The Invar tensor package: Differential invariants of Riemann

TL;DR

This paper introduces the Invar tensor package, a computational tool that efficiently finds relations among scalar invariants of the Riemann tensor up to high derivatives, enabling canonical forms and simplifying complex tensor expressions.

Contribution

The paper presents an extension of the Invar system capable of producing canonical forms for Riemann invariants with high derivatives, incorporating a comprehensive database of relations and symmetries.

Findings

Successfully computes relations for all invariants up to 12 derivatives

Produces canonical forms within seconds for complex tensor objects

Integrates with existing tensor algebra systems for practical use

Abstract

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. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica) and Canon (for Maple).