Moving between XTensor and CTensor

[Rhiannon Gwyn]

Hi,

I wonder if anyone can help. I am trying to evaluate variations with respect to the metric for specific metrics. I can calculate them in general using XTensor and VarL:

VarL[metric[-a, -b], cd][Expression]

but once I specify a metric using CTensor I can't use VarL anymore. So that's fine, I can copy across the expressions I get, like

-Riccicd[a, b] + (metric[a, b]*RicciScalarcd[])/2

But this is not the right form for CTensor calculations, I need to change things as in
CTensorconvert = {Riccicd -> Ricci[cd],
  RicciScalarcd[] -> RicciScalar[cd][], Riccicd -> Ricci[cd]}

But while this seems to work sometimes, for things that require contraction of indices I get nonsense, even though the answer is simple if I simply rewrite them by hand:

 metric[a, c]   metric[b, d]   Riccicd[-c, -d]  RicciScalarcd[]^3 /. CTensorconvert

does not give the expected

metric[a, c]metric[b, d]Ricci[cd][-c, -d] RicciScalar[cd][]^3

I must be doing something wrong - can anyone help? Alternatively is there a way to use VarL with CTensor?

Many thanks

Rhiannon

[Jose]

Hi,

Could you post code for a complete example that gives a wrong result using the CTensorconvert rules?

I don't expect VarL or VarD to give correct results once you are using CTensor expressions, but replacing abstract tensors with CTensor objects using rules should work.

Cheers,

Jose.

[Rhiannon Gwyn]

Hi

Here is a shortened file... The example at the end (and more complicated expressions) doesn't work as expected, presumably because I need to tell it to contract the indices.

Thanks!

Rhiannon

[Leo Stein]

Hi Rhiannon,

I have edited your notebook a bit. I hope the edits explain everything that's going on.

Best,

Leo

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[Leo Stein]

Oops, I sent the previous version of the file a bit prematurely. Hope you can spot the difference :)

Leo

[Rhiannon Gwyn]

Hi Leo,

Thanks, that is very helpful! I'm aware I was writing the metric initially without using tensors - that is because for the metrics I am actually interested in, writing things like

met = CTensor[   DiagonalMatrix[{-1 + 2 mass/r[], 1/(1 - 2 mass/r[]), r[]^2,  r[]^2 Sin[\[Theta][]]^2}], {-ch, -ch}];

for a metric which is not necessarily diagonal, was not very user friendly. The example in the file was just a simple example using older code.

But the redefinitions work perfectly. It makes sense that it's the order of evaluation that was the problem. Thanks!

Rhiannon