Plane Wave Analysis:
$\frac{d}{dt} \alpha = \alpha_{,t} - \mathcal{L}_\vec\beta \alpha = -\alpha^2 Q$
$\frac{d}{dt} \gamma_{ij} = \gamma_{ij,k} - \mathcal{L}_\vec\beta \gamma_{ij} = -2 \alpha K_{ij}$
$\frac{d}{dt} K_{ij} = K_{ij,t} - \mathcal{L}_\vec\beta K_{ij} = -D_i D_j \alpha + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
expanding $D_i$'s:
$\frac{d}{dt} K_{ij} = -\alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
expanding $\frac{d}{dt}$'s:
$\alpha_{,t} - \alpha_{,i} \beta^i = -\alpha^2 Q$
$\gamma_{ij,t} - \gamma_{ij,k} \beta^k - \gamma_{kj} {\beta^k}_{,i} - \gamma_{ik} {\beta^k}_{,j} = -2 \alpha K_{ij}$
$K_{ij,t} - K_{ij,k} \beta^k - K_{kj} {\beta^k}_{,i} - K_{ik} {\beta^k}_{,j} = -\alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
Let...
$\alpha = \hat\alpha exp(i(\vec{k} \cdot \vec{x} - \omega t))$
$\beta = \hat\beta exp(i(\vec{k} \cdot \vec{x} - \omega t))$
$\gamma_{ij} = \hat\gamma_{ij} exp(i(\vec{k} \cdot \vec{x} - \omega t))$
$K_{ij} = \hat{K}_{ij} exp(i(\vec{k} \cdot \vec{x} - \omega t))$
Then...
$\alpha_{,t} = -i \omega \alpha$
$\alpha_{,i} = i k_i \alpha$
$\alpha_{,ij} = -k_i k_j \alpha$
${\beta^i}_{,t} = -i \omega \beta^i$
${\beta^i}_{,j} = i k_j \beta^i$
$\gamma_{ij,t} = -i \omega \gamma_{ij,t}$
$\gamma_{ij,k} = i k_k \gamma_{ij}$
$K_{ij,t} = -i \omega K_{ij}$
$K_{ij,k} = i k_k K_{ij}$
replacing derivatives:
${\Gamma^i}_{jk} = \frac{1}{2} \gamma^{il} (\gamma_{lj,k} + \gamma_{lk,j} - \gamma_{jk,l})$
${\Gamma^i}_{jk} = \frac{1}{2} \gamma^{il} (i k_k \gamma_{lj} + i k_j \gamma_{lk} - i k_l \gamma_{jk})$
${\Gamma^i}_{jk} = i \frac{1}{2} (k_k \delta^i_j + k_j \delta^i_k - k^i \gamma_{jk})$
${\Gamma^i}_{jk} = i \frac{1}{2} k_l (\delta^l_k \delta^i_j + \delta^l_j \delta^i_k - \gamma^{il} \gamma_{jk})$
$\Gamma^i = {\Gamma^i}_{jk} \gamma^{jk}$
$\Gamma^i = i \frac{1}{2} (k_k \delta^i_j + k_j \delta^i_k - k^i \gamma_{jk}) \gamma^{jk}$
$\Gamma^i = -i \frac{1}{2} k^i$
${\Gamma^i}_{jk,l} = i \frac{1}{2} (k_k \delta^i_j + k_j \delta^i_k - k^i \gamma_{jk})_{,l}$
${\Gamma^i}_{jk,l} = -i \frac{1}{2} k^i \gamma_{jk,l}$
${\Gamma^i}_{jk,l} = \frac{1}{2} k^i \gamma_{jk} k_l$
${R^i}_{jkl} = {\Gamma^i}_{jl,k} - {\Gamma^i}_{jk,l} + {\Gamma^i}_{mk} {\Gamma^m}_{jl} - {\Gamma^i}_{ml} {\Gamma^m}_{jk}$
${R^i}_{jkl} = \frac{1}{2} k^i k_k \gamma_{jl} - \frac{1}{2} k^i k_l \gamma_{jk}
- \frac{1}{4} (k_k \delta^i_m + k_m \delta^i_k - k^i \gamma_{mk}) (k_l \delta^m_j + k_j \delta^m_l - k^m \gamma_{jl})
+ \frac{1}{4} (k_l \delta^i_m + k_m \delta^i_l - k^i \gamma_{ml}) (k_k \delta^m_j + k_j \delta^m_k - k^m \gamma_{jk})
$
