- #1

ergospherical

- 976

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##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.