Wolfram Cloud Document

In this installment of the  series, we address a central issue of differential geometry: how to take a derivative in (or on) a curved manifold without breaking coordinate invariance. Ordinary partial differentiation is not tensorial, producing garbage terms that don’t transform properly. The solution is the covariant derivative and its correction term, the connection coefficient

. We begin with the non-tensorial transformation law of Γ. Then, applying the constraints of metric compatibility

(∇g=0)

and torsion freedom (symmetry on lower indices of Γ), we derive the particular connection coefficients needed for general relativity, the Christoffel Symbols. This derivation specializes differential geometry to Riemannian geometry as needed for the general relativistic theory of gravitation. Step-by-step derivation in  automates tiresome and risky algebraic operations for high assurance in the results. At this early stage of development, we expose  as a script rather than hiding details in a package. This glass-box approach delivers better pedagogy and incremental validation. Coming up in near-future installments are derivations of the Riemann curvature tensor, a compiler proof-of-concept for translating UD expressions onto existing functions in the Wolfram Function Repository, and numerical demonstrations of the Schwarzschild solution.

Audience

This series of notebooks is targeted at readers with an undergraduate’s STEM education—engineering, computer science, mathematics. Exposure to introductory courses in Classical Mechanics (e.g., Goldstein), Electrodynamics (e.g., Jackson), and Relativity (e.g., Schutz) will help.

◼

By and large, I do not put standard textbooks in the References section, but offer clickable (non-affiliate!) links instead in the prose of this notebook.

I also invite readers to “think like a physicist,” that is, to demand conceptual cogency, but not always full mathematical rigour.

Transparent Implementation

In developing , we made a conscious decision, for the time being, to expose the implementation as a glass-box script rather than to hide it in a Package, prioritizing transparency over encapsulation at this stage of development. When the toolkit is finalized in the future, we’ll address encapsulating it in a Package.

Let’s load ’s implementation, the RelativityToolkit, directly into your notebook context.

Load the Code

The code is hosted on GitHub. The implementation, tests, and visualizations are not explained in this notebook. Explanations appear in an earlier notebook. But we run them here, for assurance.

Run the first two cells below. Expect to see test results followed by visualizations and a final gallery result.

◼

Note, if you call Quit in a notebook, evaluation may hang. If you want to “Evaluate Notebook,” mark the following cell “Evaluatable” in the Cell

Property menu, call Quit once, then comment out the Quit cell, or mark it unevaluatable again. Alternatively, you can quit the kernel via the Evaluation menu (last item). At that point, you can “Evaluate Notebook” in a clean environment. Otherwise, you can evaluate cells in this notebook individually.

Cell not evaluatable by default

Quit[]

In[]:=

(*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](*nosemicolononpurposetoseethegallery*)]

================================================================

RUNNING REGRESSION SUITE v1.6.1

================================================================

--- SECTION 1: TYPE CHECKING (VALENCE) ---

[PASS] Vector

⟶ {{μ},{}}

[PASS] Covector

⟶ {{},{ν}}

[PASS] Mixed Tensor

⟶ {{μ},{ν}}

[PASS] Mixed Tensor

μν

⟶ {{},{μ,ν}}

[PASS] Mixed Tensor

μν

⟶ {{μ,ν},{}}

[PASS] Mixed Tensor

⟶ {{ν},{μ}}

[PASS] Scalar Contraction

⟶ {{},{}}

[PASS] Scalar Contraction

⟶ {{},{}}

[PASS] Tensor-Vector

⟶ {{μ},{}}

[PASS] Tensor-Vector

⟶ {{},{ν}}

[PASS] Power of Scalar ⟶ {{},{}}

[PASS] Scalar Multiplication ⟶ {{μ},{}}

[PASS] Gradient d(f)/

-> Down[α] ⟶ {{},{α}}

Valence Mismatch

expect error message above{{},{}}

[PASS] Mismatch Handled Gracefully ⟶ {{},{}}

--- SECTION 2: ALGEBRAIC CANONICALIZATION ---

[PASS] Dummy Renaming 

 ⟶ p

i.1

(x

i.1

)

[PASS] Commutativity 

 ⟶ p

i.1

(x

i.1

)

[PASS] Index Coalescing 

== 2

 ⟶ 2p

i.1

(x

i.1

)

[PASS] Double Dummy Pairs 

 ⟶ (P

i.1

)Q

i.1

(S

i.2

)T

i.2



[PASS] Free Indices Preserved ⟶ (A

)(B

),(A

i.1

)(B

i.1

)

