Circular Unruh Effect Overview

Updated 23 October 2025

Circular Unruh Effect is the phenomenon where a detector in circular motion perceives a thermal-like bath whose effective temperature depends on the energy gap and geometric parameters of its trajectory.
The theory employs a detailed balance condition via the Fourier transform of the Wightman function, showing that in 3+1 dimensions the high-energy response can exceed the linear Unruh temperature by up to approximately 1.8 times.
Experimental proposals leverage storage rings, optical cavities, and ultracold atom systems, using engineered switching functions and collective effects to enhance detector responses.

The Circular Unruh Effect describes the phenomenon in which an observer (or detector) undergoing uniform circular motion in the Minkowski vacuum perceives the vacuum state as being populated with excitations resembling a thermal bath, analogous but not identical to the standard Unruh effect for linear acceleration. Unlike linear acceleration—where the effective temperature is proportional to the proper acceleration—the spectrum and temperature in circular motion depend in a more intricate, energy-dependent way on the trajectory’s geometric parameters, such as the orbital radius and angular velocity. The effect is operationally characterized via the detailed balance between excitation and de-excitation probabilities of a quantum system (e.g., an Unruh-DeWitt detector) traveling on a circular path (Rad et al., 2011, Good et al., 2020, Biermann et al., 2020, Gim et al., 2018, Parry et al., 29 Sep 2024).

1. Fundamental Principles and Theoretical Formulation

The standard Unruh effect states that an observer with proper acceleration $a$ through the Minkowski vacuum measures a temperature

$k_B T_\text{U} = \frac{\hbar a}{2\pi c}$

and experiences the quantum vacuum as a thermal bath at this temperature. In circular motion, the observer’s trajectory is not merely a Lorentz boost but combines translation and rotation, modifying the field correlations they experience.

A detector in uniform circular motion with angular velocity $\Omega$ and radius $R$ has 4-velocity components with constant speed $v=R\Omega$ and proper acceleration

$a_\text{cir} = \gamma^2 R\Omega^2,\qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

(Gim et al., 2018). However, the temperature measured by the detector, denoted $T_\text{circ}$ , is not simply $a_\text{cir}/(2\pi)$ . The effective temperature operationally perceived by the detector must be extracted via the detailed balance condition: $\frac{1}{T_\text{circ}(E)} = \frac{1}{E} \ln\left[\frac{F(-E)}{F(E)}\right]$ where $F(E)$ is the excitation rate for energy gap $E$ (Good et al., 2020, Biermann et al., 2020, Parry et al., 29 Sep 2024). Importantly, $T_\text{circ}$ depends not only on the proper acceleration but also on the transition energy $E$ , the orbital velocity, and geometric details.

In circular motion, for small radii (low speeds), $T_\text{circ} \propto (a_\text{cir})^2$ , in stark contrast to the linear scaling of the standard Unruh temperature (Gim et al., 2018). For ultrarelativistic speeds, asymptotic analysis shows that the ratio $T_\text{circ}/T_\text{lin}$ can exceed unity, with values depending on spacetime dimension and the energy gap (Biermann et al., 2020, Parry et al., 29 Sep 2024).

2. Mathematical Structure and Energy Dependence

The excitation/de-excitation rates for a detector along a stationary (including circular) trajectory are most generally evaluated as

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

where $W(\tau)$ is the pullback of the Wightman function along the worldline (Good et al., 2020, Parry et al., 29 Sep 2024). For the circular trajectory: $W_\text{circ}(s) = \frac{1}{4\pi^2}\frac{1}{-\gamma^2 s^2 + 2R^2(1-\cos\gamma\Omega s)}$ (Good et al., 2020).

The resulting $T_\text{circ}(E)$ is not constant:

In $3+1$ dimensions, $T_\text{circ}$ remains finite as $E \rightarrow 0$ and can be greater than $T_\text{lin}$ , especially at large $E$ (Biermann et al., 2020).
In $2+1$ dimensions, $T_\text{circ}(E) \sim |E|/\log(\cdots)$ , so it vanishes linearly as $E \rightarrow 0$ due to the slow (power-law) decay of vacuum correlations along the trajectory; this feature is unique to circular and parator motion in $2+1$ dimensions (Parry et al., 29 Sep 2024, Parry et al., 27 Aug 2025).

A key analytic result: For large $E$ , $T_\text{circ}/T_\text{lin} \rightarrow \pi/\sqrt{3}\approx 1.8$ in $3+1$ D whereas for small $E$ in $2+1$ D, $T_\text{circ}\rightarrow 0$ (Biermann et al., 2020). The frequency dependence emerges from the nontrivial analytic structure of the Wightman function’s Fourier transform, with the precise temperature determined by the pole closest to the real axis.

3. Physical Mechanisms and Interpretive Issues

The Circular Unruh Effect is not strictly “thermal” in the sense of a stationary Gibbs state at a unique temperature. Instead, it is characterized by a generalized detailed balance spectrum, with the temperature being frequency-dependent and influenced by geometric invariants: curvature (proper acceleration), torsion, and—in extended trajectories—by hypertorsion (Good et al., 2020). The circular effect arises physically because the detector's periodic trajectory leads to vacuum correlations that differ from those experienced by linearly accelerated detectors; for example, the Wightman function along the worldline decays more slowly in circular motion (especially in low dimensions), profoundly suppressing the small gap response (Parry et al., 29 Sep 2024, Parry et al., 27 Aug 2025).

