Question about eliminate metric

Question

Hi,

I had a question about the command eliminate metric. I have a metric which depends upon the coordinates and I declared the indices position to be fixed and therefore the raising and lowering are carried out using the metric.

In a part of expression of mine which looks like

$$ g^{a b}\partial_{c} g_{b d}$$

When I am trying to use the command "eliminatemetric" to the above expression it is giving $$ \partial_{c} g^{a}{}\{d}$$

which is not true because since the metric is coordinate dependant one cannot pull the metric inside the partial derivative. How can I put a rule such that I can avoid such raising and lowering while keeping the other raising and lowering indices outside the partial derivatives allowed.

Answer 1

I can't reproduce that problem; the following returns the expression unchanged:

{a,b,c,d}::Indices(position=fixed);
\partial{#}::PartialDerivative;
g{#}::Depends(\partial{#});
g^{a b}::InverseMetric; 
g_{a b}::Metric;
ex:=g^{a b}\partial_{c}{g_{b d}};
eliminate_metric(ex);

Can you post a self-contained example that shows the problem?

Comments

Hi,

This is the code I tried {u, r, z1 , z2}::Coordinate: {a , b , c , d , e , f#}::Indices(values={z1 , z2}, position=fixed): {a,b,c,d}::Indices(position=fixed); g^{a b}::InverseMetric; g{a b}::Metric; g{a b}::Depends( a , b , c , d); g^{a b}::Depends( a , b , c , d); \partial{#}::PartialDerivative;

ex4:=g^{a b} \partial{c}{g{b d}}; eliminate_metric(ex4);

I guess the problem is that I have not included the partial dependence in the metric and the error might be due to that. Can you please confirm?

Also is it possible that in this case I can restrict the dependance of the metric only on the {a,b,c,d} indices which runs over z1 and z2 only and make the metric independant of "u" and "r" coordinate. so that when i get the following contraction

\begin{align} g^{a b}\partial{r}g{b c}=\partial{r} g^{a}{c} \end{align}

but not the contraction in the following expression \begin{align} g^{a b}\partial{c} g{b c} \end{align}

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That's correct, it is not possible to catch this with a dependence on indices.