--- SECTION 3: PHYSICS, METRIC & TRANSFORMATIONS ---

[PASS] Lowering Index 

μν



 ⟶ A

[PASS] Raising Index (

μν



) ⟶ p

[PASS] Inverse Metric (

μα

αν



) ⟶

[PASS] Alpha-Conversion: distinct indices) ⟶ {μ11,μ12}

[PASS] Triple Metric Sandwich (g.g.g -> g) ⟶ g

[PASS] Delta-Gamma Contraction ⟶

--- SECTION 4: METRIC EQUIVALENCE 

 ---

[PASS] Structural Distinction

initially ⟶ (A

i.1

)B

i.1

,A

i.1

(B

i.1

)

 will rearrange terms

[PASS] Metric Equivalence 

reduces to

 ⟶ (A

i.1

)B

i.1



--- SECTION 5: SECOND DERIVATIVE Box Structure ---

[PASS] Found FractionBox + ^2 ⟶ True

--- SECTION 6: CALCULUS ENGINE ---

[PASS] Leibniz Product Rule ⟶ g

,μ

[PASS] Covariant Derivative Expansion ⟶ True

[PASS] Differentiation Linearity ⟶ True

[PASS] CD of Scalar reduces to Partial ⟶

phi$8017

,μ

[PASS] Scalar CD contains no Connection Coefficients ⟶ True

[PASS] Nested CD Alpha-Conversion (Unique Indices Generated) ⟶ True

--- SECTION 7: QUOTIENT THEOREM (Extraction) ---

[PASS] Quotient Extraction (Simple) ⟶ T

[PASS] Quotient Extraction (Distributed Dummies) ⟶ R

[PASS] Quotient Extraction (Zero Case) ⟶ 0

--- SECTION 8: TENSOR FORM ROBUSTNESS ---

[PASS] TensorForm maps arbitrary formals (Q, Z, I99) to Greek ⟶ True

--- SECTION 9: COMMA NOTATION DISPLAY LOGIC ---

[PASS] Atomic Tensor ⟶

,ν

[PASS] Sealed Product (Gamma Glue) ⟶

(A

)



,ν

[PASS] Sealed Sum ⟶

(A+B)

,ν

[PASS] Sealed Power ⟶

(

)

,ν

[PASS] Arbitrary Function (Sin) ⟶

Sin[theta]

,ν

[PASS] Nested Math (A*(B+C)) ⟶

(A(B+C))

,ν

----------------------------------------------------------------

REGRESSION SUITE COMPLETE

PASSED: 44

FAILED: 0

STATUS: GREEN (Ready for Publication)

----------------------------------------------------------------

================================================================

VISUAL SHOWCASE: FORMAL DIFFERENTIAL GEOMETRY

v1.6.1

================================================================

1. The Tensor Zoo (Standard Objects)

These objects carry valence metadata hidden inside their structure.

Object Name	Wolfram Input	Textbook Output
Contravariant Vector	x[[μ]]	x μ
Covariant Covector	p[[ν]]	p ν
Mixed Tensor	T[[μ], [ν]]	T μ ν
The Metric	g[[μ], [ν]]	g μ ν
Inverse Metric	g[[μ], [ν]]	g μ ν
Kronecker Delta	δ[[μ], [ν]]	μ δ ν
Jacobian (Partial)	Partials[x[[α]], x[[β]]]	∂x α ∂x β

Covariant Derivative

◼

A .wl block-version of code for this section, minus the prose, is available at https://github.com/rebcabin/RelativityToolkit/blob/main/covariant_derivative.wl.

We Want a Derivative!

Parallel Transport

Coordinate-Independence

To answer this conundrum, we insist, as always, that the intrinsic geometry and physics not depend on choice of coordinates. But what does intrinsic mean when all we have are numerical components whose values depend on choice of coordinates?

Intrinsic Means Tensorial

◼

intrinsic geometric and physical properties of a tensor

◼

extrinsic artifacts arising from choice of coordinates

◼

Think of changing from Cartesian to spherical coordinates in flat Euclidean 3-space: the numbers change but the geometry and physics don’t.

Plus, we will be able to separate intrinsic from extrinsic quantities cleanly.

◼

The first couple of chapters of Lovelock & Rund address this issue of intrinsic, geometry-dependent effects versus extrinsic, coordinate-dependent effects rigorously. Thinking like physicists, let’s be satisfied with the conceptual description above for now.

Tensorial and Linear Requirements

It turns out that simply demanding that our new derivative be tensorial and linear suffices to solve all the problems above, including optimality.

◼

TODO: This claim of optimality may not be good enough, even for a physicist happy with mathematical hand-waving. Misner, Thorne, and Wheeler [MTW] devote many chapters (8-15) to a more careful, physicist’s level of understanding, including demonstrations that the covariant derivative perform parallel transport! Those are missing pieces in this notebook.

