Cadabra
a field-theory motivated approach to computer algebra

eliminate_metric

Eliminate metrics by raising or lowering indices.
Eliminate metric and inverse metric objects by raising or lowering indices.
{m, n, p, q, r}::Indices(vector, position=fixed). {m, n, p, q, r}::Integer(0..9). g_{m n}::Metric. g^{m n}::InverseMetric. g_{m}^{n}::KroneckerDelta. g^{m}_{n}::KroneckerDelta. ex:=g_{m p} g^{p m}; eliminate_metric(_);
\(\displaystyle{}g_{m p} g^{p m}\)
g_{m p} g^{p m}
\(\displaystyle{}g^{p}\,_{p}\)
g^{p}_{p}
eliminate_kronecker(_);
\(\displaystyle{}10\)
10
Related algorithms are eliminate_kronecker and eliminate_vielbein.
It is sometimes useful to eliminate only those metrics which have two dummy indices (so as to avoid changing indices on non-metric factors), as in the following example:
{a,b,c,d,e,f}::Indices(position=fixed); g_{a b}::Metric; g^{a b}::InverseMetric; ex:=X_{a} g^{a b} g_{b c} g^{c d} g_{d e} g^{e f}; eliminate_metric(ex, repeat=True, redundant=True);
\(\displaystyle{}\text{Property Indices(position=fixed) attached to }\left[a, b, c, d, e, f\right].\)
\(\displaystyle{}\text{Property Metric attached to }g_{a b}.\)
\(\displaystyle{}\text{Property TableauSymmetry attached to }g^{a b}.\)
\(\displaystyle{}X_{a} g^{a b} g_{b c} g^{c d} g_{d e} g^{e f}\)
X_{a} g^{a b} g_{b c} g^{c d} g_{d e} g^{e f}
\(\displaystyle{}X_{e} g^{e f}\)
X_{e} g^{e f}
Without the redundant=True option, this would have reduced the expression to $X^{f}$.
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