Symbolic tensor calculus on manifolds Éric Gourgoulhon Laboratoire Univers et Théories (LUTH) CNRS / Observatoire de Paris / Université Paris Diderot Université Paris Sciences et Lettres 92190 Meudon, France http://sagemanifolds.obspm.fr/jncf2018/ Journées Nationales de Calcul Formel Centre International de Rencontres Mathématiques Luminy, Marseille, France 22-26 January 2018 Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 1 / 38
Outline Introduction 1 Smooth manifolds 2 Scalar fields 3 Vector fields 4 Tensor fields 5 Conclusion and perspectives 6 http://sagemanifolds.obspm.fr/jncf2018/ Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 2 / 38
Introduction Outline Introduction 1 Smooth manifolds 2 Scalar fields 3 Vector fields 4 Tensor fields 5 Conclusion and perspectives 6 Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 3 / 38
Introduction What is tensor calculus on manifolds? By tensor calculus it is usually meant arithmetics of tensor fields tensor product, contraction (anti)symmetrization Lie derivative along a vector field pullback and pushforward associated to a smooth manifold map exterior calculus on differential forms ... Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 4 / 38
Introduction What is tensor calculus on manifolds? By tensor calculus it is usually meant arithmetics of tensor fields tensor product, contraction (anti)symmetrization Lie derivative along a vector field pullback and pushforward associated to a smooth manifold map exterior calculus on differential forms ... On pseudo-Riemannian manifolds: musical isomorphisms Levi-Civita connection curvature tensor Hodge duality ... Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 4 / 38
Introduction A few words about history Symbolic tensor calculus is almost as old as computer algebra: Computer algebra system started to be developed in the 1960’s; for instance Macsyma (to become Maxima in 1998) was initiated in 1968 at MIT Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 5 / 38
Introduction A few words about history Symbolic tensor calculus is almost as old as computer algebra: Computer algebra system started to be developed in the 1960’s; for instance Macsyma (to become Maxima in 1998) was initiated in 1968 at MIT In 1965, J.G. Fletcher developed the GEOM program, to compute the Riemann tensor of a given metric Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 5 / 38
Introduction A few words about history Symbolic tensor calculus is almost as old as computer algebra: Computer algebra system started to be developed in the 1960’s; for instance Macsyma (to become Maxima in 1998) was initiated in 1968 at MIT In 1965, J.G. Fletcher developed the GEOM program, to compute the Riemann tensor of a given metric In 1969, during his PhD under Pirani supervision, Ray d’Inverno wrote ALAM (Atlas Lisp Algebraic Manipulator) and used it to compute the Riemann tensor of Bondi metric. The original calculations took Bondi and his collaborators 6 months to go. The computation with ALAM took 4 minutes and yielded to the discovery of 6 errors in the original paper [J.E.F. Skea, Applications of SHEEP (1994)] Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 5 / 38
Introduction A few words about history Symbolic tensor calculus is almost as old as computer algebra: Computer algebra system started to be developed in the 1960’s; for instance Macsyma (to become Maxima in 1998) was initiated in 1968 at MIT In 1965, J.G. Fletcher developed the GEOM program, to compute the Riemann tensor of a given metric In 1969, during his PhD under Pirani supervision, Ray d’Inverno wrote ALAM (Atlas Lisp Algebraic Manipulator) and used it to compute the Riemann tensor of Bondi metric. The original calculations took Bondi and his collaborators 6 months to go. The computation with ALAM took 4 minutes and yielded to the discovery of 6 errors in the original paper [J.E.F. Skea, Applications of SHEEP (1994)] Since then, many software tools for tensor calculus have been developed... A rather exhaustive list: http://www.xact.es/links.html Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 5 / 38
