Diffeomorphisms and boundary conditions

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:1007.1031) 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 Near-Horizon Extremal Kerr (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...

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Cross-posted to math.stackexchange.com/q/289891/11127 – Qmechanic Feb 1 '13 at 18:37
Related question by OP: physics.stackexchange.com/q/45723/2451 – Qmechanic Feb 1 '13 at 18:43
I posted that on math.stackexchange because I hadn't find an answer here... – dingo_d Feb 2 '13 at 11:53

The most difficult part is to actually get a set of consistent boundary conditions in the first place - this requires a combination of educated guessing, physical insights, prior experience with related problems, detailed calculations and trial-and-error. In short, it is a bit of an art.

However, once you have a set of boundary conditions (as in your case the NHEK boundary conditions) things are fairly straightforward.

Let me denote the asymptotic background metric by $g$ and the state-dependent fluctuations by $h$, so that any metric of the form $g+O(h)$ is allowed by the boundary conditions.

Your goal is to check what are the gauge transformations that preserve your boundary conditions. In pure gravity, these must be some diffeomorphisms generated by some vector field $\xi$, such that $$L_\xi (g+h) = O(h)$$ where $L_\xi$ is the Lie-derivative. Since this is an equation for a symmetric tensor in $D$ dimensions you get $D(D+1)/2$ independent linear first order PDEs for the vector field $\xi$.

If you tried to solve the equation above you were precisely doing the correct thing, which hopefully answers part of your question. Let me now address the other part, namely how to solve these PDEs.

In many examples you can solve for the most general vector field $\xi$ compatible with the condition above simply by guessing a suitable Ansatz and then showing that it works.

In most applications you have some series expansion of the asymptotic metric in powers of some radial coordinate $r$ (or in exponentials of $r$, depends on your gauge choice for the radial coordinate).

To reduce clutter let me assume that the various tensor components of $g$ and $h$ are expressed in some Laurent series of $r$, and that $r\to\infty$ corresponds to the asymptotic boundary.

Then you just make the same kind of power series Ansatz for the vector field $\xi$. Typically, all components of the vector field start at $O(1)$ or smaller, but this need not be the case. If you have no clue at all, then just make the Ansatz $$\xi^0 = r^{n_0} (\xi^0_0 + \xi^0_1/r + ...)$$ where the coefficient functions $\xi^0_i$ are allowed to depend on all boundary coordinates a priori. You make a similar Ansatz for all the other components of the vector field.

Evaluating the conditions from the Lie variation then determines the exponents $n_i$ and may put restrictions on the functions $\xi^i_j$.

See for instance exercise (17.1) in exercises of week 7 on my teaching webpage http://quark.itp.tuwien.ac.at/~grumil/teaching.shtml for a guided tour through the standard AdS$_3$ example. If you have never done this sort of calculation before I recommend you start with that example before breaking the NHEK.

[On a sidenote, I have never tried to implement this algorithm in Mathematica, since I usually work in 3 dimensions where a calculation by hand is fairly quick, but I do not see any reason why it should not work in Mathematica.]

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Thank you for the detailed answer! :) I will work on this problem on my master thesis, and I tried to get the results of $AdS_3$ metric, that I've found on the web by applying the Brown-Henneaux boundary conditions, but I didn't have much time to go through with it, since I had other exams and I was working on a seminar, but I'll definitely try to work that out before starting with the NHEK. Thanks again for guiding me in the right direction :) – dingo_d Feb 7 '13 at 21:43
Hi, I'm trying to solve the problem you've stated, in your exercises, but I don't think I am doing this right. First, I don't seem to know where that $O(1)$ comes from in calculating the Lie derivative of the ++ component. Sure, I get where I get the $2g_{+-}\partial_+\xi^-$, that's straight forward, but then the $O(1)$ appears :\ Plus if I differentiate the $\xi^-$ wouldn't I get third derivative of $\varepsilon^+(x^+)$? Is there any literature or solved exercises I could see? That would help a lot... – dingo_d Feb 22 '13 at 17:09
There are various ${\cal O}(1)$ contributions in the ++ component of the Lie derivative - e.g. the first term $\xi^\mu\partial_\mu g_{++} = {\cal O}(1)$. Maybe I do not understand your question in the comment - you can e-mail me if you want, since I do not check physics stackexchange too frequently. BTW, the third derivative term appears in the next exercise (17.2), but it does not play any important role for the boundary condition preserving gauge transformations. – Daniel Grumiller Mar 21 '13 at 22:16