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Finkelstein's unidirectional membrane paper

  1. Jun 16, 2011 #1

    bcrowell

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    Finkelstein's "unidirectional membrane" paper

    The "unidirectional membrane" interpretation of black-hole event horizons originated with this paper:

    Finkelstein, Phys. Rev. 110, 965–967 (1958), "Past-Future Asymmetry of the Gravitational Field of a Point Particle," downloadable from his web page at https://www.physics.gatech.edu/user/david-finkelstein

    The basic idea is that if you impose a bunch of reasonable conditions on the Schwarzschild spacetime, and try to eliminate all coordinate singularities, you have to break time-reversal symmetry. This was a point that I hadn't understood properly before -- I'd thought that the Schwarzschild spacetime was time-reversal symmetric, as it appears to be when you write the metric in the Schwarzschild coordinates.

    I have a few questions about the paper:

    1. He seems to be claiming uniqueness of the solution, but I don't see where he proves that...? He uses the term "analytic," and certainly for an analytic function defined in part of the complex plane, any analytic extension to the whole plane is unique. But this is calculus on a manifold, so ...?

    2. He describes two classes of spacetimes that differ by time reversal, and speculates that "it is possible that the gravitational equations imply that all particles in one universe belong to the same class." Was his conjecture right? He also speculates about particle-antiparticle interpretations...was there any validity to that?

    3. He ignores negative-mass solutions. Is there any other physical ground for ignoring them, besides the fact that they would violate energy conditions? I guess they couldn't form by collapse of known forms of matter. Could they conceivably have been produced in the Big Bang?

    Thanks in advance!

    -Ben
     
  2. jcsd
  3. Jun 16, 2011 #2

    Bill_K

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    Re: Finkelstein's "unidirectional membrane" paper

    Finkelstein's coordinates break time-reversal symmetry, but the complete analytic extension of Schwarzschild is given by Kruskal coordinates, which are time symmetric.
    Solutions of elliptic equations have to be analytic, but solutions of hyperbolic ones such as Einstein's equations do not. Discontinuities can occur along the characteristics, which in our case are light rays. For example there could be matter inside the hole which does not escape, which would modify the inner solution. When talking about extensions we restrict ourselves to analytic extensions, otherwise the extension would not be unique.
    Aside from other arguments, the negative mass Schwarzchild solution has a naked singularity at r = 0 and cannot be extended further.
     
  4. Jun 16, 2011 #3
    Re: Finkelstein's "unidirectional membrane" paper

    Time-reversal symmetry is a coordinate-independent feature of static manifolds such as Schwarzschild's, its irrotational timelike killing vector assures this since imposes that the Lie derivative of the metric wrt this Killing vector field vanishes, and this condition is expressed in covariant form.
    So how could Finkelstein possibly want to break that symmetry? He would have to change differential geometry rules first, wouldn't he?
     
  5. Jun 16, 2011 #4

    bcrowell

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    Re: Finkelstein's "unidirectional membrane" paper

    I see...maybe...So did I mischaracterize Finkelstein's result, or did he mischaracterize it, or ...?
     
  6. Jun 16, 2011 #5

    Bill_K

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    Re: Finkelstein's "unidirectional membrane" paper

    He mentions it as a "Note added in proof" at the end of his paper.
     
  7. Jun 16, 2011 #6

    bcrowell

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    Re: Finkelstein's "unidirectional membrane" paper

    "Note added in proof: Schild points out that M is still incomplete, it possesses a nonterminating geodesic of finite length in one direction. Kruskal has sketched for me a manifold M* that is complete and contains M. M* is time-symmetric and violates one of the conditions on M: it does not have the topological structure of all of 4-space less a line. Kruskal obtained M* some years ago (unpublished)."

    So visualizing this on a conformal diagram like http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html", did Finkelstein basically find possibilities like I+II and I+IV?

    What's confusing me about the conformal diagram is the interpretation of the two singularities. The white hole singularity is in the past light cone of every event, so it must have existed for an infinite time in the past. Also, it can emit photons that can be observed by any observer arbitrarily far in the future, so it seems like it must exist for an infinite time in the future. The same seems to be true for the black hole singularity. But then that doesn't make sense, because you can draw a spacelike hypersurface that separates them from each other...? I guess this would explain the part of the note added in proof about the topology. But I don't understand how it can make sense that each singularity is permanent, but you can separate them with a spacelike surface !?!?

