= Schwarzschild photon escape cone
{c}
{title2=$\alpha_{\rm crit}=\arcsin\left[\frac{3\sqrt3\mu}{r}\sqrt{1-\frac{2\mu}{r}}\right]$}
At $2\mu<r\le3\mu$, measure the emission <angle> $\alpha$ from the outward radial direction in a static observer's <orthonormal tetrad>. Local <photon> energy is $E_{\rm loc}=E_\infty/\sqrt{1-2\mu/r}$, and its tangential <momentum> is $E_{\rm loc}\sin\alpha/c$. 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 $0\le\alpha<\alpha_{\rm crit}$; the limiting ray approaches the <photon sphere> and does not escape.
As $r\downarrow2\mu$ 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>.
Back to article page
