Scalar shear potential ($\chi =\frac{a^2}{\bar{N}}(\dot{E}-B)$)

For the negative extrinsic curvature convention and a shift vector entering the line element as $d{x}^i-N^{i}dt$, the trace-free scalar curvature perturbation is $\delta K^i{}_j{}_{\mathrm{TF}}=-({\partial }^i{\partial }_j-{\delta }_j^i\Delta /3)\chi$. Here spatial raised derivatives use $a^{-2}{\delta }^{ij}$, so $\Delta =a^{-2}{\nabla }^2$. Changing the sign of the metric shift vector changes the corresponding definition of $\chi$.