Response of Unruh-DeWitt detector with time-dependent acceleration

Abstract

It is well known that a detector, coupled linearly to a quantum field and accelerating through the inertial vacuum with a constant acceleration $g$, will behave as though it is immersed in a radiation field with temperature $T=(g/2\pi )$. We study a generalization of this result for detectors moving with a time-dependent acceleration $g(\tau )$ along a given direction. After defining the rate of excitation of the detector appropriately, we evaluate this rate for time-dependent acceleration, $g(\tau )$, to linear order in the parameter $\eta =\dot{g}/g^2$. In this case, we have three length scales in the problem: $g^{-1},(\dot{g}/g{)}^{-1}$ and ${\omega }^{-1}$ where $\omega$ is the energy difference between the two levels of the detector at which the spectrum is probed. We show that: (a) When ${\omega }^{-1}\ll g^{-1}\ll (\dot{g}/g{)}^{-1}$, the rate of transition of the detector corresponds to a slowly varying temperature $T(\tau )=g(\tau )/2\pi$, as one would have expected. (b) However, when $g^{-1}\ll {\omega }^{-1}\ll (\dot{g}/g{)}^{-1}$, we find that the spectrum is modified \textit{even at the order $\mathcal{O}(\eta )$}. This is counter-intuitive because, in this case, the relevant frequency does not probe the rate of change of the acceleration since $(\dot{g}/g)\ll \omega$ and we certainly do not have deviation from the thermal spectrum when $\dot{g}=0$. This result shows that there is a subtle discontinuity in the behaviour of detectors with $\dot{g}=0$ and $\dot{g}/g^2$ being arbitrarily small. We corroborate this result by evaluating the detector response for a particular trajectory which admits an analytic expression for the poles of the Wightman function.

v1, 7 pages, no figures; v2, an Acknowledgment and some clarifying comments added, matches version accepted for publication in Physics Letters B

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• The Unruh effect and its applications
• Dark Energy and Gravity
• Transition rate of the Unruh-DeWitt detector in curved spacetime
• Entropy of Null Surfaces and Dynamics of Spacetime
• Einstein's equations as a thermodynamic identity: The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons
• Unruh Effect for General Trajectories

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