Unruh–DeWitt Detector Model

• The Unruh–DeWitt detector is a two-level quantum probe coupled to a scalar field that probes local vacuum and thermal effects in various spacetime geometries.
• It computes excitation probabilities through transition rates, effectively bridging quantum field theory with observable phenomena in static and evolving backgrounds.
• The model contrasts semi-classical tunneling with detector response, highlighting the role of surface gravity and observer-dependent features in horizon thermality.

An Unruh–DeWitt detector is a simplified, phenomenological model of a localized quantum probe (commonly envisioned as a two-level system or “qubit”) that interacts with a quantum field along a specified spacetime trajectory. By monitoring the detector’s transition rates or excitation probabilities, one can probe the local and global properties of quantum fields in arbitrary geometries. This model, introduced to bridge quantum field theory and the operational notion of particle detection, provides a fundamental diagnostic tool for analyzing effects such as the Unruh effect and Hawking radiation in curved or dynamical spacetimes. The formalism is central for investigating quantum thermality, vacuum structure, and horizon phenomena.

1. Detector Formalism and Operational Role

The Unruh–DeWitt detector is typically modeled as a pointlike two-level system carrying an internal Hamiltonian, $H_{0,\mathrm{det}} = \Omega \sigma_z/2$, coupling to a real scalar field $\phi(x)$ through a monopole moment $m(\tau)$:

$$H_{\mathrm{int}} = \lambda\, \chi(\tau)\, m(\tau) \phi[x(\tau)]$$

where $\lambda$ is the coupling constant, $\chi(\tau)$ is a switching function, and $x(\tau)$ is the detector worldline. The detector’s response function is determined by the Wightman two-point correlator of the field evaluated on the detector trajectory:

$$W(\tau, \tau') = \langle 0 | \phi[x(\tau)] \phi[x(\tau')] | 0\rangle$$

For dynamical or curved backgrounds, $W$ reflects both spacetime geometry and quantum state.

The probability for excitation (to leading order in $\lambda$) is

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

where $E$ is the detector’s energy gap. The instantaneous transition rate can be similarly defined and is sensitive to trajectory, field state, switching protocol, and spacetime geometry.

The Unruh–DeWitt detector serves as a “quantum thermometer,” making operational contact between quantum field theory and observer-dependent phenomena such as the Unruh and Hawking effects.

2. Probing Temperature and Thermality in Curved and Dynamical Spacetimes

The paper (1111.6389) employs Unruh–DeWitt detectors to probe the emergence of effective temperature in spherically symmetric, dynamical spacetimes, notably black holes and cosmological (FRW) scenarios. In static backgrounds, the detector on an appropriate trajectory records a transition rate with a Boltzmann thermal factor,

$\mathcal{F}(E) \propto E\, e^{-E/T_H}$

where $T_H$ is set by a locally defined surface gravity (e.g., the Hayward/Kodama surface gravity for spherically symmetric horizons) and includes the Tolman redshift correction:

$T_{\text{local}} = \frac{\kappa}{2\pi\sqrt{V(r)}}$

for a static spherically symmetric metric $ds^2 = -V(r) dt^2 + dr^2/V(r) + r^2 d\Omega^2$.

In dynamical spacetimes—such as a cosmological FRW universe interpolating between matter-dominated and asymptotic de Sitter epochs—the response splits:

• A steady de Sitter–like part with Gibbons–Hawking temperature $T_{\mathrm{GH}} = H_0/2\pi$ (where $H_0$ is the asymptotic Hubble expansion rate)
• A transient, fluctuating “tail” not of thermal/Boltzmann form, reflecting the evolution toward (but not instant establishment of) thermal equilibrium.

Thus, while stationary observers in static black holes recover the conventional Hawking temperature, the concept of a unique temperature becomes ambiguous in dynamical cosmologies.

3. Semi-Classical Tunneling Method vs. Unruh–DeWitt Detector Formalism

The semi-classical tunneling framework and the Unruh–DeWitt detector method provide two conceptually distinct, yet fundamentally related, approaches to extracting thermal properties from horizon spacetimes:

• Tunneling method: Uses a WKB approximation for particle tunneling through a horizon; yields an invariant Boltzmann factor $\exp(-2\pi \omega/\kappa)$ for invariant energy $\omega$ (often defined via the Kodama vector), leading to

$T_{\text{tunnel}} = \frac{\kappa}{2\pi}$

It is manifestly covariant and provides consistency with Tolman’s law.

• Unruh–DeWitt detector: Directly computes the quantum field theoretic transition probability of a localized detector. In static or stationary settings, the detector confirms the same Hawking/Gibbons–Hawking temperature; in dynamical cases, the response includes significant time dependence, “memory” effects, and relaxation towards (not instantaneous) thermality.

Both methods yield identical thermal spectra for static black holes or pure de Sitter backgrounds, but diverge in dynamical regimes. For co-moving observers in time-dependent FRW backgrounds, the detector does not register a clean temperature: the response contains non-thermal (non-Boltzmann) corrections and only reduces to a thermal form in the late-time (asymptotic) limit.

