Example xAct notebooks
This is a collection of example xAct notebooks that contain common xAct usage patterns. Right click on a file name below and hit "Save link as..." to download a notebook.
General function to vary a Lagrangian
Example of angular integration on multiple direction vectors (adapted from Post-Newtonian computations)
Bianchi I Einstein equation using the CTensor framework of xCoba
Equations for Cauchy-characteristic extraction, in 1+1+2 form. The
angular sector is treated with spin-weighted scalars and
raising/lowering operators eth (or edth). A large number of features
are demonstrated, including product manifolds and metrics, and multiple
metrics on the same manifold. Needs more documentation.
Clifford algebra construction in all integer dimensions and fast evaluations of traces of products of Dirac matrices.
The tracing function is applied to weak-interaction reaction rates.
Implementation of the Friedrich's conformal field equations in dimension 4
(metric equations and extended equations).
Tools to perform conformal transformations and examples. Adapted from a subpart of xPand's code.
Simple example of coordinate change using the CTensor framework.
Examples of 3+1 decomposition applied to the Weyl tensor
Getting equations of motion and simplifying "C-tensors" for Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons theories
It shows how to define charts and specify the components of the metric so as to obtain the components of the related curvature tensors. Shows also how to change between different charts. Applied to homogeneous cosmology.
Gamma matrices (Dirac algebra).nb:
How to treat the algebra of gamma matrices with DefProduct, and canonicalize them to derive many standard identities.
Some computations for the Kerr metric in Boyer Lindqvist coordinates and the null tetrad. Computation of the surface gravity.
A common usage pattern for varying a Lagrangian with respect to the metric to get Einstein equations
Follows section 9.2 of Wald's book (General Relativity) on the Raychaudhuri equation for a congruence of timelike geodesics
Application of the Spinors and SymManipulator package to show the Lovelock tensor is non-dynamical.
Construction of the Schwarzschild metric.
Eddington Finkelstein coordinates and surface gravity as simple applications of the CTensor tools of xCoba.
Change of coordinates between Cartesian and spherical coordinates in flat 3D.
Can be used as a template for setting up change of coordinates.
1+3 decomposition of the Ricci and Bianchi identities in dimension 4 with respect to an
arbitrary unit timelike vector field. The kinematical quantities and the spatial parts of the Riemann
tensor are used to obtain the decompositions.
Minimal example 1) creating a warped product geometry and 2) breaking down the Einstein-Hilbert Lagrangian into the effective lower-dimensional Lagrangian.