Specifying components of metric tensor then getting inverse

[John Conrad]

Hi I am trying to self-learn xAct and have really no one to ask. So I have an effective metric G[a,b] that is dependent on a background metric g[-a,-b]. This is basically those acoustic metrics in curved spacetime discussed in papers such as this https://journals.aps.org/prd/abstract/10.1103/PhysRevD.99.104047. 

How do I define G[a,b] in xAct so that I may then get its inverse G[-a,-b]? All of the tutorials on xCoba that I see require to put explicit expressions when defining a chart. But I want the background g[-a,-b] to be diagonal but have arbitrary components. I've attached a screencap of the object that I want to construct 

[Jose]

Hi,

There is currently no automated way in xAct to compute the inverse of an object like this. For this type of computation, it's better if you construct the objects yourself (both the metric and its inverse) and then use xAct to check that they are indeed inverse of each other. You can construct an ansatz with free parameters and then solve equations for them. I attach a small notebook with an example of a simplified case.

Cheers,

Jose.