What exactly does "equal up to principal part" mean? Lots of papers use the term.
Here I'm looking at just how the terms equate:
$a_i = (ln \alpha)_{,i} = \alpha_{,i} / \alpha$
...so $\alpha a_i = \alpha_{,i}$
lapse evolution:
$\alpha_{,t} - \alpha_{,i} \beta^i = -\alpha^2 Q$
$0 = (\delta^i_j)_{,k} = (\gamma^{il} \gamma_{lj})_{,k} = {\gamma^{il}}_{,k} \gamma_{lj} + \gamma^{il} \gamma_{lj,k}$
Therefore ${\gamma^{il}}_{,k} \gamma_{lj} = -\gamma^{il} \gamma_{lj,k}$
Ricci curvature:
$R_{ij} = {R^k}_{ikj} $
$= {\Gamma^k}_{ij,k} - {\Gamma^k}_{ik,j} + {\Gamma^k}_{lk} {\Gamma^l}_{ij} - {\Gamma^k}_{lj} {\Gamma^l}_{ik}$
$= (\gamma^{kl} \Gamma_{lij})_{,k}
- (\gamma^{kl} \Gamma_{lik})_{,j}
+ \gamma^{km} \Gamma_{mlk} \gamma^{ln} \Gamma_{nij}
- \gamma^{km} \Gamma_{mlj} \gamma^{ln} \Gamma_{nik}
$
$= {\gamma^{kl}}_{,k} \Gamma_{lij}
+ \gamma^{kl} \Gamma_{lij,k}
- {\gamma^{kl}}_{,j} \Gamma_{lik}
- \gamma^{kl} \Gamma_{lik,j}
+ \frac{1}{2} \gamma^{km} (\gamma_{ml,k} + \gamma_{mk,l} - \gamma_{lk,m}) \gamma^{ln} \Gamma_{nij}
- \frac{1}{2} \gamma^{km} (\gamma_{ml,j} + \gamma_{mj,l} - \gamma_{lj,m}) \gamma^{ln} \Gamma_{nik}
$
$= - \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} (\gamma_{li,j} + \gamma_{lj,i} - \gamma_{ij,l})
+ \frac{1}{2} \gamma^{kl} (\gamma_{li,jk} + \gamma_{lj,ik} - \gamma_{ij,kl})
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} (\gamma_{li,k} + \gamma_{lk,i} - \gamma_{ik,l})
- \frac{1}{2} \gamma^{kl} (\gamma_{il,jk} + \gamma_{kl,ij} - \gamma_{ik,lj})
+ \frac{1}{2} \gamma^{km} \gamma_{ml,k} \gamma^{ln} \Gamma_{nij}
+ \frac{1}{2} \gamma^{km} \gamma_{mk,l} \gamma^{ln} \Gamma_{nij}
- \frac{1}{2} \gamma^{km} \gamma_{lk,m} \gamma^{ln} \Gamma_{nij}
- \frac{1}{2} \gamma^{km} \gamma_{ml,j} \gamma^{ln} \Gamma_{nik}
- \frac{1}{2} \gamma^{km} \gamma_{mj,l} \gamma^{ln} \Gamma_{nik}
+ \frac{1}{2} \gamma^{km} \gamma_{lj,m} \gamma^{ln} \Gamma_{nik}
$
$=
- \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} \gamma_{li,j}
- \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} \gamma_{lj,i}
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} \gamma_{ij,l}
+ \frac{1}{2} \gamma^{kl} \gamma_{li,jk}
+ \frac{1}{2} \gamma^{kl} \gamma_{lj,ik}
- \frac{1}{2} \gamma^{kl} \gamma_{ij,kl}
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} \gamma_{li,k}
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} \gamma_{lk,i}
- \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} \gamma_{ik,l}
- \frac{1}{2} \gamma^{kl} \gamma_{il,jk}
- \frac{1}{2} \gamma^{kl} \gamma_{kl,ij}
+ \frac{1}{2} \gamma^{kl} \gamma_{ik,lj}
+ \frac{1}{4} \gamma^{km} \gamma_{ml,k} \gamma^{ln} (\gamma_{ni,j} + \gamma_{nj,i} - \gamma_{ij,n})
+ \frac{1}{4} \gamma^{km} \gamma_{mk,l} \gamma^{ln} (\gamma_{ni,j} + \gamma_{nj,i} - \gamma_{ij,n})
- \frac{1}{4} \gamma^{km} \gamma_{lk,m} \gamma^{ln} (\gamma_{ni,j} + \gamma_{nj,i} - \gamma_{ij,n})
- \frac{1}{4} \gamma^{km} \gamma_{ml,j} \gamma^{ln} (\gamma_{ni,k} + \gamma_{nk,i} - \gamma_{ik,n})
