 Gravitational field of an infinite plane sheet

An infinite plane sheet of surface mass density $\Sigma$ has constant gravitational-field magnitude $2\pi G\Sigma$ on either side, directed toward the sheet. A Gaussian pillbox has equal inward flux on its two caps and enclosed mass $\Sigma A$, so $-2gA=-4\pi G\Sigma A$. A uniform infinite slab has the same exterior formula with $\Sigma =\rho h$. Translating an infinite sheet does not change its exterior field, whereas adding a slab changes its surface mass density.