A Generalized Momentum/Complexity Correspondence

TL;DR

This work extends the holographic momentum/complexity correspondence beyond matter-driven infall by replacing the GR momentum constraint with the Codazzi equation, introducing a tensor infall field $M^{abc}$ and a Weyl-based contribution $W_M$. The authors derive a generalized PVC: $\frac{d\mathcal{C}}{dt} = P_C[\Sigma] + W_M[\Sigma] + R_M[\Sigma]$, with exactness achieved under suitable boundary and transversality conditions; they explicitly verify the proposal in an exact pp-wave solution, showing the Weyl momentum matches the bulk volume growth. The results demonstrate that purely gravitational dynamics can drive complexity growth through Weyl curvature and highlight the quasi-local nature of these relations, suggesting connections to Bel–Robinson energy and implications for boundary holography in AdS/CFT and other complexity proposals.

Abstract

Holographic complexity, in the guise of the Complexity = Volume prescription, comes equipped with a natural correspondence between its rate of growth and the average infall momentum of matter in the bulk. This Momentum/Complexity correspondence can be related to an integrated version of the momentum constraint of general relativity. In this paper we propose a generalization, using the full Codazzi equations as a starting point, which successfully accounts for purely gravitational contributions to infall momentum. The proposed formula is explicitly checked in an exact pp-wave solution of the vacuum Einstein equations.

Figures (3)

• Figure 1: Schematic representation of the region $X$, whose spatial boundary $\partial X$ has cylindrical topology.
• Figure 2: The canonical $C$-field in 2+1 dimensions is orthogonal to surfaces conformal to circles.
• Figure 3: Schematic representation of the extremal surface for a pp-wave pulse traversing the box of size $2\ell$ from left to right. The transverse ${\bf R}^2$ plane is not shown.