Tensors and the metric on a curved manifold

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.

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.