 Solution

The second law of black-hole mechanics, or Hawking's area theorem, states that the total area of future-event-horizon cross-sections cannot decrease toward the future under the null energy condition and the usual predictability assumptions. If any horizon generator had $\theta <0$, the twist-free Null Raychaudhuri equation and the null focusing theorem would force a conjugate point at finite affine parameter. A generator complete to the future cannot pass through such a point while remaining on the achronal boundary of $J^{-}({\mathcal{I}}^{+})$. Hence $\theta \ge 0$ everywhere on the regular future horizon, and $dA/d\lambda =\theta A\ge 0$ proves the area law.