Circular Motion Unruh Effect

• Circular motion Unruh effect is the noninertial response of a detector in uniform rotation that reveals an effective, energy-dependent temperature rather than an exact thermal spectrum.
• The analysis uses Unruh–DeWitt detectors on circular trajectories to compare deviations from thermal behavior seen in linear acceleration, emphasizing orbital radius and angular frequency effects.
• Key studies highlight how factors like detector gap, spacetime dimensionality, finite-time switching, and interference contributions shape the observed response and challenge thermal duality.

Searching arXiv for recent and foundational papers on the circular motion Unruh effect. The circular motion Unruh effect is the noninertial response of a detector undergoing uniform rotation through the Minkowski vacuum. In contrast to the linear Unruh effect, where a uniformly linearly accelerated Unruh–DeWitt detector in $3+1$-dimensional Minkowski spacetime sees an exactly thermal spectrum at $T_U=a/(2\pi)$, circular motion generally yields only an effective, energy-dependent temperature defined through detailed balance. The response depends not only on the proper acceleration but also on the orbital radius, angular velocity, detector gap, spacetime dimension, and, in finite-time settings, the switching protocol. Recent work has sharpened this distinction by showing that exact thermal dualities familiar from uniform linear acceleration fail once circular motion is superposed, even when ambient temperature and ambient acceleration are matched by $T=a/(2\pi)$ (Biermann et al., 2020, Bunney et al., 2023).

1. Detector-theoretic formulation

The standard operational probe is the Unruh–DeWitt detector: a two-level system linearly coupled to a quantum field along a prescribed worldline. For detector energy gap $E$, coupling $\lambda$, switching $\chi(\tau)$, and trajectory $z(\tau)$, the leading-order transition probability is

$$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$

with $W$ the Wightman two-point function. For stationary trajectories and stationary states, the stationary-limit transition rate is

$$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$

and detailed balance defines an energy-dependent effective temperature through

$T_U=a/(2\pi)$0

This is exact thermality only when $T_U=a/(2\pi)$1 is constant, as for uniform linear acceleration in $T_U=a/(2\pi)$2 dimensions (Bunney et al., 2023).

For the linearly accelerated Rindler worldline,

$T_U=a/(2\pi)$3

the pullback of the Minkowski-vacuum Wightman function is

$T_U=a/(2\pi)$4

and the detector rate is exactly Planckian,

$T_U=a/(2\pi)$5

This benchmark underlies essentially all circular-motion comparisons (Bunney et al., 2023).

For circular motion, the same formalism applies, but the ratio $T_U=a/(2\pi)$6 depends on $T_U=a/(2\pi)$7. A useful shorthand is therefore the effective circular Unruh temperature $T_U=a/(2\pi)$8, defined operationally by detailed balance rather than by an exact KMS law (Biermann et al., 2020).

2. Circular worldlines and pullback correlators

Uniform circular motion in Minkowski spacetime is described by

$T_U=a/(2\pi)$9

with tangential speed $T=a/(2\pi)$0, Lorentz factor $T=a/(2\pi)$1, and proper acceleration

$T=a/(2\pi)$2

The invariant interval along the trajectory is

$T=a/(2\pi)$3

so the pullback Wightman function for a massless scalar in $T=a/(2\pi)$4 dimensions becomes

$T=a/(2\pi)$5

In $T=a/(2\pi)$6 dimensions,

$T=a/(2\pi)$7

The square-root branch structure in $T=a/(2\pi)$8 dimensions is a decisive source of low-energy differences from $T=a/(2\pi)$9 (Biermann et al., 2020).

Circular motion is stationary but not Rindler-like. The Wightman pullback contains trigonometric dependence from the orbital motion rather than the pure hyperbolic dependence characteristic of boosts. In more geometric terms, the circular worldline is an integral curve of a timelike Killing field, but it does not generate the KMS structure associated with the boost Hamiltonian. This is why a circular detector can be stationary without being exactly thermal (Parry et al., 2024).

A broader stationary-motion classification places circular motion alongside drifted Rindler motion and parator motion. Circular motion is generated by a time translation plus a rotation; drifted Rindler motion by a boost plus a spacelike translation; parator motion interpolates between them. This framework is useful because it isolates which properties of the response are tied to stationarity alone and which are special to boosts (Parry et al., 2024).

