# Extra (boundary?) term in Brans Dicke field equations

• A
• ergospherical
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.

ergospherical said:
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##.

Last edited:
ergospherical
Interesting - thank you.

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