# About parametrizing quadratic fluctuations in the metric about $AdS_2 \times S^2$

I am referring to the contents of page 20-23 of the paper, http://arxiv.org/abs/1108.3842.pdf

• Equation 4.5 seems to suggest that one wants to restrict the metric fluctuations $h$ to a subset such that, $D^\mu h_{\mu \rho} = (1/2)D_\rho h^\mu _ \mu$. (where the $D$ is presumably calculated on the background metric and the indices of $h$ are raised and lowered w.r.t the background metric) What is the motivation for this and how does one see that this fixes all the gauge freedom?

• Can someone explain (or give reference) to how equation 4.6 and 4.7 were derived? (I tried a lot but couldn't get this simplification)

• In equation 4.6 and 4.7 is it possible to impose the condition that the background metric is whatever is standard on $AdS_2 \times S^2$? (..is there a covariant way of putting in this background?..)

[...the following questions can be asked about equation 4.22, 4.23 and 4.30 too .. ]

• In equation 4.14 it seems that one parametrized the 10 components of the 4 dimensional metric $h$ via 10 scalars $B_0,..,B_9$. How has this been done?

There $u$ seems to be a chosen scalar harmonic on $AdS_2 \times S^2$ and it seems to suggest that somehow the metric components can be expressed in terms of derivatives of such a harmonic scalar. How? Can someone help derive this $4.14$?

In 4.14 the variables used $\kappa_1$ and $\kappa_2$ are defined in 4.11 and its not well motivated to me. Also these have such a singular dependence on the variable "a" - which as can be seen in 4.12 parametrizes the Abelian gauge field strength. Then is it obvious from here as to what will the parametrization be if it were just pure gravity?

It looks mysterious to me that when 4.14 is substituted into 4.6 all dependency on $u$ seems to be have vanished to give a far simpler looking 4.15, How has this "magic" happened?

• Also if the base manifold had been just $AdS_2$ then what would have been the parametrization? Is that easily readable from 4.14?

• Is there some general principle at play here by which one can automatically generate such 4.14 like parametrizations for say any $AdS_n \times S^m$?

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/1108.3842 – Qmechanic Jun 8 '14 at 20:06