the stellar matter is finite", what does the O-S model say about what comes inmediately after that finite time? IOW, I'm only referring to the future direction after finite time for the comoving observer in the O-S model.




    AFAIK this is not correct, numerical and perturbation methods alows us to use the extended Schwarzschild spacetime in the absence of perfect spherical symmetry.



    Note you are all the time using the extended Schwarzschild spacetime as a template in wich to cut the O-S model. That's the kind of dependency I'm referring to.
     
  12. Nov 12, 2012 #11

    OS collapsing star is a static object. It's exterior geometry is Schwarzschild, so it has existed forever before its contraction started ("even before the big bang") too. What has existed for a finite time is its collapsing process for the comoving observer POV.
     
  13. Nov 12, 2012 #12

    PeterDonis

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    So what? They're a legitimate slicing of the spacetime; whether that slicing was known when O-S wrote their paper is irrelevant.

    By which they mean "proper time as experienced by an observer comoving with the stellar matter".

    Nothing "comes after" it; the infalling observer gets destroyed in the singularity along with the stellar matter. Their worldlines simply end at the singularity; there is nothing "after" it.

    Remember that the singularity is spacelike; that is, it is "an instant of time", not "a place in space". My point about different slicings is simply that there is a slicing of the spacetime according to which the singularity is the instant of time "at the end of time", i.e., there is *no* time that is "after" the singularity.

    If you don't have perfect spherical symmetry, then the spacetime you're working with is only approximately Schwarzschild; how good the approximation is depends on how close you are to spherical symmetry. But that's irrelevant to the question of what *portion* of the maximally extended spacetime is actually used in the physical model.

    That's one way of describing what I'm doing, yes. But it's not the only way. Here's another: I start with the assumption of perfect spherical symmetry and solve the vacuum EFE on an "initial value" spacelike slice. I do the solution locally, starting at spatial infinity and working inwards. Eventually I reach the surface of the matter, which I assume is at rest in this initial spacelike slice; at that point my solution is no longer vacuum, but I can ensure that the switch is smooth by imposing appropriate junction conditions at the boundary. Once I'm inside the matter, I continue to assume spherical symmetry, and I also assume homogeneity because it's the only assumption that's simple enough to allow me to find an analytical solution, locally, to the EFE. I continue working inwards until I reach r = 0, the center of the collapsing matter. I now have a description of a spacelike slice on which the matter is instantaneously at rest.

    I then work the solution forward in time, using the EFE to evolve things from one spacelike slice to the next. I find that the matter is collapsing inward; then I find that a horizon forms; then I find that the matter collapses to r = 0 and forms a singularity. If I try to iterate further "forward" in time, I find that the singularity is actually spacelike; depending on exactly how I cut my spacelike slices, it may even be that the singularity *is* the single spacelike slice that I am at when the matter reaches r = 0, so there is nothing "after" it--it is the future endpoint of my solution. In any case, I can obtain a complete spacetime geometry to the future of the initial spacelike slice I started with. Notice that nowhere did I do any "cutting" of anything out of anything else; I simply used the EFE to build my solution point by point.

    After I have done all this, of course, I can discover that my solution is isometric to what I described in previous posts: a portion of regions I and II of the extended Schwarzschild spacetime, joined to a collapsing FRW geometry with appropriate junction conditions. But I didn't have to *assume* that, or start with that, or construct my solution from those pieces. The description of the solution in terms of those pieces joined together is just a helpful aid to visualizing what's going on; it is not logically or physically essential to deriving the actual solution. The knowledge that a solution could be constructed out of those pieces may have helped in arriving at the O-S model, but that's not the same as saying it is necessary to the O-S model.

    (There is also, of course, the question of what lies to the past of the initial spacelike slice. That's a separate question from how we construct the solution; it depends on what assumptions we make about what the star was doing before it started to collapse.)
     
  14. Nov 12, 2012 #13
    Thanks, that was all I meant when saying "up to the instant previous to the singularity being the only entity inside the Schwarzschild radius of the star (so there is no longer FRW isotropic dust, and therefore no more O-S model)." to wich you replied: "No, this is not correct."

    Once this is clarified, it is probably not very useful to keep debating your claim about what is more or less physically plausible, it depends on the subjective priorities one might want to consider.
    In fact all the EFE solutions even the most practical, empirically or computationally have tremendous idealization that are far from what is usually considered physical.

    BTW here's the citation history of the OS paper, curious to say the least. It was ignored until 1964, but it wasn't really until the last 5 years that it took off.
    http://libra.msra.cn/Publication/19921235/on-continued-gravitational-contraction
     
  15. Nov 12, 2012 #14

    Dale

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    This is not correct, at least as far as how I understand it. To my knowledge, neither the original dust cloud, nor the singularity, nor the event horizon has existed forever in the OS spacetime, which starts with the collapse. I.e. the OS manifold only covers back in time until the initiation of collapse.

    By contrast, in the Schwarzschild solution the exterior of the horizon, the horizon itself, the interior of the horizon, and the singularity are all considered to have existed forever. I should think it is obvious to any reasonable person that that is much less plausible.
     
  16. Nov 12, 2012 #15

    PeterDonis

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    I thought you were saying that there must be something "after" the singularity forms. If you agree that there are slicings in which there is nothing "after" the singularity forms, then yes, we're in agreement.

    All of the *analytical* solutions, yes. Numerical solutions can be far more realistic.

    Thanks! I'll take a look.
     
  17. Nov 12, 2012 #16

    PeterDonis

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    This is my understanding too, at least based on all the presentations of the O-S model that I've seen; none of them talk about what happens before the initial instant of time at which the dust cloud is instantaneously at rest.

    However, this obviously leaves the manifold incomplete; there has to be *something* to the past of that initial spacelike slice. I think it's reasonable to ask what the possibilities are for that past region of the complete manifold; and I also think it's reasonable, physically, to say that a white hole is *not* one of those possibilities.
     
  18. Nov 12, 2012 #17

    PeterDonis

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    Actually, since the O-S dust has zero pressure, it can't be static. The most straightforward extension into the past of the O-S model, keeping the assumption of zero pressure, would be the time reverse of the extension into the future; i.e., an expanding FRW region with zero pressure and starting from an initial singularity, joined to a portion of regions IV and I of the maximally extended Schwarzschild spacetime (including the white hole spacelike singularity, which could be thought of as the past endpoint of the spacetime). However, I don't see that as physically reasonable for the same reasons that a white hole in general is not physically reasonable.

    I suspect that what O-S had in mind was something like a static star with positive pressure, in equilibrium, in which the pressure suddenly declines to zero (or at least to some negligible value compared to its previous one) over a very short time (due to, say, the stoppage of nuclear reactions in its core and a consequent sharp decline in temperature).
     
  19. Nov 12, 2012 #18
    Certainly but I was referring precisely to the static star you depict below in my answer to DS.

     
  20. Nov 12, 2012 #19

    Dale

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    All the OS model insists is that at some moment there is a spherical dust cloud which is momentarily at rest. There is certainly no implication in the OS model that the "static star" has been in such a state since before the big bang.

    Plausibility is definitely subjective, so you can choose to disagree. However, to me it is clear that a model which begins from a momentarily stationary sphere of dust is more plausible than a model which begins from a singularity. We have direct experience with things that approximate a momentarily stationary sphere of dust, but not with singularities. So the opposite stance seems tenuous.
     
  21. Nov 12, 2012 #20
    Well, it's not for me to defend singularities' plausibility, or GR's for that matter, you are free to have whatever opinion, I can only refer you to the Hawking-Penrose singularity theorems.
     
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