# A tale of two superpotentials: Stability and instability in designer gravity

## Abstract

We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The boundary conditions in these 'designer gravity' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if (i) W has a global minimum and (ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P{sub {+-}}. When there is a P{sub -} branch, we rigorously prove a lower bound on the energy; the P{sub +} branch alone is not sufficient. Our numerical investigations (i) confirm this picture, (ii) confirm other critical aspects of the (complicated) proofs, and (iii) suggest that the existence of P{sub -} may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below.

- Authors:

- Physics Department, UCSB, Santa Barbara, California 93106 (United States)
- Theory Division, CERN, CH-1211 Geneva 23 (Switzerland) and APC, 10 rue Alice Domon et Leonie Duquet, 75205 Paris (France)
- Inst. f. Theor. Physik, Georg-August-Universitaet, D-37077, Goettingen (Germany)

- Publication Date:

- OSTI Identifier:
- 21020377

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 8; Other Information: DOI: 10.1103/PhysRevD.75.084008; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BOUNDARY CONDITIONS; COSMOLOGY; DE SITTER GROUP; GRAVITATION; INSTABILITY; MASS; MATHEMATICAL SOLUTIONS; POTENTIALS; SCALAR FIELDS; SCALARS; STABILITY; TACHYONS

### Citation Formats

```
Amsel, Aaron J., Marolf, Donald, Hertog, Thomas, and Hollands, Stefan.
```*A tale of two superpotentials: Stability and instability in designer gravity*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.084008.

```
Amsel, Aaron J., Marolf, Donald, Hertog, Thomas, & Hollands, Stefan.
```*A tale of two superpotentials: Stability and instability in designer gravity*. United States. doi:10.1103/PHYSREVD.75.084008.

```
Amsel, Aaron J., Marolf, Donald, Hertog, Thomas, and Hollands, Stefan. Sun .
"A tale of two superpotentials: Stability and instability in designer gravity". United States.
doi:10.1103/PHYSREVD.75.084008.
```

```
@article{osti_21020377,
```

title = {A tale of two superpotentials: Stability and instability in designer gravity},

author = {Amsel, Aaron J. and Marolf, Donald and Hertog, Thomas and Hollands, Stefan},

abstractNote = {We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The boundary conditions in these 'designer gravity' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if (i) W has a global minimum and (ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P{sub {+-}}. When there is a P{sub -} branch, we rigorously prove a lower bound on the energy; the P{sub +} branch alone is not sufficient. Our numerical investigations (i) confirm this picture, (ii) confirm other critical aspects of the (complicated) proofs, and (iii) suggest that the existence of P{sub -} may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below.},

doi = {10.1103/PHYSREVD.75.084008},

journal = {Physical Review. D, Particles Fields},

number = 8,

volume = 75,

place = {United States},

year = {Sun Apr 15 00:00:00 EDT 2007},

month = {Sun Apr 15 00:00:00 EDT 2007}

}