A generalized Momentum/Complexity correspondence

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.

General bounds on holographic complexity

“…It would be interesting to see if our methods could be refined to derive stronger bounds on complexity growth when charge is present, given that our proof amounted to writing a formula (d − 1)| Ċ| = 8πM − ∆ where there pure gauge field terms always contribute positively to ∆. 24 A strictly positive lower bound on their contribution to ∆ in terms of JHEP01(2022)040 charge would give a strengthening of our bound. It is also a possibility that our bound on complexity growth is not strict, and that the bound is true with f (M ) = 1.…”

“…Although not entirely absent: see[93] for example 24. More precisely, we have ±(d − 1) ĊV = 8πM − ∆± where ∆± is proportional to the intergral of the quantity r d−1 δ± given by(3.20).…”

“…Here we have given the name C V to the geometric quantity for convenience. This proposal has led to numerous insights on spacetime emergence in the deep interior, from the late-time behavior of wormhole volumes [10,20], to the connection between bulk depth and momentum [21][22][23][24], to the late-time breakdown of classical GR [25], among others (e.g. [26]).…”

Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity

“…(2.7) is referred to as the momentum/volume/complexity (PVC) relation. More generally, it was proven (2.7) is essentially the momentum constraint of general relativity, such that one has the 'generalized PVC' relation [115,116]…”

“…The above rate allows us to make contact with the PVC correspondence. Specifically, since in the limit B → 0 the bulk Cauchy slices Σ(A) and Σ(AB) tend to the same maximal volume surface Σ, we may relate the rate of maximal flux through B to the momentum flux through Σ upon invoking the generalized PVC relation (2.8) [115,116],…”