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+2 votes


After a computation I ended up in expression that involves partial derivatives. I want to simplofy that to covariant derivative expression . I am expressing a prototype of the problem here

Suppose after a computation i ended up in a expression like

\begin{align} \partial_{a}A^{b} + \frac{1}{2}A^{c} q^{b d}\partial_{a} q_{d c} + \frac{1}{2} A^{c} q^{b d}\partial_{c}q_{a d} - \frac{1}{2} A^{c} q^{b d}\partial_{d}q_{a c} \end{align}

I want to simplify this to $\nabla_{a}A^{b}$. For this I used the following code

{u, r, z1 , z2}::Coordinate;
{a , b , c , d , e , f , g , h , i , j , k , l , m , n#}::Indices(values={(z1),(z2)}, position=fixed);
q_{a b}::Metric;
q^{a b}::InverseMetric;
q{#}::Depends(u , a , b , c , d , e , f , g , h , i , j , k , l , m , n# ,\partial{#});
\Gamma^{a}_{b c}::TableauSymmetry(shape={2}, indices={1,2});
A_{a}::Depends(u, r , a , b , c , d , e , f , g , h , i , j , k , l , m , n#);

rule:={\partial_{a}{A^{b}} -> \nabla_{a}{A^{b}} - \Gamma^{b}_{a c} A^{c} , \partial_{a}{q_{b c}} -> \Gamma^{d}_{a b} q_{d c} + \Gamma^{d}_{a c} q_{b d}}  };

test:=\partial_{a}{A^{b}} + (1/2)*A^{c} q^{b d}\partial_{a}{q_{d c}} + (1/2)*A^{c} q^{b d}\partial_{c}{q_{a d}} - (1/2)*A^{c} q^{b d}\partial_{d}{q_{a c}};

substitute(test, rule);

But i ended up in a completely different expression . I think the issue might be that CHristoffel symbols are raised and lowered w.r.t metric. Can you please help me out to simplify such expressions.

in General questions by (650 points)

1 Answer

+3 votes

If you want to prevent canonicalise from raising or lowering indices, declare those indices with the position=independent property (not position=fixed). That seems to work in your example.

You will still need some eliminate_metric, eliminate_kronecker and sort_product after that.

by (70.9k points)