Extra (boundary?) term in Brans Dicke field equations

[ergospherical]

Here is the action:
$S = \frac{1}{16\pi} \int d^4 x \sqrt{-g} (R\phi - \frac{\omega}{\phi} g^{ab} \phi_{,a} \phi_{,b} + 16\pi L_m)$
the ordinary matter is included via $L_m$. Zeroing the variation $\delta/\delta g^{\mu \nu}$ in the usual way gives

$\frac{\delta}{\delta g^{\mu \nu}}[R\phi - \frac{\omega}{\phi} g^{ab} \phi_{,a} \phi_{,b}] + \frac{1}{\sqrt{-g}}(R\phi - \frac{\omega}{\phi} g^{ab} \phi_{,a} \phi_{,b}) \frac{\delta(\sqrt{-g})}{\delta g^{\mu \nu}} - 8\pi T_{\mu \nu} = 0$

where $T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_m)}{\delta g^{\mu \nu}}$ is the stress energy of the matter. Inserting the variations of $R$ and $\sqrt{-g}$ (which are $R_{\mu \nu}$ and $-\frac{1}{2} \sqrt{-g} g_{\mu \nu}$ respectively) gives

$G_{\mu \nu} + \frac{\omega}{\phi^2}(\frac{1}{2}g^{ab} \phi_{,a} \phi_{,b} g_{\mu \nu} - \phi_{,\mu} \phi_{,\nu}) = 8\pi T_{\mu \nu}/\phi$

On Wikipedia (https://en.wikipedia.org/wiki/Brans–Dicke_theory#The_field_equations) there is another term $\frac{1}{\phi}(\nabla_a \nabla_b \phi - g_{ab} \square \phi)$. I suspect it is a boundary term? Where did it come from.

 

[renormalize]

I suspect it is a boundary term? Where did it come from.

Your suspicion is correct. By the Palatini identity, the general variation of the Ricci scalar is:$$\delta R=R_{\mu\nu}\delta g^{\mu\nu}+\nabla_{\sigma}\left(g^{\mu\nu}\delta\Gamma_{\mu\nu}^{\sigma}-g^{\mu\sigma}\delta\Gamma_{\lambda\mu}^{\lambda}\right)\tag{1}$$For the Einstein-Hilbert action, the last term on the right is dropped because it's a total divergence that integrates to an ignorable boundary term. But for the Brans-Dicke action, we multiply the full variation (1) by $\phi$ to get:$$\phi\delta R=\phi R_{\mu\nu}\delta g^{\mu\nu}+\phi\nabla_{\sigma}\left(g^{\mu\nu}\delta\Gamma_{\mu\nu}^{\sigma}-g^{\mu\sigma}\delta\Gamma_{\lambda\mu}^{\lambda}\right)$$$$=\phi R_{\mu\nu}\delta g^{\mu\nu}-\left(\nabla_{\sigma}\phi\right)\left(g^{\mu\nu}\delta\Gamma_{\mu\nu}^{\sigma}-g^{\mu\sigma}\delta\Gamma_{\lambda\mu}^{\lambda}\right)+\nabla_{\sigma}\left(\phi\left(g^{\mu\nu}\delta\Gamma_{\mu\nu}^{\sigma}-g^{\mu\sigma}\delta\Gamma_{\lambda\mu}^{\lambda}\right)\right)$$Again we ignore the last term as a divergence, but the second part is non-zero whenever $\phi \neq \text{constant}$, and is responsible for the additional terms in the Brans-Dicke field-equations that involve second-derivatives of $\phi$.

 

[ergospherical]

Interesting - thank you.