= Synchronous-to-Newtonian density transformation with a lapse
{title2=$\delta_N=\delta_S+\frac{a^2\dot E_S}{\bar N^2}\frac{\dot{\bar\rho}}{\bar\rho}$}
For spatial scalar convention $g_{ij}=a^2[(1-2\Psi)\delta_{ij}-2E_{,ij}]$ and $g_{0i}=a^2B_{,i}$, a passive gauge displacement gives $\widetilde E=E+\lambda$ and $\widetilde B=B-\dot\lambda+\bar N^2\xi^0/a^2$. Moving from <synchronous gauge> to <Newtonian gauge> therefore requires $\lambda=-E_S$ and $\xi^0=-a^2\dot E_S/\bar N^2$. A scalar density transforms by $\delta\widetilde\rho=\delta\rho-\dot{\bar\rho}\xi^0$, giving the displayed <density contrast> relation. Keeping the background <lapse function> explicit prevents cosmic-time and conformal-time factors from being confused.
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