\section*{ABSTRACT}

\noindent
\textit{
A fundamental Doppler-like but asymmetric wave effect
that shifts
	received signals in frequency
in proportion to
	their respective source distances,
was recently described as means for
	a whole new generation of communication technology
using angle and distance,
	potentially replacing
	TDM, FDM or CDMA,
		for multiplexing.

It is equivalent to
	wave packet compression
by scaling of time
	at the receiver,
converting
	path-dependent phase into
		distance-dependent shifts,
and can multiply
	the capacity of physical channels.
The effect was
	hitherto unsuspected in physics,
appears to be responsible for both
	the cosmological acceleration
and
	the Pioneer 10/11 anomaly,
and is exhibited
	in audio data.
This paper discusses
	how it may be exploited for
instant, passive ranging of signal sources, for
	verification, rescue and navigation;
incoherent aperture synthesis
	for smaller, yet more accurate radars;
universal immunity to
	jamming or interference;
and
	precision frequency scaling of
		radiant energy in general.
}

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\Section{intro}{Introduction}

A previously unsuspected result of the wave equation,
enabling a receiver to get frequency shifts
	in an incoming signal
in proportion to
	the physical distance of its source,
		was recently described
	\cite{Prasad2005}.
Its main premise,
that signals from real sources
	must have nonzero spectral spread,
seems to be well supported by
	the cosmological and
	the Pioneer 10/11 anomalous acceleration datasets,
and
	a remarkably large gamut of terrestrial mysteries
		that can be mundanely resolved.
This perceived support actually concerns
a further inference that the mechanism
	could occur naturally in our instruments
on the order of magnitude of
	$
	10^{-18}
	~\reciprocal{\second}
	$.
This is a decay corresponding, as half-life, to
	the age of the solar system,
and is too small for
	purely terrestrial applications.
The wave effect is consistently demonstrated with
	acoustic samples, however,
testifying to
	its fundamental and generic nature.
While a general exposition deserves to be made in due course
	in a physics forum,
it presents fundamental new opportunities for
	intelligence and military technologies.

Section \ref{s:theory} contains a brief review of
	the theory of the effect,
showing that
	all real signals necessarily carry source distance information
		in the spectral distribution of phase,
analogous to
	the ordinary spatial curvature of wavefronts,
and that by scanning this phase spectrum,
	each frequency in a received waveform
			comprising multiple signals

	would be shifted in proportion to
		its own source distance and the scanning rate.
Section \ref{s:impl} presents
	the general principles of realization
by both
	spectrometry
and
	digital means.
Section \ref{s:formalism} shows
	how this enables separation of signals by physics
		instead of modulation,
whose implications for information theory
	have been discussed [\ibid].
Intelligence and military possibilities
	are considered for the first time
		in Section \ref{s:mil}.

\Section{theory}{Hubble's law shifts, temporal parallax}

The sole premise for the effect, as mentioned, is
	the nonzero bandwidth of a real signal.
The Green's function for the general wave equation
	concerns an impulse function
		$\delta(\mathbf{x},t)$
	as the elemental source.
Its Fourier transform is
\begin{equation} \label{e:impulseFT}
	F[\delta(t)] =
		\int \delta(t) \, e^{-i \omega t} \, dt
	=
		1
	\quad
	,
\end{equation}
which says that
\emph{all spectral components of an impulse
	start with the same phase}.
The source itself is the only common reference
	across any continuous set of frequencies,
hence the \emph{spectral phase contours}
	indicate its distance
	(Fig.~\ref{f:PhaseGradient}).

\begin{figure}[ht]
	\centering
	\psfig{file=phasegrad-2.eps}
	\caption{Phase contours and gradient for a source impulse}
	\label{f:PhaseGradient}
\end{figure}

The main difficulty in tapping this available source
	of source distance information
is that it lies in
	differences of phase across incoming frequencies,
and
	phases are hard to measure accurately.
However,
if we scanned the spectrum at rate $d \widehat{k}/dt$,
then,
	for each contour $\phi$
	and source distance $r$,
we should encounter
	an increasing or decreasing phase
proportional to
	$r$ and
	the integration interval $\Delta \widehat{k}$,
as shown by
	the shaded areas in the figure.
The trick that yields
	this remarkable asymmetric effect
is in the definition that
	\emph{a rate of change of phase is a frequency},
so that
	the instantaneous measures of the spectral phase contours
		obtained from the scanning
have the form of
	frequency shifts proportional to the slopes,
and therefore to
	the respective source distances.
These distance-dependent shifts
	resemble Hubble's law in astronomy
but are strictly linear
	due to their nonrelativistic, mundane origin,
and are rigorously predicted
	by basic wave theory,
		as follows.
The instantaneous phase of a sinusoidal wave
	at $(\bold{r}, t)$,
from a source at $\mathbf{r} = 0$,
	would be
\begin{equation} \label{e:twPhase}
	\phi(\mathbf{k}, \omega, t) =
		\mathbf{k} \cdot \mathbf{r} - \omega \; t
	\quad
	,
\end{equation}
and leads to
	the differential relation
\begin{equation} \label{e:dphase}
	\left. \Delta \phi \right|_{\omega,t}
	=
		\Delta (\omega \; t)
	+
		\mathbf{k} \cdot \Delta \mathbf{r}
	+
		\Delta \mathbf{k} \cdot \mathbf{r}
	\quad
	.
\end{equation}
The first term
	$\Delta (\omega \; t)$
clearly concerns the signal content if any,
	and has no immediate relevance.
The second term
	$\mathbf{k} \cdot \Delta \mathbf{r}$
expresses
	the path phase differences
		at any individual frequency,
and is involved in
	the ordinary Doppler effect,
as it describes phase change
	due to changing source distance,
as well as
	image reconstruction in holography
		and synthetic aperture radar (SAR),
as the reconstructed image concerns
	information of incremental distances of
		the spatial features of the image.
The third term,
	clearly orthogonal to other two,
represents
	phase differences across frequencies
		for any given (fixed) source distance.
Then, by varying a frequency selection $\widehat{k}$
	at the receiver,
we should see
	an incremental frequency, or shift,
\begin{equation} \label{e:fshift}
	\delta \omega =
		\lim_{\Delta t \rightarrow 0}
		\left[
			\frac{
				\left.
				\Delta \phi(\omega)
				\right|_{\omega,r,t}
			}
			{ \Delta k }
		\times
			\frac{ \Delta \widehat{k} }
			{ \Delta t }
		\right]
	=
		\left.
			\frac{\partial \phi}{\partial k}
		\right|_{\omega,r,t}
		.
			\frac{d \widehat{k}}{dt}
	\quad
	.
\end{equation}
(Note that $k$ can only refer to the incoming wave vector
	in the denominator.)
As this holds for each individual $\omega$,
	from equation (\ref{e:twPhase}),
the signal spectrum would be uniformly shifted by
	the normalized shift factor
\begin{equation} \label{e:zshift}
\begin{split}
	z(r) \equiv
		\frac{\delta \omega}{\widehat{\omega}}
	&=
		\frac{1}{\widehat{\omega}}
		\left.
			\frac{\partial \phi}{\partial k}
		\right|_{\omega,r,t}
		.
			\frac{d \widehat{k}}{dt}
	=
		\frac{r}{\widehat{k} c}
		.
			\frac{d \widehat{k}}{dt}
	=
		\frac{\beta r}{c}
	=
		\alpha \, r
	\quad
	,
\\
\text{with}\quad
	\beta &\equiv \widehat{k}^{-1} (d \widehat{k}/dt)
\quad
\text{and}
\quad
	\alpha = \beta / c
	\quad
	,
\end{split}
\end{equation}
	where $t$ denotes time kept
		by the receiver's clock.
Further,
the shifts reflect only
	the instantaneous value of $\beta$,
which is the normalized rate of change
		at the receiver.

Fig.~\ref{f:Parallax} shows
that $\delta \omega$ is
	a temporal analogue of spatial parallax,
with the instantaneous value of $\alpha \equiv \beta/c$
serving in the role of
	lateral displacement of the observer's eyes:
at each value
	$\alpha_1$ or $\alpha_2$,
the signal spectrum $\mathcal{F}(\omega)$ shifts to
	$\mathcal{F}(\omega_1)$ or
	$\mathcal{F}(\omega_2)$,
	respectively,
and if the distance to the source were increased to
	$r + \delta r$,
the spectrum would further shift to
	$\mathcal{F}(\omega_3)$.
Unlike with ordinary (spatial) parallax,
	the receiver can be fixed \emph{and monostatic},
and yet exploit
	the physical information
		of source distance.

\begin{figure}[ht]
	\centering
	\psfig{file=phenv-2.eps}
	\caption{Temporal parallax}
	\label{f:Parallax}
\end{figure}

\Section{impl}{Physics of realization}

There are three basic ways of accomplishing
	Fourier decomposition or frequency selection
in a receiver: using
		diffraction or refraction,
		a resonant circuit,
	or
		by sampling and digital signal processing (DSP).
Realization of the shifts in all three methods
	has been described,
along with how this ``virtual Hubble'' effect
	had remained unnoticed so long
	[\ibid].
A review of
	the diffraction and DSP methods
is necessary as theoretical foundation for
	the application ideas to be discussed.
These methods would also be applicable for
	array antennas and software-defined radio,
		respectively.

\Subsection{diffract}{Diffractive implementation}

Diffractive selection of
	a wavelength $\widehat{k}$
concerns using a diffraction grating
	to deflect normally incident rays
to an angle $\theta$
	corresponding to the selected $\widehat{k}$,
as determined by the grating equation
	$n \lambda = l \sin \theta $,
	where $n$ denotes the order of diffraction,
		and $l$ is the grating interval,

The property exploited is that
rays arriving at one end of the grating
combine with rays
	that arrived at the other end
		\emph{a little earlier}.
If we could change the grating intervals $l$
	inbetween,
the rays that get summed at
	a diffraction angle $\theta$
		would correspond to changing
			grating intervals $l(t)$,
as depicted in
	Fig.~\ref{f:GratingInstants}.
Their wavelengths must then relate to
	the grating interval as
	$
	n \, d \lambda/dt = (dl/dt) \, \sin \theta
	$.
Upon dividing this by
	the grating equation,
we obtain
	the modified relation
	$
	\widehat \lambda^{-1} (d\widehat \lambda/dt)
	=
	\widehat l^{-1} (d\widehat l/dt)
	\equiv
		- \beta .
	$

\begin{figure}[ht]
	\centering
	\psfig{file=grating-instants.eps}
	\caption{Time-varying diffraction method}
	\label{f:GratingInstants}
\end{figure}

This result is not merely
	a coincidence of different wavelengths
		at the focal point,
as would be obtained
	with a nonuniform grating.
Each of the summed contributions itself
	has a time-varying $\lambda$,
so the variation over the sum is consistent
	with the component variations.
A grating gathers more total power
	than, say, a two-slit device,
and the waves arrive with
	instantaneously varying phases
\begin{equation} \label{e:phaserate}
		\frac{d \phi}{dt}
	\equiv
		- \widehat\omega
	=
		\frac{\partial \phi}{\partial t}
	+
		\nabla_\mathbf{r} \phi \cdot \mathbf{\dot{r}}
	+
		\nabla_\mathbf{r'} \phi \cdot \mathbf{\dot{r}'}
	+
		\nabla_{\mathbf{k}} \phi \cdot
			\mathbf{\dot{\widehat{k} }}
	\quad
	,
\end{equation}
where
	the first term $\partial \phi/\partial t$ is
		the signal ($\omega \; t$) contribution
			in equation (\ref{e:twPhase})
		and is equal to $-\omega$;
	the second term (in $\mathbf{\dot{r}}$) concerns
		the real motion and Doppler effect if any,
			to be ignored hereon;
	the third term (in $\mathbf{\dot{r}'}$) represents
		a similar Doppler shift from
			longitudinal motion of
				either end of the receiver,
		which would be generally negligible
			for $r' \ll r$;
	and
	only the last term concerns
		equation (\ref{e:fshift}).