- \frac{1}{4} \gamma^{km} \gamma_{mj,l} \gamma^{ln} (\gamma_{ni,k} + \gamma_{nk,i} - \gamma_{ik,n})
+ \frac{1}{4} \gamma^{km} \gamma_{lj,m} \gamma^{ln} (\gamma_{ni,k} + \gamma_{nk,i} - \gamma_{ik,n})
$
$=
- \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} \gamma_{li,j}
- \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} \gamma_{lj,i}
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,k} \gamma^{ql} \gamma_{ij,l}
+ \frac{1}{2} \gamma^{kl} \gamma_{li,jk}
+ \frac{1}{2} \gamma^{kl} \gamma_{lj,ik}
- \frac{1}{2} \gamma^{kl} \gamma_{ij,kl}
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} \gamma_{li,k}
+ \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} \gamma_{lk,i}
- \frac{1}{2} \gamma^{kp} \gamma_{pq,j} \gamma^{ql} \gamma_{ik,l}
- \frac{1}{2} \gamma^{kl} \gamma_{il,jk}
- \frac{1}{2} \gamma^{kl} \gamma_{kl,ij}
+ \frac{1}{2} \gamma^{kl} \gamma_{ik,lj}
+ \frac{1}{4} \gamma^{km} \gamma_{ml,k} \gamma^{ln} \gamma_{ni,j}
+ \frac{1}{4} \gamma^{km} \gamma_{ml,k} \gamma^{ln} \gamma_{nj,i}
- \frac{1}{4} \gamma^{km} \gamma_{ml,k} \gamma^{ln} \gamma_{ij,n}
+ \frac{1}{4} \gamma^{km} \gamma_{mk,l} \gamma^{ln} \gamma_{ni,j}
+ \frac{1}{4} \gamma^{km} \gamma_{mk,l} \gamma^{ln} \gamma_{nj,i}
- \frac{1}{4} \gamma^{km} \gamma_{mk,l} \gamma^{ln} \gamma_{ij,n}
- \frac{1}{4} \gamma^{km} \gamma_{lk,m} \gamma^{ln} \gamma_{ni,j}
- \frac{1}{4} \gamma^{km} \gamma_{lk,m} \gamma^{ln} \gamma_{nj,i}
+ \frac{1}{4} \gamma^{km} \gamma_{lk,m} \gamma^{ln} \gamma_{ij,n}
- \frac{1}{4} \gamma^{km} \gamma_{ml,j} \gamma^{ln} \gamma_{ni,k}
- \frac{1}{4} \gamma^{km} \gamma_{ml,j} \gamma^{ln} \gamma_{nk,i}
+ \frac{1}{4} \gamma^{km} \gamma_{ml,j} \gamma^{ln} \gamma_{ik,n}
- \frac{1}{4} \gamma^{km} \gamma_{mj,l} \gamma^{ln} \gamma_{ni,k}
- \frac{1}{4} \gamma^{km} \gamma_{mj,l} \gamma^{ln} \gamma_{nk,i}
+ \frac{1}{4} \gamma^{km} \gamma_{mj,l} \gamma^{ln} \gamma_{ik,n}
+ \frac{1}{4} \gamma^{km} \gamma_{lj,m} \gamma^{ln} \gamma_{ni,k}
+ \frac{1}{4} \gamma^{km} \gamma_{lj,m} \gamma^{ln} \gamma_{nk,i}
- \frac{1}{4} \gamma^{km} \gamma_{lj,m} \gamma^{ln} \gamma_{ik,n}
$
$=
- \frac{1}{2} \gamma^{kl} \gamma_{ij,kl}
- \frac{1}{2} \gamma^{kl} \gamma_{kl,ij}
+ \frac{1}{2} \gamma^{kl} \gamma_{ik,lj}
+ \frac{1}{2} \gamma^{kl} \gamma_{jk,li}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{ni,j}
+ \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{ni,j}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{nj,i}
+ \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{nj,i}
+ \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{kn,i} \gamma_{lm,j}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{ij,n}
- \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{ij,n}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{ik,n} \gamma_{jm,l}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{in,k} \gamma_{jm,l}
$
now we equate things "up to principal part", namely the Ricci tensors and the evolution equations.
