Energy bounds in designer gravity

We consider asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound in d≥4 spacetime dimensions. The boundary conditions in these ``designer gravity'' theories are defined in terms of an arbitrary function W. We give a general argument that the Hamiltonian generators of asymptotic symmetries for such systems will be finite, and proceed to construct these generators using the covariant phase space method. The direct calculation confirms that the generators are finite and shows that they take the form of the pure gravity result plus additional contributions from the scalar fields. By comparing the generators to the spinor charge, we derive a lower bound on the gravitational energy when W has a global minimum and the Breitenlohner-Freedman bound is not saturated.