${R^i}_{jkl} = \frac{1}{2} k^i k_k \gamma_{jl} - \frac{1}{2} k^i k_l \gamma_{jk}
+ \frac{1}{4} (
- k_k \delta^i_m k_l \delta^m_j
- k_m \delta^i_k k_l \delta^m_j
+ k^i \gamma_{mk} k_l \delta^m_j
- k_k \delta^i_m k_j \delta^m_l
- k_m \delta^i_k k_j \delta^m_l
+ k^i \gamma_{mk} k_j \delta^m_l
+ k_k \delta^i_m k^m \gamma_{jl}
+ k_m \delta^i_k k^m \gamma_{jl}
- k^i \gamma_{mk} k^m \gamma_{jl}
+ k_l \delta^i_m k_k \delta^m_j
+ k_m \delta^i_l k_k \delta^m_j
- k^i \gamma_{ml} k_k \delta^m_j
+ k_l \delta^i_m k_j \delta^m_k
+ k_m \delta^i_l k_j \delta^m_k
- k^i \gamma_{ml} k_j \delta^m_k
- k_l \delta^i_m k^m \gamma_{jk}
- k_m \delta^i_l k^m \gamma_{jk}
+ k^i \gamma_{ml} k^m \gamma_{jk}
)
$
${R^i}_{jkl} = \frac{1}{4} (\delta^i_l (k_j k_k - k^2 \gamma_{jk}) - \delta^i_k (k_j k_l - k^2 \gamma_{jl}) - k^i \gamma_{jk} k_l + k^i \gamma_{jl} k_k)$
$R_{ij} = {R^k}_{ikj}$
$R_{ij} = \frac{1}{4} (\delta^k_j (k_i k_k - k^2 \gamma_{ik}) - \delta^k_k (k_i k_j - k^2 \gamma_{ij}) - k^k \gamma_{ik} k_j + k^k \gamma_{ij} k_k)$
$R_{ij} = \frac{1}{4} (k_i k_j - k^2 \gamma_{ij} - 3 k_i k_j + 3 k^2 \gamma_{ij} - k_i k_j + k^2 \gamma_{ij})$
$R_{ij} = \frac{3}{4} (k^2 \gamma_{ij} - k_i k_j)$
$R = R_{ij} \gamma^{ij}$
$R = \frac{3}{4} (k^2 \gamma_{ij} - k_i k_j) \gamma^{ij}$
$R = \frac{3}{4} (3 k^2 - k^2)$
$R = \frac{3}{2} k^2$
considering $\alpha$ and $\beta$
$\alpha_{,t} - \alpha_{,i} \beta^i = -\alpha^2 Q$
$-i \omega \alpha - i k_i \alpha \beta^i = -\alpha^2 Q$
$\omega = -k_i \beta^i - i \alpha Q$
$\alpha = i \frac{1}{Q} ( \omega + k_i \beta^i)$
$k_i \beta^i = -i Q \alpha - \omega$
$\gamma_{ij,t} - \gamma_{ij,k} \beta^k - \gamma_{kj} {\beta^k}_{,i} - \gamma_{ik} {\beta^k}_{,j} = -2 \alpha K_{ij}$
$\gamma_{mn} ((-i \omega - i k_l \beta^l) \delta^m_i \delta^n_j - i k_i \delta^n_j \beta^m - i k_j \delta^m_i \beta^n) = -2 \alpha K_{ij}$
$\gamma_{mn} ((\omega + k_l \beta^l) \delta^m_i \delta^n_j + k_i \delta^n_j \beta^m + k_j \delta^m_i \beta^n) = -i 2 \alpha K_{ij}$
substitute $\alpha$
$\gamma_{mn} ((\omega + k_l \beta^l) \delta^m_i \delta^n_j + k_i \delta^n_j \beta^m + k_j \delta^m_i \beta^n) = -i 2 (i \frac{1}{Q} (\omega + k_l \beta^l)) K_{ij}$
$\gamma_{mn} ((\omega + k_l \beta^l) \delta^m_i \delta^n_j + k_i \delta^n_j \beta^m + k_j \delta^m_i \beta^n) = 2 \frac{1}{Q} (\omega + k_l \beta^l) K_{ij}$
$((\omega + k_k \beta^k) \gamma_{ij} + k_i \beta_j + k_j \beta_i) = 2 \frac{1}{Q} (\omega + k_l \beta^l) K_{ij}$
