Energy Bounds in Designer Gravity

TL;DR

This work addresses the problem of defining finite conserved charges in asymptotically AdS spacetimes with tachyonic scalars near the Breitenlohner-Freedman bound under general boundary conditions specified by a function $W$. It employs the covariant phase space formalism to derive explicit Hamiltonian generators for asymptotic symmetries, showing they decompose into the pure gravity Weyl term plus scalar-field contributions, and demonstrates finiteness via a finite symplectic structure. By relating these charges to a positive spinor charge, the authors establish a lower energy bound when $W$ is bounded below, $m^2>m^2_{BF}$, and $V(\phi)$ admits a suitable superpotential; they also treat special cases with logarithmic branches and BF saturation. The results extend prior $d=3$ and higher-dimensional analyses, clarify the role of boundary data in AdS stability, and have implications for the dual CFT, confirming stability under broad designer-gravity boundary conditions.

Abstract

We consider asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound in d greater than or equal to 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 i) W has a global minimum, ii) the Breitenlohner-Freedman bound is not saturated, and iii) the scalar potential V admits a certain type of "superpotential."