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\begin{document}
\title{Review of Alcubierre Drive Experimentation and Proposals for Tabletop Generation of Gravitational Waves}
\author{Justin Long}
\affil{Affiliation not available}
\author{Lucius G Meredith}
\affil{Affiliation not available}
\date{\today}
\maketitle
\selectlanguage{english}
\begin{abstract}
\section{Abstract}
There is renewed interest in the Alcubierre Drive prompted by interferometer experimentation conducted by Harold White in NASA Eagleworks Lab. We attempt to replicate the research used by White to develop the same experiment and attempt to achieve similar results. Although the metric proposed by Alcubierre appears to be valid, it is dogged by a number of questions regarding its physical realizability. Because White does not publish his calculations for space-time warping caused by electromagnetic energy, we research and derive our own sensitivity requirements for detecting warp fields. We provide a basis for modeling warp field experiments. Examination of methods included gravitational wave detection through the use of laser, atom, and neutron interferometry as well as resonant-mass detectors while maintaining the primary requirement that the experiment must fit in a tabletop. We found that a resonant-mass detector made of a high frequency phonon trapping acoustic cavity (Goryachev & Tobar, 2014; sensitivity of $10^{-22}$ Hz) was the most feasible instrument to perform tabletop experimentation. Measuring high-frequency gravitational waves was the simplest approach to measuring results. Additionally, using the Gertsenshtein equation for converting electromagnetic energy to gravitational waves, we calculate that the magnetic flux density of an electric clock is well within the sensitivity range of Goryachev & Tobar's device (approximately $10^{-18}$ Hz), opening the possibility for small scale experimentation. After establishing experimental requirements, we propose two experiments to further advance Alcubierre experimentation: 1) a proof of spacetime warping sensitivity using an electromagnetic source; and 2) a scalable quantum optical squeezing technique for producing negative energy densities built on work by E. Davis et al., O. Firstenberg et al., and L. H. Ford et al.%
\end{abstract}%
\section{Introduction}
\emph{adjust this so it talks about how experimentation needs to be taken to a realistic tabletop level}
Since the 1994 publication of the Alcubierre solution to the Einstein Field Equations - the Alcubierre Drive (Alcubierre, 1994) - a handful of proposals and experiments have been published attempting to demonstrate the realizability of faster-than-light travel. Early proposals included a focus on zero-point energy (H.E. Puthoff, 1998). In the late 2000s, organizations such as Icarus Interstellar and the Tau Zero foundation have charted fields of interest for faster-than-light travel. A class of recent experiments, EM warping and measurement using laser interferometry (White, 2015), are of particular interest and examining their feasibility is particularly useful for advancing Alcubierre drive experimentation.
Harold White is using several methods of experimentation (including Twyman-Green and Fabry-Perot interferometry) to measure spacetime warping in a similar geometry to the Alcubierre solution. However, many objections have been raised against the instrumentation and results, including interference from a change in the refractive air index (Lee & Cleaver, 2014). The authors of this paper were not able to obtain White's calculations that would allow examination of his predicted warping - and resulting phase shift using laser interferometry - that White is attempting to measure. Given the lack of evidence to support positive results, there's reason to attempt replication of White's experiments and methods.
Because gravity is a property of spacetime geometry, a foundation for Alcubierre experimentation can be built on existing lines of research in gravitational instrumentation. The COW experiment demonstrated remarkable pioneering in neutron interferometry and gravitational measurement (Colella, Overhauser, and Werner, 1974). Since the 1960s, continued improvements have been made to Weber's original 1966 proposal for a resonant-mass detector to measure gravitational waves (Aguiar, 2014). It's important to make the distinction that White's experiment is attempting to directly measure spacetime distortion on a laser's path, whereas other laser interferometers such as LIGO are designed to detect residual gravitational phenomenon, or gravitational waves. However, the method of detection is irrelevant since a positive result requires verification of the Alcubierre geometry and not necessarily replication of White's experimental design.
