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Casimir energy in multiply connected static hyperbolic universes
Open accessAbstract
We generalize a previously obtained result for the case of a few other static hyperbolic universes with manifolds of nontrivial topology as spatial sections.
Overview
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TL;DR
The paper numerically computes the Casimir energy density of a conformally coupled massive scalar field in static, multiply connected hyperbolic universes with compact spatial sections. It extends earlier work on the Weeks manifold to the Thurston, Best, and Seifert-Weber manifolds.
Key findings
- The Casimir energy density \(C=\langle 0|T_{\mu\nu}|0\rangle u^\mu u^\nu\) was evaluated for the Weeks, Thurston, Best, and Seifert-Weber manifolds.
- Reported manifold data include Weeks: \(V=0.942707\ldots a^3\), \(R_{\text{inradius}}=0.519162\ldots a\); Thurston: \(V=0.981369\ldots a^3\), \(R_{\text{inradius}}=0.535437\ldots a\); Best: \(V=4.686034\ldots a^3\), \(R_{\text{inradius}}=0.868298\ldots a\).
- For a scalar field with mass \(m=0.4\) and scale factor \(a=10\), the distribution of \(C\) on a sphere inside each fundamental region shows spontaneous generation of low multipolar components, and the effect becomes weaker for increasing-volume universes.
Methods
- The authors solve the Einstein equations for a static hyperbolic universe with a radiation fluid \(p=\rho/3\) and negative cosmological constant, obtaining \(a=\sqrt{3/(2|\Lambda|)}\), \(\rho=\Lambda/(8\pi G)\), with quantum back reaction disregarded.
- They use point-splitting in the covering space \(R\times H^3\), with an exact Feynman/Hadamard propagator, and compute the Casimir density from a sum over geodesics via a Poincaré series over the holonomy group \(\Gamma\), excluding the direct path as a renormalization of the cosmological constant.
- Numerical evaluation uses generators obtained with SNAPPEA; the infinite summation is truncated when the geodesic distance from the origin exceeds \(d=a\,5.29834\ldots\), and additional contributions were checked to be unimportant.
Limitations
- The paper states that the infinite summation is not physically valid because the universe is expanding; the mass term naturally introduces a cutoff.
- The paper states that zeta-function techniques do not yield an analytical result for hyperbolic compact manifolds because the Laplace-Beltrami spectrum can only be determined numerically, and quantum back reaction is disregarded.
- Evident from the text: results are numerical for specific choices \(m=0.4\), \(a=10\), selected manifolds, and a chosen sphere inside each fundamental region, with no observational comparison or broader error analysis beyond the truncation check.
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0 Valid · 2 Not valid · 11 Not checked · 16 Own results and interpretations
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In all these cases it can be seen that there is a spontaneous generation of low multipolar components.”
§ Casimir energy in multiply connected static hyperbolic universes > V. CONCLUSIONS
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In the paper
“which shows that the direct path (␥ϭ1) corresponds to a divergent cosmological term ⌳ ϱ .”
§ Casimir energy in multiply connected static hyperbolic universes > ͑18͒
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In the paper
“We explicitly evaluated the distribution of the vacuum energy density of a conformally coupled massive scalar field, for static universes with compact spatial sections of negative curvature and increasing volume: Weeks, Thurston, Best, and Seifert-Weber manifolds.”
§ Casimir energy in multiply connected static hyperbolic universes > V. CONCLUSIONS
plain- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption not shown here.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The presence of the mass term, however, naturally introduces a cutoff.”
§ Casimir energy in multiply connected static hyperbolic universes > ͑18͒
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As expected, the effect becomes weaker for increasing volume universes.”
§ Casimir energy in multiply connected static hyperbolic universes > V. CONCLUSIONS
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Casimir energy in multiply connected static hyperbolic universes > II. QUANTUM FIELD THEORY IN THE STATIC UNIVERSE RÃH 3
“where we interchangeably write (x 0 ,x 1 ,x 2 ,x 2 )↔(t,x,y,z).”
§ Casimir energy in multiply connected static hyperbolic universes > IV. CASIMIR DENSITY C IN A FEW UNIVERSES
“the origin (x 1 ,x 2 ,x 3 )ϭ(0,0,0)”
Citations
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(2009) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199238743.002.0004
Metadata
- Authors
- Müller, D., Fagundes, H. V., & Opher, R.
- Affiliations
- Universidade de Brasília (BR)
- ORCID
- Müller, Daniel
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2002-10-18
- Volume
- 66
- Issue
- 8
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W1996526526
- Cited by
- 46
- References
- 28
- Citations by year
- 2024: 1 · 2023: 1 · 2021: 1 · 2020: 1 · 2019: 1 · 2017: 1 · 2016: 2 · 2013: 1
- FWCI
- 0.41
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Fundação de Amparo à Pesquisa do Estado de São Paulo
- Conselho Nacional de Desenvolvimento Científico e Tecnológico
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Electrodynamics and Casimir Effect
- 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
- Casimir energy, nontrivial topology, quantum field theory in curved spacetime, vacuum energy
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