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Roy Kerr on the Kerr vacuum

  1. Jun 11, 2007 #1

    Chris Hillman

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    A new eprint by Roy Kerr has just appeared on the arXiv, which should be of extraordinary interest to all PF members interested in black holes:

    Picking up the theme that Wikipedia may be destroying the very notion of truth as I know it, Kerr explicitly repudiates a rumor
    http://ifs.massey.ac.nz/mathnews/centrefolds/58/Aug1993.shtml [Broken]
    which I unfortunately repeated in an earlier version
    of his wikibio :frown:
    Apologies to all!

    With that out of the way, I turn to trying to interpret for a general audience some remarks which I fear may lead to further misunderstanding on the crankweb:

    On. p. 24-25, Kerr discusses the problem of "realistic" interiors. As he notes, some confusion has resulted from the fact that the phrase "interior solution" is used in the research literature to mean either "matter-filled region" in a model of an isolated massive object such as a star, or the interior of a black hole (which is a vacuum region, perhaps with some infalling massless radiation); here we mean the latter.

    Readers need to know that the causal structure of the maximal analytic extension of the Kerr vacuum looks like the right hand diagram in Slide 4 from a talk by Beverly Berger:
    (Well, almost: that diagram depicts the Reissner-Nordstrom electrovacuum and the Kerr vacuum is a bit different in a way I actually need to try to explain, darn it all.)

    However, the mainstream "best guess" on what the causal structure of the interior of a real black hole (formed by gravitational collapse) might look like is quite different, as shown in Slide 65 from the same talk:
    This is basically the Poisson-Israel model of black hole interiors which I have frequently discussed in detail on other occasions. This model is not an exact solution, but rather a semi-heuristic argument that the global structure of generic exact solutions (which are generally assumed to be impossible to obtain in a useful analytic form) which yield black hole models should have certain features as suggested by the diagram in Slide 65.

    These two slides depict Carter-Penrose diagrams, which are two dimensional diagrams often used to illustrate the global structure of Lorentzian manifolds, especially in the case of spacetimes which have asymptotically flat branches, which is the case here. In these diagrams, each point represents an ordinary two sphere. An ingoing "timelike geodesic" represents the world sheet of an infalling shell of matter, while an "ingoing null geodesic" represents the world sheet of an infalling shell of massless radiation (e.g. EM radiation).

    For example, in the diagram for the Kerr vacuum, the blue curve represents a spherical assemblage of observers who fall past the event horizon ("outer horizon" in Kerr's eprint), then past a Cauchy horizon (CH, also IH, "inner horizon" in Kerr's eprint). The vertical bars represent timelike strong scalar curvature singularities, which are thought to be unrealistic. The actual diagram for the Kerr vacuum includes a further asymptotically flat region in which one seems to have a distant object with negative mass. This is tricky and I'd prefer to avoid it, but Kerr refers to it on p. 22 , so I need to at least mention it here. I should perhaps add that the inner and outer ergospheres are distinct from the inner and outer horizons, and this is again hard to visualize from the Carter-Penrose diagram, but fortunately I don't think this distinction is relevant here.

    The diagram for the Poisson-Israel model shows that the IH has been partially converted to a weak curvature singularity and the timelike strong scalar curvature singularity has been replaced by a spacelike strong scalar curvature singularity (horizontal bar). As I noted in the previous thread, the features of this weak curvature singularity and Cauchy horizon are that (1) an enounter with it might be survivable and (2) if so, gtr refuses to predict what happens after that. (See the discussion by Israel in the article in the book I cited in the previous thread.)

    Kerr writes, regarding the ring singularity (let's say the righthand vertical bar in the Penrose-Carter diagram for the Kerr vacuum):
    This is because, in a suitable chart covering the interior region, the coordinate "disk" spanning the nominal locus of the ring singularity (remember that the singularity is not really part of the Lorentzian manifold) acts as a branch cut. This is not easily visualized from the Carter-Penrose diagram, but see Hawking & Ellis for a nicely illustrated discussion of this phenomenon.

    Kerr continues:
    This is somewhat ambiguous, but I think he is referring to the replacement of the IH in the Kerr vacuum by the CH in the Poisson-Israel model. Then he says
    i.e. might have some kind of geometric singularity. Next, Kerr adds some critical remarks concerning the singularity theorems:
    These remarks are unfortunately rather ambiguous, particularly in this context. My guess is that they imply a viewpoint similar to something I have said myself on previous occasions, in discussing the "no hair theorems" in relation to the problem of realistic black hole interiors (anyone interested can Google for that). I would be surprised if they imply a view at variance with the general expectations of Israel and Poisson concerning the question of what gtr predicts should occur in generic black hole interiors, but I can't be sure without requesting clarification.

    Another cryptic remark, from p. 24:
    I can think of at least three possible interpretations; I have no idea which if any is intended.

    [PF note: I moved my post from another thread once I realized that this will likely prove of wider interest than I first anticipated. This new thread also continues a discussion in the Astrophysics subforum, in the thread "Singularity one-dimensional?"]
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jun 11, 2007 #2
    An interesting description on the Kerr vacuum interiors using the Doran coordinate chart is given by Andrew Hamilton in his paper http://arxiv.org/PS_cache/gr-qc/pdf/0411/0411060v2.pdf" [Broken].

    I remember seeing a video clip of a presentation some time ago giving an overview of this document. Must be out somewhere on the web. He is using some really cool software in that presentation, I suppose it is not GNU licensed?

    Note: whatever is physical and what not might be clear to some, I think such statements are premature. A long time ago black holes were supposed to be unphysical as well!
    Last edited by a moderator: May 2, 2017
  4. Jun 12, 2007 #3
    Thanks again Chris.
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