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An affinely parameterized geodesic $x^{\mu }(\lambda )$ satisfiesFor the conformally flat metric $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$, its Christoffel symbols areLet $u^{\mu }=d{x}^{\mu }/d\lambda$. Along a null geodesic, ${\eta }_{\alpha \beta }{u}^{\alpha }{u}^{\beta }=0$, so the equation reduces toThe acceleration is parallel to the tangent. A change of parameter removes it, leaving a straight null line of the Minkowski metric. Thus conformal rescaling changes the affine parameter but not the unparametrized light ray.

Consequently Nordstrom theory of gravitation predictsfor light passing a star. This contradicts the observed nonzero deflection and the successful general-relativistic value $4GM/(b{c}^2)$.

For a spatial initial slice with vanishing extrinsic curvature, the vacuum momentum constraint is automatic and the Hamiltonian constraint requires zero Ricci scalar. In three spatial dimensions, the indicated conformally flat metric has $\mathcal{R}=-8{\psi }^{-5}\Delta \psi$, with a nonzero conformal factor. Thus a harmonic function supplies the data on its domain. A positive example is $\psi =1+a/r$ on a punctured or exterior region. There is no nonconstant globally smooth example on all of Euclidean three-space tending to one at infinity, by the maximum principle for harmonic functions. Time symmetry of an initial slice does not imply that its entire evolved spacetime is static.

Every conformally flat metric in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.