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New blackhole solution to Einstein eqn, no info paradox (Krzysztof Meissner)

  1. Jan 6, 2009 #1


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    Horizons and the cosmological constant
    Krzysztof A. Meissner
    6 pages
    (Submitted on 6 Jan 2009)
    "A new solution of the Einstein equations for the point mass immersed in the de Sitter Universe is presented. The properties of the metric are very different from both the Schwarzschild black hole and the de Sitter Universe: it is everywhere smooth, light can propagate outward through the horizon, there is an antitrapped surface enclosing the point mass and there is necessarily an initial singularity. The solution for any positive cosmological constant is qualitatively different from the Schwarzschild solution and is not its continuous deformation."

    "4. Conclusions
    In the paper we have shown that the solution of the Einstein equations (5) for a point mass immersed in the universe with the positive cosmological constant has very special properties: the metric is everywhere smooth, light can propagate outward through the horizon, there is an antitrapped surface enclosing the point mass and there is necessarily an initial singularity. Although with extremely small value of H such an object for all practical purposes looks like a usual black hole the conceptual difference resulting from the fact that there is no horizon for the outward propagation of light can be far-reaching

    – first, one should rethink a notion of a black hole entropy as proportional to the area of the horizon and second, there seems to be no information loss even classically since the communication of the inside with the outside is extremely weak but nonvanishing.

    It is also interesting to note that in the presence of such objects there is necessarily an initial singularity in distinction to the pure de Sitter universe and there is no continuous deformation connecting Λ > 0 solution described in this paper and the Schwarzschild metric."

    What interests me here is how this deSitter black hole solution will serve as a basis for LQG research. We know of Chris Meissner already from his LQG papers:
    Eigenvalues of the volume operator in loop quantum gravity
    Krzysztof A. Meissner
    12 pages, Class.Quant.Grav. 23 (2006) 617-626

    Black hole entropy in Loop Quantum Gravity
    Krzysztof A. Meissner
    10 pages, Class.Quant.Grav. 21 (2004) 5245-5252

    And there is already a considerable number of papers using LQG to resolve the black hole singularity---typically finding a bounce. Now I'm wondering what will happen when the same researchers go after this new solution, a black hole in a universe with positive cosmological constant.

    One could argue that this solution of Meissner's has added realism at least in the sense that the universe does seem to have a positive cosmological constant. In that case, according to Meissner, there is no horizon. Light carrying information can gradually escape from the interior. This is not the same as the thermal Hawking radiation originating just outside the horizon.
    Last edited: Jan 7, 2009
  2. jcsd
  3. Jan 7, 2009 #2
    Hi Marcus

    Interesting stuff. Just to confirm, is the H used in the metric in the first equation the inverse of Hubble time (~2.26e-18 s^-1) and when they refer to H being related to the cosmological constant [itex](\Lambda)[/itex] by [itex]\Lambda=3H^2/(8\pi G)[/itex], if I'm not mistaken, this is actually the equation for critical density (the symbol of [itex]\rho_c[/itex] being replaced with [itex]\Lambda[/itex]) of which the cosmological constant contributes to (about 71%). The equation I normally see for the cosmological constant is [itex]\Lambda=\rho_{\Lambda}(8\pi G)/c^2[/itex].

  4. Jan 7, 2009 #3


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    Meissner's H is certainly an inverse time. Somewhat confusingly (to me) he uses the same letter H as usually denotes the conventional Hubble rate which is also an inverse time! So are we supposed to think of Meissner H as a Hubble-like parameter? He doesn't say.

    Meissner doesn't explicitly spell out what units/conventions he is using. Pretty clearly he has set c = 1, and therefore does not include c in the equations. But since he includes (Newton's) G in the equations, I imagine he has not set G = 1. On the other hand that would mean his Lambda is a density. As you point out.

    Lambda is normally treated as inverse area, or curvature, and a different symbol rhoLambda is used for the associated density. Again as you point out. So he is using his own conventions, or maybe they are Polish conventions that I just don't know about.

    None of this bothers me, however! It's just notation and my past experience of Meissner assures me that he knows what he's doing. I wish he had been more explicit, made the paper longer, put in some figures, sketched the topology, etc. But I'm OK with it at least for now.

    Can you help me picture the topology? He says it is R2 x S2.
    He also says there must be an initial singularity---which is different from the usual (no blackhole) deSitter.

    The paper is definitely interesting, but I see a possible drawback: it may be insufficiently realistic in the sense that the blackhole may need to have always been in existence. The model does not include the formation of the blackhole AFAICS.

    At this point the paper is new and I'm struggling. Please let me know anything that occurs to you about the paper. It might help.
  5. Jan 8, 2009 #4
    Hi Marcus

    I haven't fully read through the links below myself but they might be of interest-

    CMC-Slicings of Kottler-Schwarzschild-de Sitter Cosmologies
    http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Agr-qc%2F0501020 [Broken]
    (which appears to include spacetime and topological diagrams)

    Standard Clocks, Orbital Precession and the Cosmological Constant
    http://arxiv1.library.cornell.edu/PS_cache/gr-qc/pdf/0301/0301057v1.pdf [Broken]
    (goes into a bit more detail regarding the quantity of Lambda and includes for rotation)

    a couple of PF posts-

    a half-decent overview by the Max Planck Institute-
    The Kottler Metric
    http://relativity.livingreviews.org/open?pubNo=lrr-2004-9&key=kottler-18 [Broken]

    In virtually all the above, [itex]H^2r^2[/itex] is replaced with [itex]\Lambda r^2/3[/itex] (and I'm certain there's a [itex]c^2[/itex] in with [itex]\Lambda r^2/3[/itex]). In the second paper, the expression [itex]\Lambda \approx H_0^2/c^2[/itex] is used where [itex]H_0[/itex] is the Hubble Parameter. Using [itex]H_0 \approx 70 km\ sec^{-1}\ Mpc^{-1}[/itex] expressed as the inverse of Hubble time produces a figure of [itex]\Lambda \approx 10^{-56} cm^{-2}[/itex] which is commonly accepted as the geometric quantity for Λ.