\begin{comment}
The foregoing analysis partly explains why
the effect has generally escaped notice
	in the past.
The distance-dependent character of the shifts
	cannot be easy to notice
	in the transient effects of
		relative motion of a prism or a grating,
especially as
	the detected transient effect would concern
		only an individual wavelength.
To be noticeable,
a substantial pattern of spectral lines
	must appear shifted in unison,
but that requires
	uniform variation of grating intervals,
which is difficult to get
	at visible wavelengths.
Mechanical or thermal stresses
	are very difficult to apply uniformly
and
	at a continuously increasing or decreasing rate.
Acousto-optic cells are advertized as
	``continuously variable'',
but the continuity only refers to
	a large number of standing wave modes
		available for Bragg diffraction
--
if the excitation frequency is changed,
the new interval takes effect only after
	the new mode stabilizes.
The effect could be produced by
	varying the length of a Fabry-Perot resonator,
		similarly to moving the detector,
but the usual notion of Fourier analysis
	concerns steady-state spectra
and the transient effects
	have been easy to overlook.
\end{comment}

\Subsection{dsp}{DSP implementation}

In DSP,
$\widehat{k}$'s are determined by
	the sampling interval $T$,
and can therefore be continuously varied
	by changing $T$.
The discrete Fourier transform (DFT)
	is defined as
\begin{equation} \label{e:dft}
\begin{split}
	F(m \omega_T)
	&=
		\sum_{n = 0}^{N - 1}
			e^{i m \omega_T T}
			\,
			f(n T)
\\
\text{with the inverse}
\quad
	f(n T)
	&=
		\frac{1}{N}
		\sum_{m = 0}^{N - 1}
			e^{i m \omega_T T}
			\,
			F(m \omega_T)
	\quad
	,
\end{split}
\end{equation}
where
	$f$ is the input signal;
	$T$ is the sampling interval;
	$N$ is the number of samples per block;
and
	$\omega_T = 2 \pi / N T$.
The inversion is governed by
	the orthogonality condition
\begin{equation} \label{e:dordorth}
	\sum_{n = 0}^{N - 1}
		e^{i m \omega_T T}
		\,
		e^{i (k r - n \omega_T T)}
	=
		\frac{
			1 - e^{i (m - n)}
		}{
			1 - e^{i (m - n)/N}
		}
	=
		N \delta_{mn}
	\quad
	,
\end{equation}
where $\delta_{mn} = 1$
	if $m = n$, else $0$.
These definitions as such suggest that
the instantaneous selections
	$\widehat\omega_T \equiv \widehat k c$
can be varied via
	the sampling interval $T$.

To verify that
	a controlled variation of $T$
will indeed yield
	the desired shifts,
observe that in equation (\ref{e:phaserate}),
both the real Doppler terms, in
	$\dot{\mathbf{r}}$
and
	$\dot{\mathbf{r}}'$,
can be ignored,
	as $r' \ll r$ for sources of practical interest.
The surviving term on the right is
\begin{comment}
\begin{equation*}
		\frac{d \phi}{dt}
	\equiv
		- \widehat\omega
	=
		\frac{\partial \phi}{\partial t}
	+
		\frac{\partial \phi}{\partial k}
		\frac{d\widehat k}{dt}
	\quad
	,
\end{equation*}
\end{comment}
$
	(\partial \phi/\partial k)
	(d\widehat k/dt)
$
where
	$\partial \phi/\partial t = - \omega$,
as before,
and
\begin{equation*}
	\frac{d\widehat k}{dt} =
		\frac{1}{c}
		\frac{
			d \widehat \omega_T
		}{dt}
	=
		\frac{1}{c}
		\frac{d}{dt}
		\left(
			\frac{2 \pi}{NT}
		\right)
	=
		- \frac{2 \pi}{N c T^2}
		\frac{dT}{dt}
	=
		- \widehat k
		\,
		\frac{1}{T}
		\frac{dT}{dt}
	\quad
	,
\end{equation*}
so that, corresponding to equation (\ref{e:zshift}),
	we do get
\begin{equation} \label{e:samplingmod}
	\widehat k^{-1} (d\widehat k/dt) \equiv
		\beta
	=
		- T^{-1} (dT/dt)
	\quad
	,
\end{equation}
confirming the desired effective variation of
	$\widehat{k}$'s.
$\bold{\Box}$

Fig.~\ref{f:TimeDomain} explains the result.
The incoming wave
presents increasing phase differences
	$\delta \phi_1$,
	$\delta \phi_2$,
	$\delta \phi_3$, ...
within the successive samples obtained from
	the diminishing intervals
	$\delta T_1 = T_1 - T_0$,
	$\delta T_2 = T_2 - T_1$,
	$\delta T_3 = T_3 - T_2$,
	\etc
From the relation
	$\widehat \omega_T \equiv \widehat k c = 2 \pi / N T$,
gradients of
	the spectral phase contours
		can be quantified as
\begin{equation*}
	\frac{\partial \phi}{\partial T} =
		\frac{\partial \phi}{\partial \widehat k}
		\frac{d \widehat k}{dT}
	=
		-
		\frac{2 \pi}{N c T^2}
		\frac{\partial \phi}{\partial \widehat k}
\end{equation*}
so that by equation (\ref{e:samplingmod}),
	each phase gradient reduces to
\begin{equation} \label{e:ksampling}
		\frac{\partial \phi}{\partial T}
		\frac{dT}{dt}
	=
		\frac{- 2 \pi}{N c T}
		\frac{\partial \phi}{\partial \widehat k}
	\cdot
		\frac{1}{T}
		\frac{dT}{dt}
	=
		\frac{\widehat \omega_T}{c}
		\frac{\partial \phi}{\partial \widehat k}
	\cdot
		\frac{1}{\widehat k}
		\frac{d\widehat k}{dt}
	=
		\frac{\partial \phi}{\partial \widehat k}
		\frac{d\widehat k}{dt}
\end{equation}
identically,
	validating the approach.

\begin{figure}[h]
	\centering
	\psfig{file=td-nopat.eps}
	\caption{Variable sampling method}
	\label{f:TimeDomain}
\end{figure}

\begin{comment}
This mechanism too is clearly rather simple,
	and we may once again wonder how it escaped notice,
especially given
	the sophisticated analytical tools available today.
Again,
the inherently nonstatic nature of the effect
	seems to be the cause,
as even wavelet analysis concerns
	strictly static time and frequency scales.
\end{comment}

\Subsection{scale}{General operating principles}

For a desired shift $z$ at a range $r$,
	equation (\ref{e:zshift}) yields
\begin{equation} \label{e:scaling}
	\frac{\delta \widehat{k}}{k}
\equiv
	- \frac{ \delta {T}}{T}
	\approx
		\frac{ c z \; \delta t}{r}
\end{equation}
for normalized incremental change of
	the receiver's grating or sampling interval
required
	over a small interval $\delta t$.
In a DSP implementation,
	appropriate at suboptical frequencies,
this determines
	the needed rate of change
		over a sampling interval $\delta T$.
As $\delta T$ would be nominally chosen
based on
	the (carrier) frequencies of interest
		and not on the range,
the nominal rate of change for achieving
	a useful $z$ at a given range $r$
would be independent of
	the operating frequencies.

For example,
for a $1~\giga\hertz$ signal,
a suitable choice of sampling interval
	is $\delta t = 100~\pico\second$,
		regardless of the range.
Chosing $z = 2$,
which is fairly large in astronomical terms
	but convenient for the present purposes,
we obtain
	$
	\delta \widehat{k}/{k}
		= 6 \times 10^{-2} / r
	$
per sample
	(taking $c = 3 \times 10^8~\metre\per\second$).
At $r = 1~\metre$,
the result is a somewhat demanding
	$6 \times 10^{-2}$,
	or $6\%$, per sample,
but for $r = 1, ~10$ or $1000 ~\kilo\metre$,
it is
	$
	6 \times 10^{-5},~
	6 \times 10^{-8}
	$
and
	$
	6 \times 10^{-11}
	$
per sample,
	respectively.

To be noticed is that
since equation (\ref{e:scaling}) prescribes
	a normalized rate of change,
the instantaneous rate of change must grow
	\emph{exponentially}
		in the course of the observation,
since by
	integrating equation (\ref{e:scaling}),
we get
\begin{equation} \label{e:exp}
	\Delta\widehat{k}
\equiv
	\Delta T ^{-1}
	=
		e^{c z \Delta t / r}
	,
\end{equation}
where $\Delta t$ denotes
	the total period of observation.
While an exponential variation is
	generally difficult to achieve
		other than with DSP,
two other problems
	must be also contended with:
the total variation over
	an arbitrarily long observation
		would be impossible anyway
and
it could exceed the source bandwidth even over
	a fairly short observation.

The only solution is to split
	the observation time into windows,
		as in DFT,
resetting the sampling interval and its variation
	at the beginning of each window,
so that
	the variation is only continuous
		within each window.

For example,
the normalized rate
	$6 \times 10^{-8}$
calculated above for
	$100~\pico\second$ sampling
and
	$10~\kilo\metre$ range
amounts to
	$3.773 \times 10^{260}$ for
		each second of observation!
Over a $1~\micro\second$ window, however,
	the sampling rate would change
		only by the factor $1.0006$.
This is also
	the spectral spread required of
the wavepackets emitted by the source,
and conversely,
	the window can be set to match
		the spread.

It is possible to \emph{cascade}
	multiple stages of gratings or DSP,
in order to multiply
	the magnitude of the shifts.
Cascading seems straightforward with
	optical gratings,
but in DSP,
as the data stream is already broken up
	into discrete samples by the first stage,
and the sampling interval is to be varied
	in each successive stage
		relative to its predecessor,
it becomes necessary to interpolate the samples
	between stages.
Interpolation is similarly needed for
	the reverse shift stage in
		distance-based signal separation
	(Section \ref{ss:ddm}).

These shifts have been verified only by simulation
for
	electromagnetic waves.
For sound data
	($c \approx 330~\metre\per\second$),
the software does reveal
	nonuniform spreading of
		subbands of acoustic samples,
consistent with
	the likely distributions of their sources.
Testing in an adequately equipped sound laboratory,
	as well as with radio waves,
is still needed.

\Subsection{diff}{Differential techniques}

The effect thus scales well
	from terrestrial to planetary distances,
but
the incremental changes required per sample
	are very small.
Errors in these changes,
	from inadvertent causes
		including thermal or mechanical stresses,
could lead to large errors
	in distance measurements using the effect.
The linearity of the effect can be exploited
to offset such errors by
	differential methods.
From equation (\ref{e:zshift}), we have
\begin{equation} \label{e:zdelta}
	\Delta z
	=
		\alpha \Delta r
	+
		r \Delta \alpha
	+
		o(\Delta r, \Delta \alpha)
	,
\end{equation}
relating first order uncertainties.
First order error due to
	an uncertainty $\Delta \alpha$
can be eliminated by measurements
	using different $\alpha$
and using the differences.

Specifically,
an error $\Delta z$ would limit
	the capability for source separation,
		presuming $r$ is known.
By designing a receiver to depend on
	the difference between two sets of shifts
		for the same incoming waves,
\eg by applying the DSP method twice
	with different values of $\alpha$,
a $\Delta z$ error can be eliminated.
Conversely,
	for ranging applications,
where $z$ is the measured variable,
from
	$
	r = z / \alpha
	$,
we get
\begin{equation} \label{e:rdelta}
	\Delta r
	=

		\Delta z / \alpha
	-
		r \Delta \alpha
	+
		o(\Delta z, \Delta \alpha)
	,
\end{equation}
so that
	the first order error $\Delta r$ can be
		once again eliminated using differences.
This is
	a temporal form of triangulation,
illustrated by the lines for
	$\alpha_1$ and $\alpha_2$
that converge at $r$
	in Fig.~\ref{f:Parallax}.
Higher order differences would yield
	more accuracy.

\Subsection{phaseshift}{Phase shifting and path length variation}

The main difficulty with
	the method of Section \ref{ss:diffract}
is ensuring uniformity of the grating intervals
	even while they are changing.
The method is not realizable by acousto-optic cells
	for this reason.
Controlled nonuniform sampling and sample interpolation
	pose similar difficulties for DSP%
	\footnote

	Radio telescopes, for instance, incorporate
		1 or 3 bit sampling at an intermediate frequency
			for correlation spectroscopy.
	Interpolation would add more noise
		to this already low phase information.
	}.\xspac

The solution is to instead vary the path length
	\emph{after} a fixed grating,
say using the longitudinal Faraday effect
	and circular polarization.
As the variation now concerns a bulk property,
	it would be easier.
The proof is straightforward.
The analogous simplification for DSP is to modify,
	instead of the sampling interval $T$,
the forward transform kernel to
\begin{equation} \label{e:kdft}
	F^{(\omega)} (m \widehat{\omega}_0)
	\equiv
	\sum_{n = 0}^{N - 1}
		e^{i m \widehat{\omega}(t) T}
		\,
		f(nT)
	,
\quad
	\widehat{\omega}(t)
	=
		\widehat{\omega}_0
		\, e^{\beta \, n T}
	,
\end{equation}
\ie
	apply changing phase shifts
		to the successive samples
	corresponding to the path length.
This is more complex,
but avoids interpolation noise
	and access to the RF frontend.