    -Ben

    [EDIT] Ah, I see. I think this explains it: http://en.wikipedia.org/wiki/White_hole#Origin
     
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  8. Jun 17, 2011 #7

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    Are you sure these deductions are valid? I agree they would be in a "normal" flat spacetime, but this isn't one. The presence of the horizons and the second exterior region "extends" the spacetime so that the white and black hole singularities can be spacelike and still be in the past/future light cones of every event. In a "normal" flat spacetime, that wouldn't be possible; only a timelike object can be in the past/future light cones of every event. That may be what the bit about the topology was referring to.
     
  9. Jun 17, 2011 #8

    bcrowell

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    Re: Finkelstein's "unidirectional membrane" paper

    Good point. In the conformal diagram, the singularities are horizontal, right? I guess operationally it might not make sense to talk about how long a singularity exists, since you can't stand close to it with a clock. Maybe it makes more sense to say that the white hole's horizon has existed for infinite time in the past, and the black hole's will exist for infinite time in the future? But even that doesn't quite make sense, because the horizons are lightlike. Gah, this makes my head hurt.
     
  10. Jun 17, 2011 #9

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    Yes. This page has a decent image:

    http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html

    Yes, since both the white and black hole singularities are spacelike, not timelike, it makes no sense (at least, not to me) to talk about "how long" they exist.

    The best way I've found to look at it comes from reading Hawking and Ellis a while back. (At least, I think that's the original source of the terms I'm about to use.) A "black hole" is present in a spacetime if the causal past of future null infinity is not the entire spacetime. The "future horizon" (or black hole horizon) is then the boundary of the region that is *not* in the causal past of future null infinity. Similarly, a "white hole" is present if the causal future of past null infinity is not the entire spacetime, and the "past horizon" is the boundary of the region that is not in the causal future of past null infinity. (One of the nice things about conformal diagrams is that all this is obvious just from looking at them.)

    So really both singularities, and both horizons, are simply not "in" the region where it makes sense to talk about "how long" something exists, because that term implies (at least to me) that you're in a region of spacetime where "how long" can be any real number, of any magnitude, which implies that you're in a region which can causally communicate with both future and past infinity. The singularities and horizons are in a separate region of spacetime where the physics doesn't quite match our intuitions about space and time.
     
  11. Jun 17, 2011 #10

    bcrowell

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    Re: Finkelstein's "unidirectional membrane" paper

    Thanks, Peter, that's very helpful!

    OK, here's a far-out question. The maximally extended Schwarzschild spacetime has two copies of Minkowski space in it, and you can't get from one to the other. However, an event inside the black hole's event horizon can have both copies of Minkowski space in its light cone. So does this mean that if you dropped through a black hole's event horizon, you could meet another suicidal explorer who had dropper through from the other universe? Or is this an unrealistic feature of black holes that didn't form by gravitational collapse?

    Also, isn't the central axis of the diagram an axis of rotational symmetry...? So in that case, why aren't the two copies of Minkowski space actually the same?
     
  12. Jun 17, 2011 #11
    Re: Finkelstein's "unidirectional membrane" paper

    This image

    [PLAIN]http://www.mathpages.com/rr/s6-04/6-04_files/image043.gif [Broken]

    casts an interesting angle on the "unidirectional" aspect of the membrane. At time -35 the particles passes up towards the event horizon and out through the horizon and at a later time passes back through the horizon, while all the while its proper time advances in the positive direction. This does not appear unidirectional at all. Now, while the passing out event is from a white hole and the passing back in event is into a black hole, the diagram makes the them appear as one object and the events are only separated by time. However there is the issue of time and space swapping roles below the event horizon so the the particle moving outward is spatially separated from the the particle moving inwards. Note that we have the same particle moving outward is simultaneous with itself moving inward so the single particle is in two places at the same time. (Below the event horizon, two coordinates with the same radial coordinate are simultaneous.) Also note that we can draw the trajectories of other particles on the same chart so that we can have other particles going outward in the same region as particles going inwards, so there is no clear demarcation between black hole and white hole here. You cannot point to one region of the chart and say "the white hole is here" or "the black hole is there" because they are completely arbitrary. I can for example draw the path of an outgoing particle starting out at coordinate time +30 which crosses the path of the incoming particle in the chart.
     
