Solving ODE with negative expansion power series

Question

Since on Mathematics stackexchange I didn't get an answer, I'll try it here, since people here are more familiar with this topic (general relativity related).

I am reading a dissertation of Porfyriadis "Boundary Conditions, Effective Action, and Virasoro Algebra for $AdS_3$", and I am trying to solve a system of DE to get the appropriate diffeomorfism (page 31 onward).

I am trying to solve a system of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this:

$\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$

which you get by taking a Lie derivative of background metric and setting it equal to certain $\mathcal{O}(r^n)$ terms.

$g_{ij}$ are given, from metric ofc. I need to assume that the solution (since I'm looking for components of $\xi^\mu$( which is a vector with components $\xi^t, \xi^r, \xi^\phi$) is given with power series of the form:

$\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$,

and this is to be seen as expansion around 1/r (expansion around $r=\infty$).

Now when I plug this in the ODE I get this

$\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where

$\xi^\mu_{n,i}$

is the derivative with the respect to i-th component.

What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my O(r) terms cancel each other out? Or?

I'm kinda stuck, at this seemingly easy point.

In the thesis he gets 6 equations with coefficients, first one should be:

$\xi^r_{n-1}+l^2\xi^t_{n,t}+\xi^t_{n-2,t}=0,\ n\ge 2$,

But I am not getting this. What am I doing wrong?

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EDIT: For further clarity: The metric is that of $AdS_3$ given with line element:

$ds^2=-\left(1+\frac{r^2}{l^2}\right)dt^2+\left(1+\frac{r^2}{l^2}\right)^{-1}dr^2+r^2 d\phi^2$,

and the differential equations in question are given by solving $\mathcal{L}_\xi g_{\mu\nu}=\mathcal{O}(r^n)$, where $\mathcal{O}(r^n)$ are the fall off conditions. In the dissertation, he took the deviation of nonzero components of the metric to be subleading, that is:

$\mathcal{L}_\xi g_{tt}=\mathcal{O}(r)$ $\mathcal{L}_\xi g_{rr}=\mathcal{O}(r^{-3})$ $\mathcal{L}_\xi g_{\phi\phi}=\mathcal{O}(r)$, while others are $\mathcal{O}(1)$.

Solving Lie derivative gives me 6 equations, which I should solve by plugging in the above ansatz ($\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$), but this is the part I get stuck.

ADDENDUM:

I was looking at other components, and have noticed that I have $g_{rr}$ factor with some of them. That term in metric is:

$g_{rr}=\left(1+\frac{r^2}{l^2}\right)^{-1}$

Now, is it legitimate thing to expand this around $r=\infty$ so that I can put $r$ terms inside the sums (assumed solution)?

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user dingo_d

Comments

@dingo_d Well, Math Overflow is generally only for research-level math. If your question is just needing an application of well-known ODE theory, it belongs on Math Stackexchange. If it calls upon physical principles in some meaningful way (beyond just "this came up while doing physics"), then it is okay here. But I doubt it will be particularly well-received on Math Overflow. In general, though, if you feel the question should be moved, you can ping a moderator and request they migrate it for you, to avoid cross-site duplication.

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user Chris White

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@DavidZ considering that that question does not have an answer and this one does, it would seem more appropriate to leave this one open and close the other one instead.

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user Antonio Vargas

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@AntonioVargas sure, perhaps, but I trust that the mods can make that decision. ;-)

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user David Z

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Well, yes, in the end, the question is just about ODEs, so maybe it won't be appropriate on MO. '

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user dimensio1n0

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There is a meta question regarding the closing of this question here. Basically, the argument is that this should not be a duplicate as it has an answer, while the "original" question does not. So the duplicate chain does not make sense!

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user user1729 PhD

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That may be, but the content of your question is all mathematics. And again, without stating your original ODE this is very hard to answer - as is any question of the form "how do I derive this equation $ ... $?" that does not provide the necessary premises.

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user episanty

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You could see i f i t gets answers on Math Overflow.

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user dimensio1n0

Answer 1

I didn't really read the question (at least the stuff about metrics or whatever), but I will write what I think is the answer anyway. You are doing an expansion around $r=\infty$. You have $$\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r).$$ This equation says that as $r \to \infty$, the LHS, which is $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}$ only goes to $\infty$ as fast as $r$ to the first power. This means that the higher powers of $r$ from the three terms cancel each other.

Well how can they cancel each other? Lets look at the $r^2$ piece. The first term gives a contribution $\frac{2}{l^2}\xi^r_1r^{2}$. The second term gives a contribution $2\xi^t_{2,t} r^2$, and the third term gives a contribution $\frac{2}{l^2}\xi^t_{0,t}r^{2}$. The sum of these contributions is $$\frac{2}{l^2}\xi^r_1r^{2}+2\xi^t_{2,t} r^2+\frac{2}{l^2}\xi^t_{0,t}r^{2} = (\frac{2}{l^2}\xi^r_1r^{2}+2\xi^t_{2,t}+\frac{2}{l^2}\xi^t_{0,t})r^2.$$ For this to be zero, we must have $$\frac{2}{l^2}\xi^r_1+2\xi^t_{2,t}+\frac{2}{l^2}\xi^t_{0,t} =0,$$ or rearranging, $$\xi^r_1+l^2\xi^t_{2,t}+\xi^t_{0,t} =0.$$

If that is good let's move on to the general $n\ge 2$. The first term gives a contribution $\frac{2}{l^2}\xi^r_{n-1}r^{n}$. The second term gives a contribution $2\xi^t_{n,t} r^n$, and the third term gives a contribution $\frac{2}{l^2}\xi^t_{n-2,t}r^{n}$. The sum of these contributions is $$\frac{2}{l^2}\xi^r_{n-1}r^{n}+2\xi^t_{n,t} r^n+\frac{2}{l^2}\xi^t_{n-2,t}r^{n} = (\frac{2}{l^2}\xi^r_{n-1}r^{2}+2\xi^t_{n,t}+\frac{2}{l^2}\xi^t_{n-2,t})r^n.$$ For this to be zero, we must have $$\frac{2}{l^2}\xi^r_{n-1}+2\xi^t_{n,t}+\frac{2}{l^2}\xi^t_{n-2,t} =0,$$ or rearranging, $$\xi^r_{n-1}+l^2\xi^t_{n,t}+\xi^t_{n-2,t} =0.$$ The $n$ where this needed to be zero were the $n$ for higher than linear terms, i.e., $n\ge2$.

This is the equation that you said you didn't understand how they got it. I hope this clears up part of the confusion.

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user NowIGetToLearnWhatAHeadIs

Comments

Finally someone who know how to explain it!!!!! THANK YOU SO MUCH!! I'll try to set up a bounty and give it to you, you deserve it :)

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user dingo_d

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Oh and, while I'm at it, do you have any material on this kind of solving method? If it has any special name so that I can google it or sth like it? I'd be really grateful, since I looked all over but had no luck...

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user dingo_d

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I don't know of a name for it besides just using power series. There probably is a name. The only other time I remember seeing something quite like this is solving the quantum harmonic oscillator using power series. Here is a link:Quantum Harmonic Oscillator. Maybe someone else will be more helpful.

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user NowIGetToLearnWhatAHeadIs

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I'll look it up, maybe I find a proper name for this :D I'll give you bounty in an hour (at least that's what M.SE says :D)

This post imported from StackExchange Mathematics at 2014-06-09 18:51 (UCT), posted by SE-user dingo_d