= Linearized Schwarzschild metric
{title2=$ds^2=(1-r_S/r)dt^2-(1+r_S/r)dr^2-r^2d\Omega^2$}
For a static spherical point source, the weak-field solution in isotropic radius $\rho$ has $g_{tt}=1-2GM/\rho$ and spatial metric $-(1+2GM/\rho)(d\rho^2+\rho^2d\Omega^2)$. The <isotropic-to-areal weak-field radius change> gives the displayed metric with $r_S=2GM$. Its validity requires $r_S/r\ll1$. Treating it as an exact metric near $r\sim r_S$ can turn omitted terms into large corrections; it does not establish horizons or strong-field photon dynamics.
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