Introduction Tensor calculus software Packages for general purpose computer algebra systems xAct free package for Mathematica [J.-M. Martin-Garcia] Ricci free package for Mathematica [J.L. Lee] MathTensor package for Mathematica [S.M. Christensen & L. Parker] GRTensor III package for Maple [P. Musgrave, D. Pollney & K. Lake] DifferentialGeometry included in Maple [I.M. Anderson & E.S. Cheb-Terrab] Atlas 2 for Maple and Mathematica SageManifolds included in SageMath Standalone applications SHEEP, Classi, STensor, based on Lisp, developed in 1970’s and 1980’s (free) [R. d’Inverno, I. Frick, J. Åman, J. Skea, et al.] Cadabra (free) [K. Peeters] Redberry (free) [D.A. Bolotin & S.V. Poslavsky] cf. the complete list at http://www.xact.es/links.html Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 6 / 38
Introduction Tensor calculus software Two types of tensor computations : Abstract calculus (index manipulations) xAct/xTensor MathTensor Ricci Cadabra Redberry Component calculus (explicit computations) xAct/xCoba Atlas 2 DifferentialGeometry SageManifolds Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 7 / 38
Introduction The purpose of this lecture Present a symbolic tensor calculus method that runs on fully specified smooth manifolds (described by an atlas) Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 8 / 38
Introduction The purpose of this lecture Present a symbolic tensor calculus method that runs on fully specified smooth manifolds (described by an atlas) is not limited to a single coordinate chart or vector frame Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 8 / 38
Introduction The purpose of this lecture Present a symbolic tensor calculus method that runs on fully specified smooth manifolds (described by an atlas) is not limited to a single coordinate chart or vector frame runs even on non-parallelizable manifolds Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 8 / 38
Introduction The purpose of this lecture Present a symbolic tensor calculus method that runs on fully specified smooth manifolds (described by an atlas) is not limited to a single coordinate chart or vector frame runs even on non-parallelizable manifolds is independent of the symbolic engine (e.g. Pynac/Maxima , SymPy ,...) used to perform calculus at the level of coordinate expressions Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 8 / 38
Introduction The purpose of this lecture Present a symbolic tensor calculus method that runs on fully specified smooth manifolds (described by an atlas) is not limited to a single coordinate chart or vector frame runs even on non-parallelizable manifolds is independent of the symbolic engine (e.g. Pynac/Maxima , SymPy ,...) used to perform calculus at the level of coordinate expressions Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 8 / 38
Introduction The purpose of this lecture Present a symbolic tensor calculus method that runs on fully specified smooth manifolds (described by an atlas) is not limited to a single coordinate chart or vector frame runs even on non-parallelizable manifolds is independent of the symbolic engine (e.g. Pynac/Maxima , SymPy ,...) used to perform calculus at the level of coordinate expressions with some details of its implementation in SageMath, which has been performed via the SageManifolds project : http://sagemanifolds.obspm.fr by these contributors: http://sagemanifolds.obspm.fr/authors.html Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 8 / 38
Smooth manifolds Outline Introduction 1 Smooth manifolds 2 Scalar fields 3 Vector fields 4 Tensor fields 5 Conclusion and perspectives 6 Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 9 / 38
Smooth manifolds Topological manifold Definition Let K be a topological field. Given an integer n ≥ 1 , a topological manifold of dimension n over K is a topological space M obeying the following properties: 1 M is a Hausdorff (separated) space 2 M has a countable base : there exists a countable family ( U k ) k ∈ N of open sets of M such that any open set of M can be written as the union of some members of this family. 3 Around each point of M , there exists a neighbourhood which is homeomorphic to an open subset of K n . Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 10 / 38
Smooth manifolds SageMath implementation See the online worksheet http://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/ blob/master/Worksheets/JNCF2018/jncf18_scalar.ipynb On CoCalc: https://cocalc.com/share/e3c2938e-d8b0-4efd-8503-cdb313ffead9/ SageManifolds/Worksheets/JNCF2018/jncf18_scalar.ipynb?viewer= share Direct links available at http://sagemanifolds.obspm.fr/jncf2018/ Éric Gourgoulhon Symbolic tensor calculus on manifolds JNCF, 25 Jan 2018 11 / 38
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