\documentclass[main.tex]{subfiles}
\begin{document}

\marginpar{Wednesday\\ 2020-11-4, \\ compiled \\ \today}

We were discussing the flow of plasma from the donor star to the compact object through the inner Lagrangian point.
The velocities of the compact object in the frame of the gas are \(v_\parallel \sim \SI{10}{km/s}\) and \(v_\perp \sim \SI{100}{km/s}\), so we neglect \(v_{\parallel}\).

Suppose we have a mass \(M\), and a particle in a bound orbit around this mass.
Its energy \(E\) and energy per unit mass \(\epsilon \) will be
%
\begin{align}
E = - \frac{GMm}{2a} && \epsilon = - \frac{GM}{2a}
\,,
\end{align}
%
while its specific angular momentum will be (in terms of the eccentricity \(e\)):
%
\begin{align}
\qty( \frac{L}{m})^2 = \ell^2 = (1 - e^2) GMa
\,.
\end{align}

Then, we can write the semimajor axis as
%
\begin{align}
\frac{1}{a} = \frac{(1-e^2) GM}{\ell^2}
\,,
\end{align}
%
so the specific energy will read
%mk
\begin{align}
\epsilon = - \frac{GM (1-e^2) GM}{2\ell^2}= - \frac{(GM)^2(1-e^2)}{2 \ell^2}
\,.
\end{align}

We can ask ourselves: what is the orbit which has the minimum energy \(\epsilon _{\text{min}}\) at fixed \(\ell\)? The only thing which can vary is the eccentricity \(e\), so the minimum energy is attained for the circular orbit, with \(e = 0\), where
%
\begin{align}
\epsilon _{\text{min}} = -\frac{(GM)^2}{2 \ell^2}
\,.
\end{align}

The stream of gas will be subjected to frictional forces, which will dissipate energy, and since the energy of circular orbits is minimum this will circularize the orbit.
We will discuss the timescale of this process later.

We can estimate the radius of circularization, \(R _{\text{circ}}\): we know that for a circular orbit the angular momentum will reach its Keplerian value, \(L_k\), and the velocity will reach its Keplerian value.
This reads
%
\begin{align}
v_K = \sqrt{ \frac{GM}{R}}
\,,
\end{align}
%
which comes from equating \(v^2 /R\) and \(GM/ R^2\).
The Keplerian (specific!) angular momentum reads
%
\begin{align}
L_K = R v_K
\,,
\end{align}
%
which we can compute at \(R _{\text{circ}}\):
%
\begin{align}
L_K (R _{\text{circ}}) = \sqrt{GM R _{\text{circ}}}
\,.
\end{align}

If we fix the specific angular momentum \(\ell\) of an incoming fluid parcel we can then determine the radius of its orbit, \(R _{\text{circ}} = \ell^2 / GM\).

We can compute this initial value of \(\ell\) since the velocity of the fluid is given by the \(v_{\perp} = \omega b_1 \), and then \(\ell = v_\perp b_1 = \omega b_1^2\).
Then, finally, we have
%
\begin{align}
R _{\text{circ}} = \frac{\omega^2 b_1^{4}}{GM} = \frac{4 \pi^2 b_1^{4}}{GM P^2}
\,.
\end{align}

In units of the orbital separation, and using Kepler's law
%
\begin{align}
\omega^2 = \frac{G (M_1 + M_2 )}{a^3}
\,
\end{align}
%
we get (now denoting, more specifically, as \(M_1 \) the mass we previously just called \(M\)):
%
\begin{align}
\frac{R _{\text{circ}}}{a} &= \frac{\omega^2 b_1^{4}}{GM_1  a}  \\
&= \frac{b_1^{4} G (M_1 + M_2 )}{GM_1 a^{4}} = \qty(\frac{b_1 }{a})^{4} (1 + q)
\,.
\end{align}

Yesterday we saw that \cite[]{frankAccretionPowerAstrophysics2002}:
%
\begin{align}
\frac{b_1 }{a} \approx \num{.5} - \num{.227} \log q
\,,
\end{align}
%
using which we get
%
\begin{align}
\frac{R _{\text{circ}}}{a} \approx \qty(\num{.5} - \num{.227} \log q)^{4} (1+q)
\,,
\end{align}
%
and we can also calculate the radius of the Roche lobe by inverting a result from yesterday (with \(q \to 1/q\)):
%
\begin{align}
\frac{R_1}{a} = \begin{cases}
    \num{.38} -\num{.2} \log q & \num{.05} < q < 2 \\
    \frac{\num{.426}}{(1 + q)^{1/3}} & q > 2
\end{cases}
\,.
\end{align}

We can then see that \(R _{\text{circ}}\) is at least 10 times smaller than the radius of the lobe.

\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{figures/roche-vs-circularization}
\caption{Roche lobe of star 1 versus circularization radius; both are plotted as \(R / a\).}
\label{fig:roche-vs-circularization}
\end{figure}

For a star we would need to ensure that \(R _{\text{circ}} > R_{*}\), but for a compact object there are no issues.

\subsection{The accretion disk}

Particles at their Keplerian velocities and radii around the compact object would keep orbiting, were it not for dissipative effects, which allow for energy and momentum to be transmitted throughout the disk.
% If we had two concentric disks, both of which

There are three characteristic times we need to account for:
%
\begin{align}
t _{\text{dyn}} < t _{\text{rad}} < t _{\text{visc}}
\,,
\end{align}
%
the dynamical, radiative and viscous timescale. Injection happens on a short \(t _{\text{dyn}}\) timescale, circularization happens on a longer \(t _{\text{rad}}\) timescale, shrinkage happens on an even longer \(t _{\text{visc}}\) timescale.

The true trajectory of a fluid element will be a spiral, which we can approximate with a succession of circles.
This is how an accretion disk forms.

