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Weighing the vacuum energy
Open accessAbstract
We discuss the weight of vacuum energy in various contexts. First, we compute the vacuum energy for flat spacetimes of the form ${\mathbb{T}}^{3}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$, where ${\mathbb{T}}^{3}$ stands for a general 3-torus. We discover a quite simple relationship between energy at radius $R$ and energy at radius $\frac{{l}_{s}^{2}}{R}$. Then we consider quantum gravity effects in the vacuum energy of a scalar field in ${\mathbb{M}}_{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ where ${\mathbb{M}}_{3}$ is a general curved spacetime, and the circle ${S}^{1}$ refers to a spacelike coordinate. We compute it for general relativity and generic transverse diffeomorphisms theories. In the particular case of unimodular gravity vacuum energy does not gravitate.
Overview
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TL;DR
The paper computes vacuum energy in compactified spacetimes and quantum gravity corrections, finding a nonlocal duality between torus radii and that gravitons contribute twice a scalar field’s vacuum energy. In unimodular gravity the vacuum energy does not gravitate, although the paper’s direct calculation fails in that singular limit.
Key findings
- For a massless scalar on \(T^3 \times R\), the Casimir energy per unit volume is \(E_0 = -\pi^2/(45L^3)\), which is negative definite.
- The effective potential at radius \(R\) and at radius \(l_s^2/R\) are related by a nonlocal integral transform, unlike string T-duality where the dual potentials are proportional.
- With dynamical gravity, the leading \(1/L^3\) vacuum energy splits as \(E_0 = -\pi^2/(15L^3) = -2\pi^2/(45L^3) - \pi^2/(45L^3)\); the graviton contribution is exactly twice that of a single scalar, and this remains true in the general backgrounds considered.
- In unimodular gravity, vacuum energy does not gravitate; the paper cites independent calculations [35,36] showing it does not couple to the gravitational field, though Bianchi identities require some coupling not proportional to \(E_0\).
Methods
- Uses background-field QFT and the one-loop effective action/potential, with heat-kernel and zeta-function regularization; Riemann theta functions handle toroidal compactification.
- Computes scalar vacuum energy on \(T^3 \times R\), then scalar-plus-graviton corrections on \(M_3 \times S^1\) by expanding around backgrounds and using De Donder gauge.
- Extends the computation to general transverse diffeomorphism theories via compensator and Weyl transformations, expanding scalar fields around constant backgrounds and using heat-kernel coefficients.
Limitations
- The paper states that its techniques fail for unimodular gravity because the Weyl transformation is singular and there is no Einstein frame; the TDiff results cannot be trusted in the unimodular limit, so the authors rely on independent calculations for that case.
- The paper states it only derives a general formula for \(T^3 \times R\) manifolds, not all flat \(F^3 \times R\) manifolds, though it conjectures the calculation could be extended.
- Evident from the text: the dynamical-gravity computation restricts the background to \(M_3 \times S^1\) and reduces gauge invariance to \(\mathrm{Diff}(M_3) \times SO(2)\); the TDiff computation uses constant scalar backgrounds; and the unimodular prediction is not experimentally verified.
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Claims the paper makes
0 Valid · 0 Not valid · 13 Not checked · 48 Own results and interpretations
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In the paper
“where we clearly see that the effect of integrating in the compact dimension is a shift in the effective mass of the scalar field.”
§ Weighing the vacuum energy > A. Review of known results
boostedstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“In the massless case, for n ¼ 4, we obtain E 0 ¼ W V 3 ¼ - π 2 45L 3 :ð12Þ”
§ Weighing the vacuum energy > A. Review of known results
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“so that the effective potential energy reads E 0 ¼ - Z dτ τ 1 ffiffiffiffiffiffiffi ffi 4πτ p e -τ P ab g ab à n a n b R a R b -M2 τ ¼ X ab g ab à n a n b R a R b þ M2 1=2 :ð34Þ”
§ Weighing the vacuum energy > II. VACUUM ENERGY INDUCED IN THREE-DIMENSIONAL TORI
plain- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption not shown here.
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In the paper
“The remarkable fact is that it is negative definite, not the most natural thing to be for an energy density.”
§ Weighing the vacuum energy > A. Review of known results
plainSingle case- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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In the paper
“let us focus on the relation between this potential for radius R and for radius R ¼ l 2 s R (where at this point l s is just a constant with dimensions of length), similar to the T-duality property in string theory”
§ Weighing the vacuum energy > A. Duality property
What the sources say
- Not matchedcites [21]
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plainstates an association- Asserted: passes — The authors put it forward as true.
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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
§ Weighing the vacuum energy > APPENDIX C: THETA FUNCTIONS
“for an exhaustive exposition, see the classical text of Mumford [20]”
§ Weighing the vacuum energy > References
“20. J. P. Serre, Cours d'Arithmetique (PUF, 1994), ISBN 978-2- 13-041835-1.”
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
§ Weighing the vacuum energy > IV. DYNAMICAL TRANSVERSE GRAVITY
“Taking the unimodular limit, this function has to be able to cancel the square root of the determinant of the metric so that the cosmological constant does not couple to gravity in the unimodular case, as it is well known.”
