# Learner lab record: Geodesic coordinate-invariance audit

Course: Tensor calculus and differential geometry

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

Which features of a computed trajectory reflect geometry, and which reflect the chosen coordinate chart?

## Setup

Use the geodesic laboratory. Integrate one physical initial condition in the baseline chart, change a coordinate parameterization, and compare invariant and coordinate-dependent diagnostics.

## Variables

| Variable | Role | Unit |
| --- | --- | --- |
| Metric parameters | geometric inputs | declared model units |
| Initial position and tangent | initial data | chart coordinates |
| Step size | numerical control | affine parameter |
| Coordinate path and norm residual | dependent diagnostics | coordinates and dimensionless |

## Predict before changing controls

1. Predict which plotted coordinates may change after a chart change.

   ________________________________________________________________

2. Predict which tangent norm should remain invariant along an affinely parameterized geodesic.

   ________________________________________________________________

## Observation table

| chart | metric setting | step | endpoint coordinates | tangent norm | constraint residual |
| --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |
|   |   |   |   |   |   |

## Analyze

1. Which differences are coordinate artifacts?

   ________________________________________________________________

2. Does the tangent-norm residual converge with step size?

   ________________________________________________________________

3. Where do connection coefficients enter the trajectory?

   ________________________________________________________________

4. Which curvature invariant would distinguish a true singularity from a chart singularity?

   ________________________________________________________________

## Evidence-bounded conclusion

Changing from chart ___ to ___ changed coordinate path ___ while invariant diagnostic ___ remained ___; the numerical residual scaled ___.

________________________________________________________________