\Section{formalism}{Source separation theory}

\Subsection{kernel}{Transformation kernel}

Fourier theory more generally involves the continuous form of
the orthogonality condition (\ref{e:dordorth}),
given by
\begin{equation} \label{e:ordorth}
	\int_t
		e^{i \widehat\omega t} \,
		e^{i (k r - \omega t) } \, dt
	=
		e^{ikr} \delta(\widehat\omega - \omega)
	\quad
	,
\end{equation}
where
	$\delta()$ is the Dirac delta function.
All of the methods described require continuously varying
	the mechanism of frequency selection.
This is equivalent, in a basic sense,
	to varying the receiver's notion of
		the scale of time,
the result being
	a change of the orthogonality condition to
\begin{equation} \label{e:modorth}
\begin{split}
	\int_t
		e^{i \widehat\omega \Delta(t)}
		e^{i [kr \Delta(r) - \omega t]}
		\, dt
	&=
	\int_t
		e^{ikr\Delta}
		\,
		e^{i (\widehat\omega \Delta - \omega t)}
		\, dt
\\
	&\equiv
		e^{ikr\Delta}
		\,
		\delta [\widehat\omega \Delta - \omega]
	\quad
	,
\end{split}
\end{equation}
where
	$\Delta \equiv \Delta(r) = (1 + \alpha r)$
	from equation (\ref{e:zshift}),

but is also equivalent to
	$\Delta(t) = (1 + \alpha ct)$,
via the relation
	$c = r/t$.

How did we get to equation (\ref{e:modorth})?
Notice that
	equation (\ref{e:ordorth})
already contains
	the frequency selection factor
		$e^{i \widehat{\omega} t}$
--
the $\Delta$ in the exponent is
	the variation of
		$\widehat{\omega}$
	provided by the methods of Section \ref{s:impl}.
The second factor
	$e^{i [kr\Delta(r) - \omega t]}$
must correspondingly represent
	the incoming component picked by
		the modified selector;
its only difference from
	equation (\ref{e:ordorth})
is the $\Delta$ multiplying
	the path contribution to phase, $kr$.
Why should this $\Delta$ multiply $kr$
	and not $\omega t$?
The answer lies in the basic premise of
	equations (\ref{e:twPhase})-(\ref{e:zshift})
that our receiver manipulates
	the phase of each incoming Fourier component
		individually,
hence $\omega t$, representing
	the signal component of the instantaneous phase,
		is not touched.

The $kr$ term is affected, however,
since the summing builds up
	amplitude at only
		that value of $k$ which matches
	$
	\widehat{\omega} \equiv \widehat{k} c
	$.
This aspect is readily verified by simulation,
and it defines a lower bound on
	the window size (Section \ref{ss:scale})
		in the optical methods,
since the window reset would break
	the photon integration in a photodetector.

Equation (\ref{e:modorth}) is more general than
	equation (\ref{e:ordorth}),
as the latter corresponds to
	the special case of $\Delta = 1$
		or $\alpha = 0$.
This property has
	a fundamental consequence
impacting
	all of physics and engineering:
\emph{
Since $\alpha$ refers to
	the physics of the instruments
and could be
	arbitrary small but nonzero,
there is fundamentally no way to rule out
	a nonzero $\alpha$ in
		any finite local set of instruments.
The only way to detect a nonzero $\alpha$
	is to look for spectral shifts
		of very distant objects
}
as a large enough $r$ would yield
	a measurable $z(r)$%
	\footnote
	\label{foot:cosmos}

	A personal hunch that something like this
		could be responsible for
			the cosmological redshifts,
	had prompted informal prediction of
		the cosmological acceleration to
			some IBM colleagues in ca.1995-1996.

	The modified eigenfunctions
		$e^{i [kr\Delta(r) - \omega t]}$
	are known in astrophysics
		as the photon and particle eigenfunctions
			over relativistic space-time
		\cite{Parker1969}.
	The natural value of
		[$\alpha \approx$] $10^{-18}~\reciprocal{\second}$
	cited in the Introduction can come from
	the probability factor
		$e^{-W/k_B T}$
	for cumulative lattice dislocations
		under centrifugal or tidal stresses
			causing creep.
	At $T \approx 300~\kelvin$,
	it is
		$3 \times 10^{-11}~\reciprocal{\second}$
			at $W = 1~{\ensuremath \mathrm{eV}\xspace}$,
		$1.5 \times 10^{-18}~\reciprocal{\second}$
			at $W = 1.7~{\ensuremath \mathrm{eV}\xspace}$,
		\etc
	The point is that
		solid state theory mandates such creep,
	but it's untreated
		in any branch of science or engineering.
	Many of these details were put together in
		mundane \texttt{arxiv.org} articles
		\cite{PrasadArxiv},
	to be eventually compiled into
		a comprehensive paper.
	Thanks to NASA/JPL's continued portrayal of the anomaly
		as acceleration,
	\emph{all} other explanations tested or offered
		have been more obvious or exotic,
			and totally fruitless
		(\cf \cite{Anderson2002}).
	}.

\begin{comment}
	The NASA/JPL researchers incidently looked for
		a systematic error in earth clocks
			to explain the Pioneer 10/11 data
		\cite{Anderson2002},
	but only local consistency
		could at all be verified.
	The only distant ``clocks'' we have ever had are
		NASA's six spin-stabilized deep space missions
		\cite{Anderson1998},
	and
	they consistently point to an inconsistency with
		``deep space time''.
	Variations in the observed error,
	and its interpretation as acceleration,
		have led to confused interpretations
			by many reseachers,
	but all of the data correlates well with
		centrifugal stresses and solar tide variations.

	Another, related set of systematic errors has been identified
		in the calibration of space telescopes
			and is documented elsewhere
		\cite{Prasad2004USPspec}.
\end{comment}

\Subsection{ddm}{Distance division multiplexing\texttrademark

Almost all the applications to be discussed
critically depend on
	the fundamental separation of sources
	enabled by
		the present wave effect.
Using the notation of quantum theory,
we may denote
	incoming signals by ``kets'' $\ket{}$,
and
	the receiver states
		as ``bras'' $\bra{}$,
and rewrite
	equation (\ref{e:ordorth}) as
	$
	\iprod{\widehat\omega}{\omega, r}
	=
		e^{ikr}
		\iprod{\widehat\omega}{\omega}
	$.
Equation (\ref{e:modorth})
then becomes
\begin{equation} \label{e:qshift}
\begin{split}
	\iprod{
		\widehat\omega,
		\frac{d \widehat\omega}{dt}
	}{\omega, r}
	&\equiv
		\bra{\widehat\omega}
		H
		\ket{\omega, r}
	=
		e^{ikr\Delta(r)}
		\,
		\iprod{
			\widehat\omega
		}
		{
			\frac{\omega}{
				\Delta(r)
				}
		}
\\
	&=
		e^{ikr\Delta(r)}
		\,
		\delta \left(
			\widehat\omega
			-
			\frac{\omega}{
				\Delta(r)
				}
			\right)
	\quad
	,
\end{split}
\end{equation}
where
	$\bra{\widehat\omega, d\widehat\omega/dt}$ and
	$\bra{\widehat\omega}$
are
	the modified and original states of the receiver,
	respectively.
The sampling clock and grating modifications
then correspond to the operator $H$
\begin{equation} \label{e:qop}
	H \ket{\omega, r}
	=
		e^{ikr\Delta(r)}
		\,
		\ket{\frac{\omega}{\Delta(r)}}
\end{equation}
of an incoming wave state
	$\ket{\omega,r}$.
These equations attribute the shift to
	the incoming value $\omega$,
instead of
	to $\widehat\omega$ selected instantaneously
since for the operator formalism,
	$\bra{\widehat\omega}$ must represent
		an eigenstate resulting from observation.

Equations (\ref{e:zshift}) and (\ref{e:qop})
both say that as in the Doppler case,
	the shift is proportional
		at each individual frequency $\omega$.
This also means that
	the spectrum expands by the factor $\Delta$,
as illustrated in Fig.~\ref{f:DDM}
for the case of
	a common signal spectrum
		$F(\omega)$
emitted by two sources
	at distances $r_1$ and $r_2 > r_1$,
		respectively.
For a rate of change factor $\alpha$
	applied at the receiver,
the signals will shift and expand by
	$\Delta_1 \equiv (1 + \alpha r_1)$
and
	$\Delta_2 \equiv (1 + \alpha r_2) > \Delta_1$,
respectively.

\begin{figure}[ht]
	\centering
	\psfig{file=ddm.eps}
	\caption{Separation by source distance}
	\label{f:DDM}
\end{figure}

Despite the increased spreading of the spectrum,
there is opportunity for
	isolating the signal of a desired source
if its shifted spectrum comes out
	substantially separated
	from its neighbours,
as shown.
We could, for instance, apply
	a band-pass filter $\widetilde{G}_1$
to the overall shifted spectrum
	$H F_1 + H F_2$,
such that
	$
	\widetilde{G}_1 ( H F_1 + H F_2 )
	\approx \widetilde{G}_1 H F_1 .
	$
Writing $H$ as a function of $\alpha$,
	by equations (\ref{e:modorth}) and (\ref{e:qop}),
we get
\begin{equation} \label{e:negH}
	H^{-1}(\alpha) = - H(\alpha)
	\quad
	.
\end{equation}
We could apply a combination of
	mixing and frequency-modulation operations
to shift and compress
	$\widetilde{G}_1 H F_1$ back to $\approx F_1(\omega)$,
or just apply
	a second $H$ with a negative $\alpha$.
In general,
a set of distance-selecting
	``projection operators''
	$H^{-1} \widetilde{G}_i H$,
can thus be defined by
	the conditions
\begin{equation} \label{e:selection}
	H^{-1} \widetilde{G}_i H
		\,
		\sum_j F(r_j, \omega)
	\approx
		F(r_j, \omega)
\text{~or~}
	H^{-1} \widetilde{G}_i H
	\approx
		\delta_{ij}
\end{equation}
With base-band prefilters
	$G_i F_i \approx F_i$,
they yield
\begin{equation}
	H^{-1} \widetilde{G}_i H
	=
		\delta_{ij} G_j
\quad\text{so that}\quad
	\widetilde{G}_i H
	=
		H G_i
	\quad
	.
\end{equation}

As $H$ is parametrized by $\alpha$
	independently of $i$,
	$
	H\widetilde{G}_i H
	$
provides spectral separation
	without prior knowledge of $r_i$.
It is also independent of signal content,
wherein
	the source coordinates could be supplied
		or included via modulation,
but since $H$ depends only on
	the path phase contribution,
it provides separation at
	a more basic level.
Coordinate-based spread spectrum coding
	could be combined,
for example,
to differentiate sources
	at the same distance,
as an alternative to or to improve over
	current phased array antennas.