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  13. Jun 17, 2011 #12

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    I'll answer this one first because the answer to your other question kind of builds on the answer to this one. No, the central axis of the diagram is *not* an axis of rotational symmetry in the case of the maximally extended Schwarzschild spacetime. It is in the case of normal Minkowski spacetime, because the central axis is the worldline of the point r = 0 (the spatial origin). In the extended Schwarzschild spacetime, though, the central axis does not correspond to a single value of r. The point on the central axis where the two horizons cross is r = 2M (since that's the value of r everywhere on both horizons). As you go up or down the central axis from that central crossing point, the value of r decreases to zero. So actually the "axis of rotational symmetry" of the spacetime is represented by the two horizontal singularity lines! (At least, I'm pretty sure that's right, although the idea of two spacelike lines representing the axis of rotational symmetry makes my head hurt too.)

    I would lean towards the latter, because the spacetime of a black hole that forms by gravitational collapse does not have the same conformal diagram as the maximally extended Schwarzschild spacetime, as you can see by looking at the example of one on the page I linked to in my earlier post. A gravitational collapse diagram only includes the first exterior region (region I) and the black hole region (region II), plus a non-vacuum portion representing the collapsing matter. Also, as you can see from the diagram, the central axis in the gravitational collapse case *is* an axis of rotational symmetry. (Actually it's a "left axis" in the example, which actually makes more sense to me, since the radial coordinate has a minimum value at zero, so a strictly correct conformal diagram for, say, Minkowski spacetime, would only have a region to the right of the "axis", since a region to the left of the "axis", such as appears in the diagram for Minkowski spacetime on the page I linked to, would represent negative values of r, which don't exist). However, there's still a twist: the horizontal singularity line (the upper left of the diagram) is *also* an axis of rotational symmetry! (It also represents r = 0, after the collapsing matter has vanished into the singularity.)
     
  14. Jun 17, 2011 #13

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    This can't be correct, because for two events to be "simultaneous" (I think you meant to say "two events with the same radial coordinate are simultaneous"), they must be spacelike separated, and the two events in question (one inside the horizon at a given r-value on the "outgoing" leg, and the other inside the horizon at a given r-value on the "ingoing" leg) are timelike separated. This indicates that things are not as simple as your diagram makes them appear to be.

    If you draw the trajectories you are talking about on the maximally extended conformal diagram, you will see that the horizon the object passes on its outgoing leg is *not* the same as the horizon it passes on its ingoing leg. The first horizon goes up and to the left at 45 degrees; the second goes up and to the right at 45 degrees. Similarly, the "r = 0" singularity line the object emerges from is *not* the same as the "r = 0" singularity the object goes into. The first singularity is at the bottom of the conformal diagram, and the second is at the top.

    So your diagram leaves something out: there should be a "break" in the bottom axis (the r = 0 line) to indicate the separation between the two horizons.
     
  15. Jun 19, 2011 #14

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    Oops, meant to say "between the two singularities".
     
  16. Jun 19, 2011 #15
    Re: Finkelstein's "unidirectional membrane" paper

    Yes, that is a better way of phrasing what I meant. :)

    Below the event horizon, spacelike becomes timelike and vice versa. In the diagram below, events B and D (or B-C or C-D) are timelike separated and events A and E are spacelike separated. Agree?

    SchwarzChart.gif

    Of course this leaves a problem because Wikipedia http://en.wikipedia.org/wiki/Spacetime states
    and we have a contradiction here. This implies that the particle rising out of the white hole is not the same as the particle entering the black hole, which seems reasonable, but there is the siren call of the particle's proper time progressing smoothly and continuously along the entire path. Lesson is, proper time progressing smoothly and continuously is not proof in itself of anything.

    That does not prove anything. The first horizon has coordinates r= 2m and t= -infinity and the second horizon has coordinates r= 2m and t= +infinity. They only differ in time, but not place. Of course that depends on how you interpret time and space. Above and below the event horizon timelike and spacelike swap roles, but happens exactly at the event horizon? Undefined I guess?