4. Kodama Trajectories and Observer Dependence

The Kodama vector plays a key role in generalizing notions of energy and preferred trajectories in spherically symmetric, nonstationary spacetimes where no globally defined timelike Killing vector exists. The Kodama vector provides a substitute for Killing energy, allowing for the construction of Kodama trajectories—observer paths along which energy fluxes and temperatures are meaningfully defined.

The paper emphasizes:

• For static black holes, Kodama observers reproduce all conventional thermodynamic relationships.
• In FRW or other dynamical cosmologies, co-moving (Kodama) observers see a detector response that consists of a steady (thermal) component plus a non-thermal, oscillating “tail.” Notably, a general co-moving detector does not directly measure any effective temperature associated to a dynamical horizon’s surface gravity except in the late-time de Sitter limit.

Importantly, the authors suggest that a detector moving along a more general Kodama trajectory—beyond strictly co-moving or inertial motion—may probe richer vacuum correlations or residual thermal features, potentially making contact with the dynamical surface gravity.

5. Surface Gravity, Horizon Temperature, and their Dynamical Relation

The identification of the temperature with the surface gravity of a horizon is uncontroversial for stationary black holes:

$T_H = \frac{\kappa}{2\pi}$

However, in a general cosmological (time-dependent) context, the dynamical surface gravity $\kappa_H$ (e.g., Hayward surface gravity at a cosmological trapping horizon) is well-defined as a local geometric scalar but does not always correspond to a directly measurable, local temperature for a detector. In FRW backgrounds evolving from Einstein–de Sitter to de Sitter, $\kappa_H$ interpolates from zero to a constant, but only in the asymptotic regime does the detector response align with a Gibbons–Hawking-like temperature matching $\kappa_H/2\pi$.

The fluctuating components of the detector response during evolution cannot generally be interpreted as signatures of thermal particle creation; rather, they reflect the complicated vacuum correlations in curved backgrounds. The implication is that in highly dynamical spacetimes, the assignment of a local temperature associated to a trapping horizon’s surface gravity may be lost, or, more subtly, may pertain to the properties of vacuum correlations rather than direct particle detection.

6. Thermalization and Approach to Equilibrium

For realistic $\Lambda$CDM cosmologies and more general interpolating FRW scenarios, the asymptotic detector response approaches a de Sitter thermal distribution:

$$\mathcal{F}(E) \rightarrow E\, e^{-E/T_{\mathrm{dS}}}$$

with

$$T_{\mathrm{dS}} = H_0/2\pi \ \text{as} \ t \to \infty.$$

However, at intermediate times or during matter-dominated eras, the detector’s excitation rate contains substantial non-thermal corrections. These corrections decay exponentially with the detector switching time (or observation duration), indicating that equilibrium is only attained at late times. The transient approach to thermality is controlled by the details of the spacetime evolution and the initial quantum state of the field (and detector).

This contrasts with black hole backgrounds, where the identification of a horizon temperature via the detector, tunneling method, or horizon stress tensor is robust for stationary observers.

7. Implications, Open Questions, and Relation to Vacuum Correlations

Thermal particle detection via the Unruh–DeWitt detector in static spacetimes supports the direct physical meaningfulness of the Hawking and Gibbons–Hawking temperatures and their relationship to horizon surface gravity (modulo Tolman red-shifting). In dynamical space-times, the detector response encodes not only local geometric data but also the history and nontrivial vacuum structure associated with quantum field evolution. The loss (for co-moving detectors) of a temperature directly tied to surface gravity in rapidly evolving cosmologies suggests that the latter, or its time-dependent generalization, may be more fundamentally related to vacuum correlation structure than straightforward particle emission.

Open questions include the detector’s response along more general (e.g., accelerated) Kodama trajectories, the universality of the equilibrium limit in generic cosmologies, and a deeper potential connection to holographic principles, where horizon surface gravity might be interpreted as quantifying nonlocal vacuum correlations.

$\begin{array}{|c|c|c|} \hline \text{Spacetime} & \text{Kodama/Observer Traj.} & \text{Detector Response}\ \hline \text{Static black hole} & \text{Kodama (static)} & \text{Thermal, }T_H = \kappa/(2\pi)\ \text{de Sitter (FRW patch)} & \text{Co-moving Kodama} & \text{Thermal, }T_{GH} = H_0/(2\pi)\ \text{Evolving FRW} & \text{Co-moving} & \text{Asymp. thermal + non-thermal tail}\ \hline \end{array}$

In summary, the Unruh–DeWitt detector formalism provides a direct, operational probe of quantum field thermality and horizon temperature in curved spacetime. It confirms the traditional temperature–surface gravity correspondence for static spacetimes, but in dynamical cosmologies it reveals that the link between horizon surface gravity and detector temperature is not generically preserved—though equilibrium may be restored at late times. These findings have significant implications for the interpretation of quantum effects near dynamical horizons, the physical meaning of surface gravity, and the nature of semiclassical thermality in general relativity (1111.6389).