$K_{ij,t} - K_{ij,k} \beta^k - K_{kj} {\beta^k}_{,i} - K_{ik} {\beta^k}_{,j} = -\alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$-i \omega K_{ij} - i k_k K_{ij} \beta^k - i k_i K_{kj} \beta^k - i k_j K_{ik} \beta^k = k_i k_j \alpha + i k_k {\Gamma^k}_{ij} \alpha + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$K_{mn} (\delta^m_i \delta^n_j (-i \omega - i k_k \beta^k) - i k_i \delta^n_j \beta^m - i k_j \delta^m_i \beta^n) = \alpha (k_i k_j + i k_k {\Gamma^k}_{ij} + R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
substitute $\alpha$, ${\Gamma^k}_{ij}$, $R_{ij}$
$K_{mn} (\delta^m_i \delta^n_j (-i \omega - i k_k \beta^k) - i k_i \delta^n_j \beta^m - i k_j \delta^m_i \beta^n)
= i \frac{1}{Q} (\omega + k_k \beta^k) (k_i k_j + i k_k (i \frac{1}{2} k_l (\delta^l_j \delta^k_i + \delta^l_i \delta^k_j - \gamma^{kl} \gamma_{ij})) + (\frac{3}{4} (k^2 \gamma_{ij} - k_i k_j)) + K K_{ij} - 2 K_{ik} {K^k}_j)$
$K_{mn} (\delta^m_i \delta^n_j (-i \omega - i k_k \beta^k) - i k_i \delta^n_j \beta^m - i k_j \delta^m_i \beta^n)
= i \frac{1}{Q} (\omega + k_k \beta^k) (\frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j + K K_{ij} - 2 K_{ik} {K^k}_j)$
$(K_{ij} (\omega + k_k \beta^k) + K_{kj} k_i \beta^k + K_{ik} k_j \beta^k)
= -\frac{1}{Q} (\omega + k_k \beta^k) (\frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j + K K_{ij} - 2 K_{ik} {K^k}_j)$
contract $K_{ij}$'s def with $K^{ij}$
$K^{ij} (K_{ij} (\omega + k_k \beta^k) + K_{kj} k_i \beta^k + K_{ik} k_j \beta^k)
= -\frac{1}{Q} (\omega + k_k \beta^k) K^{ij} (\frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j + K K_{ij} - 2 K_{ik} {K^k}_j)$
$(K_{ij} K^{ij} (\omega + k_k \beta^k) + K_{kj} K^{ij} k_i \beta^k + K_{ik} K^{ij} k_j \beta^k)
= -\frac{1}{Q} (\omega + k_k \beta^k) (\frac{5}{4} k^2 K^{ij} \gamma_{ij} - \frac{3}{4} K^{ij} k_i k_j + K K^{ij} K_{ij} - 2 K^{ij} K_{ik} {K^k}_j)$
$(K_{ij} K^{ij} (\omega + k_k \beta^k) + K_{kj} K^{ij} k_i \beta^k + K_{ik} K^{ij} k_j \beta^k)
= -\frac{1}{Q} (\omega + k_k \beta^k) (\frac{5}{4} k^2 K - \frac{3}{4} K^{ij} k_i k_j + K K^{ij} K_{ij} - 2 K^{ij} K_{jk} {K^k}_i )$
contract $K_{ij}$'s def with $\gamma^{ij}$
$\gamma^{ij} (K_{ij} (\omega + k_k \beta^k) + K_{kj} k_i \beta^k + K_{ik} k_j \beta^k)
= -\gamma^{ij} \frac{1}{Q} (\omega + k_k \beta^k) (\frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j + K K_{ij} - 2 K_{ik} {K^k}_j)$
$(\gamma^{ij} K_{ij} (\omega + k_k \beta^k) + \gamma^{ij} K_{kj} k_i \beta^k + \gamma^{ij} K_{ik} k_j \beta^k)
= -\frac{1}{Q} (\omega + k_k \beta^k) (\frac{5}{4} k^2 \gamma^{ij} \gamma_{ij} - \frac{3}{4} \gamma^{ij} k_i k_j + K K_{ij} - 2 \gamma^{ij} K_{ik} {K^k}_j)$
$2 K_{ij} k^i \beta^j = -\frac{1}{Q} (\omega + k_k \beta^k) (K^2 + Q K + 3 k^2 - 2 K_{ij} K^{ij})$
