xAct for the General Theory of Relativity
A very introductory set of video lectures on using xTensor, xCoba and xTerior in General Relativity
xAct is an efficient tensor algebra set of Wolfram Mathematica packages, used and trusted by many researchers in theoretical physics. It is written in the Wolfram language and it is given under a free, open source GPL license. You can download it from here, and find the documentation here. Installing it in your system is easy, and brief instructions are also given in the videos and the notebooks below.
You can also find the videos in a youtube video list. A separate list with computer exercises using Mathematica, can be found here. The full list of the instructor's videos related to the course can be found here.
- Introduction to xTensor for symbolic calculations in General Relativity. Downloading, installing, finding documentation. Defining a Manifold with metric and its Christoffel connection. Defining tensors. Tensorial expressions and contractions. \(\delta_\mu^\nu, \delta_{\mu_1\ldots\mu_n}^{\nu_1,\ldots,\nu_n}, \epsilon_{\mu_1,\ldots,\mu_n}\). Covariant derivative \(\nabla_\mu \) and \(\partial_\mu\). Lie derivatives and Lie brackets. Exercise: Prove that \({\cal L}_{[u,v]} w = [{\cal L}_u,{\cal L}_v]w \). (Math nb, Math pdf)
- Using xTensor for symbolic computations involving the Riemann tensor and its contractions. Symmetries of \( R^\mu{}_{\nu\rho\sigma}, R_{\mu\nu}, G_{\mu\nu},\ldots \). Manipulations using RiemannToChristoffel, EinsteinToRicci, WeylToRiemann, TFRicciToRicci. Commutator \([\nabla_\mu,\nabla_\nu] \) and the functions SortCovDs, CommuteCovDs. Rules and substitutions: MakeRule, IndexSet, IndexRule. Exercises: Bianchi identities, geodesic deviation equation, \(\nabla_\mu\nabla_\nu \xi_\lambda = R_{\lambda\nu\mu\rho} \xi^\rho \) (Math nb, Math pdf)
- Introduction to xCoba. Use of the xCoba Mathematica package to do tensor computations with components in General Relativity. (Math nb, Math pdf)
- Use of the xCoba Mathematica package to compute the components of curvature tensors, Christoffel symbols, and the geodesic equations. Download the template notebook. (Friedmann [pdf], Schwarzschild [pdf], Rindler [pdf])
- Differential forms: The xTerior differential forms' computer algebra package (part of xAct). Wedge product, exterior derivative, coordinate transformations. (Math nb, Math pdf)
- Variations of the metric with xTras: (only Math nb, Math pdf, documentation)