It has been further suggested \cite{Prasad2005}
that DDM raises
	the theoretical capacity of a channel to
	$\sim 2 \mathcal{C} L / \lambda \gg \mathcal{C}$,
where
	$\mathcal{C}$ is the traditional Shannon capacity
		for a source-receiver pair using the channel,
by admitting
	additional sources into the channel
		at intermediate distances.
Further,
if combined with
	directional selectivity say from
		phased array antennae,
we would get true
	\emph{space-division multiplexing}

		[\ibid].
\begin{comment}
	as illustrated in Fig.~\ref{f:sdm}.
The figure shows
	how a source $A$ may be isolated by
		its spatial coordinates
	within the region
	$(r \pm \delta r, \theta \pm \delta \theta)$,
from interference from other sources
using
	the same frequencies, codes, \etc

\begin{figure}[ht]
	\centering
	\psfig{file=idealrcv.eps}
	\caption{Source separation and space division multiplexing}
	\label{f:sdm}
\end{figure}
\end{comment}

\Section{mil}{Military and intelligence applications}

The above theory represents a long overdue
	rigorous analysis of
		the spectral decomposition of waveforms
	by a real, and therefore imperfect, instrument.
The nature of the error was inferred, as mentioned,
from an immense gamut of
	astronomical,
	planetary
and
	terrestrial data,
spanning all scales of range
	ever measured.
A natural occurrence of the effect
	subsequently deduced from creep theory
		(footnote, page \pageref{foot:cosmos})
could have been contradicted
	in at least one set of data,
but remains totally consistent%
	\footnote

	Consistency with GPS datasets
		has also been recently verified.
	Calculations from GPS base stations data indicate
		an ongoing rising of land
	at a median rate of
		$
		1.6
		~\milli\metre\per{\ensuremath {\mathrm y}\xspace}
		$
		(see \texttt{http://ray.tomes.biz}),
	large enough to easily accommodate
		a nontectonic apparent expansion of the earth
	at $H_0 \times 6.371~\kilo\metre
		\approx 0.437~\milli\metre\per{\ensuremath {\mathrm y}\xspace}$,
	for the natural occurrence.
	}.
Instead, we clearly face
	a converse problem in physics itself,
that its patterns
	have been so well explored on earth
that only astronomical tests can expose
	remaining shortcomings
	\cite{Anderson1998}.
The known laws of physics only reflect
	understood mechanisms,
like
	interference and the Doppler effect
and only partially%
	\footnote

	E.g.,
	diffractive corrections in
		astrophysics and quantum field theories
	are limited to
		the Fraunhofer and Fresnel approximations,
	both being limited, by definition, to
		total deflections of $\le \pi/2$.
	Cumulative diffractive scattering would contribute,
		as known in microwave theory,
		a decay $e^{-\sigma r}$ to the propagation.
	This makes \emph{all wavefunctions}
		in the \emph{real universe} Klein-Gordon eigenfunctions,
	and gives light
		a rest-mass...
	}.
We once again face a need to lead physics
	by engineering%
	\footnote

	Recalling the classic case of thermodynamics.
	The present effect
	implies that photons,
		traditionally viewed as indivisible and immutable,
			are reconstituted
		by every real, imperfect, receiver.
	Though it is yet to be demonstrated with light,
	the theory of Sections \ref{s:theory} and \ref{s:impl}
		is fundamental and does not permit
			a different result for quanta.
	}.\xspac

\Subsection{shift}{Precision, power-independent frequency shifting}

This application would itself be
	a direct, visible test of the effect,
the idea being that
	the method could be applied also to
a proximal source
	at a precisely known distance
in order to uniformly scale its spectrum
	with great accuracy.
Accuracy is expected because
	the source distance
can be accurately set
	by simple mechanical means,
independently of
	the exponential control of $\alpha$.
The mechanism would be
	the first means for shifting frequencies
that is independent of
	the nature and energy of the waves,
		and their frequencies and bandwidth,
yet realizable by
	a mundane, static means.
The main difficulty is the magnitude of $\alpha$
	needed for source distances of under $1~\metre$,
but it should be
	eventually solvable.

Likely applications include
transformation of high power
	modulated $\giga\hertz$ carriers to
		$\tera\hertz$ or optical bands,
and	efficient, tunable
		visible light, UV or even X- or $\gamma$-rays,
all
	with controlled coherence
and
	without nonlinear media.

\ifthenelse{\boolean{extended}}{
\cbstart

Two forms of implementation are being explored,
	the variable grating scheme of
		Section \ref{ss:diffract},
	exploiting
		the magnetostrictive ``smart material'' Terfenol-D,
and
	the path variation method of
		Section \ref{ss:phaseshift}
	using a liquid crystal or a photorefractive medium
		for a faster modulation
	than seems possible with
		the Faraday effect in glasses.
Another aspect being studied is
	folding of the source path
using
	an optical fibre or a transmission line,
so as to ensure realizability of
	possibly several metres in a compact package
and
	at least $1~\metre$ in a single chip.
The Terfenol-D design,
	prepared for NSF SBIR/STTR Solicitation 05-557,
envisaged
	a grating etched directly on
		the side of a Terfenol-D medium,
and was expected to provide
	shift factors $z \ge 2$
at least with
	a strong, distant, commonly available source,
	\viz the sun,
as a first step of
	optical validation.
\cbend
}{}

\Subsection{radar}{Monostatic ranging and passive radar}

This application was immediately envisaged
	when the effect was first actually suspected%
	\footnote

	The idea of such an effect had been disclosed,
	by appointment, to an IP attorney
		on the morning of 2001.9.11.
	The mechanism itself (Section \ref{s:theory})
		was uncovered only much later in 2004.
	},
taking from the well known notion of
	the cosmological distance scale
		available from the Hubble redshifts.
By providing
	a ``virtual Hubble flow'' view
that can be activated and scaled
	at the receiver's discretion,
the present methods enable
	ranging, or distance measurement,
of any source
	that can be seen or received,
	at only half
		the round-trip delay,
and
	zero transmitted power.

To compare,
as traditional active radar depends on
	round trip times (RTT),
its range is fundamentally limited
	by the transmitter power.
While Venus and Mars have been explored by radar,
for example,
the ranging of other planets
	and of astronomical objects in general
is largely limited
	to spatial triangulation,
including
using the earth's orbit itself
	as the baseline.
Tracking of orbiting satellites
places similar demands on
	the radar transmitter power $P$,
as the range $R$ is governed by
	the ``radar power law''
	$P \propto R^4$.
This makes
	current (active) radars generally bulky.
The power law is relaxed for
	cooperative targets with transponders,
		to $P \propto R^2$,
where $P$ denotes
	transponder power,
and the range is obtained from the RTT of
	a transponded signal%
	\footnote

	This is generally what's used for deep space probes
		\cite{Bender1989,Vincent1990}.
	}.
The present methods make this lower power law
	available for all targets,
and also eliminate the need for
	a spatial baseline for parallax,
as the baseline is now given by
	the range of variation of $\alpha$
		at the receiver.
As shown in Section \ref{ss:scale},
a very large baseline seems to be indeed possible
	with DSP alone
for both terrestrial
	and earth orbit distances.

Current passive radar technology,
	like Lockheed's Silent Sentry,
involves
	coherent processing of the received scatter
of energy from
	television broadcast
	(see US Patent 3,812,493, 1974)
and
	cellular base stations.
The accuracy is again dependent on
	information representing overall trip times,
and hence entails
	coherent processing.

Accurate ranging is now possible via
	triangulation using temporal parallax,
as explained
	in Section \ref{ss:diff}.
The difference is that
	coherent processing can provide
		imaging
	with subwavelength precision,
but
	triangulation would be simpler
	for ranging and tracking.
We still need to illuminate silent targets,
but the processing could be simplified
	for the same accuracy.
The accuracy of tracking could improve
	for radiating targets
given
	the lower overall trip time.
We can also use
	temporal triangulation
to simplify traditional radar
	and as a fast, efficient cross-check.

\Subsection{jam}{Overcoming interference, jamming and noise}

The capability for separating sources
	regardless of the signal content or modulation
is also a guarantee that
	a desired signal can be isolated
		from any interfering signals
at the same frequencies and
	bearing the same modulations or spread-spectrum codes,
so long as
	the interfering sources are
		at other directions or distances
	from the receiver.

Simulation shows that
	even signals in destructive interference
		can be separated.
The criterion for separation is simply
	the bandwidth to distance ratio:
denoting the low and high frequency bounds of
the signal bandwidth $W$ by
	$\mathcal{L}$ and $\mathcal{H}$, respectively,
each of equations (\ref{e:zshift}) or (\ref{e:modorth})
	implies
\begin{equation}
	(1 + \alpha r_i) \mathcal{H}
		\le
	(1 + \alpha r_{i+1}) \mathcal{L}
\end{equation}
as the condition for separation between
	the $i$th and $(i+1)$th sources,
		assuming $r_i \le r_{i+1}$
(see Fig. \ref{f:DDM}).
This means
\begin{equation}
	\alpha r_i
	\ge
		\frac{W}{(\delta r_i/r_i) \mathcal{L} - W}
	=
	\left[
		\frac{\delta r_i}{r_i}
		\frac{\mathcal{L}}{W}
		-
		1
	\right]
\end{equation}
where $W = \mathcal{H} - \mathcal{L}$
	and $\delta r_i = r_{i+1} - r_i$.
As separation is only possible for
	$\alpha > 0$,
we need to have
	$\delta r_i > r_i W / \mathcal{L}$.
It then follows, however,
that sufficiently narrow subbands of
	the total received signal
	would be ``source-separable''
even when the condition should fail
	for $W$ as a whole.

All of the limitations are thus purely technological,
\eg
	how many and how narrow subbands
		we can construct,
	how linear we can make
		the subband and band-pass ($\widehat{G}$) filters
	(since phase envelope distortions will alter the shift),
and
	how large the stop-band rejection
		we can get.
The stop-band rejection ratio is important
	in overcoming jamming,
and the separation as such should suffice, in principle,
	for filtering out truly extraneous noise
		like lightnings.

\Subsection{isar}{Incoherent aperture synthesis}

In a conventional SAR,
multiple targets or features are imaged
	from the echos received on a moving platform,
		typically an aircraft or satellite,
	as depicted in Fig.~\ref{f:sar}.

\begin{figure}[ht]
	\centering
	\psfig{file=sar.eps}
	\caption{Aperture synthesis}
	\label{f:sar}
\end{figure}

If the onboard oscillator could maintain coherence for
	the duration of the flight,
\ie not suffer random phase shifts,
an entire target region could be imaged
	by a simple Fourier transform,
after correcting for
	the Doppler shifts in the received echos
		due to the radar's own motion.
At each point $Q$
	on the flight path,
echos are received from multiple features
	$A$, $B$, \etc
and corresponding to
	multiple previous locations
		$P_1$, $P_2$, ...
	of the transmitter.
Fourier techniques and
	their optical holographic counterparts
are most efficient
	for unravelling the information.

As stated in Section \ref{ss:radar},
the fundamental reason for coherent processing
	is our current dependence on RTT of radar echos
		for the range information.
The dependence is obviated by
	the wave effect.
In the example scenario,
echos received at $Q$
	from target features differing in direction,
	like $A$ and $B$,
can be separated by the phase information
	from an antenna array.
The echos from the same direction,
	such as from $A$ and $D$,
can now be similarly separated
	using the methods of
		Section \ref{s:impl}
--
in particular,
the changing delay transform
	of equation (\ref{e:kdft})
can be applied to
	recordings of the received echo waveform.

Coherent processing ordinarily has
	two advantages
that would seem to be impossible
	in any other approach:
it tends to curtail noise and is generally precise
	to within a wavelength.
We would get suppression of external noise,
	as discussed in Section \ref{ss:jam},
and subwavelength resolution,
	as shown in Section \ref{ss:diff},
so that applications including
	hand-held short-range carrier-deck operation,
for which coherent processing is difficult,
	become feasible.