Since \(M _{\text{disc}} \ll M_1 \), the self-gravity of the accretion disk is negligible. Therefore, the azimuthal velocity of matter in the disk will closely match the Keplerian velocity
%
\begin{align}
v_{\phi } = v_K = \sqrt{ \frac{GM_1}{R}}
\,.
\end{align}

We can already estimate the efficiency of the accretion process: the specific energy of the gas at the inner radius of the disk, \(R _{\text{in}}\), which is the star radius for a NS and the ISCO for a BH.
The specific energy is
%
\begin{align}
\epsilon (R _{\text{in}}) = - \frac{GM_1 }{R _{\text{in}}}
+ \frac{1}{2} v_K^2 = - \frac{1}{2} \frac{GM_1 }{R _{\text{in}}}
\,.
\end{align}

The variation of the energy can be calculated starting from infinity since \(R_1 \gg R _{\text{in}}\):\footnote{This is a classical estimate, but it works well enough: for example, for a Schwarzschild BH we have calculated explicitly \(\epsilon _\infty - \epsilon (R _{\text{in}}) = 1 - \sqrt{8/9} \approx \SI{6}{\percent}\), while this expression would give \(1/12 \approx \SI{8}{\percent}\). Not strictly correct, but in the right ballpark.}
%
\begin{align}
\epsilon_{\infty } - \epsilon (R _{\text{in}}) = \frac{1}{2} \frac{GM_1}{R _{\text{in}}}
\,,
\end{align}
%
therefore the luminosity of the disk will be
%
\begin{align}
L _{\text{disc}}= \frac{1}{2} \frac{GM_1 }{R _{\text{circ}}} \dot{M} c^2
\,,
\end{align}
%
only half of the accretion luminosity, defined as
%
\begin{align}
L _{\text{acc}} = \frac{GM_1 }{R _{\text{in}}} \dot{M} c^2
\,.
\end{align}

Now we want to make more detailed predictions.
A key point is viscosity: friction between the various gas elements.

Let us consider two layers of the disk.
They will have a macroscopic bulk motion, with \(v_\phi = R \Omega(R) \),
superimposed with a microscopic motion which can be at very small, up to mesoscopic scales.
We can have micro-scale motion of ions, but also
medium-scale structures can form: turbulent eddies, since the Reynolds number can be shown to be very large.

\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{figures/turbulent-eddies-accretion}
\caption{Turbulent eddies moving.}
\label{fig:turbulent-eddies-accretion}
\end{figure}

Suppose we have an eddy which starts in \(A\), moves radially, and then dissipates in \(A'\).
Further, let us say that the length scale of its motion is \(\lambda \), and the typical velocity of its motion across the disk is \(\overline{v}\).

Its radius and velocity at \(A\) will be \(R, \Omega(R) \times  R\); at \(A'\) they will be \(R + \lambda \) and still \(\Omega (R) \times R\).

In terms of specific angular momentum, when the eddy dies it will dissipate angular momentum \((R+\lambda) R \Omega (R) \).

For an eddy moving from \(B\) to \(B'\) in the opposite direction we will have \(R (R + \lambda ) \Omega (R + \lambda )\).
Since the motion is thermal, on average there will be as many particles going in both direction.

% \todo[inline]{But there is more volume at higher \(R\), so more matter!}

Suppose that the height of the disk is \(H\): then the mass per unit time carried by the eddies will be \(H 2 \pi R \overline{v} \rho \).

Then, the variation of angular momentum will be
%
\begin{align}
\frac{\Delta L}{\Delta t} &= 2 \pi R H \rho \overline{v}
\qty[R (R + \lambda ) \Omega (R) - R (R+\lambda ) \Omega (R + \lambda )]  \\
&\approx - 2 \pi R^2 (R+ \lambda ) H \rho \overline{v} \dv{\Omega }{R} \lambda  \\
&\approx - 2\pi R^3 H \rho \overline{v} \lambda \dv{\Omega }{R}
\,,
\end{align}
%
and if we introduce the surface density of the disk:
%
\begin{align}
\Sigma = \int_{- H/2 }^{H/2} \rho \dd{z} \approx \rho H
\,
\end{align}
%
we can write this torque as
%
\begin{align}
\frac{ \Delta L}{\Delta t} = \tau_{\text{partial}} \approx -2 \pi R \Sigma \qty(\overline{v} \lambda ) R^2 \dv{\Omega }{R}
\,.
\end{align}

For a Keplerian accretion disk we always have \(\dv*{\Omega }{R} < 0\), since \(\Omega _K = \sqrt{GM / R^3}\): this means that the torque is positive.
Note, however, that we are still only considering the effect on a certain layer of the one above it --- this is not the full picture yet.

This is the torque which the inner part of the disk exerts on the outer part, decelerating it.
We can then introduce a function \(G(R) = - \tau \), the torque exerted by the outer part of the disk on the inner part, accelerating it.

For a given layer rotating at \(R \Omega (R)\), the layers above it will try to accelerate it, while the ones below it will try to decelerate it. What will be the net effect? It will be\footnote{We use the same letter \(\tau \) as before, but we should specify that this time it accounts for all the contributions to a certain layer of the disk, while before it was only one-way.}
%
\begin{align}
\tau =  G(R + \dd{R}) - G(R) = \dv{G}{R} \dd{R}
\,.
\end{align}

These opposite effects will dissipate heat.
The differential work dissipated will be given by
%
\begin{align}
\dd{W} = \tau \dd{\phi } = \dv{G}{R} \dd{R} \dd{\phi }
\,,
\end{align}
%
so, since to a first approximation \(G\) can be taken to be a constant, the power will be
%
\begin{align}
\dv{W}{t} = \dv{G}{R} \dd{R} \Omega
\,.
\end{align}

Integrating to find the total power we get
%
\begin{align}
\dot{E} &= \int_{R _{\text{in}}}^{R _{\text{out}} } \dv{G}{R} \Omega \dd{R}
\,,
\end{align}
%
but we can integrate by parts to find
%
\begin{align}
\dot{E} &= \eval{G \Omega }_{R _{\text{in}}}^{R _{\text{out}}}
- \int_{R _{\text{in}}}^{R _{\text{out}}} G \dv{\Omega }{R} \dd{R}
\,,
\end{align}
%
so we can identify a global, \emph{convective term}: the variation of \(G \Omega \). On the other hand \(G \dv*{\Omega }{R} \dd{R}\) is a local dissipation term.

