Linearized Schwarzschild metric ($d{s}^2=(1-r_S/r)d{t}^2-(1+r_S/r)d{r}^2-r^{2}d{\Omega }^2$)

For a static spherical point source, the weak-field solution in isotropic radius $\rho$ has $g_{tt}=1-2GM/\rho$ and spatial metric $-(1+2GM/\rho )(d{\rho }^2+{\rho }^{2}d{\Omega }^2)$. 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.

The first-order null-orbit equation $(u^{\prime }{)}^2=r_S^2{E}^2/h^2-u^2(1-u)$ formally has a circular solution at $u=2/3$, with $|h/E|=3\sqrt{3}{r}_S/2$. Its second-order equation gives a perturbation $\delta u^{\prime \prime }=\delta u$, hence instability. However, $u=2/3$ is outside the weak-field expansion. The exact Schwarzschild spacetime has a corresponding unstable photon sphere, but that physical result needs the exact geometry; extrapolating the linearized metric is not a controlled proof.

The area of a symmetry sphere in the weak-field isotropic metric is $4\pi {\rho }^2(1+2GM/\rho )$. Equating it to $4\pi r^2$ defines the areal radius and gives the displayed change. To first order $dr=d\rho$, and replacing $1/\rho$ by $1/r$ changes the perturbation only at second order. This explains why the isotropic angular correction disappears in the linearized Schwarzschild metric without discarding spatial curvature.