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Violation of Energy Bounds in Designer Gravity

Thomas Hertog

TL;DR

The paper addresses the stability of designer gravity theories—AdS gravity with tachyonic scalars under boundary conditions defined by a function $W$—and questions a proposed energy bound tied to the minimum of $W$ when the scalar potential derives from a superpotential $P$. By analyzing asymptotics, spinor charges, and taking conformal rescalings of AdS-invariant solitons, the authors construct explicit counterexamples where negative-energy initial data exist even for $V$ arising from $P$ and $eta(\alpha)$ determined by $W$. The results show that the positive-energy guarantees depend critically on both $P$ and the boundary data; only certain combinations yield a stable ground state, while others admit arbitrarily negative mass solitons. In the AdS/CFT context, these findings imply that while many boundary deformations preserve positivity, a universal PET does not hold for all designer-gravity boundary conditions, and criteria linking $P$ and $W$ must be identified for spinor-based positivity to apply.

Abstract

We continue our study of the stability of designer gravity theories, where one considers anti-de Sitter gravity coupled to certain tachyonic scalars with boundary conditions defined by a smooth function W. It has recently been argued there is a lower bound on the conserved energy in terms of the global minimum of W, if the scalar potential arises from a superpotential P and the scalar reaches an extremum of P at infinity. We show, however, there are superpotentials for which these bounds do not hold.

Violation of Energy Bounds in Designer Gravity

TL;DR

The paper addresses the stability of designer gravity theories—AdS gravity with tachyonic scalars under boundary conditions defined by a function —and questions a proposed energy bound tied to the minimum of when the scalar potential derives from a superpotential . By analyzing asymptotics, spinor charges, and taking conformal rescalings of AdS-invariant solitons, the authors construct explicit counterexamples where negative-energy initial data exist even for arising from and determined by . The results show that the positive-energy guarantees depend critically on both and the boundary data; only certain combinations yield a stable ground state, while others admit arbitrarily negative mass solitons. In the AdS/CFT context, these findings imply that while many boundary deformations preserve positivity, a universal PET does not hold for all designer-gravity boundary conditions, and criteria linking and must be identified for spinor-based positivity to apply.

Abstract

We continue our study of the stability of designer gravity theories, where one considers anti-de Sitter gravity coupled to certain tachyonic scalars with boundary conditions defined by a smooth function W. It has recently been argued there is a lower bound on the conserved energy in terms of the global minimum of W, if the scalar potential arises from a superpotential P and the scalar reaches an extremum of P at infinity. We show, however, there are superpotentials for which these bounds do not hold.

Paper Structure

This paper contains 11 sections, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Designer Gravity
  3. Tachyonic Scalars in AdS
  4. Asymptotics and Conserved Charges
  5. Stability and Ground State
  6. AdS-invariant boundary conditions
  7. Violation of Energy Bounds
  8. Asymptotically AdS Solitons
  9. AdS solitons imply negative energy
  10. Further Examples
  11. Conclusion

Figures (4)

  • Figure 1: Scalar potential $V$ that can be written in terms of a superpotential $P$ with $P'(0)=0$.
  • Figure 2: Soliton solution $\phi (r)$ with boundary conditions specified by $\beta=0$.
  • Figure 3: The dashed line corresponds to a critical scalar potential $V$ that is on the verge of violating the Positive Energy Theorem for standard scalar AdS boundary conditions. The full line gives a potential that arises from a superpotential, yet violates the PET with $W>0$ designer gravity boundary conditions
  • Figure 4: The function $\beta_{s}(\alpha)$ obtained from the solitons.