Candelas and Deutsch (1977) computed the renormalized vacuum stress of the massless scalar and electromagnetic fields in the Minkowski vacuum outside an infinite plane conductor that accelerates uniformly along its normal. Far from the plane the stress is local and thermal-like (black-body but not Planckian, with negative energy density scaling as −11/(240π2ξ4)-11/(240\pi^{2}\xi^{4}) in the Rindler coordinate ξ\xi), while near the surface it diverges as the inverse cube of the distance — a pathology the authors attribute to the acausality of ideal perfect-conductor boundary conditions, arguing only heuristically (via a skin-depth cutoff δ\delta and the dimensional estimate ℏc−3a−2δ−2\hbar c^{-3}a^{-2}\delta^{-2}) that finite conductivity regularizes it, and that the induced pressure is likely unmeasurable. The open problem is the controlled computation of the electromagnetic vacuum stress for an explicit frequency-dependent (dispersive, possibly absorptive) material response, the realistic magnitudes and observability following from it, and the conditions under which a finite renormalized vacuum energy per unit transverse area arises for the accelerated geometry; no controlled dispersive calculation of this stress is known. The physical origin of the thermal-like spectrum is not part of the open core: it is accounted for by the standard theory of the Fulling–Davies–Unruh effect.

Background

In flat spacetime, an observer with constant proper acceleration can be described in Rindler coordinates t=ξsinh⁡τt=\xi\sinh\tau, x=ξcosh⁡τx=\xi\cosh\tau (with y,zy,z unchanged), which cover the right Rindler wedge x>∣t∣x>|t|; the hyperbolae ξ=const\xi=\mathrm{const} are worldlines of proper acceleration 1/ξ1/\xi. The Minkowski vacuum — the ground state of the quantum field in flat spacetime — restricted to a Rindler wedge, is seen by such accelerated observers as a thermal state at the local Unruh temperature (2πξ)−1(2\pi\xi)^{-1} (the Fulling–Davies–Unruh effect), so acceleration and horizons endow vacuum observables with thermal-looking spectra.

Candelas and Deutsch (1977) computed the renormalized vacuum expectation value ⟨0∣T^μν∣0⟩\langle 0|\hat T_{\mu\nu}|0\rangle of the stress tensor (renormalized meaning that the divergent contribution already present in empty space, without the conductor, is subtracted) for the massless scalar field (Dirichlet and Neumann boundary conditions, in which the field, respectively its derivative normal to the surface, vanishes at the surface) and for the electromagnetic field in the Minkowski vacuum outside an infinite plane conductor accelerating uniformly, with proper acceleration 1/a1/a, along its normal — a configuration equivalent to a plane 'supporting the ether' against the uniform gravitational field of an infinite flat earth. They built Feynman Green functions — the two-point propagation amplitudes of the quantum field — in Rindler coordinates from modes involving modified Bessel functions KiνK_{\mathrm{i}\nu} of imaginary order (the order ν\nu being the Rindler frequency), subtracted the free Minkowski singularity, and evaluated the stress. Far from the plane the renormalized stress is 'local', proportional to ξ−4\xi^{-4} and independent of the plane's acceleration, and corresponds to the absence from the vacuum of black-body radiation at the local temperature (2πξ)−1(2\pi\xi)^{-1}; for the electromagnetic field the far-field energy density is negative, with leading form proportional to −11/(240π2ξ4)-11/(240\pi^{2}\xi^{4}), and the Rindler-frequency spectral weight of the diagonal components is (ν3+ν)/(e2πν−1)(\nu^{3}+\nu)/(e^{2\pi\nu}-1), i.e. black-body-like but not exactly Planckian. Approaching the plane, the renormalized stress components diverge as inverse powers of ϵ=ξ−a\epsilon=\xi-a (the electromagnetic energy density like [30π2aϵ3]−1[30\pi^{2}a\epsilon^{3}]^{-1}). The authors attribute this surface divergence to the acausal character of ideal 'perfect conductor' boundary conditions, which require instantaneous rearrangement of surface charges, and argue that the finite conductivity of any real material — characterized by a frequency-dependent response and a skin depth δ\delta, the penetration depth of an electromagnetic field of a given frequency into the conductor — cuts off the high-frequency contributions and renders the stress finite. They support this only by dimensional estimates (restricting the Rindler-frequency and transverse-wave-number integrals to ν≲a/δ\nu\lesssim a/\delta and k≲1/δk\lesssim 1/\delta, giving surface pressures of order ℏc−3a−2δ−2\hbar c^{-3}a^{-2}\delta^{-2}) and conclude heuristically that the effect is likely too small to measure; no explicit dispersive or absorptive material model is computed.

