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Integrable Systems and Spacetime Dynamics.

Marcela Cárdenas1, Francisco Correa2, Kristiansen Lara1

  • 1Departamento de Física, Universidad de Santiago de Chile, Avenida Ecuador 3493, Estación Central, 9170124 Santiago, Chile.

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Summary

The Ablowitz-Kaup-Newell-Segur (AKNS) system

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Area of Science:

  • Theoretical Physics
  • General Relativity
  • Mathematical Physics

Background:

  • The Ablowitz-Kaup-Newell-Segur (AKNS) integrable hierarchy is a complex mathematical system.
  • General relativity describes gravity as a property of spacetime.
  • Negative cosmological constants have implications for the expansion of the universe.

Purpose of the Study:

  • To demonstrate a novel connection between the AKNS integrable hierarchy and three-dimensional general relativity.
  • To explore the geometrization of the AKNS system through gravitational field boundary conditions.
  • To investigate the role of asymptotic symmetry and conserved charges in this framework.

Main Methods:

  • Formulating three-dimensional general relativity with a negative cosmological constant.
  • Constructing novel, invariant boundary conditions for the gravitational field.
  • Utilizing SL(2,R) conjugacy classes to study gravitational configurations.
  • Incorporating conical singularities and black hole solutions within the boundary conditions.

Main Results:

  • The dynamical equations of the specified general relativity framework yield the AKNS integrable hierarchy.
  • The constructed boundary conditions are invariant under an asymptotic symmetry group.
  • This symmetry group is characterized by an infinite set of AKNS commuting conserved charges.
  • The study successfully includes gravitational configurations like conical singularities and black holes.

Conclusions:

  • A profound link exists between the AKNS integrable hierarchy and 3D general relativity with a negative cosmological constant.
  • This geometrization is achieved via specific, symmetry-invariant boundary conditions.
  • The findings offer new perspectives on integrable systems and their geometric interpretations in gravity.