xCoba and 3+1 decomposition

[Jolyon Bloomfield]

Hi,

I'm interested in getting xCoba to do some calculations in a 3+1
decomposition. In particular, I'd like to get it to calculate the
extrinsic curvature tensor, and the acceleration vector. However, I
can't seem to get the system to calculate these quantities, and so I
find myself resorting to using rules to replace them with their
definitions in terms of the normal and the metric, which I can supply
values for in xCoba, and which will then be calculated. Is there a
better way to do this? (or even just a way to convert those two
quantities automatically, without resorting to dodgy rules?)

Thanks,
Jolyon

[JMM]

Hi Jolyon,

There is quite a lot of functionality in xTensor to handle n+1
decomposition in abstract covariant form (section 7.8 of xTensorDoc),
in particular the definitions of the ExtrinsicK and AccelerationA
tensors, and the GaussCodazzi equations. However, there is no special
support in xCoba. As in the previous thread on the geodesic equation,
the problem is how to actually define the surface or foliation you are
working with. I guess you would have to define a chart (hopefully
adapted), and then use scalar-functions to give some coordinates in
terms of others, feed this to MetricCompute to calculate things of the
ambient metric, and finally project or compute what you want using the
formulas given by xTensor.

Cheers,
Jose.