That means all 2nd-and-higher derivatives?
Most papers have "de Donder-Fock decomposition":
$R_{ij} \approx -{d^m}_{ij,m} + 2 {d^m}_{m(j,i)} - {{d_{(j|m}}^m}_{,|i)}$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm} + \gamma_{k(i} {\Gamma^k}_{,j)}$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
+ \frac{1}{2} \gamma_{ik} ({\Gamma^k}_{lm} \gamma^{lm})_{,j}
+ \frac{1}{2} \gamma_{jk} ({\Gamma^k}_{lm} \gamma^{lm})_{,i}$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
+ \frac{1}{2} \gamma_{ik} {\Gamma^k}_{lm,j} \gamma^{lm}
+ \frac{1}{2} \gamma_{ik} {\Gamma^k}_{lm} {\gamma^{lm}}_{,j}
+ \frac{1}{2} \gamma_{jk} {\Gamma^k}_{lm,i} \gamma^{lm}
+ \frac{1}{2} \gamma_{jk} {\Gamma^k}_{lm} {\gamma^{lm}}_{,i}$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
+ \frac{1}{2} \gamma_{ik} (\gamma^{kn} \Gamma_{nlm})_{,j} \gamma^{lm}
+ \frac{1}{2} \gamma_{jk} (\gamma^{kn} \Gamma_{nlm})_{,i} \gamma^{lm}
- \frac{1}{2} \gamma_{ik} \gamma^{kn} \Gamma_{nlm} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
- \frac{1}{2} \gamma_{jk} \gamma^{kn} \Gamma_{nlm} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
+ \frac{1}{2} \gamma_{ik} {\gamma^{kn}}_{,j} \Gamma_{nlm} \gamma^{lm}
+ \frac{1}{2} \gamma_{jk} {\gamma^{kn}}_{,i} \Gamma_{nlm} \gamma^{lm}
+ \frac{1}{2} \gamma_{ik} \gamma^{kn} \Gamma_{nlm,j} \gamma^{lm}
+ \frac{1}{2} \gamma_{jk} \gamma^{kn} \Gamma_{nlm,i} \gamma^{lm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} (\gamma_{nl,m} + \gamma_{nm,l} - \gamma_{lm,n}) \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} (\gamma_{nl,m} + \gamma_{nm,l} - \gamma_{lm,n}) \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
- \frac{1}{4} \gamma_{ik} (\gamma_{nl,m} + \gamma_{nm,l} - \gamma_{lm,n}) \gamma^{lm} \gamma^{kp} \gamma_{pq,j} \gamma^{qn}
- \frac{1}{4} \gamma_{jk} (\gamma_{nl,m} + \gamma_{nm,l} - \gamma_{lm,n}) \gamma^{lm} \gamma^{kp} \gamma_{pq,i} \gamma^{qn}
+ \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nl,mj} \gamma^{lm}
+ \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nm,lj} \gamma^{lm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{lm,nj} \gamma^{lm}
+ \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nl,mi} \gamma^{lm}
+ \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nm,li} \gamma^{lm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{lm,ni} \gamma^{lm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nl,m} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nm,l} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
+ \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{lm,n} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nl,m} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nm,l} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
+ \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{lm,n} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
$
$\approx -\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
- \frac{1}{4} \gamma_{ik} \gamma_{nl,m} \gamma^{lm} \gamma^{kp} \gamma_{pq,j} \gamma^{qn}
- \frac{1}{4} \gamma_{ik} \gamma_{nm,l} \gamma^{lm} \gamma^{kp} \gamma_{pq,j} \gamma^{qn}
+ \frac{1}{4} \gamma_{ik} \gamma_{lm,n} \gamma^{lm} \gamma^{kp} \gamma_{pq,j} \gamma^{qn}
