Comoving Unruh–DeWitt Detector

• Comoving Unruh–DeWitt detectors are pointlike two-level quantum probes that measure effective temperature and particle content along an observer’s trajectory in curved spacetime.
• They evaluate the response function via the Wightman function, linking semiclassical tunneling methods with Planckian thermal spectra in static backgrounds.
• In dynamic FRW cosmologies, the detector reveals a mix of thermal and nonthermal signatures, highlighting the complex interplay between horizon dynamics and local measurements.

A comoving Unruh–DeWitt (UDW) detector is a local quantum probe—typically modeled as a pointlike two-level system—coupled to a field along its (comoving) trajectory in a curved or dynamical spacetime. Through calculation of excitation probabilities or transition rates, its interaction with quantum fields provides information about the effective “temperature,” particle content, and local geometry as perceived by the observer’s proper motion. The comoving UDW detector plays an essential role in connecting semiclassical and quantum field theoretic frameworks for the interpretation of horizon thermality in black hole, de Sitter, and generic Friedmann–Robertson–Walker (FRW) cosmologies.

1. Unruh–DeWitt Detector Framework in Spherically Symmetric Spacetimes

The UDW detector is idealized as a pointlike system locally coupled to a conformally coupled scalar field. Specifically, the transition rate for excitation per unit proper time is determined by evaluating the response function, which integrates the Wightman function of the quantum field along the detector's trajectory. In spherically symmetric spacetimes lacking global timelike Killing vectors (such as dynamical black holes or FRW cosmologies), the Kodama vector field replaces the conventional Killing flow as the generator of “energy measurements.” Kodama trajectories—integral curves of the Kodama vector—serve as operationally well-defined reference motions for stationary observers in such backgrounds.

The semi-classical tunneling method—based on computing the imaginary part of an action $I$ for a tunneling trajectory via a WKB (Hamilton–Jacobi) ansatz—yields an emission probability proportional to $\exp(-\operatorname{Im} I)$. For static horizons (e.g., Schwarzschild, de Sitter) this reduces to a Boltzmann factor $\exp(-E/T_H)$, with $T_H$ the standard Hawking/Gibbons–Hawking temperature, $T_H = \kappa/(2\pi)$, $\kappa$ being the surface gravity.

The quantum field theoretic (UDW) detector along a Kodama trajectory and the semiclassical tunneling method are found to yield consistent results for temperature in static black hole and de Sitter spacetimes, incorporating all requisite redshift and observer-dependent effects (Acquaviva et al., 2011).

2. Tolman Factor and Local Temperature for Comoving Observers

For a static horizon, the global equilibrium temperature $T_H$ must be related to what is locally measured by a detector at radius $r$ via the Tolman relation

$T_{\text{local}} = \frac{T_H}{\sqrt{V(r)}},$

where $V(r)$ is the redshift function extracted from the metric ($\sqrt{-g_{tt}}$). The transition rate derived from the UDW detector then contains a Boltzmann factor, yielding the excitation spectrum

$$\mathcal{F}(E) \sim E\, \exp\left(-\frac{E}{T_{\text{local}}}\right).$$

In de Sitter space, the analogous Tolman–de Sitter correction ensures $T_{\text{local}} = T_{dS}/\sqrt{V}$, with $T_{dS} = H_0/(2\pi)$ ($H_0$ is the Hubble constant).

The full response function formula for a detector along a radial Kodama trajectory is

$$\frac{d\mathcal{F}}{dT} = \frac{1}{4\pi^2} \int_0^\infty ds\, \cos(Es)\, [\sigma^2(T,s)]^{-1} + (\text{nonthermal corrections}),$$

with $\sigma^2(T,s)$ the geodesic interval squared for the given trajectory and metric.

3. Comoving UDW Detector in Dynamical FRW Cosmologies

In a dynamical, spatially flat FRW universe, such as one interpolating between de Sitter and Einstein–de Sitter phases as in realistic $\Lambda$CDM cosmologies, the situation changes qualitatively. The detector response function no longer yields a Boltzmann factor, and thus cannot always be interpreted as the outcome of a thermal bath. Explicitly, the excitation probability splits into two contributions:

• A stationary “de Sitter” term: Corresponds to the residual Gibbons–Hawking–like effect, giving a time-independent thermal response.
• A fluctuating (tail) term: Contains oscillatory (e.g., cosine and sine) proper-time dependences, lacks Boltzmann behavior, and describes the approach to equilibrium as a transient phenomenon.

