Abstract
In a recent note [1] I argued that the holographic origin of ordinary gravitational attraction is the quantum mechanical tendency for operators to grow under time evolution. In a follow-up [2] 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 of [2], 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 [3] constructing symmetry generators for the approximate symmetries of the SYK model.
Highlights
In a recent note [1], I argued that the holographic origin of ordinary gravitational attraction is the quantum mechanical tendency for operators to grow under time evolution
What is it that takes place in the holographic representation of a theory when an object in the bulk is gravitationally attracted to a massive body? Consider a holographic theory representing a region of empty space
It is plausible that the holographic representation of gravitational attraction has something to do with the tendency for operators to grow and become more complex [1]
Summary
What is it that takes place in the holographic representation of a theory when an object in the bulk is gravitationally attracted to a massive body? Consider a holographic theory representing a region of empty space. We might expect that the complexity is a good holographic indicator of how far from the boundary the particle is located. It seems plausible that velocity is related to the rate of change of size This is oversimplified but it roughly captures the idea that size, and its rate of change, holographically encode the motion of the particle. The gravitational pull of the heavy mass will accelerate the particle away from the boundary. It is plausible that the holographic representation of gravitational attraction has something to do with the tendency for operators to grow and become more complex [1]. The quantitative equivalence of size and complexity continues for times of order the scrambling time, but the connection between size and the motion of an infalling particle breaks down as the particle reaches the stretched horizon. I will use the symbol ≈ to indicate that an equation is correct up to such numerical factors
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