Let us introduce the radiated power per unit area of the disk (which is positive, we leave the minus sign out):
%
\begin{align}
D(R) = \frac{ \eval{\dd{(\dot{E})}}_{\text{local}}}{2 \times 2 \pi R \dd{R}}
&= G \dv{\Omega }{R} \frac{ \dd{R}}{2 \times 2 \pi R \dd{R}} = \frac{G}{4 \pi R} \dv{\Omega }{R} \marginnote{Divided by 2 since the disk has two faces.}
\,.
\end{align}

This is written as
%
\begin{align}
D(R) = \frac{G}{4 \pi R} \dv{\Omega }{R}
= \frac{1}{2} R^2 \overline{v} \lambda \Sigma \qty(\dv{\Omega }{R})^2
\,.
\end{align}

In order to dissipate energy the differential rotation \(\dv*{\Omega }{R}\) is crucial.
% We see next time that  \(\overline{v} \lambda = \nu \), the kinematic viscosity coefficient.

\end{document}


% !TEX root = ../thesis.tex

\chapter{Tests and Results}
% \label{cha:eva_tests_results}

In this chapter I will first present a few benchmarks. It is important for any simulation to be able to reproduce results that are predicted by theory. For this, I will test collision rates depending on temperature and scattering length as well as the relaxation of the cloud from an initial delta velocity distribution to a Maxwell-Boltzmann distribution.

Then I am going to show how our proposed solution of a compressing box potential compares to similar setups with harmonic trapping configurations in terms of evaporation efficiency and final atom numbers.

\section{Benchmarks}

\subsection{Collision Rates}
% Our first test is if the simulation can reproduce accurate collision rates that match the theoretical predictions. For the cross section given in \cref{eq:crosssection}, we have to take the average \meanProb over the Maxwell-Boltzmann distributions for both collision partners to find the mean collision rate.
% \begin{align*}
%     \meanProb &= \iint \multidiff{3}{c_1} \multidiff{3}{c_2}\; \frac{8\pi a^2\crel}{1 + a^2 k^2} \times n(\vec{c}_1) n(\vec{c}_2) \\
%     &= \frac{8\pi a^2}{(2\pi m \kB T)^3}\iint \multidiff{3}{c_1} \multidiff{3}{c_2}\; \frac{\crel}{1 + a^2 k^2} \exp\!\left(-\frac{m(\vec{c}_1^2 + \vec{c}_2^2)}{2\kB T}\right)
%     \intertext{We can now change variables to centre-of-mass and relative coordinates and integrate out the angular part}
%     &= (4\pi)^2 \frac{8\pi a^2}{(2\pi m \kB T)^3}\int_0^\infty \diff{\crel}\; \crel^2 \frac{\crel}{1 + a^2 k^2} \exp\!\left(-\frac{m\crel^2}{4\kB T}\right) \nonumber \\
%     &\phantom{=} \quad \times \int_0^\infty \diff{c_\text{COM}}\; c_\text{COM}^2 \exp\!\left(-\frac{mc_\text{COM}^2}{\kB T}\right) \\
%     &= 4\sqrt{\pi} a^2 \left(\frac{m}{\kB T}\right)^\frac{3}{2} \int_0^\infty \diff{\crel} \frac{\crel^2}{1 + a^2 \frac{m^2 \crel^2}{4\hbar^2}} \exp\!\left(-\frac{m\crel^2}{4\kB T}\right)
%     \intertext{where we have performed the integration over $c_\text{COM}$ and used the relation $k = \frac{m\crel}{2\hbar}$. Making the integral dimensionless and introducing the (also dimensionless) parameter $\chi = \sqrt{\frac{\hbar^2}{a^2 m \kB T}}$ we find}
%     &= 64\sqrt{\pi} \cdot \frac{a\hbar}{m} \cdot \chi \int_0^\infty \diff{u}\; \frac{u^3}{\chi^2 + u^2} \exp(-u^2) \\
%     &= 32\sqrt{\pi} \cdot \frac{a\hbar}{m} \cdot \chi \left(1 + \chi^2\exp(\chi^2)\text{Ei}(-\chi^2)\right)
%     \intertext{where Ei is the exponential integral \cite[Eq.~6.2.5]{NIST:DLMF}. Because $\chi^2$ can be much larger than 1, for calculation on a computer it is better to express this using the confluent hypergeometric function $U$ \cite[Eq.~13.2.6]{NIST:DLMF}:}
%     \meanProb &= 32\sqrt{\pi} \cdot \frac{a\hbar}{m} \cdot \chi \left(1 + \chi^2U(1,1,\chi^2)\right).
% \end{align*}
% We can now use our previous result for the mean spatial density in an arbitrary power-law trap from \cref{eq:spatial_density,eq:mean_density} to find the mean collision rate $\langle\Rcoll\rangle$ as
% \begin{align*}
%     \langle\Rcoll\rangle &= \langle n\rangle \meanProb \\
%     &= 4\sqrt{\pi} \frac{N}{x_0y_0z_0} \cdot \frac{a\hbar}{m} \cdot \chi \left(1 + \chi^2U(1,1,\chi^2)\right) \times \prod_{i\in\{x,y,z\}}\frac{1}{\Gamma\!\left(1 + \frac{1}{\alpha_i}\right)}\left(\frac{U_i}{2 \kB T}\right)^{\frac{1}{\alpha_i}}.
% \end{align*}
Our first test is if the simulation can reproduce accurate collision rates that match the theoretical predictions. For the cross section given in \cref{eq:crosssection}, we have to take the average \meanProb over the Maxwell-Boltzmann distributions for both collision partners to find the mean collision rate. This calculation is done in \cref{sec:appendix_average} and the result is
\begin{equation*}
    \meanProb = 32\sqrt{\pi} \cdot \frac{a\hbar}{m} \cdot \chi \left(1 + \chi^2U(1,1,\chi^2)\right),
\end{equation*}
where
\begin{equation*}
    \chi = \sqrt{\frac{\hbar^2}{a^2 m \kB T}}
\end{equation*}
is a dimensionless parameter that combines the scattering length $a$, the atomic mass $m$ and the temperature $T$. $U(1,1,\chi^2)$ is one of the confluent hypergeometric functions \cite[Eq.~13.2.6]{NIST:DLMF}.

