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Gravitons in a Casimir box
Open accessAbstract
A bstract: The partition function of gravitons with Casimir-type boundary conditions is worked out. The simplest box that allows one to achieve full analytical control consists of a slab geometry with two infinite parallel planes separated by a distance d . In this setting, linearized gravity, like electromagnetism, is equivalent to two free massless scalar fields, one with Dirichlet and one with Neumann boundary conditions, which in turn may be combined into a single massless scalar with periodic boundary conditions on an interval of length 2 d . When turning on a chemical potential for suitably adapted spin angular momentum, the partition function is modular covariant and expressed in terms of an Eisenstein series. It coincides with that for photons. At high temperature, the result provides in closed form all sub-leading finite-size corrections to the standard (gravitational) black body result. More interesting is the low-temperature/small distance expansion where the leading contribution to the partition function is linear in inverse temperature and given in terms of the Casimir energy of the system, whereas the leading contribution to the entropy is proportional to the area and originates from gravitons propagating parallel to the plates.
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TL;DR
The paper computes the partition function of gravitons with Casimir-type boundary conditions in a slab geometry and shows it coincides with that of photons. It derives modular-covariant exact results, with high-temperature black-body corrections and a low-temperature entropy proportional to the plate area.
Key findings
- In a slab with two infinite parallel planes separated by distance \(d\), linearized gravity with Casimir-type boundary conditions is equivalent to two free massless scalar fields (one Dirichlet, one Neumann), or to a single massless scalar with periodic boundary conditions on an interval of length \(2d\); the graviton partition function coincides with the photon one.
- With a chemical potential for suitably adapted spin angular momentum, the partition function is modular covariant and expressed through the real analytic Eisenstein series: \(\ln Z(\tau,\bar\tau)=\frac{1}{2\pi^2}\frac{L_1L_2}{L_3^2}\frac{f_2(\tau,\bar\tau)}{\operatorname{Im}\tau}\), with \(\tau=\alpha+i\beta/L_3\), \(L_3=2d\).
- At high temperature, the result gives all sub-leading finite-size corrections to the black-body result; for \(\alpha=0\), \(S(\beta,0)=\frac{8\xi(4)L_1L_2 d}{\beta^3}+\cdots\), where \(\xi(4)=\pi^2/90\).
- At low temperature/small distance, the leading partition-function contribution is linear in inverse temperature and given by the Casimir energy \(E_0(0)=-\xi(4)L_1L_2/L_3^3\); the leading entropy is \(S(\beta,\alpha)=\frac{3\xi(3)L_1L_2}{\beta^2}+\cdots\), proportional to the area and originating from gravitons propagating parallel to the plates.
Methods
- Hamiltonian/Pauli-Fierz analysis of free massless spin-2 fields in a slab geometry, imposing Casimir-type boundary conditions: \(h_{ab}=0=\pi_{ab}\), \(h_{33}=0=\pi_{33}\), and \(\partial_3 h_{a3}=0=\partial_3\pi_{a3}\) at the plates.
- Mode expansions and reduced phase-space decomposition isolate the particle sector, modes propagating parallel to the plates (\(n_3=0\)), and modes with normal propagation (\(n_3>0\)); these are reorganized into Dirichlet/Neumann scalar fields and then a single periodic scalar on length \(2d\).
- Partition function computed via functional methods (Eisenstein series), canonical quantum-statistical expansion (modified Bessel functions), and method of images, with modular parameter \(\tau=\alpha+i\beta/L_3\).
Limitations
- The paper states it does not address whether gravitons can be confined to a box to achieve thermal equilibrium, does not speculate on the physical meaning of perfect conductors for gravitons, and does not discuss full nonlinear general relativity, black-hole physics, or cosmological models.
- Stated open questions include ADM surface charges for this setup, extension to more general boundary geometries and curved backgrounds, and the need for interactions/Newton’s constant to distinguish linearized gravity from electromagnetism.
- Evident from the text: the analysis is restricted to free linearized gravity in an idealized slab geometry, with no interactions included.
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“for large |z|, all other terms are exponentially suppressed.”
§ Gravitons in a Casimir box > JHEP02(2021)216
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“For the second of these transformations, we get ln Z(β , α ) = α 2 + β 2 L 2 3 ln Z(β, α).”
§ Gravitons in a Casimir box > JHEP02(2021)216
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“More generally, the leading contributions are given by the first two terms, while the others are exponentially suppressed.”
§ Gravitons in a Casimir box > JHEP02(2021)216
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“For α = 0, this gives the standard black body result (B.17) when taking into account that the volume of the system is V = L 1 L 2 d.”
§ Gravitons in a Casimir box > JHEP02(2021)216
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“In the expression of the partition function in terms of the real analytic Eisenstein (5.5), this term comes from the modes with vanishing Matsubara frequencies n 4 = 0.”
§ Gravitons in a Casimir box > JHEP02(2021)216
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Metadata
- Authors
- Alessio, F., Barnich, G., & Bronte, M.
- Affiliations
- Université Libre de Bruxelles (BE)
- Venue
- Journal of High Energy Physics
- Published
- 2021-02
- Volume
- 2021
- Issue
- 2
- ISSN
- 1029-8479 (ISSN-L)
- OpenAlex
- W3109090483
- Semantic Scholar
- 0414da306cefd338315f69da14b66b4c5d2e9c8b
- Cited by
- 8
- References
- 48
- Citations by year
- 2026: 1 · 2023: 4 · 2021: 3
- FWCI
- 0.73
- Type
- journal article
- Language
- en
- Update notices
- erratum (2021-03-24)
- Open access
- diamond
- doi.org
- Licence
- cc-by
- creativecommons.org/licenses/by/4.0 (tdm)
- creativecommons.org/licenses/by/4.0 (vor, 24 d)
- Locations
- Journal of High Energy Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Fonds De La Recherche Scientifique - FNRS (FRFC PDRT.1025.14, IISN 4.4503.15, IISN 4.4503, T.1025.14, PDR T.1025.14, 4.4503, 4.4503.15)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Electrodynamics and Casimir Effect
- 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
- graviton, Casimir effect, partition function, linearized gravity, slab geometry, periodic boundary conditions, modular covariance, Eisenstein series, blackbody radiation, finite-size corrections, Casimir energy, gravitational entropy, parallel plates
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