OGRe: An Object-Oriented General Relativity Package for Mathematica

Barak Shoshany
Department of Physics, Brock University,
1812 Sir Isaac Brock Way, St. Catharines, Ontario, L2S 3A1, Canada
[email protected] | https://baraksh.com/
DOI: 10.21105/joss.03416

• Summary
• Features
• Installing and loading the package
• Documentation
• Issue and pull request policy
• Copyright and citing

Summary

OGRe is a modern Mathematica package for differential geometry and tensor calculus, designed to be both powerful and user-friendly. It can be used in a variety of contexts where tensor calculations are needed, in both mathematics and physics, but it is especially suitable for general relativity.

Tensors are abstract objects, which can be represented as multi-dimensional arrays once a choice of index configuration and coordinate system is made. OGRe stays true to this definition, but takes away the complexities that come with combining tensors in different representations. This is done using an object-oriented programming approach, taking advantage of principles such as encapsulation and class invariants to eliminate the possibility of user error.

The user initially defines each tensor in OGRe using its explicit components in any single representation. Operations on this tensor are then done abstractly, without needing to specify which representation to use. Possible operations include addition of tensors, multiplication of tensor by scalar, trace, contraction, and partial and covariant derivatives.

OGRe will automatically choose which representation to use for each tensor based on how the tensors are combined. For example, if two tensors are added, then OGRe will automatically use the same index configuration for both. Similarly, if two tensors are contracted, then OGRe will automatically ensure that the contracted indices are one upper (contravariant) and one lower (covariant). OGRe will also automatically transform all tensors being operated on to the same coordinate system.

Transformations between representations are done behind the scenes; all the user has to do is specify which metric to use for raising and lowering indices, and how to transform between the coordinate systems being used. This also means that there is no room for user error. The user cannot mistakenly perform "illegal" operations such as 2A^μν+B_μλC_λν. Instead, the user simply inputs the names of the tensors, the order (but not the configuration) of indices for each, and the operations to perform - and the correct combination 2A^μν+B^μ_λC^λν will be automatically deduced.

I initially created OGRe for use in my own research, so I made it as flexible and powerful as possible. I also wanted my students to be able to use it easily and efficiently, even if they only have minimal experience with Mathematica and/or general relativity, so I made it simple to learn and easy to use. As a result, this package is equally suitable for both experienced and novice researchers.

Features

• Define coordinate systems and the transformation rules between them. Tensor components are then transformed automatically between coordinates behind the scenes as needed.
• Each tensor is associated with a specific metric. Tensor components are then transformed automatically between different index configurations, raising and lowering indices behind the scenes as needed.
• Display any tensor in any index configuration and coordinate system, either in vector/matrix form or as a list of all unique non-zero elements. Metrics can also be displayed as a line element.
• Automatically simplify tensor components, optionally with user-defined simplification assumptions. Simplifications can be parallelized for a significant performance boost. The user may utilize the package's simplification algorithm for any Mathematica expression, not just tensors.
• Export tensors to a Mathematica notebook or to a file, so they can later be imported into another Mathematica session without having to redefine them from scratch.
• Highly customizable. User settings are exported and imported along with the tensors. Some settings are persistent between sessions.
• Easily calculate arbitrary tensor formulas using any combination of addition, multiplication by scalar, trace, contraction, partial derivative, and covariant derivative.
• Built-in modules for calculating the Christoffel symbols (Levi-Civita connection), Riemann tensor, Ricci tensor and scalar, Einstein tensor, curve Lagrangian, and volume element from a metric.
• Calculate the norm-squared of tensors of any rank.
• Calculate the geodesic equations in terms of a user-defined affine curve parameter, in two different ways: from the Christoffel symbols or from the curve Lagrangian. For spacetime metrics, the geodesic equations can be calculated in terms of the time coordinate.
• Designed with speed and performance in mind, using optimized algorithms designed specifically for this package.
• Fully portable. Can be imported directly from the web into any Mathematica notebook, without downloading or installing anything. Integrates seamlessly with the Wolfram Cloud.
• Clear and detailed documentation, with many examples, in both Mathematica notebook and PDF format. Detailed usage messages are also provided.
• Open source. The code is extensively documented; please feel free to fork and modify it as you see fit.
• Under continuous and active development. Bug reports and feature requests are welcome, and should be made via GitHub issues.

