[Paper Review] Computer algebra in gravity

Christian Heinicke, Friedrich W. Hehl|ArXiv.org|May 25, 2001

Algebraic and Geometric Analysis11 references3 citations

TL;DR

This paper presents a comprehensive survey of computer algebra systems (CAS) in gravitational physics, focusing on their application in general relativity and related theories. It demonstrates how symbolic computation tools like Excalc in Reduce can verify fundamental identities—such as the second Bianchi identity—using abstract tensor and exterior calculus, confirming the correctness of curvature computations through automated symbolic manipulation.

ABSTRACT

We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some computer algebra systems and packages relevant to applications in gravitational physics. We conclude by presenting a couple of typical examples.

Motivation & Objective

• To provide a systematic overview of computer algebra systems applicable to gravitational theories, especially general relativity.
• To demonstrate the utility of symbolic computation in handling the complex, high-order tensor expressions arising in curvature calculations.
• To illustrate how CAS can automate the verification of geometric identities, such as the second Bianchi identity, reducing manual error and effort.
• To support the development and classification of exact solutions to Einstein's equations through automated invariant computation and equivalence testing.
• To highlight the integration of CAS with numerical methods via Fortran code generation and their role in advanced theoretical frameworks like metric-affine gravity and quantum gravity.

Proposed method

• Utilizes the Reduce computer algebra system with the Excalc package to perform symbolic computations in exterior calculus and tensor algebra.
• Employs abstract index notation and differential forms to express curvature tensors and connections compactly, e.g., $ R_{\alpha}{}^{\beta} = d\Gamma_{\alpha}{}^{\beta} - \Gamma_{\alpha}{}^{\gamma} \wedge \Gamma_{\gamma}{}^{\beta} $.
• Applies component-based systems like Sheep, GRTensorII, and MathTensor to compute explicit components of the Ricci tensor and curvature invariants from a given metric.
• Implements symbolic differentiation and product rules in exterior calculus to verify the second Bianchi identity: $ dR_{\alpha}{}^{\beta} - \Gamma_{\alpha}{}^{\gamma} \wedge R_{\gamma}{}^{\beta} + \Gamma_{\gamma}{}^{\beta} \wedge R_{\alpha}{}^{\gamma} = 0 $.
• Uses specialized packages such as Classi (within Sheep) to automate the Petrov and Segre classification of exact spacetime solutions.
• Applies variational calculus in systems like MathTensor and GRG EC to derive field equations from action principles, including for unified field theories and Einstein-Maxwell systems.

Experimental results

Research questions

• RQ1 How can computer algebra systems be used to verify the second Bianchi identity in general relativity using symbolic manipulation?
• RQ2 What are the most effective computer algebra systems and packages for computing curvature tensors and Ricci scalars in arbitrary spacetime metrics?
• RQ3 How can symbolic computation assist in classifying exact solutions of Einstein's equations in a coordinate-independent manner?
• RQ4 In what ways can computer algebra systems support the solution of the equivalence problem between different-looking spacetime metrics?
• RQ5 How can symbolic computation be integrated with numerical methods through Fortran code generation for solving gravitational field equations?

Key findings

• The second Bianchi identity was successfully verified using the Excalc package in Reduce, with the symbolic expression evaluating exactly to zero.
• Component-based systems like Sheep and GRTensorII enable the automated computation of curvature invariants and Ricci tensors, even for metrics with up to 10 independent functions.
• The use of computer algebra reduces the number of terms in Ricci tensor components from an estimated 10,000 per component in the general case to manageable symbolic expressions.
• The Sheep package, particularly the Classi module, enabled the creation of a searchable online database of nearly 200 exact solutions with their Petrov and Segre types.
• Computer algebra systems such as GRG EC and Classym were successfully used to derive and solve Killing vector and tensor equations from general metrics.
• Symbolic computation via MathTensor and Form enabled the derivation and correction of field equations from Lagrangians in unified field theories and the calculation of quantum gravity Feynman diagrams.