\section{Supporting and Relevant Theory}
\subsection{Post-Newtonian Approximation of Spacetime Warping by Electromagnetic Source}
Because the Einstein Field Equations are computationally difficult and a tabletop experiment involves measurement of weak fields, Post-Newtonian approximations are helpful to understanding the scale of measurements of laboratory EM fields. Applicable approximations will also describe the nature of the EM field and give and indicate the geometry of gravitational warping. We found it necessary to focus on Post-Newtonian approximations that directly correlated an EM field to gravitational effects.
Mathematics describing the connection between neutron stars and gravitational waves is applicable to understanding similar relationships at a tabletop scale. Preston Jones has shown that a time varying spacetime metric gives rise to electromagnetic radiation (Jones, 2007)
\begin{equation}
\nabla\cdot{S}+\frac{\partial U}{\partial t} = \frac{4}{c^4}U\frac{\partial \phi^2}{\partial t},
\end{equation}
where use has been made of the Poynting vector, $S = \selectlanguage{greek}\frac{c}{4π}\selectlanguage{english}E \selectlanguage{ngerman}× B$, and the factor of $c$ has been reinstated to illustrate the order of magnitude of this effect. This demonstrates that in a region of nonzero internal energy density $U$ and rapidly time varying gravitational potential, $\phi$, electromagnetic radiation will be generated. Note, the approximation above only uses the covariant relationships.
We assume that the relationship can be reversed. By solving for the time varying gravitational potential, $\phi$, we show that electromagnetic radiation gives rise to a time varying spacetime metric.
\begin{equation}
\phi = \frac{c^2}{2}\sqrt{\int\frac{\nabla\cdot{S}}{U}\frac{\partial U}{\partial t}}
\end{equation}
By having an equation that readily describes the relationship between spacetime warping and EM radiation, we can accurately predict results in experimental setups. This is also important because the calculations for White's Alcubierre experiments were not readily available to assist estimations of laser warping and phase shift.
\subsection{Gertsenshtein and Conversion of Gravitational Waves to Electromagnetic Waves}
Gertsenshtein gives a clear equation describing the relationship between electromagnetic energy and gravitational waves. The equation used to calculate the amplitude of the gravitational wave is (Gertsenshtein, 1962)
\begin{equation}
\frac{a(x)}{b} = \sqrt{\frac{\gamma}{\pi c^3} F^{(0)2} R_{0} T},
\end{equation}
where $F$ represents electromagnetic energy in gauss, $R_{0}$ represents the distance from the EM source in meters, and $T$ is the time allotted for gravitational radiation in years.
There is supporting mathematics that the relationship can be reversed. When an EMW $(E, H)$ propagates in the field $H_0$ there appears a stress tensor proportional to $HH_0$ which is variable in space and time. This tensor is the source of GW. When a GW propagates through the field $H_0$ there occurs a stretching and compression of the magnetic field, accompanied by the appearance of an alternating magnetic field $h(x, t)H_0$, where h is the variation of the metric in the GW. The field $hH_o$ is the source for the EMW and it is obvious a medium mediates the conversion process (Zel'dovich, 1973; Mitskevich et al., 1969; Boccaletti et al., 1970; Dubrovich et al., 1972).
The investigation into $H_0$ and the reversal of Gerstenshtein's equation also supports our earlier assumption that we could derive an equation in the Post-Newtonian form describing time varying gravitational potential, $\phi$, from an electromagnetic field. Interestingly, Zel'dovich considered the conversion of EMW to GW to be so trivial that in the presence of matter it was undetectable.
