Complexity and Newton's Laws

TL;DR

The paper argues that gravitational attraction in holographic settings emerges from the quantum tendency of operators to grow in size and complexity under time evolution, a link tested in the SYK/NERN framework. It refines the size–momentum correspondence and shows that momentum corresponds to the rate of complexity growth, with Newtonian dynamics arising in the throat of near-extremal black holes and confirmed by Qi–Streicher’s finite-temperature growth results. The analysis leverages AdS2 symmetry, Schwarzian boundary dynamics, and a CV-based comparison to connect boundary motion, bulk forces, and complexity, offering a concrete holographic mechanism for gravity as an emergent consequence of complexity increase. The work also explores the behavior in empty AdS2 and discusses broader implications and potential connections to entropic gravity and traversable wormholes, while acknowledging the role of gauge choices and finite-throat effects.

Abstract

In a recent note I argued that the holographic origin of gravitational attraction is the quantum mechanical tendency for operators to grow under time evolution. In a followup the claim was tested in the context of the SYK theory and its bulk dual---the theory of near-extremal black holes. In this paper I give an improved version of the size-momentum correspondence and show that Newton's laws of motion are a consequence. Operator size is closely related to complexity. Therefore one may say that gravitational attraction is a manifestation of the tendency for complexity to increase. The improved version of the size-momentum correspondence can be justified by the arguments of Lin, Maldacena, and Zhao constructing symmetry generators for the approximate symmetries of the SYK model.

Figures (13)

• Figure 1: The three regions outside a near-extremal charged black hole. Unlike for uncharged black holes, there is now a 'throat' separating the Rindler and far regions.
• Figure 2: Penrose diagram for a NERN black hole. The curved lines represent the trajectory of the black hole boundary at $r=2r_+.$ On the left side the boundary is shown in its equilibrium location while on the right it is moving in reaction to some matter.
• Figure 3: A particle is introduced at the top of the potential, and subsequently rolls down the potential,
• Figure 4: The boundary recoils when the particle is accelerated. At all times the particle and the boundary have equal and opposite momentum.
• Figure 5: Tree-like operator growth. The size at at any circuit-depth is the final number of fermions while the complexity is the number of vertices in the diagram. In this figure the size is $81$ and the complexity is $40$. The complexity at the next step would be $40+81 = 121.$ The time scale for a unit change in depth is $\Delta t.$ In general $\Delta t$ may itself be time dependent.