Horizon flux inequality for causal matter ($E-{\Omega }_{H}L\ge 0$)

For a future-directed causal four-momentum $p$ crossing a future Killing horizon with generator $\xi =t+{\Omega }_H\phi$, the Lorentzian inner product obeys $p\cdot \xi \le 0$. With $E=-p\cdot t$ and $L=p\cdot \phi$, this is $E\ge {\Omega }_{H}L$. A massive particle has strict inequality. The same bound follows in the nonextremal quasistationary regime from the Physical-process first law for a rotating black hole and the second law of black-hole mechanics. Negative Killing energy in a Penrose process is allowed, but its angular momentum must satisfy this bound.