In text "Covariant theory of asymptotic symmetries, conservation laws and central charges" is given an example of finding central charges and superpotential (among other things).

I am interested in $AdS_3$ case, since there is a lot of literature on this space-time, and I'm going to need to preform the same analysis for 4 dimensional Near-Horizon Extremal Kerr (NHEK) metric, so I would like to know how to do that in a simpler case.

In a given example, on page 48, after specifying boundary conditions, they go on to find the linear part of $\delta[(1/16\pi)\sqrt{-g}(R-2\Lambda)/\delta g_{\mu\nu}]$, which is, as far as I understand it, a kind of variation of Lagrangian.

The formula is given:

$\mathcal{H}^{\mu\nu}[h;\bar{g}]:=\frac{\sqrt{-g}}{32\pi}\left[-h\bar{R}^{\mu\nu}+\frac{1}{2}h\bar{R}\bar{g}^{\mu\nu}+2h^{\mu\alpha}\bar{R}_\alpha^\nu+2h^{\nu\beta}\bar{R}_\beta^\mu-h^{\mu\nu}\bar{R}-h^{\alpha\beta}\bar{R}_{\alpha\beta}\bar{g}^{\mu\nu}+\bar{D}^\mu\bar{D}^\nu h+\bar{D}^\lambda\bar{D}_\lambda h^{\mu\nu}-2\bar{D}_\lambda\bar{D}^{(\mu}h^{\nu)\lambda}-\bar{g}^{\mu\nu}(\bar{D}^\lambda\bar{D}_\lambda h-\bar{D}_\lambda\bar{D}_\rho h^{\rho\lambda})+2\Lambda h^{\mu\nu}-\Lambda \bar{g}^{\mu\nu}h\right]$

I also have the background metric ($\bar{g}_{\mu\nu}$) which is that of $AdS_3$, $\bar{D}$ is covariant derivative, $h$ is trace, given by $h=\bar{g}^{\mu\nu}h_{\mu\nu}$. Ricci tensor and scalar are known.

Now, what confuses me is: how did they get the results (for example $\mathcal{H}^{tt}\to\mathcal{O}(r^-{3})$)? I don't understand this, since they only give the boundary conditions as leading orders in $r$ ($h_{\mu\nu}\to \mathcal{O}(r^{m})$, $m\in\mathbb{Z}$). I can raise and lower indices, I know about summation, and I can find out what terms need to be present in this expression.

But how do I preform calculation with $\mathcal{O}$ notation?

I've been baffled by this every time I read similar articles. They all do these calculations, but I cannot find a single example where everything is explained in detail :\

So any help on clarifying this is welcomed. Any mathematics books that explain this or something...

This post imported from StackExchange Physics at 2014-03-07 13:47 (UCT), posted by SE-user dingo_d