considering $\alpha$ and $\beta$ but only in the x direction ($k_i = \delta^x_i k$)
$\alpha = i \frac{1}{Q} ( \omega + k \beta^x)$
$(\omega + k_l \beta^l) \gamma_{ij} + \gamma_{kj} k_i \beta^k + \gamma_{ik} k_j \beta^k = -i 2 \alpha K_{ij}$
breakdown of $\gamma_{ij}$
$(\omega + 3 k \beta^x) \gamma_{xx} + 2 k \gamma_{xp} \beta^p = -i 2 \alpha K_{xx}$
$(\omega + k \beta^x) \gamma_{xp} + k (\gamma_{xp} \beta^x + \gamma_{pq} \beta^q) = -i 2 \alpha K_{xp}$
$(\omega + k \beta^x) \gamma_{pq} = -i 2 \alpha K_{pq}$
substitute...
$\gamma_{pq} = -\frac{2 i \alpha}{\omega + k \beta^x} K_{pq}$
$(\omega + k \beta^x) \gamma_{xp} + k (\gamma_{xp} \beta^x - \frac{2 i \alpha}{\omega + k \beta^x} K_{pq} \beta^q) = -2 i \alpha K_{xp}$
$\gamma_{xp} = \frac{2 i \alpha}{\omega + 2 k \beta^x} (-K_{xp} + \frac{k}{\omega + k \beta^x} \beta^q K_{pq})$
$(\omega + 3 k \beta^x) \gamma_{xx} + 2 k \frac{2 i \alpha}{\omega + 2 k \beta^x} (-K_{xp} + \frac{k}{\omega + k \beta^x} \beta^q K_{pq}) \beta^p = -i 2 \alpha K_{xx}$
$\gamma_{xx} =
- \frac{2 i \alpha}{\omega + 3 k \beta^x} K_{xx}
+ \frac{4 i k \alpha }{(\omega + 2 k \beta^x)(\omega + 3 k \beta^x)} K_{xp} \beta^p
- \frac{4 i k^2 \alpha }{(\omega + k \beta^x)(\omega + 2 k \beta^x)(\omega + 3 k \beta^x)} K_{pq} \beta^p \beta^q
$
solve for $K_{ij}$ using $\gamma_{ij}$'s definition:
$K_{pq} = i \frac{\omega + k \beta^x}{2 \alpha} \gamma_{pq}$
$-K_{xp} + \frac{k}{\omega + k \beta^x} \beta^q (i \frac{\omega + k \beta^x}{2 \alpha} \gamma_{pq}) = -i \frac{\omega + 2 k \beta^x}{2 \alpha} \gamma_{xp}$
$K_{xp} = i \frac{1}{2 \alpha}((\omega + 2 k \beta^x) \gamma_{xp} + \beta^q \gamma_{pq})$
$K_{xx} = i \frac{1}{2 \alpha}(
(\omega + 3 k \beta^x) \gamma_{xx}
+ 2 k \gamma_{xp} \beta^p
+ (1 - k) \frac{2 k}{\omega + 2 k \beta^x} \gamma_{pq} \beta^p \beta^q
)$
breakdown of $K_{ij}$
$K_{xx} (\omega + 3 k \beta^x) + 2 k K_{xp} \beta^p
= i \alpha (
\frac{5}{4} k^2 \gamma_{xx}
- \frac{3}{4} k^2
- \gamma^{xx} K_{xx}^2
- 2 \gamma^{xp} K_{xp} K_{xx}
+ \gamma^{pq} K_{xx} K_{pq}
- 2 \gamma^{pq} K_{xp} K_{xq}
)$
$K_{xp} (\omega + 2 k \beta^x) + k K_{pq} \beta^q
= i \alpha (
\frac{5}{4} k^2 \gamma_{xp}
- \gamma^{xx} K_{xx} K_{xp}
+ 2 \gamma^{xq} K_{xp} K_{xq}
+ \gamma^{qr} K_{xp} K_{qr}
- 4 \gamma^{xq} K_{xx} K_{pq}
- 2 \gamma^{qr} K_{xq} K_{pr}
)$
$K_{pq} (\omega + k \beta^x)
= i \alpha (
\frac{5}{4} k^2 \gamma_{pq}
+ \gamma^{xx} K_{xx} K_{pq}
+ 2 \gamma^{xr} K_{xr} K_{pq}
+ \gamma^{rs} K_{rs} K_{pq}
- 2 K_{px} \gamma^{xx} K_{xq}
- 4 K_{px} \gamma^{xr} K_{rq}
- 2 K_{pr} \gamma^{rs} K_{sq}
)$
substitute $\gamma_{ij}$...