We can now use our previous result for the mean spatial density in an arbitrary power-law trap from \cref{eq:spatial_density,eq:mean_density} to find the mean collision rate $\langle\Rcoll\rangle$ as
\begin{equation}\label{eq:meancollisionrate}
    \langle\Rcoll\rangle = \langle n\rangle \meanProb = 4\sqrt{\pi} \frac{N}{x_0y_0z_0} \cdot \frac{a\hbar}{m} \cdot \chi \left(1 + \chi^2U(1,1,\chi^2)\right) \times \prod_{i\in\{x,y,z\}}\frac{1}{\Gamma\!\left(1 + \frac{1}{\alpha_i}\right)}\left(\frac{U_i}{2 \kB T}\right)^{\frac{1}{\alpha_i}}.
\end{equation}

To validate the accuracy of the simulation, we test this for different scattering lengths and temperatures. We choose a (near) box potential with $\alpha_x = \alpha_y = \alpha_z = 1000$, side lengths of \SI{1}{mm} in all directions and a potential depth of $U_0 = \SI{e-26}{J} \sim \SI{725}{\micro\kelvin}\times\kB$. The atomic mass is set to \SI{87}{u}. We start each of the simulations with \num{4e8} atoms in a $(\num{0.995} \times \num{0.995} \times \num{0.995})\, \si{\milli\meter\cubed}$ cube with uniform density distribution, let them reach a steady state by waiting for 3 initial mean collision times and then let the simulation continue until 100 initial mean collision times have been elapsed.

The statistic weight is \num{32768}, meaning \num{12207} particles are simulated. The timestep is 1/50 of the mean collision time and for this test, the loss coefficients $K_{1,2,3}$ are all set to 0. The results are shown in \cref{fig:evap_test_rates} and the measured rates fit the prediction very well. Over all conducted tests, the maximum deviation of the measured from the theoretically predicted rate was \SI{.784}{\percent}, with an average of just \SI{.414}{\percent}.

\vfill
\begin{figure}[hbp]
    \centering
    \includegraphics[]{Evap/CollisionRates}
    \caption[Collision rate variation with temperature and scattering length]{\num{4e8} atoms are simulated in an approximate box potential with a volume of $\sim\!\SI{1}{\milli\meter\cubed}$ at different values of the scattering length $a$ and temperature $T$. The collision time is averaged over 100 initial mean collision times and compared to the theoretical result from \cref{eq:meancollisionrate}. The maximum deviation was measured to be \SI{.784}{\percent}.}
    \label{fig:evap_test_rates}
\end{figure}
\vfill

\subsection{Thermal Relaxation}
Next, we initialise a cloud of \num{3e8} atoms in the same potential as before and with a scattering length of $a = 100\, a_0$. The kinetic temperature is set to $T=\SI{20}{\micro\kelvin}$, but every particle is given the same speed initially. We record the speeds of every particle present for 10 initial mean collision times ${\langle \tau\rangle}_\text{i}$ and average the resulting speed distribution over 50 separate runs. The results are shown in \cref{fig:evap_relaxation}. After $\num{8.5}\,{\langle \tau\rangle}_\text{i}$, the resulting speed distribution matches the expected Maxwell-Boltzmann distribution.
\begin{figure}[htbp]
    \centering
    \includegraphics[]{Evap/Relaxation}
    \caption[Relaxation to a Maxwell-Boltzmann distribution]{The simulation is initialised with \num{3e8} atoms that all start with the same speed. Over a total of 10 initial mean collision times, the relaxation of the speeds to a Maxwell-Boltzmann distribution is observed.}
    \label{fig:evap_relaxation}
\end{figure}

% \subsection{Internal Energy per Atom}
% Our last test analyses the energy per particle. From the theoretical analysis, we expect \[\tilde{E} = \xi kB T, \quad \xi = \frac{3}{2} + \frac{1}{\alpha_x} + \frac{1}{\alpha_y} + \frac{1}{\alpha_z}\] per particle in a power-law potential with exponents $\alpha_{x,y,z}$.
% We let \[\alpha_x = \alpha_y = \alpha_z \equiv \alpha,\quad 1 \leq \alpha \leq 100.\] The starting conditions are: \todo[inline]{atom number, density, temperature} and we let the system reach equlibrium in \dots initial mean collision times, before we measure the energy per particle. The results of this can be seen in Figure~\todo{ref}. Again, we see excellent agreement with the theoretical prediction.

\section{Efficient Evaporation in a Box}
\label{sec:eva_test}
We now move on to test the proposed evaporation in a box potential. The simulation is partly based on the assumption that the atoms have been cooled to a high starting phase space density before the evaporation sequence. This goes hand in hand with our initial concept for evaporation in a box. Usually, atoms are pre-cooled before evaporative cooling in an optical dipole trap is carried out. It has been shown that $\Lambda$-enhanced grey molasses cooling is possible on the D2 line of $^{87}$Rb \cite{Rosi2018enhancedGM}. Here, a final phase space density of \num{4e-6} was achieved at a temperature of \SI{4}{\micro\kelvin}, corresponding to a density of \SI{4.9e9}{\per\centi\meter\cubed}. Starting from a density this low would be disadvantageous for evaporative cooling as it would incur high thermalisation times. Compressing the cloud however would heat the atoms again. Grey molasses cooling is mainly limited by time as the cloud can expand freely and is subject to gravity. With our proposed concept of a box potential, the cloud would not be able to expand indefinitely, given that the trap depth is high enough to contain the falling atoms. We could then repeatedly compress the gas and cool it again. Such an alternating cycle of compression and cooling has already been demonstrated with $^{85}$Rb using a near-resonant dark optical lattice, leading to a density of \SI{1.2e12}{\per\centi\meter\cubed} with a temperature of \SI{10}{\micro\kelvin}, equivalent to a phase space density of \num{2.6e-4} \cite{PhysRevA.72.043410}.

