The Principia Machina/Tools/xAct

xAct: Abstract-Index Tensor Computer Algebra for Gravity

Added to the survey on 2026-10-06 by the Research Toolkit's research pass. The scores below are that pass's assessment from the evidence cited under References, on the survey's four axes and scale; they are not the original survey's (2026-09-26), and no rule of the Principia Machina reads them. See the additions on the front page.

Overview & Architectural Fit

xAct is a suite of packages for tensor computer algebra in the Wolfram Language. Its core (xCore, xPerm, xTensor, xCoba) canonicalizes tensor expressions with their symmetries; around it some 27 packages add perturbation theory (xPert), curvature invariants (Invar), spinors, differential forms (xTerior) and utilities (xTras). It is the reference tensor system of general relativity, and complements Cadabra2.

Mathematical Evaluation for TPM

• Lagrangian / Action (7.5/10): The standard tool for the step $\delta S/\delta g^{\mu \nu }$: varying a covariant action, such as $S=\frac{1}{16\pi G}\int R\sqrt{-g}{d}^{4}x$ plus matter, with respect to the metric or the fields; xPert takes the same action to second order. The variational-derivative function names were not re-checked in this pass.
• Field / PDE (6/10): Derives covariant field and perturbation equations; does no numerics.
• Relativity (9.5/10): Canonicalization, curvature invariants, perturbation theory and spinors: the reference system for relativity.
• Quantum / Data (2.5/10): No quantum or data focus.

Integration Pipeline

1. Declare the manifold, the metric and the fields with their index symmetries.
2. Vary the action to get the field equations.
3. Check identities off shell: the contracted Bianchi identity ${\mathrm{\nabla }}^{\mu }{G}_{\mu \nu }=0$ and the identities that diffeomorphism invariance implies.
4. Perturb with xPert to second order, $S^{(2)}[h]$, for stability and the propagating degrees of freedom.
5. Cross-check components against SageManifolds, which is free.

Status, Licence and Limits

• Status: Active. Version 1.3.0, released 29 December 2025.
• Licence: GPL; the Mathematica host is commercial.
• Home: http://www.xact.es
• Limits: Needs a Mathematica licence, which limits who can reproduce a result. High-order perturbations need care with performance and memory. Results depend on declaring index symmetries correctly. xAct's site asks users to cite each package's own paper as well.

On the Principia Machina

A tool runs only where the sponsor installed and bound it, and what it returns is untrusted input, vetted before use; its numbers are never values of the site's calculator, TPC. An equation it derives is transcribed into the kernel's form grammar before anything checks it, and lands as a RESULT of regime L with EQUATION and ROLE. See Agent/Tools and the operator's guide.

References

• XPERM2008 – Martín-García J.M. (2008). xPerm: fast index canonicalization for tensor computer algebra. Comput. Phys. Commun. 179(8), 597–603. doi:10.1016/j.cpc.2008.05.009.
• XACT – Martín-García J.M. et al. (2025). xAct: Efficient tensor computer algebra for the Wolfram Language, v1.3.0. www.xact.es.