"factor_out" algorithm and unequal indices

Question

Hi, Let me explain my question using the following example:

{a, b, c, d, e, f}::Indices.
\eta_{a b}::Metric.
\eta_{a}^{b}::KroneckerDelta.
F_{a}^{b} G_{b c} + F^{b d}G_{d a c b};
@factor_out!(%){F_{a b}};

In this case the @factor_out doesn't work because there is no F_{a b} in the expression. On solution might be transform the F_{a}^{b} and F^{b d} into F_{a b} using KroneckerDelta and the @substitute command and then factor it out. But this solution for a long expression with many indices is really cumbersome. I would like to know whether exists any more genuine solution.

Cheers!

Answer 1

With the rewrite_indices which is now in the version on github, you can do

{a, b, c, d, e, f}::Indices(position=fixed).
\eta_{a b}::Metric.
ex:=F_{a}^{b} G_{b c} + F^{b d}G_{d a c b};
rewrite_indices(_, $F_{a b}$, $\eta^{a b}$);

This inserts as many $\eta^{a b}$ (with the right indices) as necessary in order to rewrite the expression using $F_{a b}$ (with lower indices) only.

Note the position of the indices in the rewrite_indices algorithm: the first argument should give the tensor with the indices in the position in which you want to have them, the second argument is the metric/conversion tensor which you want to use for this. The names of the indices on these two arguments does not matter (just use the right index type).

Answer 2

I could solve my problem as follow. But, still may exist more simple solutions.

{a, b, c, d, e, f, g, h,}::Indices(vector, position=independent).
{\mu, \nu}::Indices(factor).
\eta_{a b}::Metric.
\eta^{a b}::InverseMetric.
\eta_{a b}::Symmetric.
\eta_{a}{b}::KroneckerDelta. 
F_{a b}::Symmetric.
F_{a}^{b} G_{b c} + F^{b d}G_{d a c b}; 
@substitute!(%)(F_{a}^{b} -> \eta^{b c}F_{c a});
@substitute!(%)(F^{a b} -> \eta^{c a} \eta^{d b} F_{c d});
@substitute!(%)(F_{c? d?} -> \eta_{\mu}^{c?} \eta_{\nu}^{d?} F_{\mu \nu});
@factor_out!(%){F_{b? d?}}; 

Comments

At the moment this is the only way to do this. As soon as rewrite_indices is back in working order again, you will be able to do this kind of rewriting in a less cumbersome form.