3. Effective temperatures, asymptotic regimes, and dimensional dependence

In $E$0 dimensions, the circular effective temperature remains of order $E$1 across broad parameter regimes, but it is not constant in the detector gap. In the ultrarelativistic limit $E$2, uniformly in $E$3,

$E$4

with limiting forms

$E$5

Hence $E$6 is Unruh-like but not Planckian: the effective temperature depends on frequency, yet stays within an $E$7 factor of $E$8 (Biermann et al., 2020).

In $E$9 dimensions, the high-energy asymptotics are similar, but the low-energy regime is qualitatively different. At fixed $\lambda$0 and $\lambda$1,

$\lambda$2

Later work traced this vanishing to the weak decay of the Wightman function along the circular trajectory and showed that, among stationary motions, it is unique to $\lambda$3 dimensions and therein to circular and parator motion (Biermann et al., 2020, Parry et al., 2024).

A different infrared limit, emphasized in a Planck-fit analysis, holds angular velocity fixed and varies the orbital radius. Near the center of rotation,

$\lambda$4

so the effective temperature grows quadratically with proper acceleration near $\lambda$5. The same analysis found that $\lambda$6 rises from zero, reaches a maximum at finite radius, and then falls to zero as $\lambda$7, even though $\lambda$8 there (Gim et al., 2018). This directly excludes the common but incorrect identification of “larger proper acceleration” with “larger detector temperature” for circular motion.

This suggests that proper acceleration alone is not a sufficient invariant descriptor of circular detector response. The orbital frequency, radius, detector gap, and spacetime dimension all enter essentially.

4. Non-KMS structure and the failure of thermal-acceleration dualities

The exact thermal character of the linear Unruh effect depends on KMS periodicity in imaginary proper time. Circular motion breaks this structure. In the comparison between two natural “circularized” generalizations of linear Unruh duality—an observer in uniform circular motion in a thermal bath at $\lambda$9, and an observer in “hypertor motion,” namely circular motion around a uniformly linearly accelerated trajectory in the Minkowski vacuum—the detector responses are generally different: $\chi(\tau)$0 The hypertor pullback contains both $\chi(\tau)$1 and $\chi(\tau)$2 terms, destroying simple imaginary-time periodicity, while the thermal-bath pullback is globally KMS in inertial time but not in detector proper time. The result is that localized measurements along the worldline distinguish ambient acceleration from ambient temperature in most regimes (Bunney et al., 2023).

There are controlled near-equivalence regimes. For small radius $\chi(\tau)$3, both scenarios reduce at leading order to the uniform linear acceleration response $\chi(\tau)$4, with differences only at $\chi(\tau)$5. The rate difference

$\chi(\tau)$6

is positive at order $\chi(\tau)$7, and the discrepancy is largest when $\chi(\tau)$8. Large-gap asymptotics can also coincide in special limits, notably small $\chi(\tau)$9 or small $z(\tau)$0, but these are approximate rather than exact dualities (Bunney et al., 2023).

Within the broader stationary-motion family, drifted Rindler motion clarifies what is special about circular trajectories. For fixed proper acceleration, the drifted Rindler large-gap temperature remains bounded, whereas the circular large-gap temperature can be arbitrarily large. In $z(\tau)$1 dimensions, drifted Rindler motion does not share the generic circular small-gap collapse unless the drift speed approaches unity; the vanishing small-gap temperature is instead tied to the weak large-$z(\tau)$2 decay of the circular and parator pullback Wightman functions (Parry et al., 2024).

5. Finite-time switching, small-gap recovery, and controlled generalizations

Finite-time switching is not merely a technical detail. In $z(\tau)$3 dimensions, if the long-time limit is taken with conventional sign-definite switching, the small-gap effective temperature vanishes. Recent work reformulated the long-time/small-gap problem using asymptotically scaled switching families (ASSFs) and isolated a precise Small Frequency Suppression condition,

$z(\tau)$4

as the criterion that restores a nonzero limiting temperature in a simultaneous long-time/small-gap limit (Parry et al., 27 Aug 2025, Parry et al., 8 Jul 2026).

