INTRODUCTION

Minkowski spacetime with its Lorentz geometry is the geometrical framework for most physical measurements, in particular those involving radiation and scattering processes. Indeed, the asymptotic ``in'' and ``out'' regions of the scattering matrix, as well as the asymptotic ``far-field'' regions of a radiator reflect this fact.

If the scatterer or radiation source is accelerated linearly and uniformly, then the standard approach is to characterize its coaccelerating coordinate frame in terms of a one-parameter family of instantaneous Lorentz frames, any one of which provides the necessary ``in'' and ``out'' or ``far-field'' regions for the measurements of the scattered and emitted radiation.

However, suppose the acceleration is extreme, i.e.

• the acceleration, say $g$, lasts long enough for the scatterering/radiation source to acquire relativistic velocities and
• is large enough to do this within one cycle of its characteristic frequency $\omega$ so that
  
  $$\frac{g}{\omega} \sim \textrm{(speed of light)}~.$$
  
  

Under such a circumstance no physicist who insists on using a given asymptotic Lorentz frame as his observation platform can escape from a number of difficulties trying to execute his measurements.

First of all, there is the distortion problem. Relative to any Lorentz frame a signal emitted by, say, an accelerated dipole would be subjected to a time-dependent Doppler shift (``Doppler chirp''). The received signal starts out with an extreme blue shift and finishes with an extreme red shift. Such a distortion prevails not only in the time domain, but also in the spatial domain of that Lorentz frame. Once this distorted signal has been acquired by the observer in his Lorentz frame, he is confronted with the task of applying a time and/or space transformation to remove this distortion. He must reconstruct the signal in order to recover with 100% fidelity the original signal emitted by the source. Such a task is tantamount to changing from his familiar set of clocks and meter rods, which make up his Lorentz frame, to a new set of clocks and units of length relative to which the signal presents itself in undistorted form with 100% fidelity.

Second, there is the problem of the trajectory of the accelerated source. In order to have a ``far field'' region, the source must be much smaller than one wavelength. If the acceleration lasts long enough, the source will reach within one oscillation the asymptotically distant observer where the measuring equipment is located and thus vitiate its status as being located in the ``far field'' region: there no longer is large sphere that surrounds the source.

Finally, during such an acceleration process the source would be emitting plenty of information about itself (in the form of spectral power, angular distribution, etc.). However, to acquire this information the Maxwell field must be measured in the radiation zone. It lies outside a sphere centered around the source with radius one wave length. (Inside this sphere the radiation field is inextricably mixed up with the ``induction'' field.) Measuring the Maxwell field consists of relating its measured amplitude to the synchronized clocks and measuring rods. But this is precisely what cannot be done if the wavelength is larger than (acceleration)$^{-1}$ of the accelerated source. In that case the far field falls outside the semi-infinite domain Misner et al. (1973a) where the events are characterized uniquely by the clocks that are synchronized with the accelerated clock of the source. Put differently, the semi-finite size of the ``local coordinate system of the accelerated source'' Misner et al. (1973b) does not allow an observer to distance himself far enough from the source to identify the radiative field in the far zone.

Aside from removing the above ambiguities, the purpose of this note is to identify the spacetime framework which accommodates Maxwell's field equations applied to a uniformly and linearly accelerated radiation source. One such application is the radiation observed in response to a dipole source. The observed radiation rate is given by the familiar Larmor formula but augmented due to the unique source-induced spacetime framework. This enhanced Larmor radiation formula is the result of a straightforward calculation based on this framework. There are no arbitrary hypotheses. The formula is given by Gerlach (2001):

$$\left( \begin{array}{c} \textrm{flow of radiant}\\ \textr... ...tau^2} \right)^2 + \left( \frac{d D}{d\tau} \right)^2 \right]$$ (1)

Here

$$D= \left( \begin{array}{c} \textrm{proper}\\ \textrm{dip... ...}\\ \textrm{of the}\\ \textrm{dipole} \end{array} \right)$$

is the geometrical dipole moment. It is the time dependent magnitude of a dipole source pointing along the direction of acceleration, which is linear and uniform. Furthermore,

$$\tau= \left( \textrm{geometrical (dimensionless, \lq\lq Rindler'') time} \right) ~,$$

and the quantity

$$\left( \begin{array}{c} \textrm{flow of radiant}\\ \textr... ...ponent}\\ \textrm{of radiated momentum} \end{array}\right)~.$$

is defined by the conserved $\tau$-momentum in boost-invariant sector $F$,

$$\begin{eqnarray*} &\displaystyle\int_{-\infty}^\infty d\tau& \overbrace{\int_{0... ...component}\\ \textrm{of radiated momentum} \end{array}\right). \end{eqnarray*}$$

Formula (1) expresses a causal link between what happens at the accelerated source and the radiant energy observed on the other side of event horizon. The $\tau$-coordinate is the key. It is a symmetry trajectory on both sides of this horizon. This enables it to serve as the same standard for reckoning changes in the source in $I$ as for reckoning changes in the location in $F$.

What is the spacetime framework, i.e. the nature of the arrangement of measuring rods and clocks which makes this formula possible? Even if the spacetime framework for the accelerated dipole source is clear, comprehending Eq.(1) entails asking: (i) What is the spacetime framework for the observer who measures the radiation? (ii) What is the relationship between his framework and that of the source?

As depicted in Figure 1, the source traces out a world line which is hyperbolic relative to a globally free-float observer, one with a system of inertial clocks in a state of relative rest to one another. However, the observed energy given by Eq.(1) is to be measured by a different observer, one whose clocks, even though also inertial, have nonzero expansion relative to one another.

We shall find that the arrangement of clocks and rods of such an observer is confined to future sector $F$ of Figure 1. This sector is separated by the history of a one-way membrane (``event horizon'') from the spacetime domain, sector $I$, of the source. The purpose of this article is to establish a physical bridge between the two, and bolt them together into a single arena appropriate for the measurement of attributes of bodies subjected to extreme acceleration, Eq.(1) being one of them.

Figure: Acceleration-induced partitioning of spacetime into four boost-invariant 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 boost (a.k.a. ``Rindler'') sectors. The meaning of the boost coordinates $\xi$ and $\tau$ is inferred from the expressions for the invariant interval $-dt^2+dz^2 =-\xi ^2d\tau ^2+d\xi ^2$ in $I$ & $II$ and $-dt^2+dz^2=\xi ^2d\tau ^2-d\xi ^2$ in $F$ & $P$. The emitted radiation given by Eq.(1) applies to a point-like source whose world line traces out the hyperbola in sector $I$.

The above questions do not deal with the inertia of bodies, nor with the dynamics of material particles, nor with the dynamics of the Maxwell field equations. Instead, they address kinematical aspects of the source and the observer by introducing geometrical clocks which are commensurable.