A chart with the point-normal first-order form whose second metric derivatives do not vanish

The components $g_{ab}$ are smooth functions on ${\mathbb{R}}^4$; at every point the matrix of components is $\mathrm{diag}(-1,1+(x^0{)}^2,1,1)$, which is symmetric with one negative and three positive eigenvalues. Hence $g$ is a smooth Lorentzian metric on $U$ with the same time orientation $N={\mathrm{\partial }}_0$ as the flat model.

At the point $p=(0,0,0,0)$ the matrix is $\mathrm{diag}(-1,1,1,1)$, so $g_{ab}(p)={\eta }_{ab}$. The only component depending on the coordinates is $g_{11}=1+(x^0{)}^2$, and ${\mathrm{\partial }}_c{g}_{11}=2x^0{\delta }^0{}_c$ vanishes at $p$; all other components are constant. Hence ${\mathrm{\partial }}_a{g}_{bc}(p)=0$, and by the Christoffel formula of [L2] also ${\mathrm{\Gamma }}^a{}_{bc}(p)=0$: the identity chart satisfies the point-normal identities of [L3] at $p$.

The second derivative in the direction of $x^0$ of the $(1,1)$ component is ${\mathrm{\partial }}_0{\mathrm{\partial }}_0{g}_{11}(0)={\mathrm{\partial }}_0(2x^0){\mathrm{\mid }}_0=2\mathrm{\ne }0$. Therefore ${\mathrm{\partial }}_a{\mathrm{\partial }}_b{g}_{cd}(p)$ does not vanish for all index choices, namely not for $(a,b,c,d)=(0,0,1,1)$, and the displayed statement is refuted.

The statement refuted concerns only second-order vanishing in a chart with the point-normal first-order identities; the example asserts nothing about the curvature of $g$ at $p$, and no such claim is used. In particular this item does not refute any claim of Point-normal and exponential coordinate charts of the metric-selected connection, which is explicitly silent at second order.