\ifthenelse{\boolean{extended}}{
\Section{sim}{Simulation and testing}

\cbstart

The DSP approach of Section \ref{ss:dsp}
	was first tried in 2001 using Octave
	(an open source equivalent of Matlab),
on several \texttt{.wav} files
	that came with Windows and Linux distributions.
The expectation of discernible shifts
	seems to hold best for
the simple waveform of \emph{pop.wav}
	(Fig.~\ref{f:PopWave}),
with a fundamental frequency
	below $100~\hertz$.
Fig.~\ref{f:PopShift} shows
	the corresponding spectra:
line 1 (in red) is
	the spectrum of the unmodified samples,
and lines 2 (in green) and 3 (in blue) are
spectra obtained by
	interpolating the samples to simulate
		time-varying sampling rates
	according to Fig.~\ref{f:TimeDomain}.
The shifted spectra are spread out
	but otherwise preserved,
as expected,
	at the lower frequency end.
The harmonic peaks
	near $310~\hertz$ and $420~\hertz$
		are lost,
and
	the shifted spectra are increasingly weaker
--
both behaviours are expected as
	the interpolation reduces the number of samples.
\cbend

\begin{figure}[htb]
	\centering
	\psfig{file=popwaveform.eps, height=1.6in}
	\caption{Sample acoustic waveform}
	\label{f:PopWave}
\end{figure}

\begin{figure}[htb]
	\centering
	\psfig{file=pop.eps, height=1.6in}
	\caption{Original and shifted spectra}
	\label{f:PopShift}
\end{figure}

\cbstart

With other audio samples,
and with a newer simulator developed in Java
	especially to study
the performance of subband filtering
	described in Section \ref{ss:jam}
		for antijamming,
the frequency scaling of acoustic spectra
	is found to be less distinct.
Rounding and aliasing problems are exacerbated
	by the nonlinear sample interpolation,
but the main cause seems to be
	the distance to the microphone
		in the recording process
being
	equal to or smaller than
		the distribution of the recorded sources,
so that
	$\Delta(\omega t)$ dominates over
		the path length contribution
	$\mathbf{k} \cdot \Delta\mathbf{r}$
	(equation \ref{e:dphase}).
This can be solved by introducing
	a fixed delay corresponding to a large source distance,
and the expected effect of increased shifts
	was verified
		in the Octave tests.
In the new simulator,
different subbands in most audio \texttt{.wav} files
	are found to scale by different extents,
consistent with
	the interpretation of source spread.
Nevertheless,
tests with actual, well-sampled sonar data
	are needed to validate and characterize
		the effect adequately
	for use with sound.
The source spread problem would be worse for
	longer radio waves,
making
	the spectral phase gradients too small
		to be usable
	at long wavelengths.
This problem should vanish at short wavelengths,
and in particular,
	at optical wavelengths
where individual photons could be traced, in principle,
	to atoms that emitted them.
The source spread would likely arise again in
	laser output,
in proportion to the spread of time spent by the photons
	within the lasing cavity,
and this too
	remains to be tested.
\cbend

\cbstart

The new simulator continues to evolve
	as the lessons learnt are incorporated
		into the code.
Development is slow because
	the research is as yet unfunded
and being done in spare time spread across
	complementary research efforts
		that led to this discovery.
There were programming errors causing noise
	and limiting the range of $\alpha$
		in the simulation,
for instance,
and were identified and corrected only after
	the initial presentation and demonstration
		at WCNC 2005.
The simulator still uses sample interpolation
	(Section \ref{ss:dsp}),
which needs to be replaced with
	the more robust path variation scheme
		of Section \ref{ss:phaseshift}.
\cbend

\cbstart

Direct validation with real RF data and live feeds,
	for the DSP methods,
and with
	a Terfenol-D or liquid crystal device
		for optical wavelengths,
remain to be done,
the main limitation in each case being
	the lack of current skill with
		electronic circuit design,
	and the time,
	to implement them.
Nevertheless,
there is an enormous gamut of
	astronomical and geological evidence
consistent with
	the effect at optical wavelengths
and the uncorrected natural cause mentioned,
as reviewed in
	the WCNC presentation,
and the capability to recover
	signals even from total destructive interference,
also demonstrated at WCNC,
	should be hopefully sufficient to motivate
		independent validation efforts
and funding
	for full scale development and exploitation
		of this new physics.
The WCNC presentation and
	a demonstration of separation of an FM signal
		from similar interfering signals
	using the new simulator
have accordingly been made available online
	at \texttt{http://www.inspiredresearch.com} .
\cbend
}{}

\section*{Acknowledgement}

I thank
Asoke K. Bhattacharyya,
	for introducing me to
		inverse scattering in pulsed radar
	in 1984,
and thus to
	a key insight leading to this discovery.

\begin{raggedright}


\section{Introduction}

\label{sec:Introd}

Many of the high energy theories of fundamental physics are formulated in
higher dimensional spacetimes. In particular, the idea of extra dimensions
has been extensively used in supergravity and superstring theories. It is
commonly assumed that the extra dimensions are compactified. From an
inflationary point of view, universes with compact spatial dimensions, under
certain conditions, should be considered a rule rather than an exception
\cite{Lind04}. The models of a compact universe with non-trivial topology
may play important roles by providing proper initial conditions for
inflation.

The compactification of spatial dimensions leads to a number of interesting
quantum field theoretic effects, which include instabilities in interacting
field theories, topological mass generation and symmetry breaking. In the
case of non-trivial topology, the boundary conditions imposed on fields give
rise to the modification of the spectrum for vacuum fluctuations and, as a
result, to the Casimir-type contributions in the vacuum expectation values
of physical observables.

The Casimir effect is arguably the most poignant demonstration of the
reality of the quantum vacuum and can be appreciated most simply
in the interaction of a pair of neutral parallel plates. The
presence of the plates modifies the quantum vacuum, and this
modification causes the plates to be pulled toward each other with
a force $F\propto 1/a^4$ where $a$ is the plate separation. The
Casimir effect is a purely quantum phenomenon. In classical
electrodynamics the force between the plates is zero. The ideal
scenario occurs at zero temperature when there are no real photons
(only virtual photons) between the plates; thus, it is the ground
state of the quantum electrodynamic vacuum which causes the
attraction. One of the most important features of the Casimir
effect is that even though it is purely quantum in nature, it
manifests itself macroscopically (Figure 1). \footnote{Illustration
courtesy of Richard Obousy Consulting LLC and AlVin, Antigravit\'{e}}

\begin{figure}
\begin{center}
\includegraphics[width=11cm, height=7.5cm]{fig1.jpg}
\caption{Due to the non-trivial boundary conditions imposed on the
quantum vacuum, the plates are pulled toward each other due to a
force that is purely quantum in nature.} \label{fig1}
\end{center}
\end{figure}

Compactifed extra dimensions introduce non-trivial boundary
conditions to the quantum vacuum similar to those in the parallel
plate example, and Casimir-type calculations become important when
calculating the resulting vacuum energy (for the topological
Casimir effect and its role in cosmology see \cite{Most97} and
references therein).

One important question in the context of theories with extra
dimensions is why additional spatial dimensions hold some fixed
size. This is commonly referred to as `modulus stabilization'.
Broadly stated, the question is as follows: if there are
additional spatial dimensions, why do they not perpetually expand,
like our familiar dimensions of space, or alternatively, why do
they not perpetually contract? What mechanism is it that allows
for this higher space to remain compact and stable? Of the handful
of theories that attempt to answer this problem, the Casimir
effect is particularly appealing due to its naturalness.

We find this effect compelling due to the fact that it is a
natural feature intrinsic to the fabric of space itself. There is
much research in the literature that demonstrates that with some
combination of fields, it is possible to generate a stable
minimum of the potential at some extra dimensional radius
\cite{ac}-\cite{Saha05}. In addition, the Casimir effect has been
used as a stabilization mechanism for moduli fields which
parameterize the size and the shape of the extra dimensions both in
models with a smooth distribution of matter in the extra
dimensions and in models with matter located on branes.

The Casimir energy can also serve as a model for dark energy
needed for the explanation of the present accelerated expansion of
the universe (see \cite{Milt03}-\cite{bcp} and references
therein). One interesting feature of these models is the strong
dependence of the dark energy density on the size of the extra
dimension.

One compelling possibility that we would like to introduce in this
paper is that at the energies accessible in the next generation of
particle accelerators, Standard Model (SM) fields might interact
with the graviton Kaluza-Klein (KK) tower which would effect the
local minimum of the potential. This would have the effect of
locally adjusting the radius of the extra dimension during this
interaction. Because the dark energy density is a function of the
size of the extra dimension, one remarkable feature of this idea
is that because the extra dimensions may be (temporarily)
adjusted, so too would the local dark energy density. This
adjustment would mean that, for the duration of the interaction,
the expansion of spacetime would be changed due to
technological intervention. What is particularly appealing about
this predicted phenomenon is its potential application as a future
exotic spacecraft propulsion mechanism.

In the present paper we shortly review the uses of the Casimir
effect in both standard Kaluza-Klein type and braneworld scenarios
for the stabilization of extra dimensions and for the generation
of dark energy. We also explore the energy requirements that would
be needed to temporarily adjust the size of the higher dimension
and hypothesize that, with some imagination, this mechanism could
be used by a sufficiently advanced technology as a means of
spacecraft propulsion.

\section{Kaluza-Klein type models}

To be consistent with the observational data, the extra dimensions in the
standard Kaluza-Klein description are assumed to be microscopic, with a size
much smaller than the scale of four dimensions. A generic prediction of
theories involving extra dimensions is that the gauge and Yukawa couplings
are in general related to the size of extra dimensions. In a cosmological
context, this implies that all couplings depend on the parameters of the
cosmological evolution. In particular, if the corresponding scale factors
are dynamical functions their time dependence induces that for the gauge
coupling constants. However, the strong cosmological constraints on the
variation of gauge couplings coming from the measurements of the quasar
absorbtion lines, the cosmic microwave background, and primordial
nucleosynthesis indicate that the extra dimensions must be not only small
but also static or nearly static. Consequently, the stabilization of extra
dimensions in multidimensional theories near their present day values is a
crucial issue and has been investigated in a number of papers. Various
mechanisms have been considered including fluxes from form-fields, one-loop
quantum effects from compact dimensions, wrapped branes and string corrections.

Let us consider the higher-dimensional action
\begin{equation}
S=\int d^{D}x\sqrt{|\det g_{MN}|}\left\{ -\frac{1}{16\pi G}R\left[ g_{MN}%
\right] +L\right\} ,  \label{action}
\end{equation}%
with the matter Lagrangian $L$ which includes also the cosmological constant
term. We take a spacetime of the form $R\times M_{0}\times \ldots \times
M_{n}$ with the corresponding line element
\begin{equation}
ds^{2}=g_{MN}dx^{M}dx^{N}=g_{\mu \nu }dx^{\mu }dx^{\nu
}+\sum_{i=1}^{n}e^{2\beta _{i}}g_{m_{i}n_{i}}dx^{m_{i}}dx^{n_{i}},
\label{ds2reduc}
\end{equation}%
where $M_{i}$, $i=0,1,\ldots ,n$, are $d_{i}$-dimensional spaces, $g_{\mu
\nu }$, $\beta _{i}$ are functions of the coordinates $x^{\mu }$ in the
subspace $R\times M_{0}$ only, and the metric tensor $g_{m_{i}n_{i}}$ in the
subspace $M_{i}$ is a function of the coordinates $x^{m_{i}}$ in this
subspace only. In order to present the effective action in the subspace $%
R\times M_{0}$ in the standard Einsteinian form we make a conformal
transformation of the $(d_{0}+1)$-dimensional metric:%
\begin{equation}
\tilde{g}_{\mu \nu }=\Omega ^{2}g_{\mu \nu },\;\Omega =\exp \left[
\sum_{j=1}^{n}d_{j}\beta _{j}/(d_{0}-1)\right] .  \label{conftrans}
\end{equation}%
Dropping the total derivatives the action is presented in the form
\begin{equation}
S=\frac{\prod_{j}\mu ^{(j)}}{16\pi G}\int d^{d_{0}+1}x\sqrt{|\det \tilde{g}%
_{\mu \nu }|}\left\{ -R[\tilde{g}_{\mu \nu }]+\sum_{i,j}G_{ij}\tilde{g}^{\mu
\nu }\partial _{\mu }\beta _{i}\partial _{\nu }\beta _{j}-2U\right\} ,
\label{action2}
\end{equation}%
where $\mu ^{(j)}=\int d^{d_{j}}x\sqrt{|\det g_{m_{j}n_{j}}|}$, $%
G_{ij}=d_{i}\delta _{ij}+d_{i}d_{j}/(d_{0}+1)$, and%
\begin{equation}
U=\frac{\Omega ^{-2}}{2\prod_{j}\mu ^{(j)}}\int d^{D-d_{0}-1}x\prod_{j}\sqrt{%
|\det g_{m_{j}n_{j}}|}\left\{ \sum_{i}e^{-2\beta
_{i}}R[g_{m_{i}n_{i}}]-16\pi GL\right\} .  \label{Vpot}
\end{equation}%
In (\ref{Vpot}), $R[g_{m_{i}n_{i}}]$ is the Ricci scalar for the metric $%
g_{m_{i}n_{i}}$.

