\,$. The on shell action takes the form $$I_{on \, shell }\,=\,   \, \int d^4q \,\,\,\Big[  {V^ 0_\mu}^\ast ( q ) {\cal F}^{\mu\nu} (z,p)  V^0_\nu ( q )
  \Big]_{_{ \! z \to z_0 }}
 \,,
\label{onshellaction3}$$ where $${\cal F^{\mu \nu} } (z,q)  \,=\,   \frac{1}{2 g_5^2}   e^{- k^2 z^ 2  }  \sqrt{-g}  \, g^{zz}   g^{\mu\nu} 
v^\ast  ( q,z)  \partial_z v (q,z) ]\,.$$

The corresponding retarded Green’s function is: $$G_R^{\mu\nu}  (q) \,=\, {\cal F}^{\mu\nu}  (z=z_0 , q)  \,,$$ and the spectral function is the imaginary part of the retarded Green’s function: $$\rho^{\mu\nu}  (q) \, =\, -  {\cal I}m  \{ G_R^{\mu\nu}  (q) \}   \,.$$ The bulk to boundary propagators $ v (q,z) $ are solutions of the equations of motion. These equations have different forms for the temporal $V_0$ and spatial $V_i$ components of the vector field. For the case of a plane wave solution with momentum $q^\mu = (\omega , \vec q )  $ they are: $$\begin{aligned}
\partial_z \Big( \frac{ e^{-k^ 2 z^ 2 }  } { z}  \partial_z V_0 (q,z) \Big) &-&
 \frac{ e^{-k^ 2 z^ 2 } }{ z f(z) } \Big(   \frac{\omega^ 2 }{ f(z) }  -\vert {\vec q}\vert^2 \Big)  
   V_0 (q,z) \,=\, 0  \cr \cr
      \partial_z \Big( \frac{ 