# Scalar Field Theory for Gravity

While reading the book Gravitation Foundation and Frontiers by Padmanabhan, I came across the Lagrangian for a scalar theory of gravity. But the coupling term consist of trace of the Energy Momentum Tensor. If we change the coupling term to $F(\phi)T_{ab}T^{ab}$, what prevents it from being a viable theory?

If we linearize the equations of motion about a $\phi = 0$ background (and assume the usual sort of kinetic term), we will find a non-relativistic limit of something like $\nabla^2 \phi \propto \rho^2$, which is inconsistent with Newtonian gravity.