$(\omega + 3 k \beta^x - \frac{5 \alpha^2 k^2}{2 (\omega + 3 k \beta^x)}) K_{xx}
+ (2 k + \frac{5 \alpha^2 k^3}{(\omega + 2 k \beta^x)(\omega + 3 k \beta^x)}) K_{xp} \beta^p
=
- i \frac{3}{4} \alpha k^2
i \alpha (
- \frac{i 5 k^4 \alpha}{(\omega + k \beta^x)(\omega + 2 k \beta^x)(\omega + 3 k \beta^x)} K_{pq} \beta^p \beta^q
- \gamma^{xx} K_{xx}^2
- 2 \gamma^{xp} K_{xx} K_{xp}
- 2 \gamma^{pq} K_{xp} K_{xq}
+ \gamma^{pq} K_{xx} K_{pq}
)$
$
(\omega + 2 k \beta^x - \frac{5 k^2 \alpha^2}{2 (\omega + 2 k \beta^x)}) K_{xp}
+ (k + \frac{5 k^3 \alpha^2}{2 (\omega + k \beta^x) (\omega + 2 k \beta^x)}) \beta^q K_{pq}
=
i \alpha (
- \gamma^{xx} K_{xx} K_{xp}
- 4 \gamma^{xq} K_{xx} K_{pq}
+ 2 \gamma^{xq} K_{xp} K_{xq}
+ \gamma^{qr} K_{xp} K_{qr}
- 2 \gamma^{qr} K_{xq} K_{pr}
)$
$ (\omega + k \beta^x - \frac{5}{4} k^2 \frac{2 \alpha^2}{\omega + k \beta^x}) K_{pq}
= i \alpha (
+ \gamma^{xx} K_{xx} K_{pq}
+ 2 \gamma^{xr} K_{xr} K_{pq}
- 2 \gamma^{xx} K_{xp} K_{xq}
- 4 \gamma^{xr} K_{xp} K_{rq}
+ \gamma^{rs} K_{pq} K_{rs}
- 2 \gamma^{rs} K_{pr} K_{sq}
)$
assuming $\beta=0$
$\alpha_{,t} - \alpha_{,i} \beta^i = -\alpha^2 Q$
$\omega = -k_i \beta^i - i \alpha Q$
$\omega = -i \alpha Q$
$\alpha = \frac{i \omega}{Q}$
$\gamma_{ij,t} - \gamma_{ij,k} \beta^k - \gamma_{kj} {\beta^k}_{,i} - \gamma_{ik} {\beta^k}_{,j} = -2 \alpha K_{ij}$
$\gamma_{ij} = -\frac{2 i \alpha}{\omega} K_{ij}$
$\gamma_{ij} = \frac{2}{Q} K_{ij}$
so $K_{ij} = \frac{Q}{2} \gamma_{ij}$
so $K = \frac{3}{2} Q$
$K_{ij,t} - K_{ij,k} \beta^k - K_{kj} {\beta^k}_{,i} - K_{ik} {\beta^k}_{,j} = -\alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$-i \omega K_{ij} = \alpha (k_i k_j + i k_k {\Gamma^k}_{ij} + R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$-i \omega K_{ij} = \frac{i \omega}{Q} (k_i k_j + i k_k {\Gamma^k}_{ij} + R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$K_{ij} = -\frac{1}{Q} (k_i k_j + i k_k {\Gamma^k}_{ij} + R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
substitute ${\Gamma^k}_{ij}$, $R_{ij}$ expansion
$K_{ij} = -\frac{1}{Q} (k_i k_j - \frac{1}{2} k_k (k_j \delta^k_i + k_i \delta^k_j - k^k \gamma_{ij}) + \frac{3}{4} (k^2 \gamma_{ij} - k_i k_j) + K K_{ij} - 2 K_{ik} {K^k}_j)$
