Complexity Growth Rate in Lovelock Gravity

Using the $\text{complexity}=\text{action}$ framework, we compute the late time growth of complexity for charged black holes in Lovelock gravity. Our calculation is facilitated by the fact that the null boundaries of the Wheeler-DeWitt patch do not contribute at late times and essential contributions coming from the joints are now understood. The late time growth rate reduces to a difference of internal energies associated with the inner and outer horizons, and in the limit where the mass is much larger than the charge, we reproduce the celebrated result of $2M/\pi$ with corrections proportional to the highest Lovelock coupling in even (boundary) dimensions. We find in some cases a minimum mass below which complexity remains effectively constant, even if the black hole contains a nondegenerate horizon.

Figure 1

The causal structure for a charged AdS black hole, with outer and inner horizons. The blue shaded region denotes the WDW patch, anchored at the boundary times $t_L=t_R=t/2$. At late times, the null boundaries of the WDW patch approach the inner and outer horizons.