Past exam of the general relativity course of the University of Oxford / 2022 / 2 / e / Solution

In standard spherical coordinates, parametrize the two-sphere byThe coordinate basis transforms from the fixed Cartesian basis according to ${\mathbf{e}}_a=(\partial x^i/\partial q^a){\mathbf{e}}_i$. ThereforeTheir inner products are ${\mathbf{e}}_{\theta }^2=R^2$, ${\mathbf{e}}_{\theta }\cdot {\mathbf{e}}_{\varphi }=0$ and ${\mathbf{e}}_{\varphi }^2=R^2{\sin }^2\theta$. Thus the induced metric isThe source's intermediate display places an extra $R^2$ outside entries already labelled $g_{\theta \theta },\dots$. Interpreting those entries as the actual metric tensor components would double-count the factor: for example, $R=2$ gives $g_{\theta \theta }=4$ directly from ${\mathbf{e}}_{\theta }^2$, not $16$. Its final matrix is the correct induced metric; the entries of its intermediate matrix must instead denote dimensionless coefficients.