How useful would a metric independant basis be?

[CPL.Luke]

if I could write down an operator that depended on the metric but always had the same eigenvalues regardless of the metric, how useful would that be?

I ask as I have an idea for how to accomplish this but want to know how much time its worth, ie is it just a curiosity or something more?

 

[0xDEADBEEF]

Maybe someone else understands better from your hints what it is that you want to do.

Just some thoughts:
If you embed your space in a hyperspace, and only operate on the hyperspace, then this is a basis independent of the subspaces "internal" metric. So if your subspace bends and curls, you can take an outside standpoint. The question is, how the hyperspace connects to reality.

 

[Hurkyl]

It's easy to classify all such operators. An operator satisfies your condition if and only if, it's a diagonal matrix when written in the coordinates given by the basis of eigenvectors. So, you just need one function of the shape "metric --> complex number" for each diagonal entry. The only thing remaining is to make sure that your different functions always take on distinct values, so that none of your operators have extra eigenvalues.

(I suppose things become more complicated if it doesn't have a complete basis of eigenvectors...)


As to how useful it would be... I couldn't say. It could be that your particular construction reveals a subtle property of the metric, or maybe some such operators collect a bunch of information together in a useful way, or maybe doing this construction with different eigenvalue sets reveals something interesting or maybe it's just be a curiousity, or maybe something entirely different is going on.

 

[CPL.Luke]

well I was thinking along the lines of an operator of the type you find in quantum mechanics, which are linked together via a metric

for instance if you take the energy momentum relation in General relativity it depends on a metric, now if I were to write such a thing such that this metric dependant operator had a single set of eigenvalues regardless of what metric was involved wheter it would be of some use. I was thinking it might form a useful basis with which to calculate, as it wouldn't matter what co-ordinate system you use but I wasn't to sure.