# Holographic complexity of the electromagnetic black hole

## TL;DR

This paper investigates the holographic complexity of electromagnetic black holes using the CA conjecture, revealing universal features for magnetic black holes and establishing how boundary terms influence late-time complexity growth.

## Contribution

It introduces a boundary term approach to define holographic complexity for electromagnetic black holes and derives a general late-time growth rate expression.

## Key findings

- Vanishing late-time growth rate is universal for static magnetic black holes.
- Boundary terms are essential for well-defined complexity in electromagnetic black holes.
- A relationship between boundary term constants and the electromagnetic Lagrangian function is established.

## Abstract

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 $et.\, al$, 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 $h(\math{F})$ of the electromagnetic field: if $h(\math{F})$ is a convergent function, the choice of the proportional constant is independent on explicit expressions of $h(\math{F})$ and it should be chosen as $4/3$; if $h(\math{F})$ is a divergent function, the proportional constant is dependent on the asymptotic index of the Lagrangian function.