Plugging in an ansatz for a metric perturbation

[Claire]

Hello,

After using "Perturbation" and "ExpandPerturbation" I have an equation that is the perturbed eom for the metric involving metric perturbations deltag^1, and I want to plug in the components for a given chart.

I created a CTensor that gives the ansatz for this perturbed metric, but when I try to plug it into the equation via  /. deltag^1-> CTensor[...], it doesn't work.

Should I be doing this differently?

Best wishes,

Claire

[Jose]

Hi,

I attach your notebook with a few changes and comments. The main problem was that you had imitated two separate implementations of the procedure to replace an abstract metric by a component metric, and they were not interacting well with each other. I just removed one of them and completed the computation with the other.

Cheers,

Jose.

[Claire]

Hi again,

I have two follow up questions.

1. I noticed that when I ask for certain components of tensors, say the Ricci tensor with one index up and one down, it returns an expression that contains the metric with raised indices, which of course makes sense. However, clearly it means these terms have not been computed, and I'm wondering if this can cause any problems in calculations, and therefore if I should compute them? It doesn't seem to have caused any issues in the equations of motion I have, however surely at some point I will encounter some of the curvature tensors with raised indices. In fact the tensor A appears in the equations of motion with a raised index, but I have only defined it with a lowered index, so why does this not cause problems?
2. The only time the covariant derivative is defined is when the metric is introduced, but I have seen some documentation where it is introduced separately. I just want to be sure that it is computing the correct Christoffel symbols in the equations of motion, because there seems to be 3 different covariant derivatives defined: {PD, CD, PDBL}. So which one is being used in the eoms?

Best wishes,

Claire

[Jose]

Hi,

Question 1.: I guess you mean computations of the form ctensor[{...}, {...}]. In principle this should be handled by the function ReplaceMetric in your notebook. I mean, if you have an expression expr with metric factors that stay symbolic, try ReplaceMetric[expr, ...] and if this doesn't work then send an example.

Question 2.: When you work with an explicit metric (given as a CTensor object met) and a chart BL, then you need to use just CD = CovDOfMetric[met] and the derivative PDBL of BL. The derivative PD is there to play the role of abstract "partial derivative" associated to a generic chart, in the same way that the metric g defined with DefMetric[-1, g[-a, -b], cd] is a generic metric. Therefore we can work with generic abstract situations with g, cd, PD and then we can replace them respectively with met, CD, PDBL for concrete cases.

Cheers,

Jose.