The recovered temperature is

$z(\tau)$5

with

$z(\tau)$6

Crucially, under technical boundedness and localisation assumptions, sign changes in the detector-field coupling are necessary: single-sign ASSFs cannot satisfy Small Frequency Suppression, so they cannot yield a nonvanishing limiting temperature (Parry et al., 8 Jul 2026). This gives a rigorous operational meaning to the idea that low-frequency spectral weight must be cancelled, not merely attenuated.

A separate recent extension places a detector on a coherent superposition of circular trajectories. In this setting, off-diagonal Wightman terms act as genuine interference contributions. With broad Gaussian switching $z(\tau)$7, vertically stacked concentric circles produce only minor deviations from single-trajectory effective thermality, whereas a superposition of a static central point and a surrounding circular trajectory produces a significant reduction in effective temperature and a stronger energy-gap dependence (Cey et al., 3 Jul 2026). This suggests that quantum control over trajectories can tune, rather than simply reveal, circular Unruh signatures.

6. Experimental and analogue directions

Analogue platforms exploit the replacement $z(\tau)$8 or $z(\tau)$9, dramatically lowering the effective “speed of light.” In a quasi-$$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$0D Bose–Einstein condensate, a localized laser probe acts as an effective Unruh–DeWitt detector for phonons. For a circular trajectory with $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$1 and $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$2, the predicted analogue circular Unruh temperature is $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$3, and for $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$4 experimental realizations with $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$5, the integrated excess signal-to-noise ratio is $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$6 (Gooding et al., 2020).

A related proposal uses third-sound modes in superfluid helium-4 thin films. There the local laser probe follows a circular path and the acceleration signal is isolated by subtracting the inertial Doppler contribution. For $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$7, $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$8, ambient temperature $$P(E)=\lambda^2\int d\tau\,d\tau' \, e^{-iE(\tau-\tau')}\,\chi(\tau)\,\chi(\tau')\,W\big(z(\tau),z(\tau')\big),$$9, and laser power $W$0, the effective circular Unruh temperature is $W$1, and the authors find the corresponding acceleration-dependent signal within experimental reach (Bunney et al., 2023).

Cavity QED proposals target dispersive rather than absorptive signatures. For a two-level atom in circular motion inside a high-$W$2 cavity, the Lamb shift acquires rotation-induced sidebands and can be enhanced, quenched, or completely screened by detuning. In one explicit parameter set, a rotation-induced shift of order $W$3 appears already at $W$4 (Peng et al., 19 Jun 2026). A complementary cylindrical-cavity proposal uses coherent driving and Dicke superradiance to enhance counter-rotating processes: with $W$5, $W$6, $W$7, and $W$8, the estimated circular acceleration is $W$9 and the peak driven counter-rotating rate is $$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$0 (Zheng et al., 2024).

Electromagnetic detectors introduce further structure. One recent analysis shows that when the rotational angular velocity exceeds the atomic transition frequency, $$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$1, the excitation rate can become comparable to the emission rate even for very small centripetal acceleration, because the leading rates remain finite as $$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$2. In that regime the effective temperature satisfies $$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$3 for $$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$4 (Zhou et al., 2024). Another multilevel-atom study finds that magnetic dipole transitions dominate electric dipole transitions, both in free space and in cavity schemes, and proposes suppressing spontaneous emission through the atom’s multilevel structure to enhance circular-Unruh detectability (Janson et al., 30 Jun 2026). An entanglement-based route has also been proposed: two atoms in circular motion can relax to an initial-state-independent asymptotically entangled state, furnishing a nonthermal signature of circular acceleration rather than a direct thermometric one (Zhou et al., 2023).

Taken together, these developments indicate that the circular motion Unruh effect is best understood not as a single temperature law but as a family of detector responses governed by stationary but non-KMS vacuum sampling. Exact thermality survives only for boost-generated motion. Circular trajectories instead exhibit effective temperatures, spectral sidebands, switching-sensitive low-gap behavior in $$F(E)=\int_{-\infty}^{\infty} d\tau\, e^{-iE\tau}\,W(\tau),$$5 dimensions, and experimentally relevant signatures in transition rates, Lamb shifts, interference patterns, and entanglement.