Define two different metrics on the same manifold

[Bogdan Ganchev]

I would like to ask whether it is possible to define two different metrics on the same manifold and then have Christoffel symbols, Riemann, Ricci and Einstein tensors for each of them.
I tried the obvious by just evaluating DefMetric a second time with a different symbol and a different covariant derivative.

DefManifold[M, 4, {a, b, c, d, e, f, i, j, k, l}]

DefMetric[-1, g[-a, -b], CD, {";", "\[EmptyDownTriangle]"}]

DefMetric[-1, h[-a, -b], Cd, {"|", "\[FilledDownTriangle]"}]

Initially it seemed as though I had what I wanted, but then I started computing things and I noticed that the second metric is not really treated properly as such.

Writing h[b,c]h[-a,-b] doesn't reduce to a delta function, no matter what I do, hence in my calculations I get things like h[i,-j] which doesn't simplify to a delta function.

I need this because I want to define a Tensor that is the difference between two connections which are with respect to two different metrics and then take its covariant derivative (with respect to one of the metrics).

Regards

[Alex Vañó Viñuales]

Hi Bogdan,

from my understanding, xAct only supports one metric as the principal metric of a manifold, the one it uses with ToCanonical, moving of indices and other simplifications. When I worked with several metrics, I had to do extra work (like careful substitutions) in the calculations to obtain the results I wanted. Maybe somebody with a deeper expertise can complement my simple explanation.

Cheers,

Alex

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[Bogdan Ganchev]

I just missed the fact that the second use of DefMetric defines the inverse of h[-a,-b] as Invh[-a,-b]. After noticing this, everything is fine.