What Hawking Radiation Looks Like as You Fall into a Black Hole

TL;DR

This work analyzes what a freely falling Unruh-DeWitt detector experiences near the horizon of a semiclassical Schwarzschild black hole in the Hartle-Hawking and Unruh vacua. By computing the detector's response along radial infall and disentangling switching effects from genuine particle detections, the authors show that near-horizon measurements are switching-dominated and do not directly reveal Hawking radiation. They introduce an operational effective local temperature $T_{\text{eff}}$ by matching the detector response to a Minkowski thermal state, observing a smooth increase from the Hawking temperature $T_{\mathrm{H}}$ far from the horizon to approximately $2T_{\mathrm{H}}$ at the horizon and continuing inward, with strong agreement to a 6D embedding prediction. The results support global embedding approaches for near-horizon thermodynamics and suggest that higher-dimensional embeddings may yield further insights into black hole spacetimes and the behavior of infalling observers.

Abstract

We study the measurements of a freely falling Unruh-DeWitt particle detector near the horizon of a semiclassical Schwarzschild black hole. Our results show that the detector's response increases smoothly as it approaches and crosses the horizon in both the Hartle-Hawking and Unruh vacua. However, these measurements are dominated by the effects of switching the detector on and off, rather than by the detection of Hawking radiation particles. We demonstrate that a freely falling Unruh-DeWitt detector cannot directly measure Hawking radiation near the horizon because the time required for thermalization is longer than the time spent near the horizon. We propose an operational definition of the effective temperature along an infalling trajectory based on measurements by a particle detector. Using this method, we find that the effective temperature measured by a freely falling observer in the Hartle-Hawking vacuum increases smoothly from the Hawking temperature far from the horizon to twice the Hawking temperature at the horizon, and continues to rise into the interior of the black hole. This effective temperature closely matches an analytical prediction derived by embedding Schwarzschild spacetime into a higher-dimensional Minkowski space, suggesting that further exploration of higher-dimensional embeddings could provide new insights into the near-horizon behavior of black holes.

Figures (9)

• Figure 1: The response $\mathcal{F}$ of a static detector in a Minkowski thermal state with temperature $T$, as a function of the measurement duration $\Delta\tau$. Left: the transition between switching-dominated and particle-dominated regimes occurs at $\Delta\tau \sim T^{-1}$ when the detector energy gap $E$ is of order $T$. Gray lines denote the asymptotic limits for small (dotted) and large (solid) $\Delta\tau$. Right:$\mathcal{F}$ does not simply decouple into its "switching component" $\mathcal{F}_\text{switch}$ plus its "particle component" $\mathcal{F}_\text{par}$. Depending on the detector energy and the switching duration, the actual response can be smaller or larger than this sum.
• Figure 2: Left: the KMS temperature $T_\mathrm{KMS}$ measured by a static detector is much larger than the temperature $T$ of the thermal state when $\Delta\tau \lesssim T^{-1}$. Right: the response $\mathcal{F}$ of a static detector is monotonic in $T$ when $E$ and $\Delta\tau$ are held fixed. Thus, $\mathcal{F}$ can be used to measure $T$, even for short switching durations where the detector has not yet thermalized.
• Figure 3: The response $\mathcal{F}$ of a detector moving with constant velocity $v$ through a Minkowski thermal state with temperature $T$, for very long measurement durations (left) and shorter durations (right). Left: in the limit of large $\Delta\tau$, the dependence of $\mathcal{F}$ on the energy gap $E$ is modified by the Doppler effect. Right: for small measurement durations, $\mathcal{F}$ is only weakly dependent on $v$ because switching effects dominate. $\mathcal{F}_\text{static}$ denotes the response measured by a static detector. The detector energy gap is $E = T^{-1}$, but the results are similar for all $E$ within an order of magnitude this value.
• Figure 4: The effective potential $V_\ell$ for $\ell \leq 4$.
• Figure 5: The response $\mathcal{F}$ of a detector freely falling from rest at infinity into a Schwarzschild black hole in the Hartle-Hawking and Unruh vacuum states. At all energies shown, the response increases smoothly as the detector approaches and crosses the horizon. The legend is the same for both panels.