For the simulation, we use a box potential created from two ring beams that intersect each other at an angle. The functional form of this potential is
\begin{equation} \label{eq:eva_potential}
    U(\vec{r}) = U_0 \left[ \left( \frac{\sqrt{(-\sin(\frac{\alpha}{2})x + \cos(\frac{\alpha}{2})y)^2 + z^2}}{R} \right) ^ P + \left( \frac{\sqrt{(\sin(\frac{\alpha}{2})x + \cos(\frac{\alpha}{2})y)^2 + z^2}}{R} \right) ^ P \right]
\end{equation}
where $\alpha$ is the angle between the beams, $R$ is the radius of the ring and the exponent $P$ is given by $P = \num{87}$. This specific value was chosen following the results of Hueck et al.~\cite{PhysRevLett.120.060402}, where it was demonstrated that a ring beam made by using an axicon can be focused to achieve a power law potential with a similar exponent.
\Cref{fig:steinmetz_solid} shows the outline of the potential for the angles $\alpha=\SI{90}{\degree}$ and $\alpha=\SI{157.5}{\degree}$. The latter case is the configuration that will be used in the experiment and therefore also in the simulation below.
\vfill
\begin{figure}[htbp]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \input{TexContents/Figures/Evap/BoxTrapEvaporation/Steinmetz1.tikz}
        \caption{$\alpha = \SI{90}{\degree}$}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \input{TexContents/Figures/Evap/BoxTrapEvaporation/Steinmetz2.tikz}
        \caption{$\alpha = \SI{157.5}{\degree}$}
    \end{subfigure}
    \caption[Common volume of two crossed cylinders]{It is important to note that the shapes shown here do not represent an equipotential surface of \cref{eq:eva_potential}. Instead, they simply show the intersection of two cylinders at the two different angles.}
    \label{fig:steinmetz_solid}
\end{figure}
\vfill

% \begin{figure}[htbp]
%     \centering
%     \begin{subfigure}[b]{.49\textwidth}
%         \centering
%         \includegraphics[trim=0 1.7ex 0 6ex,clip]{Evap/SteinMetzPython}
%         \caption{\SI{90}{\degree}}
%     \end{subfigure}
%     \begin{subfigure}[b]{.49\textwidth}
%         \centering
%         \includegraphics[trim=0 1.7ex 0 6ex,clip]{Evap/SteinMetzPython_Angle}
%         \caption{\SI{157.5}{\degree}}
%     \end{subfigure}
%     \caption{Common volume of two crossed cylinders}
%     \label{fig:steinmetz_solid}
% \end{figure}

\subsubsection*{Parameter Optimisation}
For all cases, we use a simple optimisation scheme.
The initial beam radius is fixed to \SI{360}{\micro\meter} and the initial trap depth is given by $\SI{50}{\micro\kelvin}\times \kB$. We then employ only one exponential ramp for the beam radius and the potential depth where we compress the beam to a final radius of \SI{25}{\micro\meter} and a final trap depth of $\SI{450}{\nano\kelvin}\times \kB$:
\begin{equation*}
    X(t) = \frac{X_\text{f} - X_\text{i}}{\euler^{-t_\text{d}/\tau} - 1} \times \left(\euler^{-t/\tau} - 1\right) + X_\text{i}.
\end{equation*}
Here, $X$ is the parameter to which the ramp is applied, $X_\text{i,f}$ are the initial and final values before and after the ramp, $t_\text{d}$ is the duration of the ramp and $\tau$ is the exponential time constant.
After an initial coarse search of the available parameter space for $t_\text{d}$ and $\tau$ for both parameters, we choose the values that yield the highest possible evaporation efficiency
\[
    \gamma = -\frac{\log(\tilde{\rho}_\text{f}/\tilde{\rho}_\text{i})}{\log(N_\text{f}/N_\text{i})},
\]
with the initial and final values for the phase space density $\tilde{\rho}_\text{i/f}$ and atom number $N_\text{i/f}$. Then we go on to vary all three parameters ($t_\text{d}$, $\tau$ for the radius $R$ and the trap depth $U_0$) in a smaller range around the previous optimum and try to find an improvement.

If a phase space density of $\PSD = 1$ is reached, the simulation is stopped prematurely because it is purely classical and results at higher phase space densities are not valid anymore.

