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 AdS3", 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:
ξr∂rgrr+2gtt∂tξt=O(r)
which you get by taking a Lie derivative of background metric and setting it equal to certain O(rn) terms.
gij are given, from metric ofc. I need to assume that the solution (since I'm looking for components of ξμ( which is a vector with components ξt,ξr,ξϕ) is given with power series of the form:
ξμ=∑nξμn(t,ϕ)rn,
and this is to be seen as expansion around 1/r (expansion around r=∞).
Now when I plug this in the ODE I get this
2l2∑nξrnrn+1+2∑nξtn,trn+2l2∑nξtn,trn+2=O(r), where
ξμn,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:
ξrn−1+l2ξtn,t+ξtn−2,t=0, n≥2,
But I am not getting this. What am I doing wrong?
EDIT: For further clarity: The metric is that of AdS3 given with line element:
ds2=−(1+r2l2)dt2+(1+r2l2)−1dr2+r2dϕ2,
and the differential equations in question are given by solving Lξgμν=O(rn), where O(rn) are the fall off conditions. In the dissertation, he took the deviation of nonzero components of the metric to be subleading, that is:
Lξgtt=O(r)
Lξgrr=O(r−3)
Lξgϕϕ=O(r), while others are O(1).
Solving Lie derivative gives me 6 equations, which I should solve by plugging in the above ansatz (ξμ=∑nξμn(t,ϕ)rn), but this is the part I get stuck.
ADDENDUM:
I was looking at other components, and have noticed that I have grr factor with some of them. That term in metric is:
grr=(1+r2l2)−1
Now, is it legitimate thing to expand this around r=∞ 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