Nordstrom theory of gravitation

Nordstrom's scalar theory restricts the spacetime metric to the conformally flat form $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$. Its null geodesics have the same unparametrized paths as those of the Minkowski metric, so the theory predicts no gravitational bending of light.

A metric is conformally flat when it is locally a positive scalar multiple of a flat metric. Multiplication by the conformal factor preserves null cones and unparametrized null geodesics.

A conformal factor is the positive scalar field multiplying a metric under a conformal rescaling. In numerical relativity one often writes a spatial metric as ${\gamma }_{ij}={\psi }^4{\tilde{\gamma }}_{ij}$.

For $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$,For a null curve, the last term drops from the geodesic equation; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.