Testing holographic conjectures of complexity with Born–Infeld black holes

Abstract In this paper, we use Born–Infeld black holes to test two recent holographic conjectures of complexity, the “Complexity = Action” (CA) duality and “Complexity = Volume 2.0” (CV) duality. The complexity of a boundary state is identified with the action of the Wheeler–deWitt patch in CA duality, while this complexity is identified with the spacetime volume of the WdW patch in CV duality. In particular, we check whether the Born–Infeld black holes violate the generalized Lloyd bound: $$\dot{\mathcal {C}}\le \frac{2}{\pi \hbar }\left[ \left( M-Q\Phi \right) -\left( M-Q\Phi \right) _{\text {gs}}\right] $$ C ˙ ≤ 2 π ħ M - Q Φ - M - Q Φ gs , where gs stands for the ground state for a given electrostatic potential. We find that the ground states are either some extremal black hole or regular spacetime with nonvanishing charges. For Born–Infeld black holes, we compute the action growth rate at the late-time limit and obtain the complexities in CA and CV dualities. Near extremality, the generalized Lloyd bound is violated in both dualities. Near the charged regular spacetime, this bound is satisfied in CV duality but violated in CA duality. When moving away from the ground state on a constant potential curve, the generalized Lloyd bound tends to be saturated from below in CA duality