Computational General Relativity in the Wolfram Language using Gravitas I: Symbolic and Analytic Computation (2308.07508v1)

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.

• The paper introduces Gravitas, an open-source framework that unifies symbolic and numerical approaches for comprehensive spacetime analysis.
• It leverages robust tensor calculus and differential geometry to automate computations of curvature and relativistic field equations.
• Its modular, extensible design paves the way for advanced simulations in astrophysics and potential breakthroughs in quantum gravity research.

Overview of "Computational General Relativity in the Wolfram Language using Gravitas I: Symbolic and Analytic Computation"

The paper introduces Gravitas, an open-source computational framework for general relativity within the Wolfram Language, aiming to unify symbolic and numerical relativity into a single coherent structure. Gravitas stands out due to its seamless integration of symbolic and numerical methods, facilitating a transition between continuous and discrete representations of spacetime geometries. The framework is highly modular and extensible, allowing comprehensive representation and manipulation of spacetime geometries, topologies, gauge conditions, coordinate systems, and matter fields.

Key Features

• Unified Symbolic and Numerical Relativity: Gravitas provides a coherent environment for representing and manipulating both symbolic/analytical and numerical/discrete models of spacetimes. Its symbolic subsystem excels in handling general and flexible representations of arbitrary geometries, bolstered by an extensive library of metrics and coordinate conditions.
• Tensor Calculus and Differential Geometry: The framework includes sophisticated capabilities for performing tensor calculus and differential geometry operations. This includes computing a wide array of curvature tensors and handling complex tensor expressions automatically, easing the computation of geometrical and physical properties of spacetime.
• Energy-Matter Field Representations: Gravitas supports a variety of energy-matter distributions and features specialized capabilities in handling relativistic electromagnetic fields. It automates solving the Einstein and Einstein-Maxwell field equations both analytically and numerically, marking a significant advancement in computational relativity methods.
• Advanced Numerical Relativity Approaches: Although primarily a symbolic subsystem in this initial exposition, Gravitas anticipates expanding its capabilities to include sophisticated numerical techniques for relativity simulations. Upcoming functionality will leverage hypergraph rewriting for adaptive refinement in numerical evolution, supporting both standard and unconventional spacetime topologies and coordinate structures.

Implications and Applications

Gravitas is set to play a crucial role in areas related to mathematical relativity, astrophysics, and potentially quantum gravity research. Its ability to handle complex simulations of spacetime dynamics positions it as a valuable tool for studying phenomena like black hole collisions or cosmological models. The dynamic transition between symbolic computations and numerical simulations is designed to enhance the efficiency and accuracy of current computational methods used in these domains.

Future Prospects

The future development of Gravitas aims to expand its libraries and functionalities, integrating support for advanced geometric and physical models, such as scalar-tensor theories and non-Riemannian geometry. This evolution will enable more comprehensive studies, potentially offering new insights into unresolved questions in theoretical physics. Additionally, its applications could extend beyond general relativity, providing computational frameworks for emergent theories like quantum gravity.

In summary, the paper presents Gravitas as a powerful tool, integrating the symbolic and numerical aspects of relativity into a singular, versatile framework. Its innovative approach promises to simplify and expand the scope of computational relativity, opening paths to new explorations in the fundamental aspects of spacetime and gravitation.