---
title: "Tensors and the Metric on a Curved Manifold"
description: "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"
canonical: https://lightmysky.com/learn/science/tensors-and-the-metric-on-a-curved-manifold-mt_9r9SG3NcZR
source: https://lightmysky.com/learn/science/tensors-and-the-metric-on-a-curved-manifold-mt_9r9SG3NcZR.md
retrieved: 2026-09-12
---

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# 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.

Subject: Science · Area: Astronomy & Astrophysics · Ages 23 to 24
Page: https://lightmysky.com/learn/science/tensors-and-the-metric-on-a-curved-manifold-mt_9r9SG3NcZR

## Ready when they can

- 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

## Lesson: Writing physics that survives relabelling

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.

**Example.** 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.

**Recap.** Labels carry no physics, the transform law sorts objects, and the metric turns coordinate steps into distances.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [General Relativity: Curved Spacetime and Its Tests](https://lightmysky.com/learn/science/general-relativity-curved-spacetime-and-its-tests-mt_-rZZxZwU2L)
- [Topological Spaces: Continuity Without Distance](https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI)
- [Radiation from an Accelerating Charge](https://lightmysky.com/learn/science/radiation-from-an-accelerating-charge-mt_Qr0rMEOYAg)

## Opens up

- [Parallel Transport, Christoffel Symbols and the Geodesic Equation](https://lightmysky.com/learn/science/parallel-transport-christoffel-symbols-and-the-geodesic-equation-mt_bUEL2R91Vy)