Raise/lower indices with xPert

[Sashwat Tanay]

Hi everybody,

In the following screenshot, var1 is the full (perturbed) metric. The perturbation is around the flat metric g_{ab}. As is usual in perturbation theory, indices are raised/powered not with the full metric but rather the flat background metric g_{ab}.

I implement it in a contrived 2-step process:

1. Contract the tensor with the flat background metric.

2. Set the partial derivatives of this metric to 0 since it's flat.

Is there a more efficient and straightforward way to achieve the same goal?

 

Regards,

Sashwat Tanay

[Jose]

Hi,

xAct assumes by default that the metric is not flat. There is an option of DefCovD and DefMetric called FlatMetric, with default value False. If your metric is flat then you should use DefMetric[..., FlatMetric -> True]. This makes the associated curvature tensors to be zero, simplifying many computations.

But this does not yet make PD[-a][g[-b, -c]] give zero. This is a coordinate condition that must be set separately with something like the upvalue

   PD[_][_g] ^:= 0

Then you will be able to contract the metric just using ContractMetric, with no additional rules needed.

Cheers,

Jose.