\subsection{Amplitude of Gravitional Waves from EM Source}
Because we know Gertsenshtein's equation yields an amplitude, we can use a Fourier Transform to calculate the frequency of gravitational waves. For a practical reference, we use known EM measurements of an electric clock (Duke Energy, 2013) to calculate the amplitude and frequency of a gravitational wave. When an electric clock is measured at 0.3048 meters (1 ft), the strength of its EM field is 15.5 gauss. Calculating for $T = 0.00011408$, we find that the amplitude of the gravitational waves of an electric clock are
\begin{equation}
\frac{a(x)}{b} = 8.11596*10^{-20} \sqrt{\frac{\text{Newton} \text{Second}^3} {\text{Kilogram}^2 \text{Meter}}},
\end{equation}
and when applying a Fourier Transform to calculate the frequency of the gravitational waves we get
\begin{equation}
\int_{-\infty}^{\infty} \frac{a(x)}{b} e^{- 2\pi i x \xi} dx = 7.59863*10^{-18} \sqrt{Hz}.
\end{equation}
Assuming EM-based Alcubierre experiments are using radiation sources similar to electric clocks, we generally know the magnitude of tabletop gravitational waves is nearly $10^{-18}$.
\subsection{General Relativity, Quantum Field Theory, Quantum Inequalities, and Negative Energy}
Developing methods and instrumentation to measure gravitational fields in the laboratory is moot if a mechanism for generating negative energy cannot be engineered. The Alcubierre solution resolves to a negative energy density, requiring negative energy and/or exotic mass (Alcubierre, 1994).
Much work remains to be done by resolving General Relativity with Quantum Field Theory. Semi-classical theory is self-inconsistent because when coupled with Relativity's stress-energy tensor, it violates the Uncertainty Principle (Riggs, 1996). Interestingly however, Quantum Field Theory allows for the suppression of vacuum fluctuations, leading to sub-vacuum phenomena. This sub-vacuum phenomena can appear as negative energy density (Ford, 1996). Further research by Krasnikov into quantum inequalities supports the physical nature of negative energy (Krasnikov, 2003). Interestingly, Dan Solomon demonstrated a negative energy density for a Dirac-Maxwell field. Negative energy density can be formulated for a Dirac field interacting with an Electromagnetic field (Solomon, 1999), hinting that negative energy experimentation may be possible with multiple techniques than something like photon squeezing. Specifically, Solomon arranged a Dirac-Maxwell field as
\begin{equation}
\usepackage{physics}
\xi_{ave}\left( \lvert{\Omega^{\prime}}\rangle \right) = \xi_{ave}\left( \lvert{\Omega}\rangle \right) - f \frac{\int_{V}\left( \vec{J}_{e} \cdot \vec{A}_{cl}\left(\vec{x};\vec{\chi}_{1}\right) \right) d\vec{x} }{V},
\end{equation}
where an $f$ can always be found that makes $\xi_{ave}\left( \lvert{\Omega^{\prime}}\rangle \right)$ a negative number with an arbitrarily large magnitude.
For the purpose of Alcubierre experimentation, we assume that negative energy densities in Quantum Field Theory are in fact similar to General Relativity's own interpretation. This assumption allows us to perform Alcubierre and negative energy experiments with the expectation that by mixing Quantum Field Theory, General Relativity, and semi-classical theory, we can obtain real results.
Although Quantum Field Theory allows for regions of negative energy density, the magnitude and duration of negative energy is limited by quantum inequalities (Pfenning et al., 1998). This becomes an issue for scaling an Alcubierre drive. Assuming the quantum inequalities apply to the scale of a spaceship, a warp bubble 200 meters across would require a total amount of negative energy equal to 10 billion times the mass of the observable universe (Ford et al., 2000). General Relativity does not have such limits similar to the quantum inequalities, with the exception of localized negative energy density potentially violating the weak energy condition. However, when energy conditions are measured in a non-localized way, it is possible to uphold the weak energy condition (Roman, 1986). It remains to be seen how the quantum inequalities would affect implementation at a larger scale, and it begs the question of whether negative energy in Quantum Field Theory is the same as its General Relativity counterpart. If the two are dissimilar, any experiment built on Quantum Field Theory aiming to produce negative energy may not actually lead to construction of an Alcubierre drive at a large scale. We accept this as a potential risk to experimentation.