\subsubsection*{Starting Conditions}
We simulate different initial phase space densities ranging from \num{5e-6} to \num{e-3} with atom numbers from \num{2e8} to \num{3e7} as well as temperatures from \SI{10}{\micro\kelvin} down to \SI{2.5}{\micro\kelvin} for the highest phase space density. We initialise the particles with a harmonic density distribution, where the RMS radius can be calculated from the peak phase space density, the atom number and the temperature.
This gives us a range of estimates on how effective this cooling method could be for different starting conditions and we will be able to tell how much pre-cooling is necessary for good results.
The cases with their respective fixed parameters are shown in \cref{tab:evap_cases}.
\begin{table}[bp]
    \centering
    \caption[Initial parameters for the simulated cases]{$\hat{\PSD}$ and $\hat{n}$ are the initial peak phase space and spatial densities, respectively. $\sigma$ is the RMS radius of the initially harmonic cloud.}
    \begin{tabular}{cS[table-format=1.1e-1]S[table-format=1.2]S[table-format=2.1]S[table-format=3]S[table-format=1.1e+2]}
        \toprule
        \# & \multicolumn{1}{c}{$\hat{\PSD}$} & \multicolumn{1}{c}{$N/\num{e8}$} & \multicolumn{1}{c}{$T$/\si{\micro\kelvin}} & \multicolumn{1}{c}{$\sigma$/\si{\micro\meter}} & \multicolumn{1}{c}{$\hat{n}$/\si[per-mode=reciprocal]{\per\meter\cubed}}\\
        \midrule
        1  & 5.0e-6 & 2.00 & 10.0  & 809 & 2.4e16 \\
        2  & 1.9e-5 & 1.20 & 7.1 & 528 & 5.4e16 \\
        3  & 7.1e-5 & 0.77 & 5.0   & 345 & 1.2e17 \\
        4  & 2.7e-4 & 0.48 & 3.5 & 225 & 2.7e17 \\
        5  & 1.0e-3 & 0.30 & 2.5 & 147 & 6.0e17 \\
        \bottomrule
    \end{tabular}
    \label{tab:evap_cases}
\end{table}
% We simulate two different scenarios for both angles: In the first scenario, we assume that the cycling scheme of compression and cooling works. In that case, we start at a phase space density of \todo{number} and a temperature of \todo{number}, corresponding to a density of \todo{number}. Our initial beam radius is given by \todo{number}, so we start with \todo{number} atoms.

% In the second scenario, we do not assume such a great amount of pre-cooling, so we instead start with a phase space density of only \todo{number} at a temperature of \todo{number}, corresponding to the results of \cite{Rosi2018enhancedGM}. At the same initial beam radius, this gives us an initial atom number of \todo{number}.
In all cases, we assume a trap lifetime of \SI{1}{min} (giving $K_1 = \SI{.017}{\per\second}$), no two-body losses and no influence of gravity. This can be achieved in an experiment with a sample prepared in the absolute ground state of $^{87}$Rb where gravity is compensated by a linear magnetic field. The three-body loss coefficient is taken from \cite{threebody} as $K_3 = \SI{1.8e-41}{m^6\per\second}$. At densities below \SI{e19}{\per\meter\cubed}, three body losses are negligible compared to the losses due to background collisions.

\subsubsection*{Findings}
In case 1, the initial density is so low that only a small fraction of the initial atom number can actually be captured in the box. Furthermore, this lower density results in a smaller collision rate and the compression is not fully able to compensate for this. We can achieve efficiencies $\gamma > 2$ only by making the evaporation ramp \SI{30}{s} long. Still, the final phase space density is $\PSD_\text{f}\ll 1$ and less than \num{200000} atoms remain in the trap after the ramp. We can conclude from this that either larger initial trapping beams or a higher initial density would be necessary.

At initial phase space densities above $\num{e-5}$, the situation improves drastically and efficiencies close to or above 3 can be achieved. However, this comes again at the expense of long ramp durations.

The results for all five cases are listed in \cref{tab:evap_results} and \cref{fig:evap_comparison} shows a comparison of the cases 2--5 against various results from other research groups. Here we can see that our achieved efficiency can compete with the compared results. In the cases 4 and 5, a large portion of the atoms remains in the trap at a phase space density of $\PSD = 1$. We also find that the achieved efficiency is very dependent on the initial atom number. This is expected as the initial atom number determines how strong the compression needs to be to achieve high densities and in turn high collision rates.

Lastly, we show the optimised trajectory for case 5 in \cref{fig:evap_trajectory}. During the first second of compression, no evaporation takes place, which is apparent by the relatively constant atom number and phase space density. Starting at a lower trap depth might reduce the total time necessary to reach $\PSD = 1$.

The trajectories for the other cases can be found in \cref{fig:eva_cases14}.
%
% \begin{table}[bp]
%     \centering
%     \caption{Results and optimised parameters for the simulated cases}
%     \begin{tabular}{cS[table-format=1.1e+1]S[table-format=1.1e+1]S[table-format=1.1e-1]S[table-format=1.3]S[table-format=2.1]S[table-format=3]S[table-format=1.2]}
%         \toprule
%         \# & \multicolumn{1}{c}{$N_\text{i}$} & \multicolumn{1}{c}{$N_\text{f}$} & \multicolumn{1}{c}{$\PSD_\text{i}$} & \multicolumn{1}{c}{$\PSD_\text{f}$} & \multicolumn{1}{c}{$T_\text{i}$/\si{\micro\kelvin}} & \multicolumn{1}{c}{$T_\text{f}$/\si{\nano\kelvin}} & \multicolumn{1}{c}{$\gamma$} \\
%         \midrule
%         1  & 1.1e7 & 1.3e5 & 3.7e-6 & 0.059  & 10 & 152 & 2.17 \\
%         2  & 1.9e7 & 5.5e5 & 1.2e-5 & 0.29 & 7.0 & 137 & 2.83 \\
%         3  & 2.8e7 & 1.3e6 & 3.9e-5 & 0.83 & 5.0 & 185 & 3.20 \\
%         4  & 3.3e7 & 2.0e6 & 1.2e-4 & 1    & 3.6 & 144 & 3.25 \\
%         5  & 2.8e7 & 2.3e6 & 3.4e-4 & 1    & 2.5 & 160 & 3.21 \\
%         \bottomrule
%     \end{tabular}
%     \newline
%     \vspace{3ex}
%     \newline
%     \begin{tabular}{cS[table-format=2.0]S[table-format=1.1]S[table-format=2.1]}
%         \toprule
%         \# & \multicolumn{1}{c}{$t_\text{d}$/s} & \multicolumn{1}{c}{$\tau_R$/s} & \multicolumn{1}{c}{$\tau_{U_0}$/s} \\
%         \midrule
%         1  & 30 & 4 & 30 \\
%         2  & 16 & 1.6 & 15 \\
%         3  & 14 & 1.4 & 13 \\
%         4  & 9 & 0.9 & 8.5 \\
%         5  & 7 & 0.7 & 6.5 \\
%         \bottomrule
%     \end{tabular}
%     \label{tab:evap_results}
% \end{table}
%