- \frac{1}{4} \gamma_{jk} \gamma_{nl,m} \gamma^{lm} \gamma^{kp} \gamma_{pq,i} \gamma^{qn}
- \frac{1}{4} \gamma_{jk} \gamma_{nm,l} \gamma^{lm} \gamma^{kp} \gamma_{pq,i} \gamma^{qn}
+ \frac{1}{4} \gamma_{jk} \gamma_{lm,n} \gamma^{lm} \gamma^{kp} \gamma_{pq,i} \gamma^{qn}
+ \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nl,mj} \gamma^{lm}
+ \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nm,lj} \gamma^{lm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{lm,nj} \gamma^{lm}
+ \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nl,mi} \gamma^{lm}
+ \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nm,li} \gamma^{lm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{lm,ni} \gamma^{lm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nl,m} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
- \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{nm,l} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
+ \frac{1}{4} \gamma_{ik} \gamma^{kn} \gamma_{lm,n} \gamma^{lp} \gamma_{pq,j} \gamma^{qm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nl,m} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
- \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{nm,l} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
+ \frac{1}{4} \gamma_{jk} \gamma^{kn} \gamma_{lm,n} \gamma^{lp} \gamma_{pq,i} \gamma^{qm}
$
$\approx
-\frac{1}{2} \gamma^{lm} \gamma_{ij,lm}
+ \frac{1}{4} \gamma^{lm} \gamma_{il,mj}
+ \frac{1}{4} \gamma^{lm} \gamma_{im,lj}
- \frac{1}{4} \gamma^{lm} \gamma_{lm,ij}
+ \frac{1}{4} \gamma^{lm} \gamma_{jl,mi}
+ \frac{1}{4} \gamma^{lm} \gamma_{jm,li}
- \frac{1}{4} \gamma^{lm} \gamma_{lm,ij}
- \frac{1}{4} \gamma^{lp} \gamma^{qm} \gamma_{jl,m} \gamma_{pq,i}
- \frac{1}{4} \gamma^{lp} \gamma^{qm} \gamma_{jm,l} \gamma_{pq,i}
+ \frac{1}{4} \gamma^{lp} \gamma^{qm} \gamma_{lm,j} \gamma_{pq,i}
- \frac{1}{4} \gamma^{lm} \gamma^{qn} \gamma_{nl,m} \gamma_{jq,i}
- \frac{1}{4} \gamma^{lm} \gamma^{qn} \gamma_{nm,l} \gamma_{jq,i}
+ \frac{1}{4} \gamma^{lm} \gamma^{qn} \gamma_{lm,n} \gamma_{jq,i}
- \frac{1}{4} \gamma^{lp} \gamma^{qm} \gamma_{il,m} \gamma_{pq,j}
- \frac{1}{4} \gamma^{lp} \gamma^{qm} \gamma_{im,l} \gamma_{pq,j}
+ \frac{1}{4} \gamma^{lp} \gamma^{qm} \gamma_{lm,i} \gamma_{pq,j}
- \frac{1}{4} \gamma^{lm} \gamma^{qn} \gamma_{nl,m} \gamma_{iq,j}
- \frac{1}{4} \gamma^{lm} \gamma^{qn} \gamma_{nm,l} \gamma_{iq,j}
+ \frac{1}{4} \gamma^{lm} \gamma^{qn} \gamma_{lm,n} \gamma_{iq,j}
$
$\approx
- \frac{1}{2} \gamma^{kl} \gamma_{ij,kl}
- \frac{1}{2} \gamma^{kl} \gamma_{kl,ij}
+ \frac{1}{2} \gamma^{kl} \gamma_{ik,lj}
+ \frac{1}{2} \gamma^{kl} \gamma_{jk,li}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{ni,j}
+ \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{ni,j}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{nj,i}
+ \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{nj,i}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{kn,i} \gamma_{lm,j}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{il,m} \gamma_{kn,j}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{jl,m} \gamma_{kn,i}
$
From the two, it looks like the 2nd derivatives are in common.