    Also consider this KS chart:
    600px-KruskalSzekeres.svg.png The green r=constant line goes first up and to the left and then curves up and two the right, just like the two event horizons you mentioned so while its direction and time coordinate is changing, its spatial location does not. The two event horizons you mentioned are just an extreme version of the green r= constant curve.

    In the first chart at the top of this post, I have added the "break" between the two singularities that you desire. It does not fix anything. For example I have placed the break at "now" (Year 2011). Now consider the particle that has its apogee (event C) at year 1911. It rises and falls back into the white hole contradicting the fact that particles are only supposed to exit from white holes. Other trajectories have particles rising out of the black hole contradicting the fact that particles can only enter but not leave a black hole. The only way to cure these contradictions is to insist that all particles have their apogees at year 2011, but of course we know from experience that that is not realistic.

    Any rational person looking deep into the scenario (and not starting out with a "Finkelstein said it must be right,so I won't look too deep into it" attitude) would agree there are multiple problems with his interpretation of black hole interiors. I can offer an alternative interpretation which makes all the problems go away, but the rules of PF forbid me from challenging the conventional interpretation, even though I or others might learn something from the discussion. As far as I know Finkelstein is one of the few well known people from the golden relativity era that is still alive and active in physics and it would be really cool if he came here to discuss it.

    P.S. Have a look at the red and orange light cones I put in the top diagram. They imply light can travel directly from the white hole to the black hole (or vice versa) without crossing an event horizon.
     
  17. Jun 19, 2011 #16

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    yuiop, I should have been a bit clearer about one point in my previous response: your diagrams are *not* coordinate charts, and you simply can't use them to reason about how events are related the way you can with (properly drawn) coordinate charts. In particular, your diagram does not properly represent the causal structure of the spacetime (which events are timelike/spacelike related).

    No, I do not. Both pairs are timelike separated. As you quoted from the Wikipedia page, both pairs of events lie on the same worldline, which is the worldline of a timelike object. Any pair of events lying on the worldline of the same timelike object must be timelike separated.

    What is going on here is, as I said in my previous post, there are two *separate* regions in the spacetime which have r < 2M (i.e., two separate "interior" regions inside the horizon). Event A is in the first region, and event E is in the second. Thus, they are timelike separated even though they have the same r coordinates. If they were both in the same interior region, then you would be correct that they would have to be spacelike separated (since r is a timelike coordinate inside the horizon). But they're not.

    The reason your diagram has difficulty representing this is that, as I said above, your diagram is not a proper coordinate chart, and it does not properly represent the causal structure of the spacetime. If you want to see how things are actually working, try drawing the "Year 2011" worldline represented in your top diagram on the Kruskal chart (I pick that one because your diagram shows that one emerging from the white hole and going back into the black hole, which is correct). You will see that the analogue of event A on that worldline is in region IV while the analogue of event E is in region II. You will also note that there are two "copies" of each r = constant curve; for r < 2M, one copy is in region IV and the other is in region II. So two events can both have the same r coordinate < 2M but lie on different r = constant curves, in different regions.

    Now try drawing the 1911 and 2111 curves, also, on the Kruskal chart. You will see that there is *no* way to have a curve both emerge from and go back into the *same* "hole" (white or black). All three curves *have* to emerge from the white hole and go back into the black hole. So they all work the same as the 2011 curve did above. That's how A and E can be timelike separated.

    No, it doesn't. It only implies, as I said above, that the white hole and the black hole are separate regions of the spacetime, each with their own copies of every r = constant curve for r values < 2M. For r > 2M, the particle is in region I the whole time, which is why your deduction that events B and D are timelike separated is correct (if B were in region I but D were in region III of the Kruskal chart--in the other exterior "universe"--then that deduction would *not* hold; the two events would be spacelike separated).

    Incorrect; t = minus infinity, r = 2M corresponds to the line going up and to the left at 45 degrees, while t = plus infinity, r = 2M corresponds to the line going up and to the right at 45 degrees. So they really are two different horizons; they do differ in "place" as well as in time. Schwarzschild coordinates do not represent this properly because of the coordinate singularity at r = 2M; that's one of the reasons why Kruskal coordinates are useful.