\clearpage
\vbox{
\noindent\begin{minipage}[c][.7518\textheight][t]{\textwidth}

% \begin{figure}[tp]
    % \centering
    % \begin{minipage}{1\textwidth}
        \input{TexContents/Figures/Evap/Comparison.pgf}
        \captionof{figure}[Comparison of the simulation results against other papers]{The atom numbers in the bottom left figure are extrapolations from the initial atom number and the average efficiency. For the comparison results, the phase space densities are peak values.}
        \label{fig:evap_comparison}
    % \end{minipage}
% \end{figure}
\end{minipage}

\nointerlineskip
\noindent\begin{minipage}[c][.1795\textheight][b]{\textwidth}
\singlespacing

% \begin{table}[b]
    \centering
    % \vspace*{3cm} % change this to align Table 10.1 and 10.2
    \captionof{table}[Results and optimised parameters for the simulated cases]{The indices i,f denote the initial and final values. $t_\text{d}$ is the duration of the simulation and $\tau_R$/$\tau_{U_0}$ the time constant for the expontential ramp of the trap radius/potential depth.}
    \label{tab:evap_results}
    \begin{tabular}{cS[table-format=1.1]S[table-format=1.2]S[table-format=1.1e-1]S[table-format=1.3]S[table-format=2.1]S[table-format=3]S[table-format=1.2]S[table-format=2.0]S[table-format=1.1]S[table-format=2.1]}
        \toprule
        \# & \multicolumn{1}{c}{$N_\text{i}/\num{e7}$} & \multicolumn{1}{c}{$N_\text{f}/\num{e6}$} & \multicolumn{1}{c}{$\PSD_\text{i}$} & \multicolumn{1}{c}{$\PSD_\text{f}$} & \multicolumn{1}{c}{$T_\text{i}$/\si{\micro\kelvin}} & \multicolumn{1}{c}{$T_\text{f}$/\si{\nano\kelvin}} & \multicolumn{1}{c}{$\gamma$} & \multicolumn{1}{c}{$t_\text{d}$/s} & \multicolumn{1}{c}{$\tau_R$/s} & \multicolumn{1}{c}{$\tau_{U_0}$/s} \\
        \midrule
        1  & 1.1 & 0.13 & 3.7e-6 & 0.059  & 10.0 & 152 & 2.17 & 30 & 4.0 & 30.0 \\
        2  & 1.9 & 0.55 & 1.2e-5 & 0.29 & 7.0 & 137 & 2.83 & 16 & 1.6 & 15.0 \\
        3  & 2.8 & 1.3 & 3.9e-5 & 0.83 & 5.0 & 185 & 3.20 & 14 & 1.4 & 13.0 \\
        4  & 3.3 & 2.0 & 1.2e-4 & 1    & 3.6 & 144 & 3.25 & 9 & 0.9 & 8.5 \\
        5  & 2.8 & 2.3 & 3.4e-4 & 1    & 2.5 & 160 & 3.21 & 7 & 0.7 & 6.5 \\
        \bottomrule
    \end{tabular}
% \end{table}
\end{minipage}
}

\clearpage
%
%
\begin{figure}[bp]
    \centering
    \includegraphics{Evap/Trajectory}
    \caption[Evaporation trajectory for case 5]{It can be seen that the first second does not contribute much to the evaporation. Here, the phase space density \PSD and atom number $N$ are nearly constant while the temperature $T$ increases due to the compression. The atom number and phase space density axes are logarithmic, the others linear.}
    \label{fig:evap_trajectory}
\end{figure}


\documentclass[main.tex]{subfiles}
\begin{document}

\marginpar{Wednesday\\ 2021-12-1}

Last time we reached the expression for the plane-wave solution to GR.

Consider a wave propagating along the \(x^{1}\) direction.
Our equation is \(\square _F \overline{h}_{\mu \nu } = 0 \), to be solved together with our gauge condition \(\partial_{\mu } \overline{h}^{\mu }{}_\nu = 0\).

The solution will only depend on \(t - x /c\), which we denote as \(w\). We will have
%
\begin{align}
\pdv{}{x} \overline{h}^{\mu }{}_{\nu } &= - \frac{1}{c} \pdv{}{w } \overline{h}^{\mu }{}_\nu \\
\pdv{}{t} \overline{h}^{\mu }{}_{\nu } &=  \pdv{}{w } \overline{h}^{\mu }{}_\nu
\,,
\end{align}
%
so, the full condition reads
%
\begin{align}
\partial_{\mu }  \overline{h}^{\mu }{}_{\nu } = \frac{1}{c} \pdv{}{t} \overline{h}^{0}{}_\nu + \pdv{}{x} \overline{h}^{x }{}_\nu
= \frac{1}{c} \pdv{}{w} \qty[h^{0}{}_{\nu } - h^{x}{}_{\nu }]
= 0
\,.
\end{align}

The constant the difference of these perturbations are equal to is inessential --- it can be recovered with a rescaling of, say, background time --- so we have
%
\begin{align}
h^{0}{}_{\nu } = h^{x}{}_{\nu }
\,.
\end{align}

The gauge we chose did not fully determine our transformation: equation \eqref{eq:christoffel-gauge-transformation} will not change if we use an additional \(\xi \) such that \(\square \xi  =0 \).