This structure reflects the nonequilibrium nature of the time-dependent FRW background. The thermalization process is dynamical; the detector sees transiently nonthermal fluctuations before eventual recovery (in the late-time limit) of the standard de Sitter thermal response. No clean identification between the trapping horizon’s dynamical surface gravity $\kappa_H$ and an effective temperature emerges for the comoving detector—thus marking a critical difference from the static case.

4. Surface Gravity, Temperature, and Kodama Versus Comoving Motion

In stationary spacetime, surface gravity $\kappa$ directly sets the temperature for both tunneling and detector-based thermometry, $T = \kappa/(2\pi)$. However, in dynamical geometric settings (e.g., nontrivial time-dependent FRW universes), the operational meaning of $\kappa_H$ as a local temperature parameter breaks down for comoving detectors. This suggests a decoupling between dynamical gravity and local thermal response in evolving backgrounds—a comoving detector will not, in general, measure a temperature $T_H = |\kappa_H|/(2\pi)$.

Nevertheless, detectors following general Kodama trajectories, as opposed to simply comoving worldlines, may probe additional features of the quantum vacuum, such as vacuum correlations, which are associated more with quantum correlations than with classical particle creation. The nuanced relation between dynamical surface gravity, detector thermometry, and vacuum structure is thus sensitive to both the detector's trajectory and the underlying spacetime dynamics.

5. Key Mathematical Results

The relationship between detector response, background geometry, and temperature is concretized by the following formulas:

• Semiclassical emission probability: $F \sim E \exp(-E/T_H)$, $T_H = \kappa/(2\pi)$.
• UDW detector response:

$$\frac{d\mathcal{F}}{dT} = \frac{1}{4\pi^2} \int_0^\infty ds\, \cos(Es)\, [\sigma^2(T,s)]^{-1} + \text{(tail correction)}$$

$\sigma^2(T,s) = a(T) a(T-s) [x(T) - x(T-s)]^2$ (for a given scale factor and trajectory).

• Tolman correction: $T_{\text{local}} = T_H/\sqrt{V(r)}$.
• FRW de Sitter case: The response becomes

$$\frac{d\mathcal{F}}{dT} \propto [\exp(2\pi E/T_{dS}) - 1]^{-1}$$

for a comoving observer, giving the familiar Gibbons–Hawking result.

6. Physical Interpretation and Implications

The synthesis of semiclassical tunneling and UDW detector analysis shows that, in static and quasi-static contexts (black hole or de Sitter), a sharply-defined thermodynamic and quantum-field-theoretic temperature emerges, compatible with the redshift-corrected Planckian response. For time-dependent, nonstationary spacetimes such as generic FRW models, the thermal character of the detector response becomes transient, with equilibrium reached dynamically—thermalization is not immediate, and intermediate states lack a simple Boltzmann interpretation. For comoving UDW detectors, this precludes a direct mapping between local horizon dynamics and thermometry.

A plausible implication is that the connection between gravitational surface gravity and locally measurable temperature is not universally valid, but is contingent upon both stationarity and the detector's adapted (Kodama or geodesic) motion. In genuinely dynamical backgrounds, the surface gravity may be more readily connected to features of the quantum vacuum, such as field correlations, rather than to classical particle production or a naive temperature assignment.

7. Summary Table: Detector Response in Different Regimes

Regime/Spacetime	Local Temperature Interpretation	Detector Response Structure
Static Black Hole, de Sitter	$T = \kappa/(2\pi)$, Tolman corrected	Planckian/Boltzmann, pure thermal spectrum
FRW (de Sitter–Einstein–de Sitter)	No simple $T$; $T$ not surface gravity	De Sitter thermal + nonthermal tail terms
General dynamical (nonstatic)	No direct thermal assignment	Nonstationary, approach to equilibrium

This framework reveals the full depth of the comoving Unruh–DeWitt detector as both a thermometric and nonequilibrium quantum probe, showing the importance of geometric, kinematic, and causal structure in the determination of temperature and quantum field response in dynamical spacetimes (Acquaviva et al., 2011).