= Positive-shift scalar shear convention
{title2=$\kappa=3(\mathcal D\Phi+H\Psi)+\Delta\chi_+$}
For $N_i=a^2B_{,i}$, $\gamma_{ij}=a^2[(1-2\Phi)\delta_{ij}+2E_{,ij}]$ and $K_{ij}=-(\dot\gamma_{ij}-D_iN_j-D_jN_i)/(2N)$, put $\chi_+=a^2(B-\dot E)/\bar N$. Then $K^i{}_j=-H\delta^i{}_j+(\dot\Phi/\bar N+H\Psi)\delta^i{}_j+\partial^i\partial_j\chi_+$ and $\kappa=3(\dot\Phi/\bar N+H\Psi)+\Delta\chi_+$. Defining $\chi=-\chi_+$ instead makes the trace-free tensor coefficient negative and the trace gradient term $-\Delta\chi$. A sign change must affect both pieces.
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