meson solutions in the black hole
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As in the zero temperature case, we take a vector field $V_m = (V_\mu,V_z)\,$ ($\mu = 0,1,2,3$) described by an action integral with the general form of eq. (\[vectorfieldactionzeroptemp\]) and soft wall background $\Phi = k^2z^2   $ . But for describing the thermal spectra one considers the geometry as the Minkowski version of the black hole metric (\[EuclideanMetric\]): $$ds^2 \,\,= \,\, \frac{R^2}{z^2}  \,  \Big(  -  f(z) dt^2 + \frac{dz^2}{f(z) }  + d\vec{x}\cdot d\vec{x}  \Big)   \,.
 \label{MinkoviskyMetric}$$ where again: $ f (z) = 1 - z^ 4/ z_h^ 4  $ and the gauge theory temperature is related to the horizon position by eq. (\[temp\]). It is important to note that this black hole geometry will be stable only for temperatures $T > T_c$. We will calculate the thermal spectral functions using this black hole metric for all temperatures with the interpretation that for $T <  T_c$ it represents a super-cooled (unstable) phase.

As in the zero temperature case, we choose the gauge $V_z = 0 $ and assume $ V^0_{\mu}(x) =
\lim_{z\to z_0} V_\mu (x,z) $ to be the sources of the correlation functions of $  J^\mu (x) \,$. Now, with the radial AdS 