Holographic complexity of the electromagnetic black hole

In this paper, we use the “complexity equals action” (CA) conjecture to evaluate the holographic complexity in some multiple-horzion black holes for F(Riemann) gravity coupled to a first-order source-free electrodynamics. Motivated by the vanishing result of the purely magnetic black hole founded by Goto $et\mathrm{.}al$, we investigate the complexity in a static charged black hole with source-free electrodynamics and find that this vanishing feature of the late-time rate is universal for a purely static magnetic black hole. However, this result shows some unexpected features of the late-time growth rate. We show how the inclusion of a boundary term for the first-order electromagnetic field to the total action can make the holographic complexity be well-defined and obtain a general expression of the late-time complexity growth rate with these boundary terms. We apply our late-time result to some explicit cases and show how to choose the proportional constant of these additional boundary terms to make the complexity be well-defined in the zero-charge limit. For the static magnetic black hole in Einstein gravity coupled to a first-order electrodynamics, we find that there is a general relationship between the proper proportional constant and the Lagrangian function $h(\text{math}F)$ of the electromagnetic field: if $h(\text{math}F)$ is a convergent function, the choice of the proportional constant is independent on explicit expressions of $h(\text{math}F)$ and it should be chosen as $4\mathrm{/}3$; if $h(\text{math}F)$ is a divergent function, the proportional constant is dependent on the asymptotic index of the Lagrangian function.