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Perturbing the Schwarzschild metric

  1. May 31, 2008 #1

    Latex source below, please click on link above, though.

    I've been working through the exercises in the Penrose book "The
    Road to Reality". There's one that I'm really puzzled about.

    He's talking about an "eternal" black hole - never created by
    collapse of a star, with the Schwarzschild or Kerr metric. It's
    simplified by a conformal map so the metric is $ds^2=dr^2-dt^2$.

    What \underline{is} O exactly? It isn't the singularity,
    is it? The metric doesn't diverge at O, apparently, so O
    isn't the singularity.

    There's a Gibbons-Perry argument that is a "derivation" of the
    black hole temperature, described above. The spacetime is
    complexified and a function on the spacetime, for example
    $\tau$, which is analytic at the origin on the spacetime O,
    is complex analytic in a neighborhood around O. So you get
    a $\theta$ which is just the angle in the real plane around
    O, and $\tau=i\beta\theta$. $\tau$ has a period of
    $i\beta 2\pi$ in a neighborhood around O.

    "He says "There appears to be no physical justification for
    assuming that physical quantities defined on the spacetime
    have a regularity at O, and it is hard to see why the
    argument provides a justification for the Hawking temperature,
    despite its mathematical elegance. Any physically
    realistic model of a black hole would possess deviations
    from the exact Schwarzschild - or Kerr - metric, and these
    deviations can reasonably be expected to get larger and
    larger, finally diverging towards infinity the closer we
    extend towards O."

    Then the exercise asks, "See if you can give an argument
    justifying this claim. Hint: Think of small linear
    perturbations. Do you expect exponential time-behavior?
    Consider eigenmodes of $\partial/\partial \tau$".

    I guess an eigenmode of $\partial/\partial \tau$ is a
    perturbation $h_{\alpha\beta}$ in the metric with
    $\partial h_{\alpha\beta}/\partial \tau=ch_{\alpha\beta}$,
    $c$ some complex number. If $c$ is pure imaginary it'd
    be a component in the fourier transform of $h_{\alpha\beta}$
    from time-space to frequency-space. And the point is that
    you don't expect $c$ to be pure imaginary, so that $e^{c\tau}$
    increases exponentially in $\tau$.

    Why would you expect $\partial/\partial \tau$ to have
    eigenmodes at all? If $h_{\alpha\beta}$ has an extension as
    a complex analytic function to a neighborhood of O, then as
    a function of $\tau$ it has to have a
    period of $i\beta 2\pi$, you can see how it would have a
    Fourier decomposition as trig functions in the
    $i\tau$ direction, which would be exponential in the
    $\tau$ direction. But nobody said that $h_{\alpha\beta}$
    is analytic around O, we're apparently trying to show it
    has a singularity at O!

    Suppose one does know that $h_{\alpha\beta}$ is exponential
    in $\tau$. The constant-$\tau$ lines are radial lines from
    O. So I guess $h_{\alpha\beta}$ \emph{would} be varying wildly
    near O. I don't see how you would see from this that it's
    actually getting larger as you extend towards O.
  2. jcsd
  3. May 31, 2008 #2


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    Use "tex" tags next time, e.g. [noparse][tex]ds^2=dr^2-dt^2[/tex][/noparse] to get


    If you want LaTeX stuff to appear in the middle of a sentence, use "itex" instead. I think it produces smaller output and doesn't mess up the distance between lines as much.
  4. May 31, 2008 #3
    The singularity is the squigly line top and bottom. The point O isn't really a point, afaik, it's just an artefact of the compactification/coordinate system. It is essentially the region inside the BH at time t=0 (or the boundary thereof). It's just that that region looks like a point at that time. They call it a 2-sphere so I assume they mean O to be the horizon at t=0.

    Unfortunately I can't comment on the rest of the question without doing a bit more digging. The euclidean argument for black hole temperature/entropy/whatnot "gives the right results" but is on such shaky grounds that it seems more coincidental (problems have to do with the fact that Lorentzian metrics and Euclidean metrics aren't simply analytic continuations of one another). I wouldn't lose too much sleep over details such as this.
  5. Jun 1, 2008 #4
    Every point on the inside of the conformal diagram represents a 2-sphere. It represents a very symmetrical 4D space.
    The horizon is the sloping diagonal line through O.
    I know that, and that was the gist of what Roger Penrose was saying. My question is about his exercise though. He said perturbations in the Schwarzschild metric would go to infinity at O, for some reason related to the euclideanization. Why would they go to infinity? I said in the .pdf what I'd been able to figure out about what he's getting at, but it's not a connected argument. Is there some way to connect [tex]\partial h_{\alpha\beta}/\partial\tau[/tex] with [tex]\partial h_{\alpha\beta}/\partial R[/tex] (R being radial "distance" from O), so as to argue that [tex]h_{\alpha\beta}[/tex] must diverge as you go towards O? I can't see how.
  6. Jun 4, 2008 #5
    Maybe this is the answer: http://camoo.freeshell.org/30.5.pdf
    I'm not completely satisfied with it, but no particular light dawned.
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