next up previous


The emission or the scattering of light from localized sources is the most effective way for information to be transferred to the human eye, the window to our mind. We can increase the size of this window, with specialized detectors, or systems of detectors. They intercept the information, record and re-encode it, before passing it on to be assimilated and digested by our consciousness.

The spacetime framework for most physical measurements, in particular those involving radiation and scattering processes, consists of inertial frames, or frames which become nearly inertial by virtue of the limited magnitude of their spatial and temporal extent. Indeed, the asymptotic ``in'' and ``out'' regions of the scattering matrix as well as the asymptotic ``far-field'' regions of a radiator reflect the inertial nature of the spacetime framework for these processes.

Should one extend these processes to accelerated frames? If so, how? Let us delay answering the first question and note that Einstein, in his path breaking 1907 paper [#!Einstein1907!#], gave us the answer to the second: View an accelerated frame as a sequence of instantaneous locally inertial frames. Thus a scattering (or any other physical) process observed relative to a lattice of accelerated clocks and equally spaced detectors can be understood in terms of the lattice of inertial clocks and equally spaced inertial detectors [#!detectors!#] of one or several of these instantaneous locally inertial frames. Accelerated frames seem to be conceptually superfluous! Acceleration can always be transformed away by replacing it with an appropriate set of inertial frames. To make observations relative to an accelerated frame comprehensible, formulating them in terms of a sequence of instantaneous inertial frames seems (at first sight) to be sufficient.

The introduction of these inertial frames into physics was one of the two historical breakthroughs [#!Einstein1907b!#] for Einstein, because mathematically they are the tangent spaces, the building blocks from which he built general relativity.

However, characterizing an accelerated frame as a one-parameter family of instantaneous Lorentz frames was only an approximation, as Einstein himself points out explicitly[#!Einstein1907!#] in his 1907 article. The approximation consists of the fact that the Lorentz frames never have relativistic velocities with respect to one another. Thus Einstein approximated a hyperbolic world line in $I$ of Figure 1 by replacing it with a finite segment having the approximate shape of a parabola. If Einstein had not made this assumption, then he would have found immediately that associated with every uniformly linearly accelerated frame there is a twin moving into the opposite direction, and causally disjoint from the first. Nowadays these twins are called Rindler sectors $I$ and $II$ as in Figure 1.

Figure 1: Acceleration-induced partitioning of spacetime into the four Rindler coordinatized sectors. They are centered around the reference event $(t_0,z_0)$ so that $U=(t-t_0)-(z-z_0)$ and $V=(t-t_0)+(z-z_0)$ are the retarded and advanced time coordinates for this particular quartet of Rindler sectors.
\begin{figure}\epsfclipon \setlength{\epsfysize}{3.5in} \setlength{\epsfxsize}{6...
...ffile[0 400 500 680]{}} \epsfverbosetrue

Thus, linearly uniformly accelerated frames always come in pairs, which (a) are causally disjoint and (b) have lightlike boundaries, their past and future horizons.

These two Rindler coordinatized sectors together with their past $P$ and future $F$ form a double-slit interferometer[#!Gerlach1999!#] relative to a spatially homogeneous but expanding coordinate frame. The two Rindler sectors $I$ and $II$ comprise the double slit portal through which wave fields propagate from $P$ to $F$. During this process the wave field interacts with sources, which due to their acceleration, are confined to, say, Rindler sectors $I$ and/or $II$. The interference between the waves coming from these two sectors is observed in $F$. There the field amplitude is sampled in space and in time.

Consider the field which is due to accelerated sources in $I$ or in $II$. A single inertial radio receiver which samples the field temporally is confronted with a metaphysically impossible task: Track and decode a signal with a Doppler chirp (time dependent Doppler shift) whose phase is logarithmic in time. The longer and more violent the acceleration of the source, the more pronounced the initial blueshift and/or the final redshift at the receiver end. Tracking the amplitude and the phase of such a chirped signal becomes a debilitating task for any receiver.