$Q K_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j = -\frac{5}{4} k^2 \gamma_{ij} + \frac{3}{4} k_i k_j$
substitute $K_{ij}$, $K$
$Q \frac{Q}{2} \gamma_{ij} + \frac{3}{2} Q \cdot \frac{1}{2} Q \gamma_{ij} - 2 \frac{Q}{2} \gamma_{im} \gamma^{mn} \frac{Q}{2} \gamma_{nj} = -\frac{5}{4} k^2 \gamma_{ij} + \frac{3}{4} k_i k_j$
$\frac{1}{2} Q^2 \gamma_{ij} + \frac{3}{4} Q^2 \gamma_{ij} - \frac{1}{2} Q^2 \gamma_{ij} + \frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j = 0$
$\frac{3}{4} Q^2 \gamma_{ij} + \frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j = 0$
trace of $K$:
$K = \gamma^{ij} K_{ij} = -\frac{1}{Q} (k^2 + i k_k {\Gamma^k}_{ij} \gamma^{ij} + R + K^2 - 2 K_{ij} K^{ij})$
$K^2 + Q K = -k^2 - i k_k \Gamma^k - R + 2 K_{ij} K^{ij}$
substitute $\Gamma^i$ and $R$ expansion
$K^2 + Q K = -\frac{3}{2} k^2 - \frac{3}{2} k^2 + 2 K_{ij} K^{ij}$
$K^2 + Q K + 3 k^2 - 2 K_{ij} K^{ij} = 0$
substitute $K_{ij}$
$K^2 + Q K + 3 k^2 - 2 \frac{Q}{2} \gamma_{ij} \gamma^{im} \gamma^{jn} \frac{Q}{2} \gamma_{mn} = 0$
$K^2 + Q K + 3 k^2 - \frac{3}{2} Q^2 = 0$
substitute $K$
$\frac{9}{4} Q^2 + \frac{3}{2} Q^2 + 3 k^2 - \frac{3}{2} Q^2 = 0$
$Q^2 = -\frac{4}{3} k^2$
$\frac{3}{4} Q^2 \gamma_{ij} + \frac{5}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j = 0$
substitute $Q$ back into the equality derived from $K_{ij,t}$...
$\gamma_{ij} = \frac{3 k_i k_j}{k^2}$
$K = \frac{3}{2} Q$
$K = \frac{3}{2} \sqrt{-\frac{4}{3} k^2}$
$K = i \sqrt{3} |k|$
$K_{ij} = \frac{Q}{2} \gamma_{ij}$
$K_{ij} = i \sqrt{3} \frac{k_i k_j}{|k|}$
$\alpha = \frac{i \omega}{Q}$
$\alpha = \frac{i \omega}{\sqrt{-\frac{4}{3} k^2}}$
$\alpha = \frac{\sqrt{3} \omega}{2 |k|}$
setting $\alpha=1$ and $\beta=0$
$\gamma_{ij,t} = -2 K_{ij}$
$-i \omega \gamma_{ij} = -2 K_{ij}$
$\gamma_{ij} = -2 i \frac{1}{\omega} K_{ij}$
$K_{ij} = i \frac{\omega}{2} \gamma_{ij}$
trace of $K_{ij}$:
$K = i \frac{3}{2} \omega$
$K_{ij,t} = \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$-i \omega K_{ij} = \alpha (\frac{3}{4} k^2 \gamma_{ij} - \frac{3}{4} k_i k_j + K K_{ij} - 2 K_{ik} {K^k}_j)$
substitute $K$, $K_{ij}$
$\frac{\omega^2}{2} \gamma_{ij} = \frac{3}{4} k^2 \gamma_{ij} - \frac{1}{4} \omega^2 \gamma_{ij} - \frac{3}{4} k_i k_j$