What's the difference?
$R_{ij} - (\approx R_{ij})$
$= (
+ \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{kn,i} \gamma_{lm,j}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{ij,n}
- \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{ij,n}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{ik,n} \gamma_{jm,l}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{in,k} \gamma_{jm,l}
) - (
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{kn,i} \gamma_{lm,j}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{il,m} \gamma_{kn,j}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{jl,m} \gamma_{kn,i}
)$
$= (
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{ik,n} \gamma_{jm,l}
- \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{in,k} \gamma_{jm,l}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{il,m} \gamma_{kn,j}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{jl,m} \gamma_{kn,i}
- \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{kn,i} \gamma_{lm,j}
+ \frac{1}{2} \gamma^{km} \gamma^{ln} \gamma_{lm,k} \gamma_{ij,n}
- \frac{1}{4} \gamma^{km} \gamma^{ln} \gamma_{km,l} \gamma_{ij,n}
$
...for $a_{i,j} = (\alpha_{,i} / \alpha)_{,j} = \alpha_{,ij} / \alpha - \alpha_{,i} \alpha_{,j} / \alpha^2$
...so $\alpha_{,ij} = \alpha a_{i,j} + \alpha_{,i} \alpha_{,j} / \alpha$
start with evolution equation:
$\alpha_{,t} - \alpha_{,k} \beta^k = -\alpha^2 Q$
... apply $\partial_i$ ...
$\alpha_{,ti} - \alpha_{,ki} \beta^k - \alpha_{,k} {\beta^k}_{,i} = -2 \alpha \alpha_{,i} Q - \alpha^2 Q_{,i}$
$\alpha_{,ti} / \alpha - \alpha_{,i} \alpha_{,t} / \alpha^2 - \alpha_{,ki} \beta^k - \beta^k \alpha_{,i} \alpha_{,k} / \alpha^2 - \alpha_{,k} {\beta^k}_{,i}
= -2 \alpha_{,i} Q - \alpha Q_{,i} - \alpha_{,i} \alpha_{,t} / \alpha^2 - \beta^k \alpha_{,i} \alpha_{,k} / \alpha^2$
... divide by $\alpha$ and add $-\alpha_{,i} \alpha_{,t} / \alpha^2$ and $ -\beta^k \alpha_{,i} \alpha_{,k} / \alpha^2$ to both sides ...
$a_{i,t} - \beta^k a_{i,k}
= - \alpha Q_{,i} - 2 \alpha_{,i} Q - \alpha_{,i} \alpha_{,t} / \alpha^2 + \alpha_{,k} {\beta^k}_{,i} - \beta^k \alpha_{,i} \alpha_{,k} / \alpha^2$
$a_{i,t} - \beta^k a_{i,k} \approx -\alpha Q_{,i}$
...for $d_{ijk,a} = \frac{1}{2} \gamma_{jk,ia}$
start with evolution equation:
$\gamma_{ij,t} - \gamma_{ij,l} \beta^l - \gamma_{il} {\beta^l}_{,j} - \gamma_{lj} {\beta^l}_{,i} = -2 \alpha K_{ij}$
... apply $\frac{1}{2} \partial_k$ ...
