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


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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

  • 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

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