Abstract
Abstract
In this notebook, we extend and formalize the Calculus, a computational frontend for unambiguous, symbolic tensor calculations in the Wolfram Language. features concise, but unambiguous input like T[[μ],[ν]] (“T, up μ, down ν”) and typeset output like .
μ
T
ν
We take the view that Wolfram’s existing Symbolic Tensors plus existing repository functions like MetricTensor and RiemannTensor furnish powerful, backend tools for differential geometry and relativity. on the other hand, focuses on structural logic of tensor algebra, thus a frontend with syntactic and semantic capabilities:
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valence checking (type safety)
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automatic index canonicalization and coalescing
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collision-free coordinate transformations
should be useful for applications of differential geometry beyond relativity. Differential geometry is enjoying rapidly growing utility in computer vision and robotics, and has long been central to classical and quantum mechanics, to field theory, and to gauge theory. We have expressed interest in these applications in many earlier notebooks and plan to revisit some of them, buttressing prior work with formal rigour via .
That said, this notebook is the first in a series dedicated specifically to general relativity by implementing Eric Poisson’s A Relativist's Toolkit in . We will verify his derivations ab initio, assuring not just correctness, but rigour of the derivation. In the next installment (in preparation), we’ll cover covariant derivatives and the connections for general relativity. After that, we’ll cover geodesics, Lie derivatives, Killing vectors, the Levi-Cevita tensor, geodesic deviation, Fermi Normal Coordinates, and reproduce famous metrics like Schwarzschild and Kerr.
All along, we’ll incrementally develop , a compiler backend: a translation layer to map expressions onto Wolfram’s existing SymbolicTensor and elements of the function repository. Our objective is that users can derive physical laws in the high-assurance environment and then seamlessly compile them to Wolfram’s computational infrastructure.
is an exercise in formal (machine-checked) mathematics with compiler-like architecture. While the Wolfram language is not, prima facie, a purely formal system like Rocq, Lean, or TLA+, Wolfram’s symbolic algebra lets us have many formal checks, not to mention numerical agility, throughout any work.
Introduction
Introduction
Wolfram Mathematica is a superpower for general-purpose algebra and calculus. The Wolfram Function Repository offers user-contributed, component-oriented tools for MetricTensor, ChristoffelSymbol, EinsteinTensor, SolveEinsteinEquations, and so on, for relativity problems.
For physicists who need to understand or reproduce derivations—the index raising, lowering, and contracting, furnishes the complementary and layered frontend services:
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Grammar: knows that adding a vector to a covector is physically meaningless and will stop with a valence check.
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Notation: has easy input forms for abstractly indexed tensors. These input forms as small as possible while being unambiguous, e.g., T[[μ],[ν]]. ’s output forms match typeset notebooks and papers as closely as possible, e.g., .
T
μ
ν
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Abstractions: manipulates tensors as whole symbolic objects, rather than as decorated arrays, preserving index notation in both input and output forms.
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Rigour: handles renaming of contraction-bound (dummy) indices via alpha-conversion, preventing the index-capture errors that plague pen-and-paper derivations.
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Bridges: will map to Wolfram’s existing support for tensors as a target virtual machine.
Our guide for this journey is Eric Poisson’s text, A Relativist’s Toolkit. We will proceed section by section over several notebooks, implementing his textual toolkit as a software toolkit.
Notes
Notes
This work builds upon my previous notebooks on special and general relativity, where (“Up/Down”) notation was first introduced to disambiguate covariant and contravariant indices.
Deployment
Deployment
To ensure this notebook is self-contained and reproducible, the entire source code for ’s relativity toolkit, with its unit tests and regression tests, is included inline here. In future installments of this series, as the codebase grows, we will transition to open-source distribution hosted on GitHub.
The Code and Tests
The Code and Tests
This code is evolving. This notebook contains an early version, formatted inline for human reading. You may also be interested in the latest version. We explain how to download and run it below.
Optional: Download and Run Latest version
Optional: Download and Run Latest version
The most up-to-date version is hosted on GitHub. This notebook contains a preliminary version inline for your inspection. To download and run the most current version, unlock and run the following cells. They’re marked “unevaluatable” by default so that they won’t interfere with those who want to evaluate the whole notebook. Unlock them via the “Cell” menu, “Properties” submenu, “Evaluatable” item. Expect to see unit tests and the following visual gallery.
In[]:=
(*1.Cleanthekerneltoensurewearen'tseeingcacheddefinitions*)Quit[]
(*2.LOADTHECODE*)Module[{base,rules,tests,viz,bust},base="https://raw.githubusercontent.com/rebcabin/RelativityToolkit/main/";bust="?t="<>ToString[Round[AbsoluteTime[]]];(*Forcefreshdownload*)(*DefineURLs*)rules=base<>"RelativityToolkit.wl"<>bust;tests=base<>"RelativityToolkit_RegressionTests.wl"<>bust;viz=base<>"RelativityToolkit_VisualShowcase.wl"<>bust;(*Loadthem*)Get[rules];Get[tests];Get[viz](*nosemicolononpurposetoseegallery*)]
Out[]=
Description | Rendered Output |
Generic Mixed Partial | 2 ∂ ∂x∂y |
Coordinate Hessian (The 'Garbage Term') | 2 ∂ ′ α ∂x α β |
Nested in an Equation | lhs 2 ∂ μ ∂x λ ν |
Multiplied by Jacobian | 2 ∂ ′ α ∂x α β ∂x β ∂x ′ β |
Comma Notation | A μ ,ν |
Covariant Derivative (Abstract) | A μ ;ν |
Covariant Derivative (Expanded) | A μ ,ν λ μ Γ νλ |
How to Read this Notebook
How to Read this Notebook
The code is presented here in reverse of the best order for human reading, because it’s necessary to evaluate the implementation before showing what it can do and how to use it (def before ref, the standard dilemma of Literate Programming, which we can’t practice).
