A Synchronous to Newtonian gauge

[ergospherical]

In the context of cosmology, you can perturb around the FRW background, conventionally:$$g = a^2(\tau)[(1+2A)d\tau^2 - 2B_a dx^a d\tau -(\delta_{ab} + h_{ab}) dx^a dx^b]$$with $a,b$ being spatial indices only (1,2,3). You can do gauge transformations $\tilde{x} = x + \xi$ of the coordinates. These are typically split into $\xi^0 \equiv T$ and $\xi^a = \partial^a L + \hat{L}^a$. You find the following first order transformations,\begin{align*}
\tilde{A} &= A - T' - \mathcal{H} T \\
\tilde{B}_a &= B_a + \partial_a T - L_a' \\
\tilde{h}_{ab} &= h_{ab} - 2\partial_{(a} L_{b)} - 2\mathcal{H} T \delta_{ab}
\end{align*}The problem is specifically, is starting from the synchronous gauge, where$$g = a^2(\tau)[d\tau^2 - (\delta_{ab} + h_{ab}) dx^a dx^b]$$with $h_{ab} = \tfrac{1}{3}\delta_{ab} h + (\hat{k}_a \hat{k}_b - \tfrac{1}{3} \delta_{ab}) h_S$ and where $h, h_S$ are functions. And then converting to the Newtonian gauge, where$$\tilde{g} = a^2(\tilde{\tau})[(1+2\Phi)d\tilde{\tau}^2 - (1-2\Psi) \delta_{ab} d\tilde{x}^a d\tilde{x}^b]$$There are supposed to be transformations of the form
$$\tilde{\tau} = \tau + \sum_k T_k(\tau) e^{i\mathbf{k} \cdot \mathbf{x}}, \quad \tilde{\mathbf{x}} = \mathbf{x} + \sum_k L_k (\tau) i\hat{\mathbf{k}} e^{i\mathbf{k} \cdot \mathbf{x}}$$The problem is to find $T_k$ and $L_k$ in terms of $h_S$ and $h$. I can re-write the second one of these with $\xi^j = \partial^j \left[ \sum_k L_k \tfrac{1}{k} e^{i\mathbf{k} \cdot \mathbf{x}} \right]$, in other words my two transformation functions are\begin{align*}
T &= \sum_k T_k e^{i\mathbf{k} \cdot \mathbf{x}} \\
L &= \sum_k L_k \frac{1}{k} e^{i\mathbf{k} \cdot \mathbf{x}}
\end{align*}I must be missing something obvious about how to determine the forms of $T_k$ and $L_k$. For example, in the synchronous gauge then $A=0$ and we get\begin{align*}
\tilde{A} = 0 - T' - \mathcal{H} T = - \sum_{k} \left[ (T_k' + \mathcal{H} T_k) e^{i\mathbf{k} \cdot \mathbf{x}} \right]
\end{align*}where $\tilde{A} = \Phi$. I'm supposed to find, apparently, that $T_k = \tfrac{h_S'}{2k^2}$ and $L_k = \tfrac{h_S}{2k}$. Would someone be able to point me in the right direction?

 

[ergospherical]

I think I've got it, after a bit of clearing things up. As before, we can find all the gauge transformations via $\Delta \delta (\mathrm{something}) = - \mathcal{L}_{\xi} (\mathrm{something})$. It's easier for now to just suppress the Fourier subscripts and work with a given mode.
$$\delta h_{0i} = -a^2(i k^i T - i\hat{k}^i \dot{L}) = -a^2(k T - \dot{L}) i \hat{k}^i$$
and also the transformation of the function $h_S$,
$$\delta h_S = -2a^2 Lk$$
In synchronous gauge you have $h_{0i} = 0$ and $h_S$ non-zero, whilst in Newtonian gauge you have $h_{0i} = 0$ still but also $h^S = 0$ (whilst $h = -6\Psi$).

So $\delta h_S = -a^2 h_S = -2a^2 Lk$ gives $L = h_S/2k$, and then $T = \dot{L}/k = \dot{h}_S/2k^2$.