ligo-comprehensive/comprehensive.tex

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\newcommand{\GWA}{\textsc{GW150914}}
\newcommand{\GWB}{\textsc{GW151226}}

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\begin{document}

\input{title.tex}

\section{Introduction}
In 1916, Einstein predicted the existence of gravity waves as a consequence of his theory of general relativity. However, despite a century of experimental effort, direct observation eluded scientists until September 2015 when the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) observed a gravity wave resulting from a binary black hole merger.

\section{The History of Gravity Waves}
\subsection{From General Relativity to Gravitational Waves}
Ever since the initial prediction of gravity waves

For the purposes of describing the detection of gravity waves, I will start by introducing the \emph{stress-energy tensor} $T^{\alpha\beta}$ which satisfies the relation \[T^{\alpha\beta}\Sigma_\beta = P^\alpha.\] Where $P^\alpha$ is the total 4-momentum that flows through the a small 3-volume $\Sigma_\beta$. Now the question is: how does one interpret the components of $T^{\alpha\beta}$? Consider first the $P^0$ equation. This can be expanded as \[P^0 = T^{00} \Sigma_0 + T^{0i} \Sigma_i, \quad i\in\left\{1,2,3\right\}.\] 

% If this is unclear, consider the 3-dimensional case where $\Sigma_i$ is a surface. This surface can equivalently be expressed as \[\Sigma_i=A\hat{n}.\] Where A is the area of the surface and $\hat{n}$ is the unit-normal to the surface. 

% Where \[\Sigma_\beta\] is a 3-volume vector in 4-dimensional lorentz space. If this is unclear, consider that a 2-volume in 3-dimensional space, i.e. a rectangle, can be combined with a vector to form a 3-volume. Similarly, a 3-volume in 4-dimensional space can be combined with another vector(tensor) to create a scalar(tensor with reduced rank).

$P^\alpha$ is the 4-momentum that flows through the spatial 
\subsection{Early Efforts at Detection}

\section{The Advanced LIGO Detector}

\subsection{Principle of Operation}
LIGO is at its heart a Michelson style interferometer. This type of interferometer is displayed schematically in \fig{interferometer_schematic}. The interferometer works by first producing a coherant light source, typically a laser, and then splitting the source into two beams that take different paths. The beams are then recombined and made to interfere with each other. The resulting interference pattern can then be used to infer the difference in the path lengths taken by the two lasers. To be more concrete, consider \fig{interferometer_fringe_formation}. The left configuration shows the case where the beams are recombined with perfect alignment. The points $S'_1$ and $S'_2$ show the virtual source points. Meaning the 

\begin{figure}[b]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/interferometer_schematic.png}
  \caption{Schematic representation of a basic interferometer. Image credit: Wikipedia}
\label{fig::interferometer_schematic}
\end{figure}

\begin{figure}[b]
  \centering
  \includegraphics[width=0.40\textwidth]{figures/interferometer_fringe_formation.png}
  \caption{Image credit: Wikipedia}
\label{fig::interferometer_fringe_formation}
\end{figure}

\section{Signal Extraction}
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Est tempus tempus ipsum viverra dignissim, senectus. Sociosqu ut magna mus morbi curabitur nam, non dignissim rutrum ultrices arcu, fermentum eget. Proin et. Lorem netus class curae primis vitae adipiscing. Etiam. Lacus ullamcorper torquent, iaculis fames ac nec. Arcu venenatis erat.

\section{Observations}
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Est tempus tempus ipsum viverra dignissim, senectus. Sociosqu ut magna mus morbi curabitur nam, non dignissim rutrum ultrices arcu, fermentum eget. Proin et. Lorem netus class curae primis vitae adipiscing. Etiam. Lacus ullamcorper torquent, iaculis fames ac nec. Arcu venenatis erat.
\subsection{\GWA}
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Est tempus tempus ipsum viverra dignissim, senectus. Sociosqu ut magna mus morbi curabitur nam, non dignissim rutrum ultrices arcu, fermentum eget. Proin et. Lorem netus class curae primis vitae adipiscing. Etiam. Lacus ullamcorper torquent, iaculis fames ac nec. Arcu venenatis erat.
\subsection{\GWB}

\section{Cosmological Implications}

\section{Conclusions}

\begin{appendices}
  \section{App 1}
  \section{App 2}
\end{appendices}

\bibliographystyle{plain}
\footnotesize
\bibliography{references}

% \begin{thebibliography}{9}
 % \bibitem{ODEBook}Edwards, C.H, \& Penney, D.E. (2008). Differential equations, computing and modeling. Upper Saddle River, NJ, USA: Prentice Hall.
 % \bibitem{Wolfram}Weisstein, Eric W. "Duffing Differential Equation." From MathWorld--A Wolfram Web Resource.\ http://mathworld.wolfram.com/DuffingDifferentialEquation.html
 % \bibitem{Moon}F.C.Moon an dP.J. Holmes, 'A Magnetoelastic Strange Attractor,' J. Sound Vib. 65, 285 (1979)
 % \bibitem{Smith}Smith, Leonard. CHAOS, A Very Short Introduction. Oxford, NY: Oxford University Press, 2007. Print.
 % \bibitem{Holmes}Florin Diacu and Philip Holmes (1996) Celestial Encounters: The Origins of Chaos and Stability, Princeton University Press.
 % \bibitem{Brennan}Kovacic, Ivana, and Michael J. The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, 2011. Print.
% \end{thebibliography}

\end{document}
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