×

Spinors: a Mathematica package for doing spinor calculus in general relativity. (English) Zbl 1296.83004

Summary: The Spinors software is a Mathematica package which implements 2-component spinor calculus as devised by Penrose for General Relativity in dimension 3+1. The Spinors software is part of the xAct system, which is a collection of Mathematica packages to do tensor analysis by computer. In this paper we give a thorough description of Spinors and present practical examples of use.
Program summary
Program title: Spinors{ }Catalogue identifier: AEMQ_v1_0{ }Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMQ_v1_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.: 117039{ }No. of bytes in distributed program, including test data, etc.: 300404{ }Distribution format: tar.gz{ }Programming language: Mathematica.{ }Computer: Any computer running Mathematica 7.0 or higher.{ }Operating system: Any operating system compatible with Mathematica 7.0 or higher.{ }RAM: 94Mb in Mathematica 8.0.{ }Classification: 1.5.{ }External routines: Mathematica packages xCore, xPerm and xTensor which are part of the xAct system. These can be obtained at http://www.xact.es.{ }Nature of problem: Manipulation and simplification of spinor expressions in General Relativity.{ }Solution method: Adaptation of the tensor functionality of the xAct system for the specific situation of spinor calculus in four dimensional Lorentzian geometry.{ }Restrictions: The software only works on 4-dimensional Lorentzian space-times with metric of signature (\(1, -1, -1, -1\)). There is no direct support for Dirac spinors.{ }Unusual features: Easy rules to transform tensor expressions into spinor ones and back. Seamless integration of abstract index manipulation of spinor expressions with component computations.{ }Running time: Under one second to handle and canonicalize standard spinorial expressions with a few dozen indices. (These expressions arise naturally in the transformation of a spinor expression into a tensor one or vice versa).

MSC:

83-04 Software, source code, etc. for problems pertaining to relativity and gravitational theory
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Penrose, R., A spinor approach to general relativity, Ann. Physics, 10, 171-201 (1960) · Zbl 0091.21404
[2] Martín-García, J. M., xAct: efficient tensor computer algebra
[3] Maître, D.; Mastrolia, P., S@m, a mathematica implementation of the spinor-helicity formalism, Comput. Phys. Comm., 179, 7, 501-534 (2008) · Zbl 1197.83007
[4] Peeters, K., Introducing Cadabra: a symbolic computer algebra system for field theory problems · Zbl 1196.68333
[5] Czapor, S. R.; McLenaghan, R. G.; Carminati, J., The automatic conversion of spinor equations to dyad form in MAPLE, Gen. Relativity Gravitation, 24, 9, 911-928 (1992) · Zbl 0758.53047
[6] Penrose, R.; Rindler, W., (Spinors and Space-Time. Spinors and Space-Time, Cambridge Monographs on Mathematical Physics, vol. 1 (1987), Cambridge University Press: Cambridge University Press Cambridge)
[7] Ashtekar, A., (Lectures on Non-Perturbative Canonical Gravity. Lectures on Non-Perturbative Canonical Gravity, Advanced Series in Astrophysics and Cosmology (1991), World Scientific: World Scientific Singapore) · Zbl 0948.83500
[8] Nakahara, M., (Geometry, Topology and Physics. Geometry, Topology and Physics, Graduate Student Series in Physics (1990), Adam Hilger Ltd.: Adam Hilger Ltd. Bristol) · Zbl 0764.53001
[9] Ashtekar, A.; Horowitz, G. T.; Magnon-Ashtekar, A., A generalization of tensor calculus and its applications to physics, Gen. Relativity Gravitation, 14, 5, 411-428 (1982) · Zbl 0491.53059
[10] Schouten, J. A., (Ricci Calculus. Ricci Calculus, Die Grundlehren der Matematischen Wissenschaften (1954), Springer Verlag: Springer Verlag Berlin)
[11] Martín-García, J. M., xPerm: fast index canonicalization for tensor computer algebra, Comput. Phys. Comm., 179, 597-603 (2008) · Zbl 1197.15002
[12] Martín-García, J. M.; Portugal, R.; Manssur, L., The Invar tensor package, Comput. Phys. Comm., 177, 8, 640-648 (2007) · Zbl 1196.15006
[13] Martín-García, J. M.; Yllanes, D.; Portugal, R., The Invar tensor package: differential invariants of Riemann, Comput. Phys. Comm., 179, 8, 586-590 (2008) · Zbl 1197.15001
[14] Brizuela, D.; Martín-García, J. M.; Mena Marugán, G. A., xPert: computer algebra for metric perturbation theory, Gen. Relativity Gravitation, 41, 10, 2415-2431 (2009) · Zbl 1176.83004
[15] Bäckdahl, T.; Valiente Kroon, J. A., On the construction of a geometric invariant measuring the deviation from Kerr data, Ann. Henri Poincaré, 11, 1225-1271 (2010) · Zbl 1208.83027
[16] Bäckdahl, T.; Valiente Kroon, J. A., Geometric invariant measuring the deviation from Kerr data, Phys. Rev. Lett., 104, 231102 (2010), 4
[17] Bäckdahl, T.; Valiente Kroon, J. A., The ‘non-Kerrness’ of domains of outer communication of black holes and exteriors of stars, Proc. R. Soc. A, 467, 1701-1718 (2011) · Zbl 1228.83056
[18] Edgar, S. B.; García-Parrado, A.; Martín-García, J. M., Petrov \(D\) vacuum spaces revisited: identities and invariant classification, Classical Quantum Gravity, 26, 10, 105022 (2009), 13 · Zbl 1166.83015
[19] Witten, E., A new proof of the positive energy theorem, Comm. Math. Phys., 80, 3, 381-402 (1981) · Zbl 1051.83532
[20] Nester, J. M., A new gravitational energy expression with a simple positivity proof, Phys. Lett. A, 83, 6, 241-242 (1981)
[21] Frauendiener, J., Geometric description of energy-momentum pseudotensors, Classical Quantum Gravity, 6, 12, L237-L241 (1989) · Zbl 0687.53071
[22] Szabados, L. B., On canonical pseudotensors, Sparlings’s form and Noether currents, Classical Quantum Gravity, 9, 11, 2521-2541 (1992) · Zbl 0776.53060
[23] Stewart, J., (Advanced General Relativity. Advanced General Relativity, Cambridge Monographs on Mathematical Physics (1991), Cambridge University Press: Cambridge University Press Cambridge)
[24] Bergqvist, G., Simplified spinorial proof of the positive energy theorem, Phys. Rev. D (3), 48, 2, 628-630 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.