# Kerr metric Christoffel symbols

I've been slaving away trying to calculate the Christoffel symbols for the Kerr metric.

Does anybody know of a link that I could compare my answers to? I've done some Google searches and all I can find are general discussions about the physical effects of the Kerr metric.

EDIT: Some great responses below. I also found this link www.scirp.org/journal/PaperDownload.aspx?paperID=25581 which lists the Christoffel symbols in its appendix.

• Two thoughts: (i) take the $J\to0$ limit should recover the Schwarzschild results, (ii) Exact Solutions of Einstein's Field Equations should contain the relevant information, if I recall correctly... Commented Mar 12, 2015 at 15:17

They are listed in here catalogue of spacetimes

• Nice. In a similar spirit (on the Robertson-Walker kind of metrics in different coordinates), I found this and this reference very helpful. Commented Mar 12, 2015 at 20:23

You can just write down

$$\Gamma^i{}_{k\ell}=\tfrac{1}{2}\,g^{im} \left(\partial_{x^\ell}g_{mk} + \partial_{x^k} g_{m\ell} - \partial_{x^m} g_{k\ell} \right)$$

in Mathematica. The example blow is for the Schwarzschild metric.

Here is the code. (You might have to patch the parts lost by my excessive use of display style.)

• Schwarzschild is easy; Kerr less so. Even changing the metric in Mathematica, one would need to manually specify substitutions in order for it to return anything remotely reasonable (i.e. not expanding the traditional $\Delta$ and $\rho^2$ abbreviations).
– user10851
Commented Mar 12, 2015 at 22:16
• @ChrisWhite: But it's easier than doing the same thing by hand. Commented Mar 12, 2015 at 22:44

Frolov's Black hole Physics (Google Books link) has an entire chapter on the Kerr metric, but states at the beginning of that chapter,

Mathematical properties of the Kerr metric and its generalization with electric charge included (the Kerr-Newman metric) are discussed in Appendix D.

Appendix D does include Christoffel symbols of the Kerr-Newman metric (obviously Kerr metric is the $Q=0$ limit) in Boyer-Lindquist coordinates and can be found at the Google books link (page 659). Sadly, the definition of $b$ and $p$ do not appear in the free preview1, so you'll have to procure a copy to find it (NB: it's pretty expensive (Amazon link) so you'll likely want to try your local University's library).

1 If someone does have a copy of the book, could you edit those two equations here?

Not a definite answer to your question, sorry, but if you absolutely can't find it on G. (or you lack the patience to dig through arxiv.org or G. for a lucky reference), one alternative that works for me when I just want confirmation is to visit amazon, stick in say "General Relativity" books and use their "look inside the book" feature. a screenshot of the amazon page might help. It's totally random that what you are looking for comes up, but it often does.

Hope this helps, if you havn't tried it yourself already

Or you could just post your derivation here, and ask, take you a while to write it out tho......