Figures

Figure 1: Acceleration-induced partitioning of spacetime into the four Rindler sectors.

Figure 2: Two interferometers. The Euclidean interferometer, which in optics is known as a Mach-Zehnder interferometer, consists of a half-silvered entrance mirror (thin line at the bottom), a half-silvered exit mirror (thin line at the top), and two fully reflecting mirrors (thick lines on the right and the left). The Lorentzian interferometer consists of the entrance region, Rindler Sector P, the exit region, Rindler Sector F, and the two reflective regions, Rindler Sectors I and II. The reflection is brought about by by the pseudo-gravitational potential (see Eq.(14)) in each sector. The thick hyperbolae separate the region of this potential where the mode is oscillatory from that region where it is evanescent. This reflection process is detailed in Figure 6.

Figure: Pair of monochromatic detectors configured to measure interference between waves from I and II, Figure (a), or to measure the amplitude of each waves from I and II separately, Figure (b) (``delayed choice experiment''). The detectors are monochromatic because they are surrounded by interference filters. Each such filter consists of a Fabry-Perot cavity with uniformly moving walls. Such a cavity has discrete transmission resonances at Rindler frequencies given by $\omega=n\pi /\tanh^{-1}\beta,~~n=1,2,\cdots$, where $\beta$ is the relative velocity of the cavity walls. Their histories delineate the evolution of a cavity's interior (lightly shaded wedge in the figure).

Figure 4: Minkowski-Bessel modes of positive and negative Minkowski frequency, and the four Rindler coordinate representatives for each. Note that the ensuing Figures 5 and 6 are mathematically equivalent to the above figures (Figs. 4a and b). All of them depict the Minkowski-Bessel modes $B^\pm _\omega$. Figure 6 is obtained by applying to Figure 4 some well-known identities between Hankel and Bessel functions and between McDonald and modified Bessel functions. Figures 5 and 6 are equivalent by virtue of the coordinate transformations in Figure 1

Figure 5: Amplitude splitting of the Minkowski-Bessel modes of positive and negative Minkowski frequency. Each mode expresses an interference process as found in the Mach-Zehnder interferometer familiar from optics. A wave in P far from the bifurcation event enters the ``interferometer'' from P. As indicated by the two heavy arrows, the wave splits into two partial waves: one propagates from P (tail of the arrow) across the past event horizon, and enters Rindler Sector I (tip of the same arrow). There it propagates to the right under the influence of the familiar boost-invariant pseudo-gravitational potential. This is shown explicitly in Figure 6. The wave becomes evanescent (exponentially decaying) in the shaded hyperbolic region and hence gets reflected by this potential. Upon propagating to the left the wave escapes from Rindler Sector I (tail of the next arrow) through its future event horizon into the future sector F (tip of that arrow). There it recombines with the other partial wave which got reflected in Rindler Sector II.

Figure 6: Reflection of partial waves from their potential barriers (shaded hyperbolae) in Rindler Sectors I and II. As in Figure 5, the four arrows connect partial waves propagating from one Rindler sector to another. In Rindler Sectors P and F the partial waves are proportional to the Bessel functions $J_{\pm i \omega }(k\xi )$ of order $\pm i \omega$, while in I and II they are proportional to the modified Bessel functions $I_{\pm i\omega }(k\xi )$. The latter oscillate for finite $k\xi$, but blow up exponentially as $k\xi \rightarrow \infty$ in each hyperbolically shaded region. The reflection process is brought about by the boundary condition that the incident wave in I ( $\propto I_{i \omega }(k\xi )$) must combine with the reflected wave ( $\propto I_{-i \omega }(k\xi )$) so as to form a total wave whose amplitude approaches zero as $k\xi \rightarrow \infty$. This wave is a standing wave ( $\propto (I_{i \omega }(k\xi ) - I_{-i \omega }(k\xi ))$ in I and thus expresses a process of reflection from the boost-invariant pseudo-gravitational potential. An analogous reflection process takes place in Rindler Sector II.