In the case of standard cosmological metric $\tilde{g}_{\mu \nu }$ with the
scale factor $\tilde{a}_{0}$ and the synchronous time coordinate $\tilde{t}$%
, the field equations for the set of fields $\beta _{i}=\beta _{i}(\tilde{t}%
) $ following from action (\ref{action2}) have the form
\begin{equation}
\sum_{j=1}^{n}G_{ij}\left( \beta _{j}^{\prime \prime }+d_{0}\tilde{\beta}%
_{0}^{\prime }\beta _{j}^{\prime }\right) =-\left( \frac{\partial U}{%
\partial \beta _{i}}\right) _{\tilde{\beta}_{0}},\;i=1,2,\ldots ,n,
\label{coseqnew2}
\end{equation}%
where $\tilde{a}_{0}=\tilde{a}_{0}^{(0)}e^{\tilde{\beta}_{0}}$, $%
a_{j}=a_{j}^{(0)}e^{\beta _{j}}$, the prime means the derivative with
respect to the time coordinate $\tilde{t}$ and the potential $U$ is defined
by the formula
\begin{equation}
U=\frac{1}{2\Omega ^{2}}\left( 16\pi G\rho +\Lambda
_{D}-\sum_{j=1}^{n}\lambda _{j}d_{j}/a_{j}^{2}\right) .  \label{U}
\end{equation}%
In (\ref{U}) $\rho $ is the matter energy density and $\Lambda _{D}$ is the $%
D$-dimensional cosmological constant, $\lambda _{j}=k_{j}(d_{j}-1)$, where $%
k_{j}=-1,0,1$ for the subspace $M_{j}$ with negative, zero, and positive
curvatures, respectively. In the case $\rho =0$ for the extrema of potential
(\ref{U}) one has the relations $\Lambda _{D}=(D-2)\lambda _{i}e^{-2\beta
_{i}}$ and $\partial ^{2}U/\partial \beta _{i}\partial \beta
_{j}=-2G_{ij}\Lambda _{D}/(D-2)$. It follows from here that for $\Lambda
_{D}>0$ the extremum is a maximum of the potential and is realized for
internal spaces with positive curvature. The corresponding effective
cosmological constant is positive. For $\Lambda _{D}<0$ the extremum is a
minimum and is realized for internal spaces with negative curvature. The
effective cosmological constant is negative.

There is a number of mechanisms giving contributions to the potential in
addition to the cosmological constant and curvature terms. An incomplete
list includes fluxes from form-fields, one-loop quantum effects from compact
dimensions (Casimir effect), wrapped branes, string corrections (loop and
classical). In the case of a single extra space for massless fields at zero
temperature the corresponding energy density is of the power-law form
\begin{equation}
\rho =\sigma a_{1}^{-q},  \label{rhocsigma}
\end{equation}%
with a constant $\sigma $ and $q$ being an integer. The values of the
parameter $q$ for various mechanisms are as follows: $q=D$ for the
contribution from the Casimir effect due to massless fields, $q=d_{1}-2p$
for fluxes of $p$-form fields, $q=p-d_{0}-2d_{1}$ for $p$-branes wrapping
the extra dimensions. The corresponding potential has the form (\ref{U})
with
\begin{equation}
U=\frac{1}{2\Omega ^{2}}\left( 16\pi G\sigma a_{1}^{-q}+\Lambda _{D}-\lambda
_{1}d_{1}a_{1}^{-2}\right) .  \label{Uc}
\end{equation}%
In accordance with Eq. (\ref{coseqnew2}), solutions with a static internal
space correspond to the extremum of the potential $U$ and the effective
cosmological constant is related to the value of the potential at the
extremum by the formula $\Lambda _{\mathrm{eff}}=2\Omega ^{2}U$.

Introducing the notations
\begin{equation}
y=ba_{1},\quad b=\left[ \frac{d_{1}(d_{1}-1)}{16\pi G|\sigma |}\right] ^{%
\frac{1}{q-2}},\;U_{0}=\frac{d_{1}(d_{1}-1)}{2}%
b^{2}(ba_{1}^{(0)})^{2d_{1}/(d_{0}-1)},  \label{Uc0}
\end{equation}%
potential (\ref{Uc}) is written in the form
\begin{equation}
U=U_{0}y^{-q-2d_{1}/(d_{0}-1)}\left[ \frac{\Lambda _{D}b^{-2}}{d_{1}(d_{1}-1)%
}y^{q}-k_{1}y^{q-2}+{\mathrm{sign}}(\sigma )\right] .  \label{Ucnew}
\end{equation}%
For the extremum with the zero effective cosmological constant one has%
\begin{equation}
k_{1}y^{q-2}=\frac{q}{2}\,{\mathrm{sign}}(\sigma ),\quad \Lambda
_{D}b^{-2}=k_{1}d_{1}(d_{1}-1)\left( 1-\frac{2}{q}\right) \left( \frac{2}{q}%
\right) ^{\frac{2}{q-2}}.  \label{Lambdab-2}
\end{equation}%
For positive values $q$ and for an internal space with positive (negative)
curvature the extremum exists only when $\sigma >0$ ($\sigma <0$). The
extremum is a minimum for $k_{1}(q-2)>0$ and a maximum for $k_{1}(q-2)<0$.
Potential (\ref{Ucnew}) is a monotonic decreasing positive function for $%
k_{1}=-1,0$, $\Lambda _{D}\geq 0$, $\sigma >0$, has a single maximum with
positive effective cosmological constant for $k_{1}=1,0$, $\Lambda _{D}>0$, $%
\sigma <0$ and $k_{1}=-1$, $\Lambda _{D}\geq 0$, $\sigma <0$. For the case $%
k_{1}=1$, $\Lambda _{D}>0$, $\sigma >0$ and in the $d_{0}=d_{1}=3$ model
where $\rho $ is generated by one-loop quantum effects ($q=D$ in Eq. (\ref%
{rhocsigma})) the potential (\ref{Ucnew}) is plotted in Figure \ref{fig2}
for various values of higher dimensional cosmological constant corresponding
to $\Lambda _{D}b^{-2}=2,(30/7)(2/7)^{2/5},3,4$. In the model with the
second value the effective $(d_{0}+1)$-dimensional cosmological constant
vanishes. The behavior of the potential (\ref{Ucnew}) in the other cases is
obtained from those described above by replacements $(k_{1},\Lambda
_{D},\sigma )\rightarrow (-k_{1},-\Lambda _{D},-\sigma )$ and $U\rightarrow
-U$.

\begin{figure}
\begin{center}
\includegraphics[width=8.5cm, height=6.5cm]{fig2.jpg}
\caption{The potential $U/U_{0}$ as a function of $ba_{1}$ in the model with
$d_{0}=d_{1}=3$ and with the Casimir effect as a source for $\protect\rho $.
The numbers near the curves correspond to the value of the parameter $%
\Lambda _{D}b^{2}$. For the curve with the zero cosmological constant $%
\Lambda _{D}b^{2}=(30/7)(2/7)^{2/5}$.} \label{fig2}
\end{center}
\end{figure}

Introducing the $D$-dimensional Planck mass $M_{D}$ in accordance with the
relation $G=M_{D}^{2-D}$, from (\ref{Uc0}) we see that in the model with
one-loop quantum effects the parameter $b$ is of the order of the higher
dimensional Planck mass. As it is seen from the graphs in figure \ref{fig2}
the stabilized value of the size for the internal space is of the order of $%
D $-dimensional Planck length $1/M_{D}$. Note that, as it follows from (\ref%
{action2}), for the effective 4-dimensional Planck mass one has $M_{\mathrm{%
Pl}}^{2}\approx (a_{1}M_{D})^{d_{1}}M_{D}^{2}$ and, hence, in this type of
models the effective and higher dimensional Planck masses have the same
order. Consequently, the size of the internal space is of the order of the
Planck length and is not accessible for the near future accelerators. Note
that our knowledge of the electroweak and strong forces extends with great
precision down to distances of order $10^{-16}$cm. Thus if standard model
fields propagate in extra dimensions, then they must be compactified at a
scale above a few hundred GeV range.

\section{Models with large extra dimensions}

In the previous section we have considered the models, where the extra
dimensions are stabilized by one-loop quantum effects with the combination
of curvature terms and the higher dimensional cosmological constant. We have
taken the Casimir energy density coming from the compact internal space in
the simplest form $\sigma a_{1}^{-D}$. The corresponding effective potential
is monotonic and cannot stabilize the internal space separately. In more
general cases, in particular for massive fields, the dependence of the
Casimir energy on the size of the internal space is more complicated and the
corresponding effective potential can stabilize the extra dimensions
separately.

As a simple model consider the case of a single extra space, $n=1$, being a
circle, $M_{1}=S^{1}$, with a scalar field $\varphi $ as a non-gravitational
source. In the discussion below we will assume that either the external
space $M_{0}$ is non-compact or the external scale factor is much greater
than the internal one. First we discuss the case of the untwisted field with
mass $M$\ satisfying the periodicity condition on $S^{1}$. The vacuum energy
density and the pressures along the uncompactified ($p_{0}$) and
compactified ($p_{1}$) directions are given by the expressions
\begin{eqnarray}
&&\rho =-\frac{2M^{d_{0}+2}}{(2\pi )^{d_{0}/2+2}}\sum_{m=1}^{\infty }\frac{%
K_{d_{0}/2+1}(Mam)}{(Mam)^{d_{0}/2+1}},\quad p_{0}=-\rho ,  \notag \\
&&p_{1}=-\rho -\frac{2M^{d_{0}+2}}{(2\pi )^{d_{0}/2+2}}\sum_{m=1}^{\infty }%
\frac{K_{d_{0}/2+2}(Mam)}{(Mam)^{d_{0}/2}},  \label{s2rhopT02}
\end{eqnarray}%
where $K_{\nu }(x)$ is the modified Bessel function of the second
kind. The corresponding effective potential has the form
\begin{equation}
U=\frac{1}{2}\left( a^{(0)}/a\right) ^{\frac{2}{d_{0}-1}}\left( 16\pi G\rho
+\Lambda _{D}\right) ,  \label{US1}
\end{equation}%
with a constant $a^{(0)}$. The condition for the presence of the static
internal space takes the form
\begin{equation}
d_{0}\rho -(d_{0}-1)p_{1}=\frac{\Lambda _{D}}{8\pi G}.  \label{s2condstat}
\end{equation}%
From formulae (\ref{s2rhopT02}) it follows that the left hand side of this
expression is positive and, hence, the static solutions are present only for
the positive cosmological constant $\Lambda _{D}$. The effective
cosmological constant is equal $\Lambda _{{\mathrm{eff}}}=-8\pi
G(d_{0}-1)(\rho +p_{1})$ and is positive. However, as it can be checked $%
(\partial /\partial a)[d_{0}\rho -(d_{0}-1)p_{1}]<0$ and the solutions with
static internal spaces are unstable.

In the case of a twisted scalar with antiperiodicity condition along the
compactified dimension the corresponding energy density and pressures are
obtained by using the expressions (\ref{s2rhopT02}) with the help of the
formula
\begin{equation}
q_{t}(a)=2q(2a)-q(a),\quad q=\rho ,p_{0},p_{1}.  \label{s2qtw}
\end{equation}%
For the twisted scalar the corresponding energy density is positive. The
solutions with static internal spaces are stable and the effective
cosmological constant is negative: $\Lambda _{{\mathrm{eff}}}<0$.

In Figure \ref{fig3} we have plotted the effective potential $U$ in units of
\begin{equation}
U_{0}=4\pi GM^{d_{0}+2}(Ma^{(0)})^{\frac{2}{d_{0}-1}},  \label{U(0)}
\end{equation}%
as a function on $Ma$ for the model with $d_{0}=3$ consisting a single
untwisted scalar with mass $M$ and two twisted scalars with masses $M_{t}$
for various values of the ratio $M_{t}/M$ (numbers near the curves) and for $%
\Lambda _{D}/(8\pi GM^{d_{0}+2})=2$. For the graph with zero effective
cosmological constant $M_{t}/M\approx 11.134$.

\begin{figure}
\begin{center}
\includegraphics[width=8.5cm, height=6.5cm]{fig3.jpg}
\caption{The potential $U/U_{0}$ in $d_{0}=3$ as a function of $Ma$ for a
model with single untwisted and two twisted scalars and for $\Lambda _{D}/(8%
\protect\pi GM^{d_{0}+2})=2$. The numbers near the curves correspond to the
values of the ratio $M_{t}/M$. For the graph with zero effective
cosmological constant $M_{t}/M=11.134$.} \label{fig3}
\end{center}
\end{figure}

As we see in this type of models the size of the internal space is of the
order $1/M$. The effective cosmological constant is of the order $\Lambda _{{%
\mathrm{eff}}}\sim 8\pi M_{\mathrm{Pl}}^{-2}\times 10^{-2}/a^{4}$. Setting
the value of the corresponding energy density equal to the density of the
dark energy, for the size of the internal space we find $a\sim 10^{-3}%
\mathrm{cm}$. Such large extra dimensions are realized in models where the
standard model fields are localized on a 4-dimensional hypersurface (brane).
Note that the value of the effective cosmological constant can be tuned by
the parameter $M_{t}/M$ and the observed value of the dark energy density
can also be obtained for smaller values for the size of the internal space.
Similar results are obtained in models with more complicated internal
spaces, in \ particular, in ADD models with the number of internal
dimensions $\geqslant 2$.

