Nonlinear horizon boundary conditions realizing horizon symmetries

Identify the full nonlinear generalization of the mixed horizon boundary conditions used for linear perturbations of JT and AdS₃ black holes that yields hydrodynamic actions realizing the horizon symmetries proposed for black-hole horizons.

Background

The paper derives quadratic hydrodynamic actions for small perturbations of JT and BTZ black holes by imposing mixed horizon boundary conditions motivated by the gravitational variational principle. These boundary conditions produce hydrodynamic actions whose zero modes realize specific horizon diffeomorphisms, including exponentially growing and decaying modes associated with maximal chaos.

The analysis is restricted to linear perturbations. Extending the construction nonlinearly requires determining boundary conditions beyond the quadratic approximation while preserving both a well-defined variational principle and the desired horizon-symmetry realization. The paper identifies this as a central unresolved issue for a full nonlinear formulation.