$\gamma_{ij,kt} - \gamma_{ij,kl} \beta^l - \gamma_{ij,l} {\beta^l}_{,k} - \gamma_{il,k} {\beta^l}_{,j} - \gamma_{il} {\beta^l}_{,jk} - \gamma_{lj,k} {\beta^l}_{,i} - \gamma_{lj} {\beta^l}_{,ik} = -2 \alpha_{,k} K_{ij} - 2 \alpha K_{ij,k}$
$d_{kij,t} - d_{kij,l} \beta^l - d_{lij} {\beta^l}_{,k} - d_{kil} {\beta^l}_{,j} - \frac{1}{2} \gamma_{il} {\beta^l}_{,jk} - d_{klj} {\beta^l}_{,i} - \frac{1}{2} \gamma_{lj} {\beta^l}_{,ik} = -\alpha_{,k} K_{ij} - \alpha K_{ij,k}$
$d_{kij,t} - \beta^l d_{kij,l} = - \alpha K_{ij,k}
- \alpha_{,k} K_{ij}
+ d_{lij} {\beta^l}_{,k} + d_{kil} {\beta^l}_{,j} + \frac{1}{2} \gamma_{il} {\beta^l}_{,jk} + d_{klj} {\beta^l}_{,i} + \frac{1}{2} \gamma_{lj} {\beta^l}_{,ik} $
$d_{kij,t} - \beta^l d_{kij,l} \approx -\alpha K_{ij,k}$
$d_{ijk,t} - \beta^l d_{ijk,l} \approx -\alpha K_{jk,i}$
start with evolution equation:
$K_{ij,t} - \beta^k K_{ij,k} - K_{kj} {\beta^k}_{,i} - K_{ik} {\beta^k}_{,j} = -\alpha_{,ij} + {\Gamma^k}_{ji} \alpha_{,k} + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j)$
$K_{ij,t} - \beta^k K_{ij,k} = \alpha R_{ij}- \alpha_{,ij} + {\Gamma^k}_{ji} \alpha_{,k} + \alpha K K_{ij} - 2 \alpha K_{ik} {K^k}_j + K_{kj} {\beta^k}_{,i} + K_{ik} {\beta^k}_{,j} $
$K_{ij,t} - \beta^k K_{ij,k} \approx \alpha R_{ij} - \alpha_{,ij} + \alpha_{,i} \alpha_{,j} / \alpha$
...substitute $R_{ij} \approx -{d^m}_{ij,m} + 2 {d^m}_{m(j,i)} - {{d_{(j|m}}^m}_{,|i)}$ ...
$K_{ij,t} - \beta^k K_{ij,k} \approx \alpha (-{d^m}_{ij,m} + 2 {d^m}_{m(j,i)} - {{d_{(j|m}}^m}_{,|i)}) - \frac{1}{2} \alpha (\alpha_{,ij} / \alpha - \alpha_{,i} \alpha_{,j} / \alpha^2 + \alpha_{,ij} / \alpha - \alpha_{,i} \alpha_{,j} / \alpha^2)$
$K_{ij,t} - \beta^k K_{ij,k} \approx -\alpha ({d^m}_{ij,m} + \frac{1}{2} ({{d_{jm}}^m}_{,i} + {{d_{im}}^m}_{,j} - 2 {d^m}_{mj,i} - 2 {d^m}_{mi,j}) ) - \frac{1}{2} \alpha (a_{i,j} + a_{j,i})$
$K_{ij,t} - \beta^k K_{ij,k} \approx -\alpha ({d^k}_{ij,k} + \frac{1}{2} (a_{j,i} + {{d_{jm}}^m}_{,i} - 2 {d^m}_{mj,i}) + \frac{1}{2} (a_{i,j} + {{d_{im}}^m}_{,j} - 2 {d^m}_{mi,j}))$
$K_{ij,t} - \beta^k K_{ij,k} \approx -\alpha ({d^k}_{ij} + \delta^k_{(i} (a_{j)} + {d_{j)m}}^m - 2 {d^m}_{mj)}))_{,k}$
...let ${\Lambda^k}_{ij} = {d^k}_{ij} + \delta^k_{(i} ( a_{j)} + {d_{j)m}}^m - 2 {d^m}_{mj)})$ ...
$K_{ij,t} - \beta^k K_{ij,k} \approx -\alpha {\Lambda^k}_{ij,k}$
Looks like "equal up to principal part" neglects any first derivative terms of non-conservation variables ($\alpha, \beta^i, \gamma_{ij}$).
In all equations other than $K_{ij,t} \approx R_{ij} ... $, it looks all first derivatives of non-conservation variables are removed.
In the case of $R_{ij}$ it looks like some are left in place, for the sake of simplifications. Are any added as well?