    Yes, in a sense they are. So what? If you look at a chart of Minkowski spacetime with a pair of "Rindler horizon" light rays (just like the r = 2M lines on the Kruskal chart), and a set of hyperbolas representing accelerating observers all sharing the same Rindler horizon, the pair of Rindler horizon light rays is similarly the limit of the accelerating hyperbolas. That doesn't make the two light rays the same.

    That is *not* where the "break" belongs, except if you remove the 1911 and 2111 curves and only look at the 2011 curve. For that one, yes, the "break" is roughly where I would have put it. But remember that "coordinate time" goes to infinity at both ends, so if you want to add 1911 and 2111 trajectories to the diagram, given that the 2011 trajectories are correct, then you would have to have all three "coordinate time" curves going to the far left and right ends of the chart at r = 2M, and they would therefore have to straddle the break just as the 2011 curve does. (For example, your chart has the r = 2M, t = plus infinity point of the 1911 "coordinate time" line to the *left* of the r = 2M, t = minus infinity point of the 2111 "coordinate time" line. That's not possible; t = plus infinity can't occur before t = minus infinity.)

    Again, I suggest that you try to draw all three of these trajectories (1911, 2011, and 2111) on the Kruskal chart. You will not be able to do it in such a way that the 1911 one both emerges from and goes back into the white hole, nor will the 2111 one be able to both emerge from and go back into the black hole.

    The only problem I'm aware of is that he thought his coordinate chart covered the entire spacetime, but then found out it didn't (as he notes in his "note added in proof"). But as a representation of just regions I and II, the "ingoing" Finkelstein chart works fine.

    Isn't there a forum for speculative posts? Post it there and I, for one, will be glad to read it and comment.

    I can't guarantee that, of course. :wink: But I agree it would be cool.

    How so? I don't understand. (Not that it would prove anything anyway, since as I've already noted, your diagram isn't a proper coordinate chart and doesn't represent the causal structure of the spacetime correctly.)
     
  18. Jun 19, 2011 #17
    Re: Finkelstein's "unidirectional membrane" paper

    It is not my diagram, it is a diagram by Prof Brown on his mathpages website. See http://www.mathpages.com/rr/s6-04/6-04.htm

    His diagram is consistent with the standard Schwarzschild coordinate chart. Also see this diagram from MTW:

    realisticBHkruskalsmall.jpg

    which is also the standard Schwarzschild coordinate chart. All I have done is added some sketch notes to the standard diagrams.
    The Kruskal-Szekeres chart is derived from Schwarzschild cordinates by substituting new coordinates into the the Schwarzschild metric. If the Schwarzschild solution is wrong then so is the KS solution. Also, any valid coordinate transformation should have a one to one relationship between events in the two coordinate systems. Any unique event on a coordinate system should have one and only one corresponding event on the transformed system. KS coordinates introduce a whole new parallel universe that was not there in the original Schwarzschild coordinates.
    The Schwarzschild coordinate system is two dimensional. One is space (the radial x coordinate) and the other is time. There is no y or z coordinate, so saying two events with the same r coordinate are in two different places does not make a lot sense on a one dimensional radial line, unless we invoke the parallel universe of course, which is what K-S have done.
    That only illustrates what I said before, there should be a one for one relationship between the two charts.
    Yes there is https://www.physicsforums.com/showthread.php?t=82301, but there are lot of hurdles to jump through to get a post in there, but I might give it a shot as time allows.
     
  19. Jun 19, 2011 #18

    bcrowell

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  20. Jun 19, 2011 #19

    PeterDonis

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    Re: Finkelstein's "unidirectional membrane" paper

    Understood, thanks for the source. I believe there is a similar figure somewhere in MTW. I didn't mean to imply by the phrase "your diagram" that you had originated it, just that you were using it to illustrate your points. Sorry if that caused confusion.

    "Consistent" in the sense that it shows curves which have relationships between coordinate/proper time and radius that are allowed by the Schwarzschild coordinate chart, yes (but with a caveat I'll get to in a moment). But that does not mean that his diagram is itself a proper coordinate chart. It isn't, and it isn't meant to be.

