Collected paper of Charles Sanders Pierce

Abstract

The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 ṡ 10 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 6 ṡ 10 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). Program summary: Program title:Invar Tensor Package v2.0. Catalogue identifier:ADZK_v2_0. Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZK_v2_0.html. Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland. Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html. No. of lines in distributed program, including test data, etc.:3 243 249. No. of bytes in distributed program, including test data, etc.:939. Distribution format:tar.gz. Programming language:Mathematica and Maple. Computer:Any computer running Mathematica versions 5.0 to 6.0 or Maple versions 9 and 11. Operating system:Linux, Unix, Windows XP, MacOS. RAM:100 Mb. Word size:64 or 32 bits. Supplementary material:The new database of relations is much larger than that for the previous version and therefore has not been included in the distribution. To obtain the Mathematica and Maple database files click on this link. Classification:1.5, 5. Does the new version supersede the previous version?:Yes. The previous version (1.0) only handled algebraic invariants. The current version (2.0) has been extended to cover differential invariants as well. Nature of problem:Manipulation and simplification of scalar polynomial expressions formed from the Riemann tensor and its covariant derivatives. Solution meth…