# Elucidation of covariant proofs in general relativity: example of the use of algebraic software in the shear-free conjecture in MAPLE

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## Abstract

In this paper we explore the use of a new algebraic software package in providing independent covariant proof of a conjecture in general relativity. We examine the proof of two sub-cases of the shear-free conjecture \(\displaystyle \sigma =0 {\displaystyle \, =>}\, {\displaystyle \omega }\,{\displaystyle \varTheta } =0\) by Senovilla et al. (Gen. Relativ. Gravit 30:389–411, 1998): case 1: for dust; case 2: for acceleration parallel to vorticity. We use TensorPack, a software package recently released for the Maple environment. In this paper, we briefly summarise the key features of the software and then demonstrate its use by providing and discussing examples of independent proofs of the paper in question. A full set of our completed proofs is available online at http://www.bach2roq.com/science/maths/GR/ShearFreeProofs.html. We are in agreeance with the equations provided in the original paper, noting that the proofs often require many steps. Furthermore, in our proofs we provide fully worked algebraic steps in such a way that the proofs can be examined systematically, and avoiding hand calculation. It is hoped that the elucidated proofs may be of use to other researchers in verifying the algebraic consistency of the expressions in the paper in question, as well as related literature. Furthermore we suggest that the appropriate use of algebraic software in covariant formalism could be useful for developing research and teaching in GR theory.

## Keywords

General relativity Covariant formalism Algebraic proof Einstein field equations Einstein summation index Shear-free conjecture Cosmology Maple Algebraic tensor software TensorPack## Notes

### Acknowledgements

We are grateful to J. Senovilla and C. Sopuerta for providing details of some of the more difficult sections of their proofs [13], and to N. Vandenbergh for his assistance in correcting some of our proofs. We acknowledge the authors of the Riemann and Canon packages.

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