Detectors, Reference Frames, and Time by Alexander R. H. Smith

Author:Alexander R. H. Smith
Language: eng
Format: epub, pdf
ISBN: 9783030110000
Publisher: Springer International Publishing

(5.30)

where

(5.31)

and

(5.32)

Defining

(5.33)

and changing the integration variable to , the second term in Eq. (5.30) becomes

(5.34)

From Eq. (5.34) we see that for large negative τ the integrand is small and . With this observation and Eq. (5.30), we conclude that the transition rate of a detector operating in the far past (large negative τ) is identical to the transition rate of the same detector in the BTZ spacetime. For τ ≳ 0, the integrand does not vanish and is significant. From these observations we conclude that the time-dependent contribution to the geon transition rate turns on around τ ≈ 0, and remains on as τ →∞. At this proper time (τ = 0), the detector is on the preferred hypersurface t = 0, which is singled out by the non-stationary features of the geon spacetime located behind its past and future horizons (see Fig. 5.1). The fact that the transition rate develops a time-dependence after the detector has crossed this surface (t = 0) demonstrates the detectors dependence on the non-stationary features of the geon spacetime.

Having simplified the expressions for the transition rate of a detector in the BTZ and geon spacetimes in Eqs. (5.27), (5.30), and (5.34), we now evaluate these expressions numerically10 and plot these transition rates as a function of the read out time of the detector and the detector’s energy gap for the field satisfying Neumann (Fig. 5.2), transparent (Fig. 5.3), and Dirichlet boundary conditions (Fig. 5.4).

Fig. 5.2The transition rate of a static detector in both the BTZ and geon spacetimes is plotted as a function of (a) the proper time πτ∕ℓβ at which the detector is read for a fixed energy gap of the detector, and (b) the energy gap of the detector ℓβ Ω for a fixed proper time at which the detector is read. In both (a) and (b) r h∕ℓ = 0.5, , and the field satisfies Neumann boundary conditions (ζ = −1) at spatial infinity. In (a) the dotted and solid lines of the same colour indicate the transition rate of an identical detector in the BTZ and geon spacetimes, respectively

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