5
$\begingroup$

What are the Christoffel symbols for the Schwarzschild spacetime, expressed in the Kruskal-Szekeres coordinates?

$\endgroup$

closed as off-topic by AccidentalFourierTransform, ZeroTheHero, Emilio Pisanty, Prahar, Jon Custer Jun 4 '18 at 20:42

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – AccidentalFourierTransform, ZeroTheHero, Emilio Pisanty, Prahar, Jon Custer
If this question can be reworded to fit the rules in the help center, please edit the question.

6
$\begingroup$

I worked these out recently, and it was a fair amount of computation to get them, so I thought they might be useful to others.

Throughout the following, let $m=1/2$, so the horizon is at $r=1$, i.e., $r$ is in units of the Schwarzschild radius. The exterior regions of the maximal extension of the Schwarzschild spacetime are at $r>1$, which is regions I and III. The interior is $r<1$, regions II and IV.

The version of the Kruskal-Szekeres coordinates I'll use are null coordinates $(V,W)$, equivalent to Hawking and Ellis's $(v'/\sqrt2,w'/\sqrt2)$.

Even when working in the Kruskal-Szekeres coordinates, it's convenient to express some things in terms of the Schwarzschild $r$ coordinate, which can be found using \begin{equation} r = 1+W(-VW/e). \end{equation} Here the function $W$ (not to be confused with the coordinate $W$ inside the parentheses) is the principal real branch of the Lambert W function, and $e$ is the base of natural logarithms. It's also convenient to define \begin{equation} B = \frac{4}{re^r}. \end{equation}

The metric is \begin{equation} ds^2 = B dVdW-r^2 d\Omega^2. \end{equation}

The Christoffel symbols are as follows: \begin{align} \Gamma^V_{VV} &= (r^{-1}+r^{-2})We^{-r} \\ \Gamma^W_{WW} &= (r^{-1}+r^{-2})Ve^{-r} \\ \Gamma^\theta_{V\theta} = \Gamma^\phi_{V\phi} &= -WB/4r \\ \Gamma^\theta_{W\theta} = \Gamma^\phi_{W\phi} &= -VB/4r \\ % \Gamma^V_{\theta\theta} &= -Vr/2 \\ \Gamma^W_{\theta\theta} &= -Wr/2 \\ \Gamma^V_{\phi\phi} &= -(Vr/2) \sin^2\theta \\ \Gamma^W_{\phi\phi} &= -(Wr/2) \sin^2\theta \\ % \Gamma^\theta_{\phi\phi} &= -\sin\theta \cos\theta \\ \Gamma^\phi_{\theta\phi} &= \cot\theta \\ \end{align} I got these by calculating them in the computer algebra system Maxima and then cleaning up the resulting expressions by hand. I checked my cleaned-up versions numerically against the raw output from Maxima to make sure they were right. They are implemented in an open-source software project called karl.

The metric and the Christoffel symbols misbehave at the $r=0$ singularities, and also at the coordinate singularities at $\theta=0$ and $\pi$.

related: Free-fall path into a black hole in Kruskal Coordinates

$\endgroup$
  • 2
    $\begingroup$ @BenCrowell: Could you, please, rename your null coordinate from $W$ (to, say, $U$) so that it does not clash with Lambert function? Because, even after your warning, $W(\ldots W)$ is confusing. $\endgroup$ – A.V.S. May 14 '18 at 4:27

Not the answer you're looking for? Browse other questions tagged or ask your own question.