General Relativity: An Introduction for Physicists · Problem 7.12

The equivalence principle and spacetime curvature — Hobson, Efstathiou & Lasenby

Physics · curvature-tensor-three-dimensional-manifold

Problem

By calculating the components of the curvature tensor

R^d{}_{abc}

in each case, show that the line element

ds^2 = \frac{a^2}{a^2 - r^2}\,dr^2 + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2

represents a curved three-dimensional manifold. Show that the manifold is flat in the limit

a \to 0