Abstract
A celebrated feature of SYK-like models is that at low energies, their dynamics reduces to that of a single variable. In many setups, this “Schwarzian” variable can be interpreted as the extremal volume of the dual black hole, and the resulting dynamics is simply that of a 1D Newtonian particle in an exponential potential. On the complexity side, geodesics on a simplified version of Nielsen’s complexity geometry also behave like a 1D particle in a potential given by the angular momentum barrier. The agreement between the effective actions of volume and complexity succinctly summarizes various strands of evidence that complexity is closely related to the dynamics of black holes.
Highlights
A celebrated feature of SYK-like models is that at low energies, their dynamics reduces to that of a single variable. This “Schwarzian” variable can be interpreted as the extremal volume of the dual black hole, and the resulting dynamics is that of a 1D Newtonian particle in an exponential potential
Even the lower curvature version of the complexity geometry defined in [17] is difficult to analyze at large distances, but the main points of complexity geometry have been illustrated in a simple “toy complexity geometry” (TCG) based on the twodimensional Poincare disc [18]
In a somewhat surprising and nontrivial way the complexity geometry approach reproduces the gravitational dynamics of the boundary Schwarzian theory
Summary
We would like to start with a microscopic definition of complexity, and derive its various properties. 1/(N J), where J is some arbitrary constant with units of inverse length, chosen so that energy has the correct dimensions In approximating the sinh as exponential, we are assuming that complexity is never small. This will be appropriate for low energy states. The particle comes in from infinity, rolls up the potential, and turns around at some ρ0 This exponential potential is nothing but the angular momentum barrier which ensures that geodesics are circles intersecting the boundary of the disk at right angles. Since the asymptotic velocity (or equivalently the kinetic energy) of the particle is fixed, the parameter that controls the distance to closest approach ρ0 is J.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
