As usual I'm asking a question about boundary conditions for AdS${}_3$, based on the thesis by Porfyriadis.

He is solving equations $\mathcal{L}_\xi g_{\mu\nu}$ for AdS${}_3$ metric, with a given boundary conditions that are basically a fall off conditions in $r$, since we're interested in the asymptotic behavior (as $r\to\infty$). I've finally gotten to the part where he, using the ansatz for diffeomorphism $\xi^\mu=\sum\limits_n \xi^\mu_n(t,\phi)r^n$ ($\mu=t,r,\phi$), gets a set of 6 equations for coefficients. I'll write the component of metric that was used to obtain the equation next to the equation, for clarification:

$$(tt)\qquad \xi^r_{n-1}+l^2\xi^t_{n,t}+\xi^t_{n-2,t}=0,\ n\ge 2$$
$$(tr)\qquad l^4(n+1)\xi^t_{n+1}-l^4\xi^r_{n,t}+3l^2(n-1)\xi^t_{n-1}+2(n-3)\xi^t_{n-3}=0,\ n\ge 3$$
$$(t\phi)\qquad l^2\xi^t_{n,\phi}+\xi^t_{n-2,\phi}-l^2\xi^\phi_{n-2,t}=0,\ n\ge 1$$
$$(rr)\qquad l^2(n+1)\xi^r_{n+1}+(n-2)\xi^r_{n-1}=0,\ n\ge 2$$
$$(r\phi)\qquad l^2(n+1)\xi^r_{n+1}+(n-3)\xi^\phi_{n-3}+l^2\xi^r_{n,\phi}=0,\ n\ge 3$$
$$(\phi\phi)\qquad \xi^r_{n-1}+\xi^\phi_{n-2,\phi}=0,\ n\ge 2$$

I got that, and I understand how the author got that. But how did he find the $\xi^r,\ \xi^t$, and $\xi^\phi$ I don't get :\

He said: for ($rr$) equation, using backwards induction, since for large $n$ the series for $\xi^r$ must truncate, we get that the components $\xi^r_{2m}=0,\ m\ge 1$, and $\xi^r_{2m+1}=0,\ m\ge 1$, so the most general form of $\xi^r$ is

$$\xi^r=\xi^r_1(t,\phi)r+\xi^r_0(t,\phi)+\mathcal{O}(r^{-1})\quad (\star)$$

How did he get that? I mean, I tried putting from n=10 towards n=2, and for n=2 I get

$$3l^2\xi^r_3+0\cdot\xi^r_1=0$$

And that means that for n=even my odd terms are 0, if $n\ge 2$?

I could kinda relate this to the fact that the $\xi^r=\sum\limits_n \xi^r_n r^n$, and for n=2 I'll get 0, so only n that are less than that will contribute, since I am making a expansion around infintiy. But I don't know if I'm right about this. And how did he make other equations? I tried using the same 'reasoning' but cannot get what he gets. In equations ($tr$) and ($r\phi$) he just drops off the $\xi^r_n$ terms. Why? :\ And then all of a sudden he gets

$$\xi^t=\xi^t_0(t,\phi)+\xi^t_{-1}(t,\phi)\frac{1}{r}+\mathcal{O}(r^{-2})$$
$$\xi^\phi=\xi^\phi_0(t,\phi)+\xi^\phi_{-1}(t,\phi)\frac{1}{r}+\mathcal{O}(r^{-2})$$

How?! :( I'm desperate :(

This post imported from StackExchange Physics at 2014-03-07 13:35 (UCT), posted by SE-user dingo_d