STM-D-0536Paper2012Published and peer-reviewed

Contrasting Classical and Quantum Vacuum States in Non-Inertial Frames

Timothy H. Boyer

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In one page

Timothy Boyer of City College of New York sets two accounts of empty space side by side and finds they agree everywhere except in one place that matters. In stochastic electrodynamics — ordinary classical physics with a real sea of random zero-point radiation added — the vacuum has a single unique description: its correlations depend only on the geodesic distance between points in spacetime, so its spectrum in any frame follows by the same tensor transformation you would use for any other field. Quantum field theory instead quantises mode by mode inside a mirror-walled box, taking no notice of whether that box’s time coordinate is a geodesic one, and so every accelerating frame ends up with a vacuum of its own. Boyer works a two-dimensional scalar field through in full, and the classical answer turns out not to care whether the box was accelerated gradually or suddenly: the radiation inside keeps the same spectrum either way, up to small Casimir corrections at the walls. His closing suggestion is that acceleration may not heat anything at all.

Why it matters hereChapter 2 asks what the vacuum actually is, and Boyer gives the strongest realist answer on offer: one physically present sea of radiation whose description in any frame follows from the geometry of spacetime alone, with no virtual anything. Chapter 3 is where it bites, because what an accelerating body feels in the zero-point field is precisely the question inertia turns on — and Boyer’s classical analysis says the promised thermal bath may not be there to find.

What it claims

01In the classical theory the vacuum is unique. The spectrum of random classical zero-point radiation follows from the symmetry properties of relativistic spacetime, so that in empty space its two-point correlation functions depend only on the geodesic separation between the spacetime points and its coordinate derivatives. In an inertial frame that spectrum is Lorentz invariant, scale invariant and conformal invariant, and its form in any non-inertial frame follows by ordinary tensor transformation. The radiation is physically present — there is no notion of virtual photons coming into and out of existence.Sect. I, Introduction; Sect. III D, Classical Zero-Point Radiation in the Rindler-Frame Box
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02Quantum field theory reaches a different answer because its procedure ignores the metric. Canonical quantisation fixes the same amplitude for the normal modes of any mirror-walled box, whether or not the box’s time coordinate is geodesic, so a box at rest in an inertial frame and a box at rest in an accelerating Rindler frame acquire very different vacuum states. Boyer notes that this non-uniqueness is not new — Fulling called attention to the non-uniqueness of canonical quantisation in Riemannian spacetime more than thirty years earlier.Sect. III E, Contrasting Classical-Quantum Viewpoints in a Rindler Frame
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03Boyer computes the classical spectrum in the accelerating frame explicitly. In the geodesic time coordinate of an inertial frame it goes as the inverse wave number, giving half a Planck quantum per normal mode; in the non-geodesic Rindler time coordinate it acquires a hyperbolic cotangent factor. Move the walls of the Rindler box out to the limits of the Rindler wedge and the random radiation inside is exactly the radiation of empty inertial space — whereas the quantum Rindler vacuum stays distinct from the Minkowski vacuum even in that same large-box limit.Sect. III D, Eq. (48); Sect. III E, the spectra f0 and F0
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04Whether the box is accelerated gradually or suddenly makes no physical difference in the classical theory: the radiation in the interior retains the same correlation function either way, and the only difference between the radiation inside the box and the radiation of empty inertial space outside it is the Casimir aspect that comes from the discreteness of a finite box’s normal modes. That removes, classically, the whole scenario in which a slowly accelerated box ends in a Rindler vacuum while a suddenly accelerated one fills with Rindler quanta.Abstract; Sect. III E, closing paragraphs
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05The hyperbolic cotangent appearing in the Rindler time spectrum is the same function that describes thermal radiation in an inertial frame, and that is the whole source of the thermal reading of acceleration. Boyer’s point is that the same spectra can be used to suggest either a temperature equal to Planck’s constant times the acceleration divided by two pi times the speed of light and Boltzmann’s constant, or zero temperature, depending on one’s point of view — and that the ambiguity exists precisely because Rindler time is not a geodesic coordinate.Sect. III E, the discussion following Eqs. (58) and (59)
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06The detectors matter as much as the vacuum does. A point harmonic oscillator or point dipole rotator accelerated through classical zero-point radiation does take on the average energy it would have in a thermal bath — but a point system responds only to correlations in time at a fixed place, so by construction it cannot separate acceleration from temperature. Boyer argues that relativistic systems carrying internal potential energy must be spatially extended: a classical hydrogen atom readjusts to an acceleration field through the droop of its Coulomb field lines, and responds to spatial as well as temporal correlations. What to watch: his conclusion is that it seems possible all the claims that acceleration through the vacuum provides a thermal bath are in error, and the observation that would settle it is a measurement made on a spatially extended relativistic system rather than on a point-detector model.Sect. III F, Detectors Accelerating through Classical Zero-Point Radiation; Sect. IV, Closing Summary
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Read it · abstract

Abstract

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. Based upon the classical analysis, it is suggested that the claimed heating effects of acceleration through the vacuum may not exist in nature.

(Abstract only — see the rights note above. The manuscript is free to read at arXiv:1204.6036, and the published version is Foundations of Physics 43, 923–947 (2013). Boyer’s survey of stochastic electrodynamics as the closest classical approximation to quantum theory is at /library/stm-1011f1af4f, and his detailed-balance result for classical zero-point radiation under scattering is at /library/stm-5909a0938f.)

The way in

https://arxiv.org/abs/1204.6036Posted to arXiv on 26 April 2012 under the arXiv.org perpetual non-exclusive distribution licence version 1.0 — checked on the arXiv abstract page on 2026-09-08, where the rights link resolves to arxiv.org/licenses/nonexclusive-distrib/1.0/. That is not a Creative Commons licence and grants no redistribution, so this page carries the summary, the claims and the author’s own abstract and sends the reader to the source. The same work was published as Foundations of Physics 43, 923–947 (2013), doi 10.1007/s10701-013-9726-4 — a title-and-author match confirmed against Crossref on 2026-09-08, and not recorded in this library’s registry entry. The claims below are read against the 29-page arXiv manuscript and cite its numbered sections and equations. Boyer wrote from the Department of Physics, City College of the City University of New York.

How to cite it

Timothy H. Boyer (2012) Contrasting Classical and Quantum Vacuum States in Non-Inertial Frames. arXiv:1204.6036

Where it sits in the curriculum

What the vacuum is Inertia and gravity from the vacuum