The best order to read is bottom-up:
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Visual Showcase: A gallery showing what can do
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Tests: Bite-sized samples of how to use
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Implementation: If you care how works
The code is formatted in .wl block style, with essential prose in comments instead of in Notebook cells, for easier upload to code repositories. Read the comments!
Implementation
Implementation
In[]:=
Relativity Toolkit Part 1 (Inline) Loaded Successfully.
Smoke Tests
Smoke Tests
In[]:=
Print["--- 1. Visualizing Tensors ---"];(*Weinputunambiguousstructure:x[U[mu]]*)(*Thesystemrenderstextbooknotation:*){"Vector (Contravariant)"->x[[μ]],"Covector (Covariant)"->p[[ν]],"Mixed Tensor T^u_v"->T[[μ],[ν]],"Mixed Tensor T_u^v"->T[[μ],[ν]]}//TableForm
--- 1. Visualizing Tensors ---
Out[]//TableForm=
Vector (Contravariant)x μ |
Covector (Covariant)p ν |
Mixed Tensor T^u_vT μ ν |
Mixed Tensor T_u^vT μ ν |
In[]:=
Print["--- PHYSICS ENGINE DIAGNOSTICS ---"];(*1.METRICRAISING&LOWERING*)Print["\n1. Metric Gymnastics:"];termLower=g[[μ],[ν]]A[[ν]];resLower=termLower/.metricRules;Echo[termLower,"Input g[[μ],[ν]] A[[ν]]: "];Echo[resLower,"Result (A_u): "];termInverse=g[[μ],[α]]g[[α],[ν]];resInverse=termInverse/.metricRules;Echo[termInverse,"Metric Product: g[[μ],[α]] g[[α],[ν]]"];Echo[resInverse,"Inverse Identity (Delta^u_v): "];(*2.VALENCECHECKER*)Print["\n2. Valence Safety Check:"];(*ShouldbeTrue*)valCheck=(valence[resLower]==={{},{μ}});Echo[valCheck,"Did A^v become A_u? "];(*ShouldbeFalse/Errorforbadphysics*)badPhysics=x[[μ]]+p[[μ]];Print["Checking Bad Physics (Expect Error Message below):"];valence[badPhysics];(*3.ALPHA-CONVERSION(CollisionTest)*)Print["\n3. Coordinate Transformation (Alpha-Conversion):"];(*TransformA^a'->(dx^a'/dx^mu)A^mu*)(*WatchforUniqueindiceslikemu123*)contraTrans=A[[Superscript["a","′"]]];contraResTrans=contraTrans/.robustTransformRules;Echo[contraTrans,"Original Contravariant: "];Echo[contraResTrans,"Transformed (Look for Unique Indices): "];covTrans=A[[Superscript["a","′"]]];covResTrans=covTrans/.robustTransformRules;Echo[covTrans,"Original Covariant: "];Echo[covResTrans,"Transformed (Look for Unique Indices): "];
Regression Tests
Regression Tests
Visual Showcase
Visual Showcase
Conclusion
Conclusion
In this first part of a planned series, we build and test ’s foundation algebraic engine. Presaging future integration to existing component-based tools, we introduce rules and functions for the structure of differential geometry, both generically and as applied to relativity.
We have implemented:
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Valence Checking: A type system that forbids illegal tensor operations
Coming Next: Covariant Derivatives and Connections
Coming Next: Covariant Derivatives and Connections
Algebra is only part of the story. In Part 2, we will tackle Poisson’s Sections 1.2 and 1.3. We will implement:
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Christoffel Symbols
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The Covariant Derivative
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The Einstein Connection that generates gravitation
The Horizon: Part 3 and Beyond
The Horizon: Part 3 and Beyond
After we’re done with a self-contained Relativity Toolkit, we plan to introduce The Compiler Backend. This will link our formal, high-assurance frontend structures to Wolfram’s SymbolicTensor library and the many support functions in the function repository, translating abstract geometric objects into concrete component arrays, e.g., mapping a formal Schwarzschild metric to its explicit 4×4 matrix representation. We also envision expanding our valence type-checker to include metadata about domains (e.g., Complex, Spinor), and dimensions for both expanded robustness and applicability, say to SU(3) gauge theory.
References
References
1
.Poisson, Eric. A Relativist's Toolkit: The Mathematics of Black Hole Mechanics. Cambridge University Press, 2004.
2
.Beckman, Brian. From the steam age to quantum fields: a unified approach via UD tensorial calculus. Wolfram Community post.
3
.Beckman, Brian. General relativity in the UD calculus. Wolfram Community post.
4
.Wolfram Function Repository. MetricTensor, RiemannTensor, EinsteinTensor. (For component-based calculations).
5
.Wolfram Research. Mathematica Documentation: Tensor Calculus.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Formal differential geometry in the calculus: part 1
by Brian Beckman
Wolfram Community, STAFF PICKS, January 13, 2026
https://community.wolfram.com/groups/-/m/t/3605564
by Brian Beckman
Wolfram Community, STAFF PICKS, January 13, 2026
https://community.wolfram.com/groups/-/m/t/3605564