Maximality of covariant nonlinear gravitational boundary conditions

Determine whether the nonlinear gravitational boundary conditions \(\left[\tilde E_{IJ}+\alpha\tilde B_{IJ}\right]_{\mathcal I}=0\) in four-dimensional Fefferman–Graham spacetimes are the most general covariant boundary conditions that admit a covariant phase space structure on the space of Einstein solutions.

Background

For four-dimensional Fefferman–Graham spacetimes, the paper studies boundary conditions relating the rescaled electric and magnetic parts of the Weyl tensor at conformal infinity. Existing results of Friedrich and Souêtre establish well-posedness of the initial-value formulation for this family, and the paper constructs a covariant phase space using a Lagrangian augmented by Gauss–Bonnet and gravitational Pontryagin terms.

The remaining classification problem is whether every covariant boundary condition supporting a covariant phase space must be of this electric-plus-magnetic form.