§ Weighing the vacuum energy > IV. DYNAMICAL TRANSVERSE GRAVITY
“f Λ ðxÞ ¼ x 2-n 2n :ð70Þ”
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
§ Weighing the vacuum energy > IV. DYNAMICAL TRANSVERSE GRAVITY
“we can rely on an independent calculation of the vacuum energy in unimodular gravity by two different groups [35,36].”
§ Weighing the vacuum energy > References
“35. L. S. Brown and G. J. Maclay, Vacuum stress between conducting plates: An image solution, Phys. Rev. 184, 1272 (1969).”
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
§ Weighing the vacuum energy > A. Review of known results
“Sð0Þ ¼ ð2πÞ n-1 ζð1 -nÞ:ð16Þ”
§ Weighing the vacuum energy > A. Review of known results
“SðMÞ ¼ ðMLÞ n-1 þ 2 X ∞ k¼0 ð n-1 2 Þ k k! ðMLÞ 2k ζð2k þ 1 -nÞ × ð2πÞ n-1-2k :ð15Þ”
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
§ Weighing the vacuum energy > II. VACUUM ENERGY INDUCED IN THREE-DIMENSIONAL TORI
“These manifols have been completely classified by Wolf in [17].”
§ Weighing the vacuum energy > References
“17. L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime (Cambridge University Press, Cambridge, England, 2009), https://doi.org/10.1017/CBO9780511813924.”
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Citations
Cites
The Influence of Retardation on the London-van der Waals Forces
Casimir, H. B. G., & Polder, D. (1948) · Physical Review · doi:10.1103/physrev.73.360 · Cited by 3,134
Quantum field theory in curved spacetime
DeWitt, B. S. (1975) · Physics Reports · doi:10.1016/0370-1573(75)90051-4 · Cited by 1,730
(2010) · Spaces of Constant Curvature · doi:10.1090/chel/372/11 · Cited by 1,724
New developments in the Casimir effect
Bordag, M., Mohideen, U., & Mostepanenko, V. M. (2001) · Physics Reports · doi:10.1016/s0370-1573(01)00015-1 · Cited by 1,629
Mumford, D. B. (2007) · Birkhäuser Boston eBooks · doi:10.1007/978-0-8176-4577-9 · Cited by 1,519
Invariance theory, the heat equation, and the Atiyah-Singer index theorem
Gilkey, P. (1995) · Cited by 1,469
Aspects of Quantum Field Theory in Curved Space-Time
Fulling, S. A. (1989) · doi:10.1017/cbo9781139172073 · Cited by 1,121
Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity
Parker, L., & Toms, D. (2009) · doi:10.1017/cbo9780511813924 · Cited by 1,108
Plunien, G. (1986) · Physics Reports · doi:10.1016/0370-1573(86)90020-7 · Cited by 797
Vacuum Stress between Conducting Plates: An Image Solution
Brown, L. S., & Maclay, G. J. (1969) · Physical Review · doi:10.1103/physrev.184.1272 · Cited by 434
Research works printed but not linked: 3
- W. Pauli, 1973, Selected Topics in Field Quantization: Volume 6 of Pauli Lectures on Physics
- J. P. Serre, 1994, Cours d’Arithmetique
- L. F. Abbott, 1982, Acta Phys. Pol. B, 13, 33
Cited by
Revisiting vacuum energy in compact spacetimes
Haridev, S. R., & Samantray, P. (2022) · Physics Letters B · doi:10.1016/j.physletb.2022.137489 · Cited by 6
Divergences of the scalar sector of quadratic gravity
Álvarez, E. A., & Anero, J. G. (2022) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ac7cb6
Self-citation
Metadata
- Authors
- Alvarez, E., Anero, J., & Santos-Garcia, R.
- Affiliations
- Instituto de Física Teórica (ES)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2021-04-19
- Volume
- 103
- Issue
- 8
- ISSN
- 2470-0010 (ISSN-L)
- 2470-0029
- OpenAlex
- W3101575585
- Semantic Scholar
- 2d9ac7c83f3da8527462809c779d1d1b754745b9
- Cited by
- 2
- References
- 43
- Citations by year
- 2022: 2
- FWCI
- 0.12
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Physical review. D/Physical review. D. (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- Biblos-e Archivo (Universidad Autónoma de Madrid) (submittedVersion, public-domain)
- Hispana (submittedVersion)
- arXiv (Cornell University)
- Funders
- Agencia Estatal de Investigación (PID2019-108892RB-I00/AEI/10.13039/501100011033)
- Horizon 2020 Framework Programme
- H2020 Marie Skłodowska-Curie Actions (674896, 690575)
- Ministerio de Ciencia, Innovación y Universidades (FPU16/01595)
- IFT Centro de Excelencia Severo Ochoa (SEV-2016-0597)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- 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
- Keywords
- vacuum energy, energy-momentum tensor, 3-torus, flat spacetime, scalar field, quantum gravity, unimodular gravity, diffeomorphism invariance, general relativity, circle compactification
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