Narrow Results

In this paper, we use the complexity equals action proposal and investigate holographic complexity for hyperscaling violating theories on different subregions of space-time enclosed by the null boundaries. We are interested in computing the onshell action for certain subregions of the intersection between the Wheeler DeWitt patch and the past, as well as, the future interior of a two-sided black brane. More precisely, we extend the results of Ref. 1 in parts, to hyperscaling violating geometries and to find the finite onshell action, we define the proper counter terms. We show that in computing the rate of complexification the dynamical exponent plays a crucial rule, but, at the late time, rate of the complexity growth is independent of the hyperscaling parameters.

We study Crofton’s formula in the Lorentzian AdS₃ and find that the area of a generic spacelike two-dimensional surface is given by the flux of spacelike geodesics. The “complexity=volume” conjecture then implies a new holographic representation of complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.