Einstein-Maxwell-dilaton neutral black holes in strong magnetic fields: Topological charge, shadows, and lensing

The light rings (LRs) topological charge (TC) of a spacetime measures the number stable LRs minus unstable LRs. It is invariant under smooth deformations obeying fixed boundary conditions. Asymptotically flat equilibrium black holes (BHs) have, generically, $\mathrm{TC}=\ensuremath{-}1$. In Einstein-Maxwell theory, however, Schwarzschild-Melvin BH---describing neutral BH immersed in strong magnetic field---has $\mathrm{TC}=0$. This allows existence BHs without and produces remarkable phenomenological features, like panoramic shadows. Here we investigate generalized solution Einstein-Maxwell-dilaton scanning effect dilaton coupling $a$. We find that TC changes discontinuously from $\mathrm{TC}=0$ to $\mathrm{TC}=\ensuremath{-}1$ precisely at Kaluza-Klein value $a=\sqrt{3}$, when (empty) Melvin corresponds twisted reduction five-dimensional spacetime, i.e., $a$ induces transition TC. relate this qualitative change asymptotics for different also study shadows lensing values $a$, relating them