How is the $\text{AdS}_3$ stress tensor in the Fefferman-Graham expansion derived?

Question

I have been reading "Lectures on black holes and the $\text{AdS}_3$/$\text{CFT}_2$ correspondence" by Per Kraus.

We consider a theory of gravity with Einstein-Hilbert action $$ \frac{1}{16\pi G}\int \left(R-\frac{2}{\ell^2}\right)\sqrt{g}\mathrm{d}^3 x + \text{boundary terms}$$

One solution is $\text{AdS}_3$. I am struggling with deriving the $\text{AdS}_3$ stress tensor $$ T_{ij} = \frac{1}{8\pi G\ell}\left(g_{ij}^{(2)} - \mathrm{Tr}(g^{(2)})g_{ij}^{(0)}\right) \tag{2.16}$$ where the metric in $\text{AdS}$-space is $\mathrm{d}s^2 = \mathrm{d}\eta^2 + g_{ij}\mathrm{d}x^i\mathrm{d}x^j$ in Gaussian normal coordinates. There's a radial coordinate introduced earlier as $$ \mathrm{d}s^2 = \left(1 + \frac{r^2}{\ell^2}\right)\mathrm{d}t + \frac{\mathrm{d}r^2}{1+r^2/\ell^2} + r^2\mathrm{d}\phi^2\tag{2.2}$$ of which I don't know how to handle it or how to relate it to the "Fefferman-Graham expansion" $$ g_{ij} = \mathrm{e}^{2\eta/\ell}g_{ij}^{(0)} + g_{ij}^{(2)} + \dots \tag{2.12}$$ where I am also not sure what role exactly the "conformal boundary metric" $g_{ij}^{(0)}$ plays and how to handle these metrics in the computation of $(2.16)$.

I think this confusion also spills to the second section, the Virasoro generators: Why does the stress tensor in terms of $w,\bar w$ defined by $g_{ij}^{(0)}\mathrm{d}x^i\mathrm{d}x^j = \mathrm{d}w\mathrm{d}\bar w$ $$ T_{ww} = \frac{1}{8\pi G\ell} g^{(2)}_{ww}\tag{2.19}$$ not contain the conformal boundary metric $g^{(0)}_{ij}$?

This post imported from StackExchange Physics at 2015-11-13 22:28 (UTC), posted by SE-user Rev SS