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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?

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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.

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    $\begingroup$ What if I take square root over TabTab. Then it reduces to Newtonian gravity. $\endgroup$ – Khushal Jul 13 '17 at 21:26
  • $\begingroup$ @Khushal: You might be able to recover the Newtonian theory in that case; I can't immediately think of an objection. However, that doesn't mean that the theory would be viable; it could fail for other reasons. In particular, relativistic scalar theories of gravity frequently have trouble describing light-bending, since most of them either have a conformally flat metric or include frame-dependent effects. If you're interested in this sort of thing, I highly recommend that you read Clifford Will's review article on the subject. $\endgroup$ – Michael Seifert Jul 14 '17 at 13:33

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