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Sign of Kretschmann Scalar in Kerr Metric

  1. Sep 23, 2015 #1
    This question is motivated by one on stack exchange, and on this paper (which comes across a bit student-y but it claims to have been reviewed, and in any case I have reproduced its results in ctensor and gnuplot).

    So: the KS (abbreviation!) conveys an overview of curvature at a given point in spacetime, which is nice, but whereas the KS for Schwarzschild has the same sign everywhere, that for the Kerr solution changes sign for various values of spin parameter (see diagrams in PDF). Does this change of sign relate to anything physical?
  2. jcsd
  3. Sep 23, 2015 #2
    This may have something to do with the prediction of closed timelike curves (which occur at [itex]g_{\phi \phi}=0[/itex]) near the ring singularity in Kerr-Newman black holes. From an old library entry-

    There is also the turnaround radius (that occurs at [itex]g_{t \phi}=0[/itex] where infall velocity becomes zero and counter-rotation occurs) which also extends to positive r in the Kerr-Newman metric.

    This paper looks at it in some detail-
    A twist in the geometry of rotating black holes: seeking the cause of acausality

    Link to old library entry-
    What are closed timelike curves?

    This paper also mentions these properties-
    http://casa.colorado.edu/~ajsh/phys5770_08/bh.pdf [Broken] page 35 (fig. 7) and section 3.14.9
    Last edited by a moderator: May 7, 2017
  4. Sep 24, 2015 #3
    Unfortunately I don't have the mathematical chops to tell if that stuff is relevant, but I have doubts. The paper looks at slow-spinning charged black holes in the equatorial plane (I am neglecting charge, and am looking at up to "maximum" spin in the whole r-theta plane).

    Strangely, this is actually quite tough to plot and interpret; see gnuplot file attached if anyone wants to try it themselves [update: you will need to change the file extension to .gp]. I suspect that the changes in sign are occurring around or within the horizon, but I'm currently trying to understand precisely where and when coordinate independence applies wrt horizons and invariants.

    Attached Files:

    Last edited: Sep 24, 2015
  5. Sep 24, 2015 #4


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    This is a great question. Looking forward to seeing any answers. The material on physics.SE seems to be of low quality.
  6. Sep 27, 2015 #5
    Latest gnuplot file, small improvements in visualization.

    I've plotted something that I think is reasonable now, and it would appear that the plus and minus curvature fluctuations do extend substantially outside the horizon, more so as the spin (a) increases. In fact it is simple to set a = 0.0 and observe the curvature outside the horizon for a Schwartzschild black hole . . . this also settles the overall sign issue.

    Probing inside is tricky ATM because the scalar blows up to rather high values, so the contour plots show adjacent regions of positive and negative curvature coming in from the outside, as it were, but on the inside there are abutments of positive and negative curvature that just don't fit on the plot. I think I need to look at clamping the z values . . . [EDIT: now done, looks very nice!]

    Anyway, if anyone finds any errors (all equations are in the file) please let me know here.

    Attached Files:

    Last edited: Sep 27, 2015
  7. Sep 28, 2015 #6
    Please excuse the bump, but I've found a sweet spot with the visualization (clamping was the answer!). Vary a between 0.0 and 1.0 and rotate the plot to see a huge richness of behaviour, including some effects which might even be detectable outside the event horizon. Do it again after "set view map" to see the contours alone. If nothing else this is a cute little "signature" for a given value of spin in the Kerr Spacetime.

    Now we wait to see if real observations of Sag A* correlate in any way with the Kretschmann scalar!

    Attached Files:

  8. Sep 29, 2015 #7
    Just stumbled across this paper, looks interesting, but complicated. I'll take what I can from it!
    [EDIT] Those diagrams at the end look familiar ;) Also the authors appear a little, er, dismissive of poor Mr Henry (the author of the paper in my OP).
    Last edited: Sep 29, 2015
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