Note that large extra dimensions accessible to all standard model fields can
also be realized. This type of extra dimensions are referred to as universal
dimensions. The key element in these models is the conservation of momentum
in the Universal Extra Dimensions which leads to the Kaluza-Klein number
conservation. In particular there are no tree-level contributions to the
electrowek observables. The Kaluza-Klein modes may be produced only in
groups of two or more and none of the known bounds on extra dimensions from
single Kaluza-Klein production at colliders applies for universal extra
dimensions. Contribution to precision electroweak observables arise at the
one loop level. In the case of a single extra dimension, recent experimental
constraints allow a compactification scale as low as TeV scale.

\section{Brane models}

Recently it has been suggested that the introduction of compactified extra
spatial dimensions may provide a solution to the hierarchy problem between
the gravitational and electroweak mass scales \cite{Arka98,Rand99}. The main
idea to resolve the large hierarchy is that the small coupling of
four-dimensional gravity is generated by the large physical volume of extra
dimensions. These theories provide a novel setting for discussing
phenomenological and cosmological issues related to extra dimensions. The
model introduced by Randall and Sundrum is particularly attractive. Their
background solution consists of two parallel flat branes, one with positive
tension and another with negative tension embedded in a five-dimensional AdS
bulk \cite{Rand99}. The fifth coordinate is compactified on $S^{1}/Z_{2}$,
and the branes are on the two fixed points. It is assumed that all matter
fields are confined on the branes and only the gravity propagates freely in
the five-dimensional bulk. In this model, the hierarchy problem is solved if
the distance between the branes is about 40 times the AdS radius and we live
on the negative tension brane. More recently, scenarios with additional bulk
fields have been considered.

For the braneworld scenario to be relevant, it is necessary to find a
mechanism for generating a potential to stabilize the distance between the
branes. The braneworld corresponds to a manifold with boundaries and all
fields which propagate in the bulk will give Casimir-type contributions to
the vacuum energy and, as a result, to the vacuum forces acting on the
branes. Casimir forces provide a natural mechanism for stabilizing the
radion field in the Randall-Sundrum model, as required for a complete
solution of the hierarchy problem. In addition, the Casimir energy gives a
contribution to both the brane and bulk cosmological constants and, hence,
has to be taken into account in the self-consistent formulation of the
braneworld dynamics.

In the $(D+1)$-dimensional generalization of the Randall-Sundrum spacetime
the background spacetime is described by the line-element%
\begin{equation}
ds^{2}=e^{-2k|y|}\eta _{\mu \nu }dx^{\mu }dx^{\nu }-dy^{2},  \label{ds2AdS}
\end{equation}%
where $\eta _{\mu \nu }$ is the metric tensor for the $D$-dimensional
Minkowski spacetime the AdS curvature radius is given by $1/k$. The fifth
dimension $y$ is compactified on an orbifold, $S^{1}/Z_{2}$ of length $a$,
with $-a<y<a$. The orbifold fixed points at $y=0$ and $y=a$ are the
locations of two $D$-branes. Consider a scalar field $\varphi (x)$\ with
curvature coupling parameter $\zeta $ obeying boundary conditions $\left(
\tilde{A}_{y}+\partial _{y}\right) \varphi (x)=0$, $y=0,a$, on the branes.
For a scalar field with brane mass terms $c_{0}$ and $c_{a}$ on the left and
right branes respectively, the coefficients in the boundary conditions are
defined by the relation $2\tilde{A}_{j}=-n^{(j)}c_{j}-4D\zeta k$ with $%
n^{(0)}=1$, $n^{(b)}=-1$ (see, for instance, \cite{Gher00,Flac01,Saha05}).
The corresponding Casimir energy is given by the expression \cite%
{Gold00,Flac01,Garr01,Saha05}
\begin{equation}
E(a)=\alpha +\beta e^{-Dka}+\frac{(4\pi )^{-D/2}k^{D}}{\Gamma \left(
D/2\right) }\int_{0}^{\infty }du\,u^{D-1}\ln \left\vert 1-\frac{\bar{I}_{\nu
}^{(a)}(u)\bar{K}_{\nu }^{(b)}(ue^{ka})}{\bar{K}_{\nu }^{(a)}(u)\bar{I}_{\nu
}^{(b)}(ue^{ka})}\right\vert ,  \label{ECasbr}
\end{equation}%
where $I_{\nu }(u)$ and $K_{\nu }(u)$ are the modified Bessel
functions,
\begin{equation}
\nu =\sqrt{(D/2)^{2}-D(D+1)\zeta +m^{2}/k^{2}},  \label{nu}
\end{equation}%
and the barred notation for a given function $F(x)$ is defined by $\bar{F}%
^{(j)}(x)=(\tilde{A}_{j}/k+D/2)F(x)+xF^{\prime }(x)$. The first and second
terms on the right of Eq. (\ref{ECasbr}) are the energies for a single brane
located at $y=0$ and $y=a$ respectively when the second brane is absent. The
coefficients $\alpha $ and $\beta $ cannot be determined from the low-energy
effective theory and can be fixed by imposing suitable renormalization
conditions which relate them to observables. The last term in (\ref{ECasbr})
is free of renormalization umbiguities and can be termed as an interaction
part. The corresponding vacuum forces acting on the branes can be either
repulsive or attractive in dependence of the coefficients in the boundary
conditions. In particular, there is a region in the parameter space where
these forces are attractive at large distances between the branes and
repulsive at small distances with an equilibrium state at some intermediate
distance $a=a_{0}$. In the original Randall-Sundrum braneworld with $D=4$,
to account for the observed hierarchy between the gravitational and
electroweak scales we need $ka_{0}\approx 37$.

In addition to the stabilization of the distance between the branes, the
quantum effects from bulk fields can also provide a mechanism for the
generation of the cosmological constant on the visible brane. On manifolds
with boundaries in addition to volume part, the vacuum energy contains a
contribution located on the boundary. For a scalar field the surface energy
density $\varepsilon _{\mathrm{v}}^{{\mathrm{(surf)}}}$\ on the visible
brane $y=a$ is presented as the sum \cite{Saha04}
\begin{equation}
\varepsilon _{\mathrm{v}}^{{\mathrm{(surf)}}}=\varepsilon _{1\mathrm{v}}^{{%
\mathrm{(surf)}}}+\Delta \varepsilon _{\mathrm{v}}^{{\mathrm{(surf)}}},
\label{emt2pl2}
\end{equation}%
where $\varepsilon _{1\mathrm{v}}^{{\mathrm{(surf)}}}$ is the surface energy
density on the visible brane when the hidden brane is absent, and the part
\begin{equation}
\Delta \varepsilon _{\mathrm{v}}^{{\mathrm{(surf)}}}=k^{D}\frac{\left(
4\zeta -1\right) \tilde{A}_{j}-2\zeta }{(4\pi )^{D/2}\Gamma \left(
D/2\right) }\int_{0}^{\infty }du\,\frac{u^{D-1}\bar{I}_{\nu
}^{(a)}(ue^{-ka})/\bar{I}_{\nu }^{(b)}(u)}{\bar{K}_{\nu }^{(a)}(ue^{-ka})%
\bar{I}_{\nu }^{(b)}(u)-\bar{I}_{\nu }^{(a)}(ue^{-ka})\bar{K}_{\nu }^{(b)}(u)%
},  \label{Delteps}
\end{equation}%
is induced by the presence of the hidden brane. The part $\varepsilon _{1%
\mathrm{v}}^{{\mathrm{(surf)}}}$ contains finite renormalization terms which
are not computable within the framework of the model under consideration and
their values should be fixed by additional renormalization conditions. The
effective cosmological constant generated by the hidden brane is determined
by the relation
\begin{equation}
\Lambda _{\mathrm{v}}=8\pi M_{\mathrm{v}}^{2-D}\Delta \varepsilon _{\mathrm{v%
}}^{{\mathrm{(surf)}}},  \label{effCC}
\end{equation}%
where $M_{\mathrm{v}}$ is the $D$-dimensional effective Planck mass scale
for an observer on the visible brane. For large interbrane distances the
quantity (\ref{effCC}) is suppressed compared with the corresponding Planck
scale quantity in the brane universe by the factor $\exp [-k(2\nu +D)a]$,
assuming that the AdS inverse radius and the fundamental Planck mass are of
the same order. In the original Randall-Sundrum model with $D=4$, for a
scalar field with the mass $|m|^{2}\lesssim k^{2}$, and interbrane distances
solving the hierarchy problem, the value of the induced cosmological
constant on the visible brane by the order of magnitude is in agreement with
the value of the dark energy density suggested by current cosmological
observations without an additional fine tuning of the parameters.

\section{Exotic Propulsion}

The idea of manipulating spacetime in some ingenious fashion to
facilitate a novel form of spacecraft propulsion has been well
explored in the literature \cite{oc1,oc2,mty,Ford:1994bj,
Ford:1996er, Pfenning:1997wh, Pfenning:1997rg, Everett:1997hb,
Everett:1995nn, Visser:1998ua, VanDenBroeck:1999sn, Lobo:2004wq,
Hiscock:1997ya, GonzalezDiaz:1999db, Puthoff:1996my,
Natario:2004zr, Lobo:2004an, Bennett:1995gp, GonzalezDiaz:2007zzb,
alc, DavisMillis:2009}. If we are to realistically entertain the
notion of interstellar exploration in timeframes of a human
life-span, a profound shift in the traditional approach to
spacecraft propulsion is clearly necessary. It is well known that
the universe imposes dramatic relativistic limitations on all
bodies moving through spacetime, and that all matter is restricted
to motion at sublight velocities ($< 3\times10^8  \rm{m/s}$, the
speed of light, or c) and that as matter approaches the speed of
light, its mass asymptotically approaches infinity. This mass
increase ensures that an infinite amount of energy would be
necessary to travel at the speed of light, and thus, this speed is
impossible to reach and represents an absolute speed limit to all
matter travelling through spacetime.

Even if an engine were designed that could propel a spacecraft to
an appreciable fraction of light-speed, travel to even the closest
stars would take many decades in the frame of reference of an
observer located on Earth. Although these lengthy transit times
would not make interstellar exploration impossible, they would
certainly reduce the enthusiasm of governments or private
individuals funding these missions. After all, a mission whose
success is, perhaps, a century away would be difficult to justify.
In recent years, however, two loop-holes to Einstein's ultimate
speed limit are known to exist: the Einstein-Rosen bridge  and the
warp-drive. Fundamentally, both ideas involve the manipulation of
spacetime itself in some exotic way that allows for superluminal
travel.

The warp drive, which is the main feature of study for this
section, involves local manipulation of the fabric of space in the
immediate vicinity of a spacecraft. The basic idea is to create an
asymmetric bubble of space that is contracting in front of the
spacecraft while expanding behind it. \footnote{Recent progress by
Jos\'{e} Nat\'{a}rio has demonstrated that, with a slightly more
complicated metric, one can dispense of the expansion.} Spacetime
is believed to have stretched at many times the speed of light in
the first second of its existence during the inflationary period.
In many ways, the idea presented in this paper is an artifical and
local re-creation of those initial conditions.

