Tensors and the Metric on a Curved Manifold
Coordinates are labels with no physical content, so physics is written with objects whose components transform in a fixed way under a change of labels. The metric holds the geometry: distances, angles, and which vectors count as parallel.
What a learner can do afterwards
- Distinguishes objects by how their components transform under a coordinate change
- Raises and lowers indices with a given metric
- Reads distances off a metric written in a non-Cartesian coordinate system
1 · Read
Coordinates are labels with no physical content: bent starlight stays the same in any labels. So you must write physics with objects that transform in a fixed way when the labels change. How components transform is what tells tensors apart from pretenders.
The metric holds the geometry: distances, angles, and which vectors count as parallel. With a given metric you raise and lower indices, moving between the two forms of the same object.
Read distances straight off the metric instead of a memorised formula. Ask how long each coordinate step is at your point, combine the steps with the metric, and the answer is the distance in those coordinates.
When coordinates look strange, blame the labels first. Check how each object transforms before trusting that an equation means what it seems to.
Labels carry no physics, the transform law sorts objects, and the metric turns coordinate steps into distances.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.