Figure: WKB-approximate formulas for the Minkowski-Bessel modes. All formulas assume that $\omega=\vert \omega \vert$. The upper (lower) signs in a picture go only with the upper (lower) signs in the corresponding boxed formula. This means, for example, that the formula (in all four Rindler sectors) for a M-B of, say, positive Minkowski frequency and of negative Rindler frequency, is designated by $B^+_{-\omega }$, and is given by the lower signed expressions in the second picture. The complete propagation process (splitting, reflection, and interference) of a M-B mode is characterized by the phase ${\mathcal S}^U$, which is continuous across the U-axis, and the phase ${\mathcal{S}}^V$, which is continuous across the V-axis. Both satisfy the Hamilton-Jacobi equation $\frac{\partial S}{\partial U}\frac{\partial S}{\partial V}=-\frac{k^2}{4}$, which is implied by Eq.(). The amplitude $\mathcal{A}$ or ( $e^{-\pi\omega} \mathcal{A}$) of each WKB wave propagating across its respective event horizon satisfies the concomitant conservation law $\frac{\partial}{\partial U}\left( {\mathcal{A}}^2\frac{\partial S} {\partial V}... ...rtial}{\partial V}\left( {\mathcal{A}}^2\frac{\partial S} {\partial U}\right)=0$, where $S={\mathcal{S}}^U$ or ${\mathcal{S}}^V$, depending on which horizon is being crossed.

Figure: Two-dimensional phase space spanned by the Minkowski frequency parameter $\theta$ ( Eq.() ) and by the Rindler frequency $\omega$. The windowed Fourier basis of orthonormal wavelets ( Eqs.()-() ) partitions this space into phase space cells of equal area $2\pi$. The support of each wavelet and its Fourier transform are concentrated horizontally and vertically within each dashed rectangle. A heavy dot locates the amplitude maximum of the corresponding wavelet and its Fourier transform. The shape of the area elements is controlled by the freely adjustable parameter $\varepsilon$. Physically it indicates the rate at which each Klein-Gordon wave packet collapses and explodes.

Figure 9: Two different partitionings of phase space. The elementary phase space areas have the same magnitude, but their shapes are different. They are characterized by the explosivity index $\varepsilon$ defined in Figure 8. The tall and skinny elements ( $\varepsilon \ll 1$) define Klein-Gordon wave packet histories with well-defined mean Minkowski frequency but indeterminate Rindler frequency. Such wave packets contract and reexpand non-relativistically in their respective frames of reference. By contrast, the short and squatty elements ( $1 \ll \varepsilon$) define Klein-Gordon wave packet histories with well-defined mean Rindler frequency but indeterminate Minkowski frequency. Such wave packets collapse and reexplode relativistically in their respective frames of reference.

Figure: Double slit interference of two waves modes entering and emerging from a pair of accelerated frames. In reality the two figures should be superimposed one on top of the other. However, for clarity, we show the two waves separately, one passing through Rindler Sector I (FIGURE B), the other through Rindler Sector II (FIGURE A). The interference occurs in Rindler Sector F, where the two waves meet and superpose to form a resultant wave.
The process of each partial wave passing through its respective spacetime sector (I and II) is a reflection process. Each figure depicts the WKB phase fronts being reflected by the pseudo-gravitational potential. Note that the direction of propagation of the wave mode is perpendicular (in the Lorentz sense) to its phase fronts. FIGURE B pictures the reflection process $P\rightarrow I \rightarrow F$, which is expressed by the wave ${\mathcal{A}} \exp [i{\mathcal{S}}^V] +{\mathcal{A}} \exp [-i{\mathcal{S}}^U]$, the WKB expression for $B^+_\omega$ in Figure 7a. The incident wave ${\mathcal{A}} \exp [-i{\mathcal{S}}^U]$, which propagates from P towards the boundary ($UV=\omega ^2$) of evanescence in Im represented in the figure by the isograms of ${\mathcal S}^U$. Similarly, the reflected wave ${\mathcal{A}} \exp [i{\mathcal{S}}^V]$, which propagates from the boundary of evanescence in I to the future F, is represented by the isograms of the phase ${\mathcal{S}}^V$. Note that the incident and the reflected phase contours intersect the boundary in a perpedicular way. In an analogous way, FIGURE A pictures the reflection process $P\rightarrow II \rightarrow F$, which is expressed by the wave $e^{-\pi \omega}{\mathcal{A}} \exp [-i{\mathcal{S}}^U] +e^{-\pi \omega}{\mathcal{A}} \exp [i{\mathcal{S}}^V]$, the WKB expression for $B^+_\omega$ in Figure 7a.
The interference occurs in Rindler Sector F, where the phase fronts in FIGURE A overlap with those in FIGURE B.