Using this form of locomotion, the spacecraft remains stationary
inside this `warp bubble,' and it is the movement of space itself
that facilitates the relative motion of the spacecraft.  The most
attractive feature of the warp drive is that the theory of
relativity places no known restrictions on the motion of space
itself, thus allowing for a convenient circumvention of the speed
of light barrier.

\begin{figure}
\begin{center}
\includegraphics[width=8cm, height=6.8cm]{fig4.jpg}
\caption{The `Top hat metric'. A bubble of asymmetric spacetime
curvature surrounds a spacecraft which would sit in the center of
the bubble. The space immediately surrounding the
spacecraft would be expanding/contracting behind/in front of the
craft. In this image the ship would `move' from left to right.} \label{fig4}
\end{center}
\end{figure}

By associating dark energy with the Casimir energy due to the KK
modes of vacuum fluctuations in higher dimensions, especially in
the context of M-theory derived or inspired models, it is possible
to form a relationship between $\Lambda$ and the radius of the
compact extra dimension.
\begin{equation}
\rho=\Lambda\propto 1/a^D.
\end{equation}
An easier way of developing the relationship between the energy
density and the expansion of space is to discuss quantities in
terms of Hubble's constant H, which describes the rate of
expansion of space per unit distance of space.
\begin{equation}
H\propto\sqrt{\Lambda} ,
\end{equation}
or in terms of the radius of the extra dimension we have
\begin{equation}
H\propto1/a^{D/2}.
\end{equation}

This result indicates that within this model, the expansion of
spacetime is a function of the size of the higher dimension. One
fascinating question to ask at this point is: could it be possible
to effect the radius of a higher dimension through some advanced
technology? If this were, indeed, possible then this technology
would provide a remarkable mechanism to locally adjust the dark
energy density, and thus the local expansion rate of spacetime
(Figure \ref{fig5}). \footnote{Illustration
courtesy of Richard Obousy Consulting LLC and AlVin, Antigravit\'{e}}

In principal, this represents an interesting way to artificially
generate the type of spacetime featured in Figure \ref{fig4}. A
spacecraft with the ability to create such a bubble would always
move inside its own local light-cone. This ship could utilize the
expansion of spacetime behind the ship to move away from some
object at any desired speed, or equivalently, to contract the
space-time in front of the ship to approach any object. The
possibility that the size of the compact geometry might, indeed,
vary depending on the location in four dimensional spacetime has
been explored in the context of string theory \cite{Giddings2005},
but never from the perspective of propulsion technology.

\begin{figure}
\begin{center}
\includegraphics[width=11.5cm, height=7.5cm]{fig5.jpg}
\caption{An extra dimension that is artifically stimulated to
contract (or expand), due to interaction with the SM fields, might
provide a way to locally adjust the dark energy density.} \label{fig5}
\end{center}
\end{figure}

To explore the types of energies that may be required to generate
this manipulation of a higher dimension, we consider that fact
that in models with large extra dimensions, the interaction of the
graviton KK tower with the Standard Model fields are suppressed
by the higher dimensional Planck scale, and the corresponding
couplings are inverse TeV in strength. This can be seen more
clearly when we consider the expansion of the metric tensor in
models with large extra dimensions computed within linearized
gravity models:
\begin{equation}
g_{MN}=\eta_{MN}+h_{MN}/M_D^{D/2-1},
\end{equation}
where $\eta_{MN}$ corresponds to flat (Minkowski) spacetime and
$h_{MN}$ corresponds to the bulk graviton fluctuations. The
graviton interaction term in the action is expressed by:
\begin{equation}
S_{\rm{int}}=1/M_D^{D/2-1}\int d^D x h_{MN}T^{MN}
\end{equation}
with $T^{MN}$ being the higher dimensional energy-momentum tensor.
The interaction of the graviton KK states with the SM fields are
obtained by integrating the action over the extra coordinates.
Because all these states are coupled with the universal strength $
1/M_{\rm{Pl}}$ this leads to the compelling possibility of the
control of the size of the extra dimensions by processes at
energies that will be accessible via the particle accelerators of
the near future. Although the coupling is extremely small, the
effective coupling is enhanced by the large number of KK states.

Referring to \ref{fig2}, additional energy in the form of matter
or radiation with the TeV energy scale can alter the shape of the
effective potential. In particular, the extrema determining the
size of the extra dimensions are modified with the change of the
Casimir energy density and hence, the dark energy density, in the
models under consideration.

\begin{figure}
\begin{center}
\includegraphics[width=7.52cm, height=4.76cm]{fig6.jpg}
\caption{By surrounding a spacecraft with false
minima an artificial inflation of the local spacetime might be
achieved.} \label{fig6}
\end{center}
\end{figure}

The key to creating a warp drive in this model is to create a
false vacuum minimum, i.e., to modify the vacuum spectrum and
inject some field which creates a deSitter minimum at the rear of
the craft and an anti-deSitter minimum at the front of the craft.
What this requires is a technology that would allow us to
artificially manipulate the field content illustrated in
\ref{fig2}, shifting the location of the minimum. In this basic
representation, the spacecraft would sit in a stable region of
space corresponding to the natural minimum of the extra dimension
(approximately flat space). At the front and rear of the craft,
regions of false minima would be artificially created via the
adjustment of the extra dimension (Figure \ref{fig6}). These modified
regions would correspond to increased and negative dark energy
densities, thus creating the warp bubble previously discussed.

\section{Extra Dimensions in General Relativity}

In the context of GR a similar phenomenology is produced for the
case of anisotropic cosmological models, in which it is the
$\textit{contraction}$ of the extra dimension that has the effect
of expanding another \cite{Levin:1994}. For example, consider a
`toy' universe with one additional spatial dimension with the
following metric
\begin{equation}
ds^2=dt^2-a^2(t)d\vec{x}^2-b^2(t)dy^2  \ .
\end{equation}
In this toy universe we will assume spacetime is empty, that there
is no cosmological constant, and that all spatial dimensions are
locally flat,
\begin{equation}
T_{\mu \nu}=\Lambda g_{\mu \nu}=0 \ .
\end{equation}
The action of the Einstein theory of gravity generalized to five dimensions will be
\begin{equation}
S^{(5)}=\int d^4x dy \sqrt{-g^{(5)}}\left(\frac{M_5^2 }{16\pi }R^{(5)}\right) \ .
\end{equation}
Solving the vacuum Einstein equations
\begin{equation}
G_{\mu\nu}=0 \ ,
\end{equation}
we obtain for the $G_{11}$ component
\begin{equation}
G_{11}=\frac{3\dot{a}(b\dot{a}+a\dot{b})}{a^2b} \ .
\end{equation}
Rewriting $\dot{a}/a=H_a$ and $\dot{b}/b=H_b$ where $H_a$ and
$H_b$ corresponds to the Hubble constant in three space and the
Hubble constant in the extra dimension respectively, we find that
solving for $G_{11}=0$ yields
\begin{equation}
H_a=-H_b .
\end{equation}
This remarkable result indicates that in a vacuum, the shear of a
contracting dimension is able to inflate the remaining dimensions.
In other words, the expansion of the 3-volume is associated with
the contraction of the one-volume.

Even in the limit of flat spacetime with zero cosmological
constant, general relativity shows that the physics of the
$\textit{compactified}$ space affects the expansion rate of the
non-compact space. The main difference to note here is that the
quantum field theoretic result demonstrates that a
$\textit{fixed}$ compactification radius can also result in
expansion of the three-volume due to the Casimir effect, whereas
the GR approach suggests that a $\textit{changing}$
compactification radius results in expansion, as is shown in (34).
What is particularly interesting about this result is that it
demonstrates that the expansion and contraction of the non-compact
space seems to be intimately related to the extra dimension in
both QFT and GR calculations.

\section{Discussion}

As we have seen, the stabilization of compact extra dimensions and the
acceleration of the other 4-dimensional part of the spacetime can be
simultaneously described by using the Casimir energy-momentum tensor as a
source in the Einstein equations. The acceleration in the 3-dimensional
subspace occurs naturally when the extra dimensions are stabilized. In this
case the Casimir energy density $\rho _{C}$ is a constant with the equation
of state $p_{C}=-\rho _{C}$, where $p_{C}$ is the Casimir pressure in the
visible universe, and an effective cosmological constant is induced. Note
that the current observational bounds for the equation of state parameter
are $-1.4<$ $p_{\mathrm{DE}}/\rho _{\mathrm{DE}}<-0.85$ and the value -1 of
this parameter is among the best fits of the recent observational data. For
the Casimir energy density to be the dark energy driving accelerated
expansion the size of the internal space should be of the order $10^{-3}%
\mathrm{cm}$. Such large extra dimensions are realized in braneworld
scenario. The value of the effective cosmological constant can be tuned by
choosing the masses of the fields and the observed value of the dark energy
density can also be obtained for smaller values for the size of the internal
space.

An important feature of both models with a smooth distribution of
matter in the extra dimensions and with branes is the dependence
of the dark energy density on the size of the extra dimensions. In
models with large extra dimensions the interaction of the graviton
Kaluza-Klein tower with the standard model fields are suppressed
by the higher dimensional Planck scale and the corresponding
couplings are inverse TeV strength. This leads to the interesting
possibility for the control of the size of extra dimensions by
processes at energies accessible at the near future particle
accelerators. This is seen from the form of the effective
potential for the scale factor of the internal subspace given
above. Additional energy density in the form of matter or
radiation with the TeV energy scale can alter the shape of the
effective potential. In particular, the extrema determining the
size of the extra dimensions are changed with the change of the
Casimir energy density and, hence, the dark energy density in the
models under consideration.

In the ADD scenario the Standard model
fields are confined to a 4D brane and a
new gravity scale $M_{D}=G^{1/(2-D)}\gtrsim \mathrm{TeV}$ is introduced in $%
4+d_{1}$ dimensions. The gravity on the brane appears as a tower of
Kaluza-Klein states with universal coupling to the Standard Model fields.
Though this coupling is small ($\sim 1/M_{\mathrm{Pl}}$), a relatively large
cross section is obtained from the large number of Kaluza-Klein states.
Kaluza-Klein graviton emission from SN1987A puts a bound $M_{D}\geqslant 30\,%
\mathrm{TeV}$ on the fundamental energy scale for ADD type models
in the case $d_{1}=2$. For the models with $d_{1}=3$ the
constraint is less restrictive, $M_{D}\geqslant
\mathrm{few\,\,TeVs}$. Hence, the latter case is still viable for
solving the hierarchy problem and accessible to being tested at
the LHC. The ATLAS experiment which will start taking data at the
LHC will be able to probe ADD type extra dimensions up to
$M_{D}\approx $ $8$ TeV. In the Randall-Sundrum scenario the
hierarchy is explained by an exponential warp factor in
$\mathrm{AdS}_{5}$ bulk geometry. Electroweak precision tests put
a severe lower bound on the lowest Kaluza-Klein mass. The masses
of the order of $1\,\mathrm{TeV}$ can be accommodated. In
universal extra dimension scenarios the current Tevatron results
constrain the mass of the compactification scale to
$M_{\mathrm{C}}>400$ GeV. The ATLAS experiment will be sensitive
to $M_{\mathrm{C}}\approx 3$ TeV. These estimates show that if the
nature is described by one of the models under consideration, the
extra dimensions can be probed and their size can be controlled at
the energies accessible at LHC. In models with the Casimir energy
in the role of the dark energy this provides a possibility for the
control of the dark energy local density through the change in the
size of the extra dimensions.

Note that the effective potential (\ref{Vpot}) contains a factor $\Omega
^{-2}\sim V_{{\mathrm{int}}}^{-2/(d_{0}-1)}$, with $V_{{\mathrm{int}}}$
being the volume of the internal space, and, hence, vanishes in the large
volume limit for the extra dimensions if the energy density grows not
sufficiently fast. This leads to the instability of a dS minimum with
respect to the decompactification of extra dimensions. This feature is
characteristic for other stabilization mechanisms as well including those
based on fluxes from form-fields and wrapped branes. The stability
properties of the metastable dS minimum depend on the height of the local
maximum separating this minimum from the minimum which corresponds to the
infinite size of the extra dimensions. Now we conclude that in models with
large extra dimensions, as in the case of the parameters for the minimum,
the height and the location of the maximum can be controlled by the
processes with the energy density of the TeV scale.

Another important point which should be touched is the following.
Though the size of the internal space is stabilized by the
one-loop effective potential, in general, there are fluctuations
around the corresponding minimum with both classical and quantum
sources which shift the size for the internal space from the fixed
value leading to the time variations of this size. This gives time
variations in the equation of state parameter for the
corresponding dark energy density. However, it should be taken
into account that the variations of the size for the internal
space around the minimum induces variation of fundamental
constants (in particular, gauge couplings) in the effective
four-dimensional theory. These variations are strongly constrained
by the observational data. As a consequence, the variation of the
equation of state parameter for the corresponding dark energy
density around the mean value $-1$ are small. It is interesting
that in the models under consideration the constancy of the dark
energy density is related to the constancy of effective physical
constants. Note that models can be constructed where the volume of
the extra dimensions is stabilized at early times, thus
guaranteeing standard four dimensional gravity, while the role of
quintessence is played by the moduli fields controlling the shape
of the extra space.