Unruh-like effects: effective temperatures along stationary worldlines

Abstract

We study the detailed balance temperatures recorded along all classes of stationary, uniformly accelerated worldlines in four-dimensional Minkowski spacetime, namely along (i) linear uniform acceleration, (ii) cusped, (iii) circular, (iv) catenary, and (v) helix worldlines, among which the Unruh temperature is the particular case for linear uniform acceleration. As a measuring device, we employ an Unruh-DeWitt detector, modeled as a qubit that interacts for a long time with a massless Klein-Gordon field in the vacuum state. The temperatures in each case (i)-(v) are functions of up to three invariant quantities: curvature or proper acceleration, κ, torsion, b, and hypertorsion, ν, and except for the case (i), they depend on the transition frequency difference of the detector, ω. We investigate numerically the behavior of the frequency-dependent temperatures for different values of κ, b, and ν along the stationary worldlines (ii)-(v) and evaluate analytically the regimes where the temperatures recorded along the different worldlines coincide with each other in terms of relevant asymptotic limits for κ, b, or ν, and discuss their physical meaning. We demonstrate that the temperatures in cases (ii)-(v) dip under the Unruh temperature at low frequencies and go above the Unruh temperature for large |ω|. It is our hope that this study will be relevant to the design of experiments seeking to verify the Unruh effect or generalizations thereof.

Interferometric Unruh Detectors for Bose-Einstein Condensates

“…Formula 14is obtained from (1) by using for a the circular motion proper acceleration in the effective BEC Minkowski geometry and adjusting the energies to be defined with respect to the laboratory time, as is appropriate for the BEC. The actual effective temperature for circular motion includes an energydependent factor of order unity [25,27,40], which we shall suppress here; a thorough analysis of this order unity factor in (2 + 1) and (3 + 1) dimensions is given in our companion paper [41].…”

Quantum Power Distribution of Relativistic Acceleration Radiation: Classical Electrodynamic Analogies with Perfectly Reflecting Moving Mirrors

“…This formula applies to very general trajectories (albeit globally asymptotically sub-light speed and time-like; although this does not stop one from being able to locally compute the power distributions of the five classes of uniform acceleration, which for an electric charge will be the same form as for uniformly accelerated moving mirrors [23]. It is unclear whether the effective temperatures will also carry-over [24]). For instance, a mirror moving on a circular arc near the speed of light will emit a synchrotrontype of radiation in the form of a narrow and intense beam directed tangent to the arc, implying that a fixed observer will see a brief flash or pulse of radiation every time the mirror moves directly toward them.…”

“…We now have picked up a π and a Fourier transform. We plug in Equation (23), which is the time-domain Equation (24), into:…”

“…Experimental verification will be facilitated by knowing the distribution of detected radiation. Recent studies have only just solved for the five classes of uniformly accelerated trajectory distribution [23] in classical electrodynamics (effective Unruh-like temperatures have also been calculated [24]). Unfortunately, mathematically, no settled upon or convincing [25] covariant expression has yet been derived to express the power distribution in a frame-independent formulation [26].…”

Unruh and analogue Unruh temperatures for circular motion in 3+1 and 2+1 dimensions

“…A well-known example is the Gibbons-Hawking effect for inertial motion in de Sitter spacetime [28], for which an analogue spacetime simulation has been proposed in [29,30]. In Minkowski spacetime, a similar effect exists for uniform accelerations that are not linear [31,32,33], including uniform circular motion [34,35,36,37,38]. The circular motion version is related to spin depolarisation in accelerator storage rings [39,40,41,42,43], which was originally predicted by different methods [44,45], and which has been observed [46], but the relation between this observation and the circular motion Unruh effect remains indirect [42,43].…”