Two further pieces of context frame the open question. First, for static (inertial) geometries, Boyer's classic calculation found a finite total Casimir energy (the shift of the vacuum energy of a quantum field induced by boundary conditions) for a conducting spherical shell even though local stress components diverge near the surface; Deutsch and Candelas (1979) later showed that these near-boundary divergences of the renormalized stress are local and universal, essentially independent of the shape of the boundary, which turns the question of finite integrated quantities into a question about cancellations in specific geometries rather than about exceptional local behavior. Second, the modern theory of the Fulling–Davies–Unruh effect and of thermal states on spacetimes with horizons explains why Rindler-frame observables in the Minkowski vacuum are thermal; the thermal character of the 1977 far-field result is a manifestation of this structure, and its non-Planckian correction reflects the specific stress-tensor observable rather than an unexplained distribution.

Problem Statement

Compute the renormalized electromagnetic vacuum stress — the renormalized expectation value ⟨0∣T^μν∣0⟩\langle 0|\hat T_{\mu\nu}|0\rangle of the electromagnetic stress tensor, written Tμν(ξ)T_{\mu\nu}(\xi) for brevity, as a function of the Rindler coordinate — in the Minkowski vacuum outside an infinite plane conductor that accelerates uniformly along its normal (proper acceleration 1/a1/a; points outside the plane labeled by the Rindler coordinate ξ≥a\xi\ge a), using an explicit realistic material model — a frequency-dependent, dispersive, and possibly absorptive conductivity or dielectric response with a finite skin depth δ\delta — in place of ideal perfect-conductor boundary conditions. The calculation should: (i) establish explicitly how the inverse-cube near-surface divergence of the perfect-conductor result is regularized by the material response, replacing the heuristic cutoffs ν∼a/δ\nu\sim a/\delta and k∼1/δk\sim1/\delta on the Rindler frequency ν\nu and the transverse wave number kk, and the dimensional estimate ℏc−3a−2δ−2\hbar c^{-3}a^{-2}\delta^{-2} with a controlled, material-specific regularized Tμν(ξ)T_{\mu\nu}(\xi); (ii) determine from that regularized stress the actual magnitudes of the surface pressure and energy density for a representative realistic conductor model at a stated acceleration, and assess whether any component is observable in principle; and (iii) establish whether, and under what conditions on the material response and geometry, a finite renormalized vacuum energy per unit transverse area arises for the accelerated-plane geometry — the free-Minkowski (empty-space) contribution subtracted and the (unbounded) transverse directions taken per unit area, so that the observable is well defined — in analogy with the finite total Casimir energy of Boyer's static conducting spherical shell, whose bounded geometry makes its mode-summed total well defined, despite universal local surface divergences. The three parts (i)–(iii) are independent, and resolving one does not resolve the others: (i) supplies the controlled material-specific regularized stress, (ii) the resulting magnitudes and observability verdict, and (iii) the finiteness/conditions criterion. For (ii) a single representative realistic dispersive (and possibly absorptive) conductor model with a finite skin depth, evaluated at a stated acceleration, is sufficient; additional material models are optional.

The physical origin of the thermal-like (black-body, non-Planckian) far-field spectrum is accounted for by the standard theory of the Fulling-Davies-Unruh effect (thermal character of the Minkowski vacuum with respect to Rindler time) and is not part of the open core; the open core is the material-specific regularization, the observability assessment, and the accelerated-geometry finiteness stated above.

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