    The caveat is that the "Schwarzschild coordinate chart" is not really a single chart. There are actually three of them. The first is the "Schwarzschild exterior chart" (I don't know if my terms are exactly the "standard" ones, but I'll try to make clear what I mean by them). It covers the region r > 2M ("region I" in the Kruskal chart). The second is what is normally referred to (more or less--again I don't know if my terms are exactly the "standard" ones) as the "Schwarzschild interior chart", which covers the region r < 2M that can be reached by freely falling through the horizon from the exterior region ("region II" in the Kruskal chart, the "black hole" region). The third could also be called a "Schwarzschild interior chart", but it covers the region r < 2M from which outgoing freely falling observers can cross into the exterior region ("region IV" in the Kruskal chart, the "white hole" region). I don't know that I've seen this third chart discussed explicitly, but it should be obvious that it is a valid chart and that it is distinct from the other two. (If any experts on the forum want to weigh in on this, please do.)

    So when you say that the diagram is "consistent with the Schwarzschild coordinate chart", that's only true if you put a "break" in the region r < 2M, as I said before, to reflect the fact that that region of the diagram is really two separate, distinct regions that do not "touch" or overlap; or, equivalently, that region does not represent a single Schwarzschild interior chart, but two disjoint ones, so the disjointness needs to be included on the diagram to make it fully consistent. And, of course, when you do that the issues you're raising go away, because it's clear that the "outgoing" and "ingoing" portions of the worldlines pass through two separate interior regions.

    You'll note, by the way, that Prof. Brown's page, which you linked to, discusses the two interior regions in the paragraphs after the diagram you refer to appears. What I'm saying is consistent with what he says. The only possible curve ball he throws is this statement:

    I haven't checked his later discussion of black holes and cosmology to see whether he talks about closed timelike curves (which is what I think he means by "closed spacetime loops"), but I believe it's been shown that there are no CTCs in Schwarzschild spacetime.

    You'll note that the text accompanying this diagram, and the correspondence between parts (a) and (b), makes it clear that in the (a) diagram, the interior region r < 2M corresponds to region II on the Kruskal chart, and that region only. It does *not* correspond to region IV. In other words, in the (a) diagram in MTW, the interior region is the "black hole" region, and that region only. If you tried to draw the entire worldline represented on Prof. Brown's diagram in part (a) of the MTW diagram, you would not be able to do it; the "outgoing" portion coming from the white hole can't be put anywhere on part (a) of the MTW diagram.

    I didn't say the Schwarzschild solution was "wrong". As I clarified above, the Schwarszschild "solution" or "chart" is not a single chart, and none of the three charts that can be called a "Schwarzschild chart" covers the entire spacetime. The Kruskal chart *does* cover the entire spacetime, and makes clear the relationship between the different Schwarzschild charts. It's all consistent.

    This is true, and it holds for each individual Schwarzschild chart. I made clear which region of the Kruskal chart each Schwarzschild chart covers above. Within each of those regions, there is indeed a one-to-one relationship as you describe. The *appearance* of a two-to-one relationship is only because you are not recognizing that region II and region IV are two separate regions, each with its own Schwarzschild chart.

    If you mean region III, you are correct. That region would actually require a *fourth* Schwarzschild chart (a second "exterior" chart). Mathematically, I believe this region has to be there to make the full analytic extension work right (but my math-fu is not good enough to give a proof of this; maybe one of the experts on the forum can give more detail about how this works). Physically, as far as I know, neither that region nor region IV is present in an spacetime that anyone believes is applicable to the real universe; in the spacetime of a black hole formed from a collapsing star, for instance, only region I and region II are present (plus a non-vacuum portion representing the collapsing matter). For regions III and IV to actually be there physically, the entire spacetime would have to be a vacuum spacetime (i.e., no actual matter present anywhere) but still somehow have a black hole (and white hole) present. As far as I know, nobody believes this is actually physically possible; there *has* to be some matter somewhere for a black hole to form, and once you have a non-vacuum region, you have the spacetime I just referred to, where the only vacuum regions are regions I and II. So in any actual physical spacetime, there would be no "parallel universe" and no "white hole".

    Actually, the full coordinate system also has two angular coordinates, theta and phi. Those are often left out because the spacetime is spherically symmetric, so nothing of physical interest depends on those coordinates. Their presence does not affect the issues we've been discussing, because there are certainly such things as radial geodesics that have constant values of the angular coordinates for their entire length; basically we've just been restricting attention to those.

