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

Choose normal coordinates ${\xi }^{\alpha }$ centered at an event $p$. At $p$, the metric tensor in these coordinates is the constant Minkowski metric ${\eta }_{\alpha \beta }=\mathrm{diag}(-1,1,1,1)$, with ${\xi }^0$ measured in length units, and its first partial derivatives vanish. In arbitrary coordinates $x^{\mu }$, the chain rule for the line element givessoThis construction is pointwise. At nearby events, the metric tensor in the chosen normal coordinates is $g_{\alpha \beta }^{(\xi )}(\xi )={\eta }_{\alpha \beta }+O(|\xi {|}^2)$, with the quadratic corrections determined by the Riemann curvature tensor. Replacing $g^{(\xi )}$ by one constant Minkowski metric throughout an open region would make that region flat, so it cannot describe general curved spacetime exactly.