Alternative Boundary Conditions in Asymptotically Anti-de Sitter Spacetimes

Abstract

Anti-de Sitter (AdS) spacetimes are not globally hyperbolic, so boundary conditions are generally needed for well-defined dynamics. Ishibashi and Wald (IW) determined boundary conditions yielding well-posed dynamics with conserved positive energy for scalar, electromagnetic, and linearized gravitational fields, using nonlocal potentials and individual spherical harmonic modes. In this paper, we determine which IW conditions are local in the primary fields and which are AdS-invariant. In four spacetime dimensions, the standard AdS-invariant boundary conditions set the rescaled magnetic field to zero for electromagnetism and the rescaled magnetic Weyl tensor to zero for Einstein gravity on the conformal boundary. We generalize these standard boundary conditions by setting arbitrary linear combinations of the electric and magnetic fields, or the corresponding Weyl tensors, to zero, and we conjecture that these exhaust the local, AdS-invariant, conservative boundary conditions in AdS$_{1,3}$. Boundary conditions with non-AdS-invariance and in different spacetime dimensions are also studied. The AdS-invariant conditions in AdS$_{1,3}$ extend to Yang-Mills theory and nonlinear gravity in a general Fefferman-Graham setting. We construct phase spaces for these general boundary conditions. The covariant phase spaces for general conditions are naturally associated with Lagrangians containing Pontryagin terms and, in gravity, a Gauss-Bonnet term. In gravity, except for the standard boundary condition, no nontrivial asymptotic gauge symmetries exist and all conserved charges such as ADM mass vanish identically, as in a closed universe. Spacetimes with Killing fields including AdS then exhibit linearization instability in the sense of Fischer and Marsden and Moncrief. For nonstandard boundary conditions, the modified symplectic structure yields new expressions for the entropy formula of black holes.