Static detectors and circular-geodesic detectors on the Schwarzschild black hole

We examine the response of an Unruh-DeWitt particle detector coupled to a massless scalar field on the ($3+1$)-dimensional Schwarzschild spacetime, in the Boulware, Hartle-Hawking and Unruh states, for static detectors and detectors on circular geodesics, by primarily numerical methods. For the static detector, the response in the Hartle-Hawking state exhibits the known thermality at the local Hawking temperature, and the response in the Unruh state is thermal at the local Hawking temperature in the limit of a large detector energy gap. For the circular-geodesic detector, we find evidence of thermality in the limit of a large energy gap for the Hartle-Hawking and Unruh states, at a temperature that exceeds the Doppler-shifted local Hawing temperature. Detailed quantitative comparisons between the three states are given. The response in the Hartle-Hawking state is compared with the response in the Minkowski vacuum and in the Minkowski thermal state for the corresponding Rindler, drifted Rindler, and circularly accelerated trajectories. The analysis takes place within first-order perturbation theory and relies in an essential way on stationarity.

Figure 1

Illustrating the “up” and “in” modes on the right-hand wedge of the Penrose diagram representing the region exterior to the four-dimensional Schwarzschild black hole. The “up” modes are shown on the left-hand side and “in” modes on the right-hand side.

Figure 2

$M\dot{\mathcal{F}}$ as a function of $E/T_{\text{loc}}$ for the static detector at various radii, showing the results for the Hartle-Hawking state (orange circles) computed from (5.5), Boulware state (blue dashed) computed from (5.11) and Unruh state (red solid) computed from (5.13).

Figure 3

$M\dot{\mathcal{F}}$ as a function of $E/T_{\text{loc}}$ for the static detector. Figure showing the results for the Hartle-Hawking state (orange circles), computed from (5.5), alongside the response rate for an inertial detector in $3+1$ Minkowski spacetime (blue dashed), $-\mathrm{\Theta }(-E)E/2\pi$, and the response rate of a Rindler detector (green solid), computed from (7.2) with a proper acceleration chosen to be (7.3).

Figure 4

Ratio of $M\dot{\mathcal{F}}$, as a function of $E/T_{\text{loc}}$, for the static detector in the Hartle-Hawking state to that of the static detector in the Unruh state. The discontinuity near the origin is caused by the numerical difficulty in computing the modes at small $\omega$.

Figure 5

Figure shows $T_a$ (8.1) as a function of $E/T_{\text{loc}}$ for a static detector. The thick, green, dashed line is the local Hawking temperature $T_{\text{loc}}$. The line with blue triangles is the numerically computed $T_a$ for the Hartle-Hawking state, showing agreement with the analytic result $T_a=T_{\text{loc}}$. The solid red curve is the numerically computed $T_a$ for the Unruh state.

Figure 6

$M\dot{\mathcal{F}}$ as a function of $EM$ for the circular detector. The figure shows the transition rate for the Hartle-Hawking state (orange circles) computed from (6.5), Boulware state (blue dashed) computed from (6.9) and Unruh state (red solid) computed from (6.13).

Figure 7

$M\dot{\mathcal{F}}$ as a function of $EM$ for the circular detector, compared with the Rindler detector with transverse drift. The figure shows the transition rate for the Hartle-Hawking state (orange circles), computed from (6.5), alongside the transition rate for a Rindler detector with transverse drift (black-dashed). The Rindler rate is computed by substituting the interval (7.14) into the regulator-free transition rate found in [12] and then numerically evaluating.

Figure 8

Transition rate of a Rindler detector with drift in the transverse direction where the transverse direction has been periodically identified. Computed from (7.18) with $|n|$ cutoff at 500.

Figure 9

Ratio of $M\dot{\mathcal{F}}$, as a function of $EM$, for the circular-geodesic detector in the Hartle-Hawking state, to the transition rate of the circular-geodesic detector in the Unruh state. The discontinuity that appears near the origin is a numerical artefact owing to the fact that solving the ODE (3.5) becomes difficult at small $\omega$.

Figure 10

Figure shows $T_a$ (8.1) as a function of $EM$ for the circular-geodesic detector, in orange circles for the Hartle-Hawking state and in solid red for the Unruh state. The horizontal dashed thick green line is the local Hawking temperature $T_{\text{loc}}$ (5.6), and the horizontal solid thick black line is the Doppler-shifted local Hawking temperarature $T_{\text{Doppler}}$ (8.2). Finally, the horizontal solid and dashed blue lines are obtained by shifting $T_{\text{loc}}$ by, respectively, the factors (c7) and (c12) that arise in similar situations involving a drift velocity in Minkowski space, as shown in Appendix pp3.

Figure 11

$\dot{\mathcal{F}}$ as a function of $E$ for a free field in a half-space, computed from (b8) at $x_0=2$ for both Dirichlet ($\eta =-1$) and Neumann ($\eta =1$) boundary conditions.

Figure 12

$\dot{\mathcal{F}}$ as a function of $E$ in the Pöschl-Teller potential, computed from (b11), in (a) with $\mu =10$ and $x_0=4$, and in (b) with $\mu =2$ and $x_0=4$.

Figure 13

The solid (blue) lines show the noninertial correction ${\dot{\mathcal{F}}}^{\text{corr}}$ (c4) to the transition rate of the Rindler detector with a transverse drift. With ${\dot{\mathcal{F}}}^{\text{corr}}/a$ on the vertical axis and $E/a$ on the horizontal axis, the dimensionful parameter $a$ becomes scaled out and the graph depends only on the dimensionless parameter $v$: the two plots show, respectively, (a) $v=0.1$ and (b) $v=0.9$. The dashed (red) curves show the asymptotic large energy approximation (c6).

Figure 14

The rotating detector’s asymptotic temperature $T_{\text{rot}}$ (c18) is plotted as a function of the ambient temperature $T$ and the detector’s angular velocity $\mathrm{\Omega }$, all expressed in units of $1/R$, where $R$ is the radius of the detector’s orbit. In the low temperature regime $T_{\text{rot}}$ is independent of $T$, and the transition between the low temperature regime and the high temperature regime is clearly visible in the plot. The limit $\mathrm{\Omega }\to 0$ at fixed $T$ is in the high temperature regime and gives $T_{\text{rot}}\to T$, visible in the plot as the straight line at $\mathrm{\Omega }=0$: this is the familiar result for an inertial detector in a co-moving thermal bath.