The idea here is to write the simplest program to carry out the computation explicit in these equations:<br>∂<sub>t</sub>g<sub>ab</sub> = –2αK<sub>ab</sub> + ∇<sub>a</sub>β<sub>b</sub> + ∇<sub>b</sub>β<sub>a</sub>
<br>∂<sub>t</sub>K<sub>ab</sub> = –∇<sub>a</sub>∇<sub>b</sub>α + α(R<sub>ab</sub> + K<sub>ab</sub>K – 2K<sub>ac</sub>K<sup>c</sup><sub>b</sub>)
+ β<sup>c</sup>∇<sub>c</sub>K<sub>ab</sub>
+ K<sub>ca</sub>∇<sub>b</sub>β<sup>c</sup>
+ K<sub>cb</sub>∇<sub>a</sub>β<sup>c</sup>
<br>By simplest I presume to minimize the effort to understand the program and see that it is indeed doing the intended calculation.
<p>We must compute the <a href=http://en.wikipedia.org/wiki/Christoffel_symbols>Christoffel symbols</a>:
<br>Γ<sup>i</sup><sub>kl</sub>
= (1/2)g<sup>im</sup>(g<sub>mk,l</sub> + g<sub>ml,k</sub> – g<sub>kl,m</sub>)
<br>and for that we need g<sup>im</sup> — the inverse of the metric tensor.
<p>With these values the <a href=http://en.wikipedia.org/wiki/Covariant_derivative>covariant derivative</a> is:
<br>∇<sub>a</sub>β<sub>b</sub> = β<sub>b;a</sub> = ∂β<sub>b</sub>/∂x<sup>a</sup> – Γ<sup>c</sup><sub>ab</sub>β<sub>c</sub>
<br>along with the other familiar equations.
<p>Taking simple derivatives of field values with respect to spacial coordinates is fraught with centering issues.
I plan to use the simplest centering schemes and document how each value is centered within the computational zone.