 Solution

Let a future-directed causal tangent in the ingoing chart beThe chosen time orientation has $\dot{v}\ge 0$; equality occurs only for the ingoing radial null direction, which has $\dot{r}<0$. Causality requiresFor $r_{-}<r<r_{+}$ one has $f<0$. The first and third terms are then nonnegative, so if $\dot{v}>0$ the inequality forces $\dot{r}<0$. If $\dot{v}=0$, it forces $\dot{\chi }=0$, and future direction again gives $\dot{r}<0$. Thus $r$ strictly decreases along every future causal curve in this region. No such curve can cross outward through $r=r_{+}$ into $r>r_{+}$, which proves the one-way causal disconnection characteristic of an event horizon.