\subsection{Fractal Physics, Spacetime, and Negative Energy}
A potential reason that General Relativity is not subject to the same extreme limitations as the quantum inequalities is that the properties of spacetime change at different scales. By exploring the geometrical properties of spacetimes by calculating the spectral dimension associated with them, it was found that a larger scale spectral dimension reaches a more classical value (Benedetti, 2009). This is a possible explanation that semi-classical models are self-inconsistent, and General Relativity and Quantum Field Theory are unresolvable. Since the generation of quantized negative energy may have fractal properties, an Alcubierre drive may require engineering at different magnitudes of scale.
An approach to Alcubierre experimentation using fractal physics may be more helpful since cosmological observations suggest existing theory based on observable phenomena is incomplete (Green, 2014). Taking survey results from Wilkinson Microwave Anisotropy Probe into account, the low amount of measurable ordinary baryonic matter, $4.628\% \pm 0.093\%$ (Beringer et al., 2014), leaves much more room to consider new theory to explain the remaining "dark energy" measured in the universe.
\section{Development of Experimental Requirements}
\subsection{Spacetime Warping & Sensitivity}
Because negative energy density has only been observed indirectly in Casimir effect and squeezed states (Ford et al., 2000), we consider it too difficult to attempt to measure it directly. To work around this, we propose using high-frequency gravitational waves as a tool to measure the effectiveness of future Alcubierre experiments. Unless Alcubierre experiments can produce specific quantum phenomenon, we propose that gravitational waves are useful for experimentation.
High-frequency gravitational waves are very difficult to detect, and require sensitive devices to measure them. Based on existing proposals and apparatuses used to detect gravitational waves such as the high frequency phonon trapping acoustic cavity by Goryachev & Tobar, we estimate that Alcubierre experiments will generate the following range of gravitational wave frequencies:\selectlanguage{english}
\begin{table}
\begin{tabular}{ l l }
\textbf{Frequency} & \textbf{Description} \\
10^{-18} \sqrt{Hz} & Largest magnitude for tabletop experimentation based on HFGW magnitude of electric clock \\
10^{-22} \sqrt{Hz} & Smallest magnitude for tabletop experimentation based on Goryachev et al.'s proposal \\
\end{tabular}
\caption{{Range of high-frequency gravitational waves in tabletop Alcubierre experiments}}
\end{table}
\subsection{Size and Containment of Instrumentation}
The relationship between EM radiation and time varying gravitational potential guides our methodology to measuring spacetime warping. Any time varying factor is suspect because spacetime warping may actually be more challenging to measure in a laboratory experiment. We assume that not all EM radiation sources can be shielded, and it is also likely that cosmic gravitational wave sources may also interfere with the experiment.
In signals processing, Lock-in Amplifiers combined with phase detection can be very powerful for detecting small signals in the presence of overwhelming noise (Armen, 2008). Depending on the type of instrument used for spacetime warping, filtering unwanted noise may be more difficult. Resonant-mass detectors may experience thermal vibration that greatly exceed gravitational wave amplitude (Blair et al., 2012). Although neutron interferometers potentially have high sensitivity, the cost is that they are highly susceptible to noise and require large, active isolated enclosures (Pushin et al., 2015). Sensitive laser interferometers may also require additional equipment to filter noise that would require more space to house.
