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.
Similar content being viewed by others
References
Einstein, A.: Die Grundlage der allgemeinen Relativatstheorie. Ann. Phys. 49, 769–822 (1916)
Huf, P.A., Carminati, J.: http://bach2roq.com/science/maths/gr/GR-software-packages.html (2014). Accessed 3 Apr 2017
MacCallum, M. A. H., Skea, J. E. F.: SHEEP: a computer algebra system for general relativity. In: Rebouças, M.J., Roque, W.L. (eds.) Algebraic Computing in General Relativity. (Proceedings of the First Brazilian School on Computer Algebra, vol. 2). Oxford University Press, Oxford (1994)
d’Inverno, R.: Introducing Einstein’s Relativity. Oxford University Press, Oxford (1992)
MacCallum, M.A.H.: http://www.maths.qmul.ac.uk/~mm/shp/ (2011). Accessed 26 Oct 2017
Hearn, A.C.: REDUCE. http://reduce-algebra.sourceforge.net/ (2009–2017). Accessed 26 Oct 2017
Garcia-Parrado Gomez-Lopez, A.: Dynamical laws of superenergy in general relativity. Class. Quantum Gravit. 25, 015006 (2008)
Huf, P.A., Carminati, J.: TensorPack: a Maple-based software package for the manipulation of algebraic expressions of tensors in general relativity. J. Phys. (Conf. Ser.) 633, 012021 (2015) (see online at http://iopscience.iop.org/article/10.1088/1742-6596/633/1/012021/meta)
Ellis, G.F.R., Maartens, R., MacCallum, M.A.H.: Relativistic Cosmology. Cambridge University Press, Cambridge (2012)
Ehlers, J.: Contributions to the relativistic mechanics of continuous media. Gen. Relativ. Gravit. 25(12), 1225–66 (1993)
Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973)
Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover, New York (1960)
Senovilla, J.M.M., Sopuerta, C.F., Szekeres, P.: Theorems on shear-free perfect fluids with their Newtonian analogues. Gen. Relativ. Gravit. 30, 389–411 (1998)
Portugal, R.: The Riemann Tensor Package. http://www.cbpf.br/~portugal/Riemann.html (2008). Accessed 3 Apr 2017
Sopuerta, C.: Applications of timelike and null congruences to the construction of cosmological and astrophysical models. Ph.D. Thesis (1996)
Treciokas, R.E., Ellis, G.F.R.: Isotropic solutions of the Einstein–Boltzmann equations. Commun. Math. Phys. 23, 1–22 (1971)
Van den Bergh, N.: The shear-free perfect fluid conjecture. Class. Quantum Gravit. 16, 13 (1999)
Collins, C.B.: Shear-free fluids in general relativity. Can. J. Phys. 64(2), 191–9 (1986)
Huf P.A., Carminati, J.: http://www.bach2roq.com/science/maths/GR/ShearFreeProofs.html (2017). Accessed 3 April 2017
Kay, D.C.: Schaums Outline of Theory and Problems in Tensor Calculus. McGraw-Hill, New York (1988)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huf, P.A., Carminati, J. Elucidation of covariant proofs in general relativity: example of the use of algebraic software in the shear-free conjecture in MAPLE. Gen Relativ Gravit 50, 5 (2018). https://doi.org/10.1007/s10714-017-2325-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-017-2325-5