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String-Generated Gravity Models
Abstract
Expansion of supersymmetric string theory suggests that the leading quadratic curvature correction to the Einstein action is the Gauss-Bonnet invariant. We show that this model has both flat and anti-de Sitter space as solutions, but that the cosmological branch is unstable, because the graviton becomes a ghost there: The theory solves its own cosmological problem. The general static spherically symmetric solution is exhibited; it is asymptotically Schwarzschild. The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the string-generated sign) or a naked singularity. We discuss the effects of higher-curvature corrections and of an explicit cosmological term on stability.
Overview
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TL;DR
The paper analyzes the effective gravity theory given by the Einstein action plus the leading quadratic curvature correction, the Gauss-Bonnet invariant, suggested by supersymmetric string expansion. It finds that the asymptotically flat branch is viable, while the anti-de Sitter cosmological branch is unstable because the graviton becomes a ghost.
Key findings
- The Gauss-Bonnet combination is the only quadratic curvature term that avoids ghosts near flat space; the model has both flat and anti-de Sitter solutions, with positive Gauss-Bonnet coefficient corresponding to anti-de Sitter space.
- The anti-de Sitter branch is intrinsically unstable: its graviton excitations are ghosts, and a positive-energy matter source leads to a negative-mass Schwarzschild–anti-de Sitter solution in which test particles are repelled.
- The general static spherically symmetric solution is exhibited and is asymptotically Schwarzschild; the sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the string-generated sign) or a naked singularity.
- With an explicit cosmological term, the effective cosmological constant is shifted; the larger value is unstable and the lower one stable, and if \(4\Lambda\kappa^2 > n(n+1)\) the theory is undefined. For \(D=5\), horizon existence depends on whether \(2GM-\kappa\) is positive.
Methods
- Theoretical analysis of the effective action in \(D=n+2\) dimensions, combining the Einstein term with the Gauss-Bonnet invariant, using form/vielbein methods and field equations.
- The paper studies small perturbations about flat and anti-de Sitter backgrounds to identify ghost behavior, and derives general static spherically symmetric solutions in Schwarzschild gauge.
- It also examines the effects of an explicit cosmological term and discusses higher-curvature corrections qualitatively; no empirical data or numerical simulations are used.
Limitations
- The paper states that unresolved matters include the effect of compactification on the \(D>4\) constructions, the role of scalar and antisymmetric tensor fields and the Chern-Simons term, the canonical formulation, and a broader understanding of allowed classical geometries.
- It states that higher-order questions are too early to speculate on, and that the scalar field neglected in the analysis will play a role in these questions.
- It also notes that it has not considered whether semiclassical tunneling modifications occur through the existence of two branches of solutions. Evident from the text: the study is a classical effective-field-theory analysis, not a full string-theoretic treatment.
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In the paper
“(For example, a cubic Gauss-Bonnet term with γ2 > α²κ/4 yields no real roots and hence no cosmological branch.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the stringgenerated sign) or a naked singularity.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“because the graviton becomes a ghost there”
- Also stated in: § S. Deser · plain
“small excitations near the anti-de Sitter branch of the Einstein plus Gauss-Bonnet theory are precisely those of ghost cosmological Einstein gravity.”
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In the paper
“We show that this model has both flat and anti-de Sitter space as solutions”
- Also stated in: § S. Deser · plain
“They are solved by the maximally symmetric spaces”
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In the paper
“The plane-fronted parallel-ray metric,10 although globally singular, is the nearest thing to a graviton in the Einstein theory; it remains a solution here.”
What the sources say
- Not matchedcites [10]
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In the paper
“The general static spherically symmetric solution is exhibited; it is asymptotically Schwarzschild.”
- Also stated in: § S. Deser · plain
“i.e., Schwarzschild with positive gravitational mass or Schwarzschild-de Sitter with negative gravitational mass”
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In the paper
“Note that if 4Λκ² is greater than n (n + 1) the theory is undefined.”
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In the paper
“Although Einstein (G Gµuv = 0) spaces are not generally solutions of (2)”
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In the paper
“but that the cosmological branch is unstable”
- Also stated in: § S. Deser · plain
“these solutions turn out to be intrinsically unstable”
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In the paper
“Expansion of supersymmetric string theory suggests that the leading quadratic curvature correction to the Einstein action is the Gauss-Bonnet invariant.”
