Proposition: An invisible second-order curvature change

Near the origin compare

Both have zero first metric variation at $\lambda =0$, the same pointwise cones, and the same first-order vacuum matching. Nevertheless,

whereas $G[{g^{(0)}}_{\lambda }]=0$.

Proof. The first parameter derivative of both metrics at zero is zero. At the coordinate origin,

Thus the connection is zero there and the curvature contains no connection-product terms. Its dependence on the second metric derivatives at that point is exactly the flat first-variation expression. Those derivatives equal the second derivatives of $2{\lambda }^2(x^0{)}^2\eta$. Equation (eq:conformal-einstein) gives (eq:second-order).