Gravitons in a Casimir box

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.