# Difeomorphisms and boundary conditions

So I asked on physics.stackexchange, but got no answer, so I'll try here:

I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism which preserve the boundary conditions in the same paper.

I found this paper (arXiv:1007.1031v1) which say that by solving $\mathcal{L}_\xi g_{\mu\nu}$, for components and equating each component with the appropriate boundary condition, I can get the most general $\xi$ (which is my goal after all).

So I took the NHEK metric which has 6 non vanishing terms ($g_{\tau\varphi}=g_{\varphi\tau}$ so that gives me 5 equations to solve), I put the boundary conditions ($\mathcal{O}(r^n)$ terms), and to simplify things a bit, I typed everything into Mathematica. But when I put my 5 differential equations in, I got the error that I have too many equations and too few variables ($\tau, r, \theta, \varphi$)!

Now I thought, did I have to include all possible $g_{\mu\nu}$? Well, that wouldn't make much sense, since all other terms of the background metric are zero, right? And even if I include them, I'll get more equations, and still only 4 variables :\ So Mathematica will probably give the same error...

So first of all, am I correct in trying to find the diffeomorphism that way? And if I'm correct, how to solve that?! It's a big system of ODE's, and it's not so trivial to solve, given how the metric looks :\

So if you have any suggestion, I'd appreciate it...

Also, I think that I should solve it by assuming the form

$$\xi^\mu=\sum_n \xi_n^\mu(\tau,\varphi)r^n$$

and maybe plugging it in, but still, I have too many equations :\

And I'm not that good with mathematica...

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Cross-posted from physics.stackexchange.com/q/51547/2451 –  Qmechanic Feb 1 '13 at 18:38