    Looking at [itex]H^2[/itex] in the 'Horizons and the cosmological constant' paper, it appears that this is a Hubble-like parameter that is relative to the cosmological constant only and should maybe be expressed as [itex]H_{(\Lambda)}^2[/itex]? Looking at [itex]\Lambda r^2/3[/itex] including for [itex]c^2[/itex] relative to [itex]H^2[/itex] in the paper-

    [tex]H_{(\Lambda)}^2=\frac{\Lambda c^2}{3}[/tex]

    [tex]H_{(\Lambda)}=\sqrt{\frac{10^{-52} \times\ (3\times10^8)^2}{3}}[/tex]

    which equals 1.73e-18 s^-1

    divide this by the inverse of Hubble time (2.26e-18 s^-1) and you get 0.765 which is pretty much the omega figure for the cosmological constant.

    Last edited by a moderator: May 3, 2017
  6. Jan 9, 2009 #5
    Hi Marcus

    Topology and metric expressed in the form of R x S is new to me. In your post you say the topology of the metric is R2 x S2. While I understand that R represents real numbers and S represents boundary (or a 'smooth spacelike Cauchy hypersurface') where R is the vacuum solution outside the event horizon and S is the event horizon (I assume the numbers inside the black hole interior are considered imaginary numbers). While reading the paper, R2 x S2 seems to imply that the metric forms a 2 dimensional vector space (x and y). While this appears straight forward, I can only assume the same applies to the boundary or 'hypersurface' and in most cases, the event horizon is S2, representing an ordinary sphere (while a circle would be R1 and R3 would represent a sphere in 4-dimensional Euclidean space. Would it be acceptable to say that a radial path into the black hole is R x S?

    A couple of sources-

    Boundary (topology)

    The product topology
    Last edited: Jan 9, 2009
  7. Jan 9, 2009 #6

    George Jones

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    Very interesting! Notice that one of PF's own was acknowledged.
    I don't know if the reasoning used to reach this is vaild for our universe. Our universe doesn't have all the reuired symmetry. Galaxies don't expand, so spacetime inside galaxies doesn't look like part of de Sitter spacetime. Consequently, I'm not sure that this argument precludes the formation of astrophysical black holes with real event horizons. Of course events horizons are teleological, so I'm not sure how to proceed.
  8. Jan 9, 2009 #7
    Friedmann equation...

    This 'Hubble-like' parameter is only a factor in the Friedmann equations relative only to the cosmological constant:

    [tex]\boxed{\Lambda = 1.112 \cdot 10^{-52} \; \text{m}^{-2}}[/tex]

    Friedmann equation:
    [tex]H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G \rho}{3} - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}[/tex]

    The Hubble parameter:
    [tex]\boxed{H_0 = \sqrt{\frac{8 \pi G \rho}{3} - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}}}[/tex]

    [tex]\boxed{H_0 = 2.26 \cdot 10^{-}^{18} \; \text{s}^{-}^{1}}[/tex]

    [tex]H_{\Lambda}^2 = \frac{\Lambda c^2}{3}[/tex]

    [tex]H_{\Lambda} = c \sqrt{\frac{\Lambda}{3}}[/tex]

    [tex]\boxed{H_{\Lambda} = 1.825 \cdot 10^{-18} \; \text{s}^{-1}}[/tex]

    http://en.wikipedia.org/wiki/Friedmann_equations" [Broken]
    Last edited by a moderator: May 3, 2017
  9. Jan 9, 2009 #8


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    As George said: it cannot make sense in generality, since blackholes can form and evaporate long before they even feel the asymptotic DeSitter conditions. You can imagine making the Hubble radius arbitrarily large, and the cosmological constant arbitrarily small.

    As for the solution itself at first glance, theres something a little weird going on b/c it appears to violate various DeSitter singularity theorems, at least morally. So theres probably an energy condition thats being tweaked somewhere.
    Last edited: Jan 9, 2009
  10. Jan 9, 2009 #9

    Why it will of course simply help continue the main feature of lqg which is that it has no connection to physical reality. For example...

    In a correct quantum theory of gravity there is no singularity but instead well-defined physics all the way up to and perhaps beyond planck scale. This is really what is meant by the notion of resolving a singularity and only a correct theory of quantum gravity can achieve it. On the other hand incorrect theories like lqg cannot resolve anything and this problem is swept under the rug by the naive idea of a bounce.
  11. Jan 9, 2009 #10


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    Steve thanks for the helpful response and links. This one looks as if it might be useful for other things besides what they have about the Kottler metric.
    http://relativity.livingreviews.org/open?pubNo=lrr-2004-9 [Broken]

    I agree it does seem interesting. I'm wondering if Meissner will be at the Valencia Black Holes/LQG workshop in March. A colleague of his at Warsaw, Jerzy Lewandowski, is one of the organizers.
    EDIT: I see Lewandowski is acknowledged in Meissner's paper.
    Last edited by a moderator: May 3, 2017
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