    As I noted above, mathematically, regions III and IV are there in the maximal analytic extension, and are separate from regions I and II. But in any actual physical spacetime, they would not be there, because physically it's not reasonable to have a curved spacetime with no matter present anywhere (as I noted above). So in an actual physical spacetime, you are correct; there is only one interior region, so any value of the r coordinate corresponds to only one "place". But also, of course, in this case there is no "white hole", so there are no freely falling worldlines that pass through the same value of r twice.
     
  21. Jun 20, 2011 #20
    Re: Finkelstein's "unidirectional membrane" paper

    Well I am glad you think that regions III and IV are unphysical, but that means we have no need for "maximally extended spacetime".

    Another problem with the maximal KS chart is illustrated below (in your favourite chart):

    p834Gravitation.jpg

    Mass or light from the white hole (region IV) can move freely into our universe and mass and light from the other universe (region III) can also move freely into the black hole (region II) in our universe. Matter or light cannot pass from our universe to any part of the "other universe" so over time, if any mass is in the other universe it will end up in our universe and gradually our universe will be inexplicably increasing in mass and energy. Has anyone detected that?
    Fully agree and thanks for picking me up on that fine point.

    As you expressed an interest only I will show you an interesting mathematical oddity that falls right out of the Schwarzschild equations (courtesy of Prof Brown)

    Copy and paste the quoted function below (based on Prof Brown's equation ref http://www.mathpages.com/rr/s6-04/6-04.htm )

    into function box (f) of this online complex function plotter java applet: http://www.digitalhermit.com/math/calc/index.html#Math [Broken] ref: Complex representation: rectangular form and then type (2,t) in function box (g) and adjust the zoom to 24 or less. Drag the small box marked a (the apogee) to a value of x=+3. This is the normal in-falling curve for a particle falling towards a black hole. Note that there is no curve below the event horizon (the red vertical line) because the "realonly" function only plots curves with a zero imaginary component and the path below the horizon is complex. (This is fixed by prof Brown by subtracting and arbitrary imaginary constant or equivalently arbitrarily adding an absolute function to the log part of his equation. OK, now drag the small (a) box so that the apogee is at x=1 (below the event horizon) and observe where the real valued curve is now. That might be a bit of an eye opener. To play around with the graph, change "realonly" to "re" or "im" to see the real and imaginary parts of the curve independently.

    Note that when the apogee is above the event horizon, that:

    1)the curve above the apogee is imaginary (so non-physical).
    2)the curve below the apogee and above the EH is pure real (so physically real).
    3)the curve below the event horizon is complex (probably not physical).
    4)the curve for negative r is complex (probably not physical).

    When the apogee is below the event horizon:

    5)The curve above the event horizon is complex (probably not physical).
    6)the curve above the apogee and below the EH is pure real (so physically real?).
    7)the curve below the apogee and above r=0 is imaginary (so non-physical).
    8)the curve for negative r is complex (probably not physical).

    (Here, complex means the real and imaginary components are both non zero, imaginary means only the imaginary component is non zero and real means only the real component is non zero.)

    With some analysis and thought there is enough information there to see that the Schwarzschild equation do not "absolutely require" that timelike becomes spacelike and vice versa below the event horizon. That only happens if you add arbitrary constants to make it happen. It would seem that if a person where to accept only real results and reject imaginary or complex results, they would remove the time-space swap below the event horizon and also remove the infinite density central singularity where the "laws of physics break down". However by adding arbitrary imaginary constants, they can conclude that space and time "must" swap over and that a singularity of infinite density that breaks the laws of physics "must" exist.

    It is also worth noting that one of the justifications given here on PF in the past for the "maximal extended KS solution" is that
    a)all wordlines should either start or end at a real singularity (black or white) or at past/future infinity. (We will gloss over that the event horizon IS at future infinity)
    b)all wordline should be smooth and continuous.

    That seems reasonable, but if you carefully plot worldlines in regions I,II, III, and IV on a KS chart, it does not actually achieve those objectives. I did it a long time ago and arrived at 16 different permutations of swapping time and space around and never achieved that objective (if I did it right).
     
    Last edited by a moderator: May 5, 2017
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