Nordstrom's theory of gravity postulates that the metric is of the form

$g_{\mu \nu} = \phi ^2 (x) \eta_{\mu \nu}.$

The field equations, in vacuum are of the form $R=0$, where $R$ is the Ricci scalar.

In the wikipedia article on <a href="https://en.wikipedia.org/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation">Nordstrom gravity</a>, it says that the Ricci scalar of the metric above is given by

$R = -6 \frac{ \Box \phi}{\phi^3},$

so the field equations should be $\Box \phi=0$.

However, later on in the article, they set $g_{\mu \nu} = \exp (2\psi) \eta_{\mu \nu},$ so that $\phi = \exp(\psi)$, and say that $\psi$ satisfies $\Box \psi=0$. However, this is not compatible with the result above, since $\Box \phi = \Box \exp(2\psi)=0$ does not imply that $\Box \psi =0$.

Later on, the article says that, for a static, spherically symmetric solution, we have $\nabla^2 \psi=0$, with $\psi$ as above. Using the usual expression for laplacian in spherical coordinates, we get $\nabla^2 \psi = \frac{d}{dr}(\frac{1}{r^2} \frac{d \psi}{dr})=0$

They then say that the metric is $g_{\mu nu} = (1-m/r) \eta_{\mu nu}$, with spherical coordinates. However, the $\psi$ needed to get this metric is not a solution of laplace's equation. And $\Box \left ( \sqrt{1-m/r} \right )$ does not equal zero. So it satisfies neither of the two field equations (which, as far as I can tell are different). What's going on here?

Anyone have a reference that discusses the spherically symmetric solution to Nordstrom gravity? The usual ones (Misner/Thorne/Wheeler etc) don't discuss the spherical vacuum solution to Norstrom gravity.

Any help is much appreciated!