It becomes evident that the choice of measurement apparatus will affect the ability of the experiment to fit on a tabletop. To keep research practical, affordable, and changeable, we propose the following guidelines for size and containment to help guide the design of instrumentation.\selectlanguage{english}
\begin{table}
\begin{tabular}{ l l l }
\textbf{Category} & \textbf{Guideline} & \textbf{Description} \\
Experimental Volume & 3 meter cuboid & Maximum area that primary instrumentation must occupy. \\
Isolation Equipment Factor & 2x experimental volume & Maximum factored size of experimental volume occupied by isolation equipment. \\
\end{tabular}
\caption{{Size and containment guidelines for spacetime warping experimentation.}}
\end{table}
\section{Instrumentation}
The guidelines and experimental requirements presented previously serve as a guide to compare different gravitational wave instruments. We can eliminate certain apparatuses that we know may be too cumbersome or noisy to implement for the purpose of tabletop experimentation.
\subsection{Comparison of Detectors}
\textbf{\textit{THIS SECTION NEEDS MORE CITATION AND EXPANSION}}
\subsubsection{Laser Interferometer}
Although a large variety of techniques and instruments exist for laser interferometry, the size of existing gravitational wave detectors that are able to respond to HFGW in the range required are at least 4km in size () and 1,333 times larger than our tabletop guideline. Laser interferometry can be immediately eliminated.
\subsubsection{Neutron Interferometer}
Neutron interferometry is particularly interesting because particles act as a wave and are subject to a range of sensitivity from gravitational potential. However, this range of sensitivity is also a source of noise (Pushin et al., 2015). A successfully isolated neutron interferometer requires thermal and acoustic isolation, secondary vibration isolation, and environmental enclosures (Wietfeldt, 2009). The amount of enclosure required to isolate an experiment come very close to established guidelines for experimental isolation. An issue remains with isolating noise in neutron interferometry. If the source of gravitational waves is time varying, a lock-in amplifier technique may not work to isolate actual gravitational waves from an EM source. For this reason, we eliminate a neutron interferometer as a viable instrument.
\subsubsection{Traditional Resonant-Mass Detectors}
Traditional resonant-mass detectors involve cooling of resonant bars to very low temperatures around 1.5K (). Resonant-mass detectors such as EXPLORER and NAUTILUS were operated until 2010, and despite their increased lifetime and joint-collaboration in analyzing datasets, the teams operating these detectors did not directly claim to measure gravitational waves (). A directly identifiable signal in the data is preferable, and we eliminate traditional resonant-mass detectors as an option for tabletop experimentation.
\subsubsection{Next Generation Detectors}
Non-traditional resonant-mass detectors may be more viable. A high frequency phonon trapping acoustic cavity proposed by Goryachev and Tobar is only 2.5cm in size - well within our tabletop guidelines - and has the added benefit of operating in a vacuum chamber. This design may easily isolate it from sources of noise and with additional shielding, also prevent EM interference with the device itself when using an EM source for HFGW. When accounting for all sources of noise, the apparatus is estimated to be capable of detecting HFGW at magnitudes up to $10^{-22} \sqrt{Hz}$ (Goryachev et al., 2014). A potential drawback is the apparatus requires a very low operating temperature of 0.01K, and it is unclear whether supporting equipment will fit within tabletop guidelines. Despite the potential drawbacks, the Goryachev & Tobar device appears to be most feasible instrument for tabletop experimentation.
\section{Harold White Experimentation Using Laser Interferometry}
[talk about sensitivity/isolation problems and inherent issues with lasers, as well as inability to determine exact optical warping factor]
\textbf{\textit{[going to make one last push to find White's calculations before critiquing lack of available material]}}
\section{Alcubierre Experimentation Proposals}
Using newly adapted theory and knowledge gained from reviewing current Alcubierre experimentation, we propose two experiments that we believe will answer questions regarding sensitivity realizability and address theoretical concerns regarding the Alcubierre metric's requirement of "negative energy".
