Unruh effect for general trajectories

We consider two-level detectors coupled to a scalar field and moving on arbitrary trajectories in Minkowski space-time. We first derive a generic expression for the response function using a (novel) regularization procedure based on the Feynman prescription that is explicitly causal, and we compare it to other expressions used in the literature. We then use this expression to study, analytically and numerically, the time dependence of the response function in various nonstationarity situations. We show that, generically, the response function decreases like a power in the detector’s level spacing, $E$, for high $E$. It is only for stationary worldlines that the response function decays faster than any power law, in keeping with the known exponential behavior for some stationary cases. Under some conditions the (time-dependent) response function for a nonstationary worldline is well approximated by the value of the response function for a stationary worldline having the same instantaneous acceleration, torsion, and hypertorsion. While we cannot offer general conditions for this to apply, we discuss special cases; in particular, the low-energy limit for linear space trajectories.

Figure 1

The ratio $R_{+}(ϵ,\tau ,\alpha ,E)/R_{+}(0,\tau ,\alpha ,E)={\tilde{R}}_{+}(aϵ,\alpha ,E/a)/R_{+}(0,\alpha ,E/a)$ as a function of $aϵ$ for $2\pi E/a={10}^{-2}$ (left panel) and $2\pi E/a=1$ (right panel) and for two values of $\alpha =5$, 100, as marked. Continuous lines correspond to the proper-time regularization, dashed curves to Schlicht’s regularization.

Figure 2

The ratio $\xi \equiv R_{+}/R_{+}^p$ for the linear trajectory Eq. (28) as a function of $E/a$, for different values of $\alpha$ (as marked).

Figure 3

The response function for the linear trajectory with acceleration $a(\tau )=2+\tanh \tau$ for $E=20$ ($g=1$). The result is given in units of ${10}^{-5}$. The dashed vertical line shows the maximum of $R_{+}^p$, ${\tau }_0=\ln [(2+\sqrt{7})/3]/2$.

Figure 4

The ratio $\xi \equiv R_{+}/R_{+}^p$ for the linear trajectory Eq. (50) with acceleration $a(\tau )=2+\tanh \tau$ and for different values of the energy.

Figure 5

The ratio $\zeta \equiv R_{+}/R_{+}^q$ as a function of $E/a$ for different values of $\alpha$ (as marked) for the linear trajectory with acceleration $a(\tau )=-\alpha /\tau$.

Figure 6

The ratio $\zeta \equiv R_{+}/R_{+}^q$ for different values of the energy for the linear trajectory with $a(\tau )=2+\tanh \tau$.

Figure 7

The ratio $\zeta =R_{+}/R_{+}^q$ as a function of $\tau /{\tau }_0$ for different values of $E/a$ (as marked) for the linear trajectory with acceleration $a(\tau )={\tau }_0^{-2/3}(-\tau {)}^{-1/3}$.

Figure 8

The ratio $\zeta =R_{+}/R_{+}^q$ as a function of $\tau /{\tau }_0$ for different values of $E{\tau }_0$ (as marked) for the linear trajectory with acceleration $a(\tau )={\tau }_0^{-2/3}(-\tau {)}^{-1/3}$.