◼

A mathematician will demand rigorous proofs. For those, see the many texts on differential geometry, perhaps starting with do Carmo's Differential Geometry of Curves and Surfaces. Other texts such as Tristan Needham, Visual Differential Geometry and Forms, and Michael Spivak's 5-volume Comprehensive Introduction to Differential Geometry are recommended.

The Partials are Not Tensors

◼

 embraces the comma notation as you can see from the Visual Showcase gallery above.

◼

Covariant tensors transform with the matching index co-wise in the numerator of the Jacobian. The mnemonical example is a gradient, say degrees of temperature per foot or inch.

Now calculate partial derivatives with respect to the primed frame:

Let’s have  do this bit of calculus. As stated, the code is maintained as a block of code rather than as alternating prose and code cells. But the comments and the Echoes tell the whole story as prose. Read The Code and the results.

Connection Coefficients

◼

A .wl block-version of code for this section, minus the prose, is available at https://github.com/rebcabin/RelativityToolkit/blob/main/law_of_gamma.wl.

The Partials are a Mess

The Way Out: Covariant Derivative

require that it transform as follows:

◼

The form of a term plus a correction is ubiquitous in linear theories such as that for the Kalman filter (see my notebook Kalman filtering as a functional fold).

◼

SPOILER: We want our new derivative ∇ (aka the semicolon) to behave like any other derivative. That means it must satisfy two properties:

Connection Coefficients

◼

In restricting connection coefficients for gravitation in general relativity, we’re saying “only certain kinds of manifold geometries and topologies are useful for describing gravitation.” Connection coefficients are more general than Christoffel symbols, pertaining to a broader class of manifolds.

Covariant Derivatives and Connections in 

’s CD functions shows off the evaluated covariant derivative:

CD’s held form exhibits semicolon notation!

As usual, Read The Code and the results. Note specifically how TensorForm does the index coalescing and simplification, right where humans go down the drain!

◼

Lovelock & Rund write dire warnings about these theorems, that misapplication is the occasion of many published errors. Here, because we know the answer is correct, even by comparison to Lovelock & Rund, we’re going to hand-wave the issue, here.

Christoffel Symbols

◼

A .wl block-version of code for this section is available at https://github.com/rebcabin/RelativityToolkit/blob/main/connections_to_christoffels.wl.

Up to this point, we have not actually used a metric tensor! MTW devote seven whole chapters to differential geometry without a metric (9-15), stressing that many general results in differential geometry—the topological results—can be derived without the metric. In particular, the covariant derivative and the connection coefficients do not, actually, depend on any metrical ability to measure distances, but only on assumptions of continuity and differentiability.

But gravitation via general relativity requires a special case: Riemannian Geometry. For that, we need a metric, and even a bit more: symmetry conditions on both the connection coefficients and on the metric. From these additions, we derive the ultra-famous gravitational Christoffel symbols:

This is equation 8.24 of MTW, after renaming some indices. It is also called “The Fundamental Theorem of Riemannian Geometry” in mathematical texts like do Carmo and Lee (see References).

The following block of code shows a step-by-step derivation via . As usual, the “prose” is in the comments and Echoes. Run it and read it!

Conclusion

The form of the Christoffel symbols shows that gravitation, manifested in the connection, is solely determined by the metric and its ordinary partial derivatives.

With a working covariant derivative in ’s Relativity Toolkit, we are equipped for future installments on geodesics, geodesic deviation, tidal forces, accelerated observers, Schwarzschild and Kerr metrics, black holes, etc.

References

Poisson, Eric . A Relativist's Toolkit : The Mathematics of Black Hole Mechanics. Cambridge University Press, 2004.

Beckman, Brian . From the steam age to quantum fields: a unified approach via UD tensorial calculus. Wolfram Community post.

Beckman, Brian . General relativity in the UD calculus. Wolfram Community post.

Beckman, Brian . Formal Differential Geometry in the UD Calculus: Part 1. Wolfram Community post.

[MTW] Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald . Gravitation. Princeton University Press, 2017.

[L&R] Lovelock, David and Rund, Hanno . Tensors, Differential Forms, and Variational Principles. Dover.

John M. Lee, Riemannian Manifolds: An Introduction to Curvature. Springer.

Manfredo P. do Carmo, Riemannian Geometry. Springer.

Wolfram Function Repository . MetricTensor, RiemannTensor, EinsteinTensor. For component-based calculations.

Wolfram Research . Mathematica Documentation. Tensor Calculus.

CITE THIS NOTEBOOK

Covariant derivatives and connections in the  calculus
by Brian Beckman
Wolfram Community, STAFF PICKS, January 19, 2026
https://community.wolfram.com/groups/-/m/t/3609079