---
title: "Tensors and the metric on a curved manifold"
description: "Changing coordinates changes component descriptions, not the physical event. Tensor transformation laws keep equations consistent. The metric gives lengths, angles and signed spacetime intervals; its "
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: 2026-11-12
---

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# Tensors and the metric on a curved manifold

Changing coordinates changes component descriptions, not the physical event. Tensor transformation laws keep equations consistent. The metric gives lengths, angles and signed spacetime intervals; its Levi-Civita connection supplies parallel transport.

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

Changing coordinates relabels the same points or events; it does not change the physics from the previous topic. Indices i and j name coordinate directions, and a prime marks new coordinates. The Jacobian ∂x′ⁱ/∂xʲ measures how a small change in xʲ changes x′ⁱ, with other old coordinates fixed. Vector components transform as V′ⁱ=(∂x′ⁱ/∂xʲ)Vʲ. Sum over a repeated index. The components may change, but the vector stays the same.

A metric gᵢⱼ gives ds²=gᵢⱼ dxⁱdxʲ. On a Riemannian surface it gives lengths and angles. In spacetime it gives a signed interval, not an ordinary positive distance. Parallel transport needs a connection. Requiring zero torsion and compatibility with the metric selects the Levi-Civita connection.

Lower an index using Vᵢ=gᵢⱼVʲ. Raise it using Vⁱ=gⁱʲVⱼ, where gⁱʲ is the inverse metric. These are vector and covector descriptions related by the metric. For g=diag(4,9) and V=(1,2), lowering gives (4,18). With a diagonal metric, multiply each component by its diagonal entry.

*(drawing: The flat plane's polar metric gives the arc length along a circle. Angles are in radians; r stays fixed.)*

The polar metric dr²+r²dθ² describes a flat plane even though its components depend on r. A curved coordinate grid does not prove curved geometry. Intrinsic curvature is a property of the metric that survives a coordinate change.

Transform every component in a calculation. In one dimension, x′=2x gives V′=2V and g′=g/4. Then g′(V′)²=gV², so the vector's squared length is unchanged. Transforming the vector but keeping the old metric would give a false change in length.

An array of numbers is not enough to identify a tensor. Its components must obey the appropriate transformation law. A scalar stays unchanged. A covector obeys W′ᵢ=(∂xʲ/∂x′ⁱ)Wⱼ, using the inverse Jacobian. Under x′=2x, a nonzero vector component doubles and a covector component halves.

**Recap.** Tensor components follow transformation laws. The metric measures geometry and relates vectors to covectors. Non-Cartesian coordinates can describe a flat space.

*Original lesson, questions and flashcards written for this dataset.*

## 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)

## Optional connections

- [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)