Cadabra manual: SatisfiesBianchi

SatisfiesBianchi

Make an object satisfy the generalised Bianchi identity.

Indicates that an object satisfies a (generalised) Bianchi identity. This is often used to link a derivative operator to a curvature tensor, as in

R_{m n p q}::RiemannTensor. D{#}::Derivative. D_{m}{ R_{n p q r} }::SatisfiesBianchi. A^{m n p q}::AntiSymmetric. ex:= D_{m}{ R_{n p q r} } A^{m n p q};

\(\displaystyle{}D_{m}\!{R_{n p q r}} A^{m n p q}\)

young_project_tensor(_);

\(\displaystyle{}D_{m}\!\left(\frac{1}{12}\,R_{n p q r} +\frac{1}{12}\,R_{q p n r} +\frac{1}{12}\,R_{n r q p} +\frac{1}{12}\,R_{q r n p} - \frac{1}{12}\,R_{p n q r} - \frac{1}{12}\,R_{q n p r} - \frac{1}{12}\,R_{p r q n} - \frac{1}{12}\,R_{q r p n} - \frac{1}{12}\,R_{n p r q} - \frac{1}{12}\,R_{r p n q} - \frac{1}{12}\,R_{n q r p} - \frac{1}{12}\,R_{r q n p} +\frac{1}{12}\,R_{p n r q} +\frac{1}{12}\,R_{r n p q} +\frac{1}{12}\,R_{p q r n} +\frac{1}{12}\,R_{r q p n}\right) \left(\frac{1}{24}\,A^{m n p q} - \frac{1}{24}\,A^{m n q p} - \frac{1}{24}\,A^{m p n q} +\frac{1}{24}\,A^{m p q n} +\frac{1}{24}\,A^{m q n p} - \frac{1}{24}\,A^{m q p n} - \frac{1}{24}\,A^{n m p q} +\frac{1}{24}\,A^{n m q p} +\frac{1}{24}\,A^{n p m q} - \frac{1}{24}\,A^{n p q m} - \frac{1}{24}\,A^{n q m p} +\frac{1}{24}\,A^{n q p m} +\frac{1}{24}\,A^{p m n q} - \frac{1}{24}\,A^{p m q n} - \frac{1}{24}\,A^{p n m q} +\frac{1}{24}\,A^{p n q m} +\frac{1}{24}\,A^{p q m n} - \frac{1}{24}\,A^{p q n m} - \frac{1}{24}\,A^{q m n p} % +\frac{1}{24}\,A^{q m p n} +\frac{1}{24}\,A^{q n m p} - \frac{1}{24}\,A^{q n p m} - \frac{1}{24}\,A^{q p m n} +\frac{1}{24}\,A^{q p n m}\right)\)

distribute(_) canonicalise(_);

\(\displaystyle{}0\,\)