Proof of a Momentum/Complexity Correspondence

TL;DR

This work investigates whether the holographic Complexity=Volume prescription encodes a Momentum/Complexity (PC) correspondence inherent to GR. It derives a PVC relation from the Momentum Constraint by using maximal-volume extremal surfaces and an infall field $C_\Sigma$, such that the complexity rate satisfies $\dot{\mathcal{C}}[\Sigma] = P_C[\Sigma] + R_C[\Sigma]$, with $P_C[\Sigma]$ representing matter momentum flux through $\Sigma$. The main finding is that in $d=2$ (i.e., $2+1$) dimensions and in any spherically symmetric spacetime the remainder $R_C$ vanishes, giving an exact PC $\dot{\mathcal{C}} = P_C[\Sigma]$, while gravitational waves and nontrivial topology can introduce nontrivial remainder terms. The authors further propose a Codazzi-based generalization that incorporates the Weyl tensor via a three-index infall tensor $M^{abc}$, suggesting a path toward PC extensions and possibly to Complexity=Action, while providing a Newtonian interpretation of infall momentum as the total derivative of a spherical clumping measure.

Abstract

We show that the holographic Complexity = Volume proposal satisfies a very general notion of Momentum/Complexity correspondence (PC), based on the Momentum Constraint of General Relativity. It relates the rate of complexity variation with an appropriate matter momentum flux through spacelike extremal surfaces. This formalizes the intuitive idea that `gravitational clumpling' of matter increases complexity, and the required notion of `infall momentum' is shown to have a Newtonian avatar which expresses this idea. The proposed form of the PC correspondence is found to be exact for any solution of Einstein's equations in 2+1 dimensions, and any spherically symmetric solution in arbitrary dimensions, generalizing all previous calculations using spherical thin shells. Gravitational radiation enters through a correction which does not have a straightforward interpretation as a PC correspondence. Other obstructions to an exact PC duality have a topological origin and arise in the presence of wormholes.

Figures (2)

• Figure 1: For each point in $S_r$, the time coordinate is chosen to properly parametrize a geodesic (dashed lines) on $Y_r$. $\vec{e}_y$ is then picked to be orthogonal to $S_r$.
• Figure 2: On the top panel, a CKV field on the Einstein--Rosen bridge is in-falling on one side and 'out-falling' on the other. On the bottom panel, insisting on being in-falling on both sides forces a discontinuous jump through a defect in the interior.