Suppose, however, the field gets intercepted by a set of mutually receding radio receivers. If they, in concert, sample the field spatially at a single instant of ``synchronous'' time, then there is no Doppler chirp whatsoever. An accelerated source which emits a sharp spectral line will produce an equally sharp spectral line in the spatial Fourier domain of the sampled space domain (in Figure 1: $UV=\xi^2=const.$) of the expanding set of radio receivers. In brief, a signal emitted by an accelerated point source is intercepted by a set of mutually receding phased radio receivers with 100 % fidelity. We shall refer to this result as the fidelity property of Rindler's spacetime geometry.

The physical reason for this result is given in Section III, the mathematical formulation in Section VIA.

The application of the fidelity property to the power emitted from an accelerated dipole oscillator is given in Section VII. This application consists of Larmor's formula[#!Landau1962!#] augmented due to the fact that the oscillator is in a state of uniform acceleration.

The fidelity property applies to the radiation from a source accelerated in Rindler $I$ as well as to a source accelerated in Rindler $II$. If the two sources have the same frequency and are coherent, then the phased array of radio receivers measures an interference pattern which is mathematically indistinguishable from that due to a standard double slit. This result is spelled out in Section VIC.

It is worth while to reiterate that the fidelity property and its two applications are statements about the Rindler coordinate neighborhoods considered jointly, with the event horizons, $\xi=0$, integral building blocks of these concepts. Some workers in the field [#!MTWch1!#], who view spacetime only in terms of ``coordinate patches'' or ``coordinate charts'' (i.e. comply with Einstein's approximation mentioned above), tend to compare the locus of events $\xi=0$ in Fig. 1 to the coordinate singularity at the North Pole of a sphere or the origin of the Euclidean plane. Such a comparison leads to a pejorative assessment of Rindler's coordinatization as ``imperfect'',``singular'', or ``poor'' at $\xi=0$[#!MTWch1!#]. This is unfortunate. As a result, this comparison diverts attention from the fact that (1) waves from $I$ and $II$ interfere in $F$ and that (2) as a consequence, the resulting interference patterns serve as a natural way of probing and measuring scattering and/or radiative sources as well as gravitational disturbances in regions $I$ and $II$.

The Rindler double-slit opens additional vistas into the role of accelerated frames $I$ and $II$. They accommodate causally disjoint but correlated radiation and scattering centers whose mutually interfering radiation is observed and measured in $F$. These measurements are mathematically equivalent to having two accelerated observers in Rindler $I$ and $II$ respectively. From these measurements one can reconstruct in all detail the location and temporal evolution of all accelerated radiation sources. The aggregate of these sources comprises what in Euclidean optics is called an object, one in Rindler $I$ the other in Rindler $II$. What is observed in $F$ is the interference of two coherent diffraction patterns of these two objects. These measurements are qualitatively different from those that can be performed in any static inertial frame. They yield the kind of information which can be gathered only in accelerated frames with event horizons. One of the virtues of the Rindler double-slit interferometer is that it quite naturally avoids an obvious metaphysical impossibility[#!MTW1973a!#], namely, have accelerating observers in Rindler sectors $I$ and $II$ which (a) have the physical robustness to withstand the high (by biological-technological standards) acceleration and/or (b) the longevity and the propulsion resources to co-accelerate for ever and never cross the future event horizon.

From the perspective of implementing measurements, the Rindler double-slit has advantages akin to those of a Mach-Zehnder interferometer [#!Born_and_Wolf!#]: it permits an interferometric examination of regions of spacetime whose expanse is spacious enough to accommodate disturbances macroscopic in extent, and it permits one to achieve this feat without putting the measuring apparatus into harm's way. However, in order to use the Rindler interferometer as a diagnostic tool one must first have the necessary conceptual infrastructure. This article provides four of its ingredients:

Nomenclature: This articlae uses repeatedly the words ``Rindler sector'', ``Rindler spacetime'', etc. This is verbal shorthand for ``Rindler coordinatized sector'', ``Rindler coordinatized spacetime'' etc. The implicit qualifier ``coordinatized'' is essential because, without it, ``Rindler sector/spacetime'' would become a mere floating abstraction, i.e. an idea severed from its observational and/or physical basis.

next up previous
Ulrich Gerlach 2001-10-09