- Also stated in: § S. Deser · hedged
“If, like the string itself, its slope expansion is to be ghost free, the quadratic term, if any, must be the Gauss-Bonnet combination, which was first proposed² for this reason and appears to be present³ in the heterotic string.4”
What the sources say
- Not matchedcites [2]
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- Not matchedcites [3]
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In the paper
“These will in general lead to ghosts near flat space (hµv≡ gμv- ηµv << 1) since their metric variations involve fourth time derivatives. By contrast, I2 only contributes terms quadratic in hμv in the field equations”
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In the paper
“For if α were negative, the metric would become imaginary at rⁿ+1 = 8||κMG. This is a real singularity, since the curvature invariants have branch singularities there; furthermore the singularity is naked when, as is possible, it occurs outside the event horizon, goo(rH) = 0.”
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In the paper
“the larger value will be unstable, the lower one stable.”
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In the paper
“Following our discussion of the quadratic case, λ will be determined by a polynomial equation, f(λ) =λ(1 +ακλ +γ₂λ²κ+ ·.·)=0, where the γ are essentially the coefficients of the higher terms.”
- Also stated in: § S. Deser · hedged
“To the extent that there are real roots, the slopes at the intercepts will have alternating signs and therefore the perand one should consider also generic turbations δG μα will appear with both signs.”
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- Also stated in: § S. Deser · hedged
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In the paper
“underlines the fact that the Gauss-Bonnet term does not affect the propagator, and in particular does not distort plane-wave solutions.”
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In the paper
“despite being quadratic in curvature, the action depends only on first time derivatives; hence the number of degrees of freedom is the same as in Einstein gravity, and only the definition of the canonical momentum will change.”
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In the paper
“For D = 5, the existence of a horizon depends on whether 2GM — κ is positive, which is the case for macroscopic masses. Otherwise, there is a naked conical singularity at the origin.”
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In the paper
“Near flat space, we simply recover the ordinary graviton,”
- Also stated in: § S. Deser · plain
“There is no contribution from the Gauss-Bonnet term because the background curvature vanishes.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § S. Deser · plain
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In the paper
“We also noted that cubic or higher corrections will only reinforce the unique physically acceptable nature of the asymptotically flat branch, since they imply unavoidable ghosts in the anti-de Sitter branch.”
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In the paper
“What we expect here is that the total energy will be positive for the asymptotically flat solutions, but not for the anti-de Sitter branch.”
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In the paper
“While the value of rH at which the horizon occurs differs from the Einstein one, the qualitative properties of the geometry are identical. The usual Kruskal completion procedure goes through just as in Schwarzschild, and Hawking radiation will still be present.”
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In the citing paper
“Lovelock theories admit static, maximally symmetric black hole solutions with metric of the form [14-17] Σ = -+ + ds f r dt f r dr r d ( ) ( ) , (5.1) 2 2 1 2 2 2”
From Causality and hyperbolicity of Lovelock theories (2014) · cites this paper as [14]
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Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
The negative-mass branch is called both de Sitter and anti-de Sitter
MinorThe paper consistently identifies the cosmological branch as anti-de Sitter, including when describing its negative-mass solution, but calls the same asymptotic solution Schwarzschild-de Sitter here.
“i.e., Schwarzschild with positive gravitational mass or Schwarzschild-de Sitter with negative gravitational mass”
“a positive-energy matter source Tµν leads to a negative-mass Schwarzschild-anti-de Sitter solution”
Citations
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Metadata
- Authors
- Boulware, D. G., & Deser, S.
- Affiliations
- University of Washington (US)
- ORCID
- Deser, S.
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1985-12-09
- Volume
- 55
- Issue
- 24
- Pages
- 2656-2660
- ISSN
- 0031-9007 (ISSN-L)
- OpenAlex
- W1985071512
- Semantic Scholar
- de20b24195f2aac5e46deee55b4a15343f186e8a
- Cited by
- 1,583
- References
- 13
- Citations by year
- 2026: 54 · 2025: 81 · 2024: 76 · 2023: 96 · 2022: 73 · 2021: 75 · 2020: 101 · 2019: 57 · 2018: 54 · 2017: 61 · 2016: 61 · 2015: 61 · 2014: 64 · 2013: 61 · 2012: 57
- FWCI
- 14.20
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- CaltechAUTHORS (California Institute of Technology) (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Keywords
- Gauss-Bonnet gravity, quadratic curvature corrections, Einstein-Hilbert action, superstring theory, anti-de Sitter space, cosmological instability, static spherically symmetric solutions, event horizon, naked singularities, higher-curvature corrections, cosmological constant
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