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...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.