https://doi.org/10.1140/epjc/s10052-020-7661-z
Regular Article - Theoretical Physics
Holographic complexity of the electromagnetic black hole
1
College of Physics and Communication Electronics, Jiangxi Normal University, 330022, Nanchang, China
2
Department of Physics, Beijing Normal University, 100875, Beijing, China
Received:
10
October
2019
Accepted:
18
January
2020
Published online:
3
February
2020
In this paper, we use the “complexity equals action” (CA) conjecture to evaluate the holographic complexity in some multiple-horzion black holes for the gravitational theory 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. But 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. However, the choice of these additional boundary terms is dependent on the specific gravitational theory as well as the black hole geometries. To show this, we apply our late-time result to some explicit cases and show how to choose the proportional constant of the additional boundary term to make the complexity be well-defined in the zero-charge limit. Typically, we investigate the static magnetic black holes in Einstein gravity coupled to a first-order electrodynamics and find that there is a general relationship between the proper proportional constant and the Lagrangian function 
of the electromagnetic field: if is a convergent function, the choice of the proportional constant is independent on explicit expressions of and it should be chosen as 4/3; if is a divergent function, the proportional constant is dependent on the asymptotic index of the Lagrangian function.
© The Author(s) 2020
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