http://camoo.freeshell.org/30.5quest.pdf" [Broken](adsbygoogle = window.adsbygoogle || []).push({});

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 $ds^2=dr^2-dt^2$.

\includegraphics{blackhole.jpg}

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.

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