The Invar tensor package: Differential invariants of Riemann

Understand

The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6x10^23 objects with up to 12 derivatives of the metric.

• This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann.
• We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis.
• The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6x10^5 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants.