# Stress-energy tensor of a scalar theory in AdS

In this review of the AdS/CFT correspondence, going through section 2.2.2 I am not quite sure where the stress-energy tensor $$T_{\mu\nu}=2\partial_\mu\phi\partial_\nu\phi-g_{\mu\nu}\left((\partial\phi)^2+m^2\phi^2\right)+\beta\left(g_{\mu\nu}\Delta-D_{\mu}D_\nu+R_{\mu\nu}\right)\phi^2$$ given in equation 2.37, comes from.

I tried deriving the tensor from the action $$S=\int{d^dx\sqrt{-g}\left(R^{\mu\nu}g_{\mu\nu}+\partial_\mu\phi\partial_\nu\phi g^{\mu\nu}+m^2\phi^2\right)}$$

since it is stated that the stress energy tensor comes from the scalar field theory in AdS space, but I don't retrieve the desired stress energy tensor.

This energy momentum tensor comes from a non-minimally coupled term in the form $$S_{\text{interaction}} = -\int d^dx\sqrt{-g}\beta R\phi^2$$ alongside the kinetic and potential term in your action. However the authors write in the paragraph below equation (2.37) that "The value of $$\beta$$ is determined by the coupling of the scalar curvature to $$\phi^2$$ , which on AdS has the same effect as the mass term in the wave equation (2.32)". On AdS space this term has the same effect with the mass term in your action.