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Cited by 1,583Black Holes and Theoretical PhysicsCosmology and Gravitation TheoriesNoncommutative and Quantum Gravity Theories

String-Generated Gravity Models

Boulware, David G., Deser, S.·1985·Physical Review Letters
DOI 10.1103/physrevlett.55.2656
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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.

AI snapshot

Written from the paper’s full text.

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.55.2656

Cites

13 of 13 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 6 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § S. Deser

    A cubic Gauss-Bonnet term with γ2 > α2κ/4 yields no real roots and hence no cosmological branch.

    “(For example, a cubic Gauss-Bonnet term with γ2 > α²κ/4 yields no real roots and hence no cosmological branch.”
    Quote in the paperNot a substantial claim
  • Findingnone given
    § S. Deser

    The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the string-generated sign) or a naked singularity.

    “The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the stringgenerated sign) or a naked singularity.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § S. Deser

    The graviton excitations about the anti-de Sitter background are ghosts.

    “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.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § S. Deser

    The Einstein plus Gauss-Bonnet model has both flat and anti-de Sitter space as solutions.

    “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”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § S. Deser

    The plane-fronted parallel-ray metric remains a solution of the Einstein plus Gauss-Bonnet model.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [10]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims. 10
  • Classified as probably not a substantial claim. 1

  • May combine two claims.

    Findingreasoning
    § S. Deser

    Generic quadratic curvature invariants RμνR^μν and R2 lead to ghosts near flat space, whereas the Gauss-Bonnet combination does not.

    “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”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § S. Deser

    For negative α, the metric becomes imaginary at r^{n+1}=8||κMG, a real singularity, and it is naked when it occurs outside the event horizon.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § S. Deser

    With an explicit cosmological term, the effective cosmological constant has two values; the larger value is unstable and the lower one stable.

    “the larger value will be unstable, the lower one stable.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § S. Deser

    For higher-order Gauss-Bonnet terms, λ is determined by a polynomial equation f(λ)=λ(1+ακλ+γ2λ2κ+...)=0; to the extent there are real roots, the slopes at the intercepts alternate signs and the perturbations δG appear with both signs.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § S. Deser

    The Gauss-Bonnet term does not affect the propagator and does not distort plane-wave solutions.

    “underlines the fact that the Gauss-Bonnet term does not affect the propagator, and in particular does not distort plane-wave solutions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § S. Deser

    The Gauss-Bonnet action depends only on first time derivatives, so the number of degrees of freedom is the same as in Einstein gravity.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § S. Deser

    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.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § S. Deser

    Near flat space, the model recovers the ordinary graviton, with no contribution from the Gauss-Bonnet term because the background curvature vanishes.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § S. Deser

    Cubic or higher corrections reinforce the unique physically acceptable nature of the asymptotically flat branch, since they imply unavoidable ghosts in the anti-de Sitter branch.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § S. Deser

    The total energy is expected to be positive for the asymptotically flat solutions, but not for the anti-de Sitter branch.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § S. Deser

    For α > 0, the qualitative properties of the geometry are identical to Einstein gravity; the usual Kruskal completion goes through and Hawking radiation will still be present.

    “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.”
    Quote in the paperSubstantial claim
11 of 11

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Lovelock theories admit static, maximally symmetric black hole solutions with metric of the form (5.1).

    “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]

    In the section headed “S. Deser” (page unavailable), the paper derives static, spherically symmetric solutions in Schwarzschild gauge for its Einstein–Gauss–Bonnet model and describes solutions with event horizons. This supports a black-hole solution of the stated general form, but the copy does not establish the claim for Lovelock theories more broadly or for maximally symmetric horizons beyond spherical ones; the metric equation itself is not decoded in our copy. (LLM, conf=0.84)

    Experimental — accuracy not yet measured

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

    Minor

    The 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.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 6

  1. M. B. Green, 1984, Phys. Lett., 149B, 177
  2. B. Zwiebach, 1985, Symposium on Anomalies
  3. R. Nepomechie, Phys. Rev. D
  4. S. Deser, Proceedings of the Workshop on Supersymmetry and its Applications, Cambridge, 1985
  5. J. Ehlers, 1962, Gravitation
  6. D. G. Boulware, Quantum Field Theory and Quantum Statistics

Cited by

Sorted by citations

Related

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)
DOI
10.1103/physrevlett.55.2656
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
Licence
link.aps.org/licenses/aps-default-license (vor)
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 › Statistical and Nonlinear 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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