$\gamma_{ij} = \frac{k_i k_j}{k^2 - \omega^2}$
substitute back into $K_{ij}$
$K_{ij} = i \frac{\omega k_i k_j}{2 (k^2 - \omega^2)}$
permuting only $\alpha$ and $\beta$ and deriving other variables:
$a_i = ln(\alpha)_{,i} = \alpha_{,i} / \alpha = i k_i$
$d_{kij} = \frac{1}{2} \gamma_{ij,k} = i \frac{1}{2} k_k \gamma_{ij}$
$K_{ij} = -\alpha ^4{\Gamma^t}_{ij}$
$= -\alpha (g^{tt} \cdot {^4}\Gamma_{tij} + g^{tk} \cdot {^4}\Gamma_{kij})$
$= -\alpha (-\frac{1}{2} \frac{1}{\alpha^2} (g_{ti,j} + g_{tj,i} - g_{ij,t}) + \frac{\beta^k}{\alpha^2} \Gamma_{kij})$
$= \frac{1}{\alpha} (\frac{1}{2} (\beta_{i,j} + \beta_{j,i} - \gamma_{ij,t}) - \beta^k \Gamma_{kij})$
substitute Fourier expansion of derivatives...
$= i \frac{1}{2} \frac{1}{\alpha} (\omega + \beta^k k_k) \gamma_{ij}$
"plane wave in x direction" according to Alcubierre, Allen, Brugmann, Seidel, Suen 2000 - section 2:
with $\alpha = 1$, $\beta^i = 0$, $\gamma_{ij} = \delta_{ij} + h_{ij}$
then permuting...
$h_{ij} = \hat{h}_{ij} exp(i(\omega t - k x))$
$K_{ij} = \hat{K}_{ij} exp(i(\omega t - k x))$
Then the evolution equations become...
$\gamma_{ij,t} = -2 K_{ij}$
$K_{ij,t} = R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j$
The derivations of the spatial metric, when permuting the linear term alone, become...
$\Gamma_{ijk} = \frac{1}{2} (\gamma_{ij,k} + \gamma_{ik,j} - \gamma_{jk,i})$
$= \frac{1}{2} ( h_{ij,k} + h_{ik,j} - h_{jk,i})$
$= \frac{1}{2} ( h_{ij} k \delta^x_k + h_{ik} k \delta^x_j - h_{jk} k \delta^x_i)$
...and...
${\Gamma^i}_{jk} = \gamma^{il} \Gamma_{ljk}$
$= \frac{1}{2} ( {h^i}_j k \delta^x_k + {h^i}_k k \delta^x_j - h_{jk} k \gamma^{xi})$
...which involves a transformed $h_{ij}$ so should get interesting...
(note Bona, Ledvinka, Palenzuela, Zacek 2004 do the same trick with Z4.
They do use the linearized metric $h_{ij}$ for $\gamma_{ij} = \delta_{ij} + h_{ij}$, but just call it $\hat{\gamma}_{ij}$ instead of $\hat{h}_{ij}$.
They also permute $\alpha$ in the same way: $\alpha = 1 + \hat{\alpha} exp(i \omega x)$.
They also only permute in space, not time. No use of $exp(i (k x - \omega t))$.
How about verifying the eigenfields of one of the formalisms.
Start with the first Bona-Masso 1997 ADM.
