 Linearized Schwarzschild metric

For a static spherical point source, the weak-field solution in isotropic radius $\rho$ has $g_{tt}=1-2GM/\rho$ and spatial metric $-(1+2GM/\rho )(d{\rho }^2+{\rho }^{2}d{\Omega }^2)$. The isotropic-to-areal weak-field radius change gives the displayed metric with $r_S=2GM$. Its validity requires $r_S/r\ll 1$. 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.