Issue - Running SeparateMetric using a frozen metric, gives a different result than using the active (first) metric

[BillN]

Hi,

I set up an "active" (first) metric g and a "frozen" (second) metric a on the tangent space of a single manifold.  Then formed the raised covariant derivative AD[c][v[b]] of a vector, which is the lowered compatible covariant derivative of the frozen metric a, then raised by the active metric g.  Then applied SeparateMetric[g] to this, as well as SeparateMetric[a], and got two different results that don't agree with each other.  A notebook giving a side-by-side comparison of the results is attached.

I'm an xAct newby, but I thought that the results should be equal in this case (they may initially look different, but would be found to be equal when manipulated in a side-by-side comparison).  Am I misinterpreting this?

If they should be equal, perhaps just like ContractMetric only working for the active metric, should SeparateMetric have the same limitation to avoid this problem?

Regards,

Bill

[Jose]

Hi,

The actions of contracting and separating a metric cannot be performed with frozen metrics. Only the first metric of each vbundle can be contracted and separated. The fact that both ContractMetric and SeparateMetric allow you to specify a metric is required for those cases in which we have several vbundles and hence potentially several first metrics.

I see that ContractMetric throws a message if you try to use a frozen metric, but SeparateMetric silently performs the (incorrect) action of separating a frozen metric (try for example SeparateMetric[a][v[-b]] in your case). I don't remember right now a strong reason to allow this (and I think I'll forbid it), but in any case you should never force contraction or separation of frozen metrics. That goes against the convention in xTensor: indices of a vbundle are raised and lowered with the first metric of that vbundle.

Cheers,

Jose.

[BillN]

Thanks Jose,

I thought what you said here was the case, so I simply now avoid applying SeparateMetric except for the first metric of a vbundle.

Regards,

Bill