coordinate defined in the region: $ z_0 \le z \le z_h $, the on shell action takes the form: $$I_{on \, shell }\,=\, - \frac{1}{2 { g}_5^2}  \, \int d^4x \,\,\,\Big[ e^{- k^2 z^ 2  } \sqrt{-g}  \, g^{zz} g^{\mu \nu}   V_\mu \partial_z V_\nu  \Big]
{\Big   \vert}_{_{ \! z \to z_0 }}^{_{ \! z \to z_h }}
 \,.
\label{onshellaction2}$$ The imaginary part of the on shell action should generate holographically the thermal spectral function. However, it was pointed out in ref. [@Son:2002sd] that for an action like (\[onshellaction2\]) the imaginary part is $z $ independent. So the contributions from the two integration limits cancel each other out. This problem can be solved following again [@Son:2002sd] and using the additional prescription that only the boundary $z= z_0$ is considered. In other words, one takes eq. (\[onshellaction2\]) with only the lower integration limit. For an interesting discussion on the interpretation of the prescription for calculating the retarded Green’s function, see [@Iqbal:2008by].

The procedure to find the retarded Green’s function involves fourier transforming the fields and decomposing the momentum space fields as it was done in the finite temperature case in eq. (\[Bulktoboundary\]): $ V_\mu (q,z) \,=\, v (q,z) V^0_\mu ( q ) 