Causal trapping between two spherical horizons ($\dot{r}\le \frac{f}{2}\dot{v}-\frac{r^2|\dot{\Omega }{|}^2}{2\dot{v}}$)

In the regular metric $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, a future causal tangent has $\dot{v}\ge 0$. For $\dot{v}>0$, causality gives $\dot{r}\le f\dot{v}/2-r^2|\dot{\Omega }{|}^2/(2\dot{v})$. If $f<0$ between two horizons, the areal radius must decrease. An inner event cannot reach the original exterior future null infinity without traversing that band in the forbidden direction.