Energy-normalized angular momentum for a Schwarzschild geodesic ($J=c^2{L}_{\infty }/E_{\infty }$)

For an unbound equatorial Schwarzschild spacetime geodesic, let $E_{\infty }$ and $L_{\infty }$ be its conserved energy and angular momentum measured relative to infinity. The ratio $J=c^2{L}_{\infty }/E_{\infty }$ has dimensions of area divided by time. For a massive test particle, $E_{\infty }=m{c}^2{\gamma }_{\infty }$ and $L_{\infty }=m\ell$, so $J=\ell /{\gamma }_{\infty }=b{v}_{\infty }$, where $\ell$ is specific angular momentum and $b$ the impact parameter. For a photon, $E_{\infty }=c{p}_{\infty }$ and $L_{\infty }=b{p}_{\infty }$, giving $J=bc$ without assigning a finite Lorentz factor or a proper time to the photon. This ratio removes the arbitrary normalization of a null affine parameter.