Photon sphere

A photon sphere is a spherical set of circular null geodesics. In Schwarzschild spacetime it occurs at $r=3M$ and is unstable.

At $2\mu <r\le 3\mu$, measure the emission angle $\alpha$ from the outward radial direction in a static observer's orthonormal tetrad. Local photon energy is $E_{\mathrm{loc}}=E_{\infty }/\sqrt{1-2\mu /r}$, and its tangential momentum is $E_{\mathrm{loc}}\sin \alpha /c$. Thus its impact parameter is $b=r\sin \alpha /\sqrt{1-2\mu /r}$. Combining this with the null effective potential in Schwarzschild spacetime gives escape precisely for outward directions $0\le \alpha <{\alpha }_{\mathrm{crit}}$; the limiting ray approaches the photon sphere and does not escape.

As $r\downarrow 2\mu$ the cone shrinks to zero opening. A static observer does not exist on the horizon itself, and no future-directed ray starting on the Schwarzschild event horizon reaches exterior infinity. At $r=3\mu$, the critical half-angle is $\pi /2$; the exactly tangential critical ray stays on the unstable circular orbit. Escape cones for a moving observer are changed by relativistic aberration.

For a Reissner-Nordstrom black hole with $|Q|\le M$, the outer unstable circular null geodesic has radius $r_{\mathrm{ph}}=(3M+\sqrt{9M^2-8Q^2})/2$. Maximizing $f/r^2$ for $f=1-2M/r+Q^2/r^2$ gives the critical impact parameter $b_c^2=r_{\mathrm{ph}}^2/f(r_{\mathrm{ph}})=2r_{\mathrm{ph}}^3/(r_{\mathrm{ph}}-M)$. The geometrical optics capture cross-section is $\pi b_c^2$, reducing to $27\pi M^2$ in Schwarzschild spacetime and $16\pi M^2$ in the extremal charged case. Finite-wavelength absorption need not equal this geometric limit.

// Target: general-relativity.bigb