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To see why the relation has to hold, you have to acknowledge that it needs to transform the line elements you have written down into each other. One can see that this is true by taking the logarithm on both sides, which yields $$\rho=\log{x^i}-\log{z}.$$ Taking the derivative of this expression and squaring it gives $\mathrm{d}\rho^2=\mathrm{d}z^2/z^2$, ...


Holographic renormalization for non-conformal branes, i.e. non-AdS/non-CFT systems, was systematically developed in this paper by Kanitscheider, Skenderis and Taylor. They even work it out for the example of the Witten model, which is the background of the Sakai-Sugimoto model. The key principle that permits one to extend the formalism of holographic ...

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