[Paper Review] Computational General Relativity in the Wolfram Language using Gravitas I: Symbolic and Analytic Computation

Jonathan Gorard|arXiv (Cornell University)|Aug 15, 2023

Algebraic and Geometric AnalysisMathematics3 citations

TL;DR

This paper introduces Gravitas, an open-source computational framework in the Wolfram Language for symbolic and analytic general relativity, enabling seamless integration of symbolic tensor calculus, arbitrary coordinate systems, and geometric structures. It automates the solution of Einstein and Einstein-Maxwell equations across diverse spacetime geometries and matter fields, with a novel hypergraph-based adaptive refinement system for numerical evolution, marking a significant advance in unified symbolic-numerical relativity computation.

ABSTRACT

We introduce a new, open-source computational general relativity framework for the Wolfram Language called Gravitas, which boasts a number of novel and distinctive features as compared to the many pre-existing computational and numerical relativity frameworks currently available within the open-source community. These include, but are not limited to: seamless integration of its powerful symbolic and numerical subsystems, and, by extension, seamless transition between analytic/continuous representations and numerical/discrete representations of arbitrary spacetime geometries; highly modular, general and extensible representations of spacetime geometries, spacetime topologies, gauge conditions, coordinate systems, matter fields, evolution equations and initial data; ability to set up and run complex numerical relativity simulations, and to perform 2D and 3D visualizations, symbolic computations and numerical analysis (including the extraction of gravitational wave signals) on the resulting data, all from within a single notebook environment; and a totally-unstructured adaptive refinement scheme based on hypergraph rewriting, allowing for exceedingly efficient discretization and numerical evolution of Cauchy initial data for a wide range of challenging computational problems involving strong relativistic field dynamics. In this first in a series of two articles covering the framework, we focus on the design and capabilities of Gravitas's symbolic subsystem, including its general and flexible handling of arbitrary geometries parametrized by arbitrary curvilinear coordinate systems (along with an in-built library of standard metrics and coordinate conditions), as well as its various high-level tensor calculus and differential geometry features. We proceed to show how this subsystem can be used to solve the Einstein field equations both analytically and numerically.

Motivation & Objective

• Address the longstanding challenge of solving Einstein's field equations analytically and numerically in arbitrary coordinate systems and spacetime geometries.
• Overcome limitations of existing frameworks by unifying symbolic computation with adaptive numerical relativity in a single notebook environment.
• Enable automated, generalizable solutions to the Einstein and Einstein-Maxwell equations across arbitrary energy-matter distributions and electromagnetic fields.
• Develop a highly extensible, modular framework capable of supporting advanced relativistic theories and future extensions in mathematical relativity and quantum gravity.
• Lay the foundation for next-generation numerical relativity simulations with adaptive mesh refinement and robust visualization capabilities.

Proposed method

• Implement a symbolic tensor calculus subsystem in the Wolfram Language capable of handling arbitrary curvilinear coordinates and general spacetime geometries.
• Integrate in-built libraries of standard metrics, gauge conditions, stress-energy tensors, and energy conditions for automated field equation setup.
• Utilize the Wolfram Language’s symbolic engine to perform high-level tensor operations, including Riemann, Ricci, and Einstein tensor computations.
• Apply hypergraph rewriting to enable totally unstructured, adaptive refinement of spacetime discretizations for numerical evolution.
• Design a modular architecture that allows seamless transition between analytic and numerical representations of spacetime geometries.
• Support integration of relativistic electromagnetism and matter field models via specialized symbolic and numerical solvers.

Experimental results

Research questions

• RQ1 How can symbolic and numerical relativity be unified within a single, extensible computational framework to solve Einstein’s equations across arbitrary geometries?
• RQ2 To what extent can a hypergraph-based adaptive refinement scheme outperform traditional structured mesh methods in simulating strong-field relativistic dynamics?
• RQ3 Can a symbolic subsystem in the Wolfram Language automate the derivation and solution of the Einstein-Maxwell equations for complex matter and field configurations?
• RQ4 How does the framework’s modular design enable extensibility to advanced theories such as Einstein-Cartan gravity or scalar-tensor theories?
• RQ5 What novel numerical relativity capabilities emerge from combining symbolic tensor calculus with adaptive hypergraph-based discretization?

Key findings

• Gravitas enables fully automated symbolic computation of the Einstein and Einstein-Maxwell field equations in arbitrary coordinate systems, with built-in support for standard metrics and stress-energy tensors.
• The framework achieves seamless transition between analytic and numerical representations of spacetime, allowing for end-to-end computation from initial data to gravitational wave extraction.
• The hypergraph-based adaptive refinement scheme allows for efficient and flexible discretization of Cauchy initial data, supporting complex, non-trivial spacetime topologies and coordinate structures.
• The symbolic subsystem natively handles tensor calculus, including Riemann, Ricci, and Einstein tensors, and supports advanced structures like frame fields and spin connections in planned extensions.
• The framework supports the automated setup and execution of numerical relativity simulations, including 2D and 3D visualizations and gravitational wave signal extraction.
• The integration of symbolic and numerical subsystems allows for high-level, generalizable workflows that reduce manual intervention in solving complex relativistic problems.