Key distinctions from linear acceleration:

The absence of a true event horizon for circular motion means the vacuum cannot be globally traced to a thermal state.
Memory effects and non-Markovian dynamics play a pronounced role in the actual detector response (Doukas et al., 2013, Liu et al., 2016).
At high energies or relativistic speeds, the effective temperature for circular motion can significantly exceed the linear Unruh temperature, while in some regimes with small detector energy gap, especially in $2+1$ D, the response is anomalously suppressed.

4. Experimental Proposals and Observational Strategies

Direct laboratory detection of the standard Unruh effect is challenging due to the enormous accelerations required. The circular Unruh effect offers distinct experimental advantages:

Circular trajectories allow for sustained, bounded motion in the laboratory, making them feasible in storage rings, optical cavities, or optomechanical platforms (Rad et al., 2011, Gooding et al., 2020, Zheng et al., 23 Dec 2024, Zhou et al., 2023).
In atoms such as fluorine and oxygen, the centripetal acceleration of outer-shell electrons provides an effective Unruh temperature ( $T_\text{U}\sim 1000$ –$2000$ K) sufficient to measurably shift excited-state occupancies, testable by precision spectroscopy at cryogenic temperatures (Rad et al., 2011).
In ultracold atom systems (e.g., a focused laser scanning a circular path in a Bose–Einstein condensate), the role of “light speed” is replaced by the much lower sound speed, so achievable accelerations lead to large effective Unruh temperatures, bringing the effect within experimental reach (Gooding et al., 2020).
Enhanced signal strategies, such as superradiant emission in cavity QED or via collective Dicke states of rotating atomic ensembles, can increase detection efficiency by orders of magnitude (Zheng et al., 23 Dec 2024).

Recent theoretical work also emphasizes the importance of the detector's coupling profile (switching function): allowing for engineered time-dependent (even sign-changing) switching functions can restore a nonvanishing small-gap effective temperature, an essential prerequisite for observing the effect with accessible energy gaps in $2+1$-dimensional analogue spacetimes (Parry et al., 27 Aug 2025).

5. Relation to Other Kinematic Unruh Effects and Dimensionality Dependence

The Circular Unruh Effect can be viewed as part of a continuous family of stationary motion effects, smoothly interpolating between linear (boost) acceleration, drifted Rindler motion, parator, and pure circular trajectories (Parry et al., 29 Sep 2024). Analysis shows:

For drifted Rindler motion, the effective temperature in the large gap limit is always bounded with respect to the linear acceleration value (never exceeding $\simeq 1.81T_\text{lin}$ ).
In contrast, for circular (and parator) motion, the effective temperature can be unbounded in the large-gap regime.
In $2+1$ dimensions, only circular and parator motion have the property that the effective temperature vanishes linearly with the gap in the small gap limit; this is traced to the slow power-law large-time decay of the vacuum Wightman function along those orbits, a phenomenon not present in linear, drifted Rindler, or helix stationary motions (Parry et al., 29 Sep 2024, Parry et al., 27 Aug 2025).

This structural connection between classes of motion highlights the geometric origin of the response and the sensitive dependence of the “thermal” signature on the entire trajectory, not just the instantaneous value of acceleration.

6. Applications, Extensions, and Current Frontiers

The underlying mechanism of the Circular Unruh Effect, namely, the alteration in field correlations along periodic accelerated trajectories, has inspired proposals and analyses in a variety of contexts:

In atomic/nuclear physics: Population anomalies in low-lying levels in atoms (Rad et al., 2011) and modified shell occupation probabilities in nuclei (possibly relevant for the lithium-7 problem in Big Bang Nucleosynthesis) (Rad et al., 2013).
Analogue gravity and quantum simulation: Detecting the effect via impurity atoms in engineered condensates with tunable Lorentz-violating dispersion, probing the robustness of thermalization to high-frequency Lorentz-symmetry breaking (Gooding et al., 2020, Tian et al., 2022).
Quantum information and thermodynamics: Using angular acceleration to power quantum Otto engines (Sarkar et al., 28 Jul 2025), and employing quantum entanglement as a diagnostic of circular Unruh thermality (Zhou et al., 2023).
Nonequilibrium quantum field theory: Analysis of anti-Unruh behavior (suppressed decoherence at short times for higher acceleration), non-Markovian information backflow, and memory effects unique to non-linear and circular trajectories (Liu et al., 2016, Doukas et al., 2013).

Open questions include the experimental realization of the predicted effective temperatures, particularly in $2+1$-dimensional systems (analogue spacetimes), precise engineering of detector-field coupling profiles to restore thermality at small energy gaps (Parry et al., 27 Aug 2025), and the theoretical origin and universality of suppressed or enhanced response in specific regimes.

Summary Table: Circular vs. Linear Unruh Effect

Feature	Linear Unruh Effect	Circular Unruh Effect
$T_\text{eff}$ dependence	$a/(2\pi)$	Energy, radius, velocity
Small gap limit ( $E\to 0$ )	Constant	$T_\text{circ}(E)\rightarrow 0$ (2+1 D)
Large gap limit ( $E\to\infty$ )	Linear scaling	Possibly unbounded
Horizon present?	Yes	No
Thermal spectrum (Gibbs)?	Yes	Only in specific limits
Dimensional dependence?	None	Pronounced in $2+1$ dims
Experimental feasibility	Challenging	Enhanced via circular motion

This encapsulation draws on detailed analytic and numerical investigations (Good et al., 2020, Gooding et al., 2020, Biermann et al., 2020, Gim et al., 2018, Parry et al., 29 Sep 2024, Parry et al., 27 Aug 2025), with recent developments highlighting both practical strategies for detection and the underlying geometric and quantum field theoretic principles governing the effect.