= Causal structure of the magnetic dilaton black hole
For $r_-<r_+$, an ingoing chart with $v=t+r_*$ has metric $-Vdv^2+2dv\,dr+R^2d\Omega_2^2$. In its future interior, $\nabla r=\partial_v+V\partial_r$ is future timelike, so every future causal curve decreases $r$ and cannot reach <future null infinity>. The singularity $r=r_-$ is spacelike. A <Kruskal–Szekeres coordinates> extension has the same qualitative <Penrose diagram> as <Schwarzschild spacetime>, with two exterior regions, a future black-hole interior and a past <white hole> interior. The past interior has the same radial interval but is not a <black hole> region; specifying the radial interval alone does not identify the future component. For $r_->r_+$, the physical exterior $r>r_-$ has $V>0$ everywhere and outgoing null rays escape from arbitrarily near a timelike <naked singularity>.
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