Installing and loading the package

This package is compatible with Mathematica 12.0 or newer. It consists of only one file, OGRe.m. There are several different ways to load the package:

• Run from local file with installation: This is the recommended option, as it allows you to permanently use the package offline from any Mathematica notebook. Download the file OGRe.m from the GitHub repository and copy it to the directory given by FileNameJoin[{$UserBaseDirectory, "Applications"}]. The package may now be loaded from any notebook by writing Needs["OGRe`"] (note the backtick ` following the word OGRe).

• Run from local file without installation: This option allows you to use the package in a portable fashion, without installing it in the Applications directory. Download the file OGRe.m from the GitHub repository, place it in the same directory as the notebook you would like to use, and use the command Get["OGRe.m", Path->NotebookDirectory[]] to load the package.

• Run from web with installation: This option allows you to automatically download and install the package on any computer. Simply run the command URLDownload["https://raw.githubusercontent.com/bshoshany/OGRe/master/OGRe.m", FileNameJoin[{$UserBaseDirectory, "Applications", "OGRe.m"}]] from any Mathematica notebook to permanently install the package. Then use Needs["OGRe`"] from any notebook to load it.

• Run from web without installation: This option allows you to use the package from any Mathematica notebook on any computer, without manually downloading or installing it, as long as you have a working Internet connection. It also ensures that you always use the latest version of the package, but be aware that updates may sometimes not be fully backwards compatible. Simply write Get["https://raw.githubusercontent.com/bshoshany/OGRe/master/OGRe.m"] in any Mathematica notebook to load the package.

To uninstall the package, just delete the file from the Applications directory, which can be done from within Mathematica using the command DeleteFile[FileNameJoin[{$UserBaseDirectory, "Applications", "OGRe.m"}]].

Documentation

The full and detailed documentation for this package may be found in the following repository files:

• OGRe_Documentation.nb: An interactive Mathematica notebook. Requires Mathematica to open.
• OGRe_Documentation.pdf: A PDF version of the notebook. Can be viewed with any PDF reader.

Once the package is loaded, the documentation can be easily accessed by executing the command TDocs[], which automatically downloads the file OGRe_Documentation.nb from GitHub and opens it in Mathematica.

Issue and pull request policy

This package is under continuous and active development. If you encounter any bugs, or if you would like to request any additional features, please feel free to open a new issue on GitHub and I will look into it as soon as I can.

Contributions are always welcome. However, I release my projects in cumulative updates after editing and testing them locally on my system, so my policy is not to accept any pull requests. If you open a pull request, and I decide to incorporate your suggestion into the project, I will first modify your code to comply with the project's coding conventions (formatting, syntax, naming, comments, programming practices, etc.), and perform some tests to ensure that the change doesn't break anything. I will then merge it into the next release of the project, possibly together with some other changes. The new release will also include a note in CHANGELOG.md with a link to your pull request, and modifications to the documentation in README.md as needed.

Copyright and citing

Copyright (c) 2021 Barak Shoshany. Licensed under the MIT license.

If you use this package in published research, please cite it as follows:

• Shoshany, B., (2021). OGRe: An Object-Oriented General Relativity Package for Mathematica. Journal of Open Source Software, 6(65), 3416, https://doi.org/10.21105/joss.03416

You can also use the following BibTeX entry:

@article{Shoshany2021_OGRe,
    author    = {Barak Shoshany},
    doi       = {10.21105/joss.03416},
    journal   = {Journal of Open Source Software},
    number    = {65},
    pages     = {3416},
    publisher = {The Open Journal},
    title     = {OGRe: An Object-Oriented General Relativity Package for Mathematica},
    url       = {https://doi.org/10.21105/joss.03416},
    volume    = {6},
    year      = {2021}
}