\subsection{Proof of Spacetime Warping Sensitivity Using an Electromagnetic Source}
One of the biggest issues with current Alcubierre experiments is the lack of proof showing direct causation between a warping source and being able to measure the gravitational effect. To solve this problem, we propose a proof of spacetime warping using an electromagnetic source. This experiment uses an electric capacitor as a source of electromagnetic radiation, and demonstrates these EMW can be directly converted to gravitational waves as predicted by the Gertsenshtein and Jones equations. The experiment will use the Goryachev & Tobar high frequency phonon trapping acoustic cavity to detect the gravitational waves themselves.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Gravitational-Wave-Detetion-with-Tabletop-EM-Source---New-Page/Gravitational-Wave-Detetion-with-Tabletop-EM-Source---New-Page}
\caption{{A diagram of gravitational wave experimentation using an EM source and a high frequency phonon trapping acoustic cavity.%
}}
\end{center}
\end{figure}
To ensure the validity of measurements, the gravitational wave detector used in the experiment must have the following isolation parameters:
\begin{itemize}
\item Vacuum isolation to prevent error from atmospheric refractive index interference
\item Electromagnetic isolation of both detector and signal processing equipment to prevent error from EM interference
\item Thermal & vibration isolation to prevent interference with resonant cavities in the detector itself
\end{itemize}
One of the challenges with this experiment is the possibility that signal processing equipment and other sources of radiation may interfere with the measurement of gravitational waves. To account for this possible source of interference, we propose measuring existing levels of background noise to assist construction of a lock-in amplifier.
A major risk of failure with this experiment is the possibility that existing matter in the laboratory may drown the signal and prevent it from being measured. Any laboratory that would measure this effect sits in the Earth's own gravity well, and it is unclear whether this will also prevent measurement of experimental signal. Taking this into account, we propose that the experiment must be attempted in multiple configurations that vary in:
\begin{itemize}
\item detector-source distance;
\item frequency of measured gravitational wave; and
\item duration of measurement.
\end{itemize}
\subsection{Scalable Quantum Optical Squeezing for Producing Negative Energy Densities}
Scalable production of negative energy is a larger problem than producing small specific amounts in laboratory experiments. Squeezed coherent states have been indirectly observed and the dynamical Casimir effect has been directly observed (Wilson, 2011). Therefore, we propose a highly experimental technique for generating negative energy based on attractive photon observations (Firstenberg et al., 2013), a quantum optical squeezing technique (Davis, 2006), and a method for direct observation of negative energy (Ford, 2009).
The experimental technique involves trapping photons by firing a laser into a super-cooled rubidium gas and accumulating a large number of photons. As observed by the Firstenberg et al. experiment, the photons will display attractive behavior and form molecule-like structures. By \textbf{\textit{exciting the photons [need to research how we can best excite them without flooding the system of positive energy - preferrably exhausting it elsewhere (think centrifuge and quantum inequalities, can an exciton process be used here?)]}}, we expect the photons to immediately squeeze their coherent states due to their close proximity and suppress vacuum fluctuations. This process will generate a large area of negative energy density.
Detection is possible by indirect observation. By firing atoms into the regions of negative energy densities, the decay rates of the atoms are expected to change (Ford et al., 2009). This particular method of observation was developed by Ford & Roman while researching negative energy observations in squeezed states and probability distributions for negative energy densities (Fewster et al., 2010).
Although the most up-to-date experimentation still upholds the quantum inequalities and their limits on negative energy (Ford et al., 2013), we push to discover scalable methods of negative energy production for the purpose of discovering new theory. Because gravity is the only direct method of observing negative energy and not enough of it can be produced in current laboratory experiments (Ford et al., 2009), we expect that scalable production of negative energy will assist development of new quantum gravity theory by allowing direct observation. It's also possible that if macro effects are observed, fractal physics theory can be further proved or disproved with data.
\textbf{\textit{need to add calculations/graphs/supporting evidence}}
\section{Conclusions}
[important: write about how we conclude faster than light travel using spacetime manipulation can be taken to a realistic level in the lab]
[talk about engineering patterns at quantum level, and how they will affect design at a marco level]
[talk about harold white's own experiments]
[talk about impacts of proposed experiments]
\selectlanguage{english}
\FloatBarrier
\end{document}