(note I'm using ${b_i}_k = \frac{1}{2} {\beta^k}_{,i}$ from the paper
3: $a_{k,t} + (-\beta^r a_k + \alpha Q \delta^r_k)_{,r} = (2 {b_k}^r - \alpha tr s \delta^r_k) a_r$
18: $d_{kij,t} + (-\beta^r d_{kij} + \alpha \delta^r_k (K_{ij} - s_{ij}))_{,r} = (2 {b_k}^r - \alpha tr s \delta^r_k) d_{rij}$
6: $K_{ij,t} + (-\beta^r K_{ij} + \alpha {\Lambda^r}_{ij})_{,r} = \alpha S_{ij}$
3: $V_{k,t} + (-\beta^r V_k + \alpha ({s^r}_k - tr s \delta^r_k))_{,r} = \alpha P_k$
expand to extract the state vector in the k'th direction:
3: $a_{k,t}
- \beta^r \delta^m_k a_{m,r}
+ \alpha \delta^r_k f \gamma^{pq} K_{pq,r}
= 2 (2 b_{j(k} a_{i)} - a_k b_{(ij)}) \gamma^{ij}
- \alpha a_k K f
- \alpha^2 a_k K f'
+ 2 \alpha f {d_k}^{ij} K_{ij}
$
18: $d_{kij,t}
- \beta^r \delta^m_k \delta^p_i \delta^q_j d_{mpq,r}
+ \alpha \delta^p_i \delta^q_j \delta^r_k K_{pq,r}
= 2 {b_k}^l d_{lij}
+ 2 {b_l}^l d_{kij}
- \alpha tr s d_{kij}
+ 2 b_{(ij),k}
- \alpha a_k K_{ij}
$
6: $K_{ij,t}
- \beta^r \delta^p_i \delta^q_j K_{pq,r}
+ \alpha \delta^r_{(i} \delta^m_{j)} a_{m,r}
+ \alpha (\delta^p_i \delta^q_j \gamma^{rm} - \gamma^{pq} \delta^r_i \delta^m_j ) d_{mpq,r}
+ 2 \alpha \delta^r_{(i} \delta^m_{j)} V_{m,r}
= \alpha S_{ij}
+ 2 {b_r}^r K_{ij}
- \alpha a_r {\Lambda^r}_{ij}
+ 2 \alpha \gamma^{lm} d_{lmk} \gamma^{kn} d_{nij}
- 2 \alpha \delta^r_{(i} d_{j)mn} {d_r}^{mn}
$
3: $V_{k,t}
- \beta^r \delta^m_k V_{m,r}
= \alpha P_k
- 2 a_i \gamma^{ij} b_{(jk)}
+ a_k \gamma^{ij} b_{(ij)}
+ \gamma^{ij} b_{(ij),k}
+ 2 {b_i}^i V_k
+ 4 {d_i}^{ij} b_{jk}
- 2 {d_k}^{ij} b_{(ij)}
- 2 \gamma^{ij} b_{jk,i}
+ 2 \gamma^{ij} b_{jk} a_i
- \gamma^{ij} b_{(ij)} a_k
$
set $\beta^i=0$...
3: $a_{k,t}
+ \alpha \delta^r_k f \gamma^{pq} K_{pq,r}
= -\alpha a_k K (f + \alpha f')
+ 2 \alpha f d_{kij} K^{ij}
$
18: $d_{kij,t}
+ \alpha \delta^p_i \delta^q_j \delta^r_k K_{pq,r}
= -\alpha a_k K_{ij}
$
6: $K_{ij,t}
+ \alpha \delta^r_{(i} \delta^m_{j)} a_{m,r}
+ \alpha (\delta^p_i \delta^q_j \gamma^{rm} - \gamma^{pq} \delta^r_i \delta^m_j) d_{mpq,r}
+ 2 \alpha \delta^r_{(i} \delta^m_{j)} V_{m,r}
= \alpha S_{ij}
- \alpha a_r {\Lambda^r}_{ij}
+ 2 \alpha {d^m}_{mk} {d^k}_{ij}
- 2 \alpha {d_(i}^{mn} d_{j)mn}
$
3: $V_{k,t} = \alpha P_k$
matrix form:
$\left[\matrix{
a_k \\ d_{kij} \\ K_{ij} \\ V_k
}\right]_{,t}
+ \left[\matrix{
0 & 0 & \alpha \delta^r_k f \gamma^{pq} & 0 \\
0 & 0 & \alpha \delta^p_i \delta^q_j \delta^r_k & 0 \\
\alpha \delta^r_{(i} \delta^m_{j)} &
\alpha (\delta^p_i \delta^q_j \gamma^{rm} - \gamma^{pq} \delta^r_i \delta^m_j) &
0 &
2 \alpha \delta^r_{(i} \delta^m_{j)} \\
0 & 0 & 0 & 0
}\right]
\left[\matrix{
a_m \\ d_{mpq} \\ K_{pq} \\ V_m
}\right]_{,r}
= Source$