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Figure 1:
Accelerationinduced partitioning of spacetime into the
four Rindler sectors.

Figure 2:
Two interferometers. The Euclidean interferometer, which
in optics is known as a MachZehnder interferometer, consists of
a halfsilvered entrance mirror (thin line at the bottom),
a halfsilvered 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 pseudogravitational 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 FabryPerot cavity with uniformly
moving walls. Such a cavity has discrete transmission resonances at
Rindler frequencies given by
,
where
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:
MinkowskiBessel 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 MinkowskiBessel modes .
Figure 6
is obtained by applying to Figure 4 some
wellknown 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 MinkowskiBessel modes of
positive and negative Minkowski frequency. Each mode expresses an interference
process as found in the MachZehnder 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 boostinvariant pseudogravitational
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
of order ,
while in I and II
they are proportional to the modified Bessel functions
.
The latter oscillate for finite ,
but blow up
exponentially as
in each hyperbolically
shaded region. The reflection process is brought about by the boundary
condition that the incident wave in I (
)
must combine with the reflected wave (
)
so
as to form a total wave whose amplitude approaches zero as
.
This wave is a standing wave (
in I and thus expresses a
process of reflection from the boostinvariant pseudogravitational
potential.
An analogous reflection process takes place in Rindler Sector II.

Figure:
WKBapproximate formulas for the MinkowskiBessel modes. All
formulas assume that
.
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 MB of, say, positive Minkowski frequency and of
negative Rindler frequency, is designated by ,
and is
given by the lower signed expressions in the second picture.
The complete propagation process (splitting, reflection, and
interference) of a MB mode is characterized by the phase
,
which
is continuous across the Uaxis, and the phase
,
which is
continuous across the Vaxis. Both satisfy the HamiltonJacobi
equation
,
which is implied by Eq.(). The amplitude
or (
)
of each WKB wave propagating across its respective event horizon satisfies the concomitant conservation law
,
where
or
,
depending on which horizon
is being crossed.

Figure:
Twodimensional phase space spanned by the Minkowski frequency
parameter
( Eq.() ) and by the
Rindler frequency .
The windowed Fourier basis of orthonormal
wavelets ( Eqs.()() ) partitions this space into phase space cells
of equal area .
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
.
Physically it indicates the rate at which each KleinGordon
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
defined in Figure 8. The tall
and skinny elements (
)
define KleinGordon wave
packet histories with welldefined mean Minkowski frequency but
indeterminate Rindler frequency. Such wave packets contract and
reexpand nonrelativistically in their respective frames of
reference. By contrast, the short and squatty elements (
)
define KleinGordon wave packet histories with
welldefined 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 pseudogravitational 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
,
which is expressed by the
wave
,
the WKB expression for
in Figure
7a. The incident wave
,
which propagates from P towards the boundary
()
of evanescence in Im represented in the figure by
the isograms of
.
Similarly, the reflected wave
,
which propagates from the
boundary of evanescence in I to the future F, is represented by
the isograms of the phase
.
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
,
which is
expressed by the wave
,
the WKB expression for
in Figure
7a.
The interference occurs in
Rindler Sector F, where the phase fronts in FIGURE A overlap with
those in FIGURE B.

Next: About this document ...
Up: PAIRED ACCELERATED FRAMES: THE
Previous: PAIRED ACCELERATED FRAMES: THE
Ulrich Gerlach
19990415