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Radion stabilization in compact hyperbolic extra dimensions
Open accessAbstract
We consider radion stabilization in hyperbolic brane-world scenarios. We demonstrate that in the context of Einstein gravity, matter fields which stabilize the extra dimensions must violate the null energy condition. This result is shown to hold even allowing for Friedmann-Robertson-Walker-like expansion on the brane. In particular, we explicitly demonstrate how one putative source of stabilizing matter fails to work, and how others violate the above condition. We speculate on a number of ways in which we may bypass this result, including the effect of Casimir energy in these spaces. A brief discussion of supersymmetry in these backgrounds is also given.
Overview
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TL;DR
In compact hyperbolic extra-dimensional brane-world models, stabilizing the radion while allowing an acceptable FRW cosmology requires bulk matter that violates the null energy condition. The authors demonstrate this in Einstein gravity, give a d-form example that fails the condition, and suggest quantum Casimir effects as a possible loophole.
Key findings
- Any bulk matter stabilizing the compact hyperbolic radion and yielding acceptable brane cosmology must violate the null energy condition; this holds even allowing FRW expansion on the brane.
- For a Minkowski brane and pure CHM, the required stress tensor gives \(T_{AB}N^A N^B = - (d-1)(d+2)M^{2+d}/(4R_h^2) < 0\) for a null vector along the hyperboloid.
- A specific stabilizing candidate—a cosmological constant plus a d-form flux—also violates the null energy condition, with \(T_{AB}N^A N^B = - (d-1)(d-2)\beta^2/8\, g_{ii}N_iN_i < 0\).
- Casimir energy may help: \(\rho_C \propto R_h^{-p}\) with \(p>0\), and if \(\rho_C<0\), the sign of energy density and pressure is “precisely right” to help stabilize \(H^3/\Gamma\); a numerical study of a minimally coupled scalar on the Thurston manifold found negative Casimir energy density.
- Supersymmetry in hyperbolic backgrounds is generally broken: for even \(d\), all supersymmetries are broken; for odd \(d\), survival depends on \(\Gamma\), and in the special case \(d=3\) no supersymmetries remain.
Methods
- The authors analyze Einstein gravity in \(4+d+n\) dimensions with a factorizable bulk \(\Sigma^{4+d+n}=F^{3+1}\times H^d/\Gamma\times S^n\), where \(H^d/\Gamma\) is a compact hyperbolic manifold and \(S^n\) is an n-sphere.
- They derive both an effective 3+1-dimensional theory with radion fields \(\phi,\psi\) and the full higher-dimensional Einstein equations, using a perfect-fluid ansatz for bulk matter and imposing the null energy condition.
- They illustrate the general result with a d-form flux plus cosmological constant example, then discuss Casimir energy and supersymmetry in compact hyperbolic backgrounds.
Limitations
- The paper states that understanding Casimir energy in the specific highly topologically complex manifolds useful for brane-world models is “extremely useful, but seems calculationally formidable and is beyond the scope of this paper.”
- The authors do not present generic results for \(\rho_C\) in specific CHMs; they only note a numerical study of a minimally coupled scalar on the Thurston manifold.
- The results apply to extra dimensions large enough that general relativity is a valid theory; the paper notes that the assumption of complete GR validity may break down, and that simple Casimir estimates are not a substitute for detailed calculation.
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0 Valid · 0 Not valid · 3 Not checked · 23 Own results and interpretations
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In the paper
“therefore when contracted twice with a null vector along one of the hyperbolic directions (i.e.N = N 0 e 0 + N i e i ), we obtain T AB N A N B = - (d -1)(d -2)β 2 8 g ii N i N i < 0 ,(37) clearly violating the null energy condition.”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > III. HYPERBOLIC BRANE WORLD COSMOLOGY
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In the paper
“We can see from this expression that the number of supersymmetries of H d is equal to the number of independent spinor components.”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > V. SUPERSYMMETRY IN HYPERBOLIC BACKGROUNDS
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In the paper
“In this study they discovered that the Casimir energy density can be negative, as needed for stabilizing the extra-dimensional manifold in our model.”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > IV. GOING BEYOND CLASSICAL MATTER : THE CASIMIR FORCE
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“As a result, these manifolds have no massless shape moduli.”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > FIG. 1: The Thurston Manifold.
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“Since nucleosynthesis occurs at a later time than this, it is clear that, even in the most optimistic case the model is not cosmologically viable.”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > III. HYPERBOLIC BRANE WORLD COSMOLOGY
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In the paper
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > I. INTRODUCTION
“ds 2 = R 2 h dχ 2 + sinh 2 (χ) dα 2 + sin 2 (β)dβ 2(3)”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > I. INTRODUCTION
“x 0 = R h cosh χ , x 1 = R h sinh χ cos α , x 2 = R h sinh χ sin α cos β , x 3 = R h sinh χ sin α sin β ,(2)”
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In the paper
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > III. HYPERBOLIC BRANE WORLD COSMOLOGY
“q = (d -1)(d -2)M 2+d 4R 2 h .(27)”
§ Radion Stabilization in Compact Hyperbolic Extra Dimensions > III. HYPERBOLIC BRANE WORLD COSMOLOGY
“q = -3 k a 2 + ȧ a 2 + ä a - 1 2 2n(n -1) R 2 s - (d -1)(d -2) R 2 h M 2+d+n”
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Metadata
- Authors
- Nasri, S., Silva, P. J., Starkman, G. D., & Trodden, M.
- Affiliations
- Syracuse University (US)
- Case Western Reserve University (US)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2002-08-28
- Volume
- 66
- Issue
- 4
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2024323291
- Semantic Scholar
- 3aa5a11ae79edf25cd2957b9a94bd02c98ba68e5
- Cited by
- 68
- References
- 31
- Citations by year
- 2026: 1 · 2021: 1 · 2020: 3 · 2019: 5 · 2018: 2 · 2017: 2 · 2016: 2 · 2015: 6 · 2014: 2 · 2013: 1 · 2012: 1
- FWCI
- 3.98
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0098747, 0094122)
- U.S. Department of Energy
- Syracuse University
- Aspen Center for Physics
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Keywords
- radion stabilization, compactified extra dimensions, null energy condition violation, general relativity, Casimir energy, supersymmetry
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