It is possible to choose four more conditions thanks to this:
%
\begin{align}
0
= \overline{h}^{0}{}_{x}
= \overline{h}^{0}{}_{y}
= \overline{h}^{0}{}_{z}
= \overline{h}^{y}{}_{y} + \overline{h}^{z}{}_{z}
\,.
\end{align}

Together with the ones from before, we find
%
\begin{align}
0
= \overline{h}^{x}{}_{x}
= \overline{h}^{x}{}_{y}
= \overline{h}^{x}{}_{z}
= \overline{h}^{0}{}_{0}
\,,
\end{align}
%
which also means \(h = 0 = \overline{h}\) (since \(h = - \overline{h}\)), therefore \(\overline{h}_{\mu \nu } = h_{\mu \nu } \).

The only two nonvanishing components left are \(h^{y}{}_{y} = - h^{z}{}_z\) and \(h^{x}{}_{y} = h^{y}{}_{x}\).

We finally have
%
\begin{align}
h_{\mu \nu } = \left[\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & h_{+} & h_{\times } \\
0 & 0 & h_{\times } & -h_{+}
\end{array}\right]
\,.
\end{align}

This is the \textbf{transverse-traceless gauge}.

\subsection{The quadrupole approximation}

We will assume that \(T_{\mu \nu } \neq 0\), but that the source of the GW is all contained within a source such that \(\abs{x'} < \epsilon\), and such that \(\epsilon \ll \lambda_{GW} = 2 \pi c / \omega \).
This means that \(\omega \epsilon / 2 \pi \sim v _{\text{source}} \ll c\).

This line of reasoning assumes that \(\omega _{\text{source}} \sim \omega _{\text{GW}} \), which we do not know yet --- we will later find that it is in fact true within a factor 2.

The GW solution reads
%
\begin{align}
\overline{h}_{\mu \nu }(t, \vec{x}) &= \frac{4 G}{c^{4}}
\int_{V} \frac{T_{\mu \nu } (t - \abs{x - x'} / c, c') \dd[3]{x'}}{\abs{x - x'}}
\,,
\end{align}
%
while in Fourier space we can expand
%
\begin{align}
T_{\mu \nu } (t, \vec{x}) = \int \widetilde{T}_{\mu \nu } (\omega , \vec{x}) e^{-i \omega t} \dd{\omega }
\,.
\end{align}

We then get
%
\begin{align}
\int \overline{h}_{\mu \nu } (\omega , \vec{x}) e^{-i \omega t} \dd{\omega } = \frac{4 G}{c^{4}}
\int_{V} \frac{ \dd[3]{x'}}{\abs{x - x'}} \int T_{\mu \nu } (\omega , x') e^{-i \omega (t - \abs{x-x'} / c) } \dd{\omega }
\,,
\end{align}
%
which means that
%
\begin{align}
h_{\mu \nu } (\omega , x) = \frac{4G}{c^{4}} \int
\frac{\dd[3]{x'}}{\abs{x - x'}}
T_{\mu \nu }(\omega , x') e^{i \omega \abs{x-x'} / c}
\,,
\end{align}
%
where we factored out the \(e^{- i \omega t}\) and the integral in \(\dd{\omega }\).

With our assumption of slow speed we can expand:
%
\begin{align}
\frac{e^{i \omega \abs{x-x'} / c}}{\abs{x-x'}} \approx
\frac{e^{i \omega r}}{r}
\,,
\end{align}
%
where \(r = \abs{x}\).
This then yields
%
\begin{align}
h_{\mu \nu }(\omega , r) = \frac{4 G}{c^{4}} \frac{e^{i\omega r}}{r}
\int_{V} T_{\mu \nu } (\omega , x') \dd[3]{x'}
\,,
\end{align}
%
so we can come back to
%
\begin{align}
\overline{h}_{\mu \nu } (t, r) = \frac{4 G}{c^{4}r} \int T_{\mu \nu } (t- r/c, x') \dd[3]{x'}
\,.
\end{align}

We can simplify this thanks to the expression \(T^{\mu \nu }{}_{, \nu } = 0\), which means we have conservation laws in the form \(\int T^{\mu 0} \dd[3]{x}\).

We can put these constants to zero (since we are not interested in any non-wavelike behavior, like the stationary Kerr-like metric due to the source).

We can do an integral of \(\partial_{\mu } T^{\mu \nu }= 0\):
%
\begin{align}
\frac{1}{c} \pdv{}{t} \int _V T^{n  0} x^{k} \dd[3]{x} &= - \int \pdv{T^{ni}}{x^i} x^{k} \dd[3]{x}  \\
&= \underbrace{\int \dd{S^{i}} (T^{ni} x^{k})}_{ \to 0} + \int T^{nk} \dd[3]{x}  \\
\frac{1}{c} \pdv{}{t} \int x^{k} T^{n0} \dd[3]{x} &= \int T^{nk} \dd[3]{x}
\,.
\end{align}

We can write this as
%
\begin{align}
\frac{1}{2} \pdv{}{t} \int \qty[ T^{n0} x^{k} + T^{k0} x^{n}] = \int T^{nk} \dd[3]{x}
\,.
\end{align}

The time component of the conservation law can be multiplied by \(x^{n} x^{k}\):
%
\begin{align}
\frac{1}{c} \pdv{}{t} \int T^{00} x^{n} x^{k} \dd[3]{x}
&= - \int \pdv{}{x^{i}} T^{0i} x^{n} x^{k} \dd[3]{x}  \\
&= + \int T^{0i} \pdv{}{x^{i}} (x^{n} x^{k}) \dd[3]{x}  \\
&= \int T^{0n} x^{k} + T^{0k} x^{n}  \dd[3]{x}
\,.
\end{align}

We then take a second derivative:
%
\begin{align}
\frac{1}{c^2} \pdv[2]{}{t} \int T^{00} x^{n} x^{k} \dd[3]{x} &=
\frac{1}{c} \pdv{}{t} \int T^{n0} x^{k} + T^{k0} x^{n} \dd[3]{x}  \\
&= 2 \int T^{nk} \dd[3]{x}
\,.
\end{align}

This is the \textbf{virial theorem} in GR.

We will assume we are working on a \(t = \text{const}\) hypersurface.
The metric is purely Euclidean there.

The object
%
\begin{align}
\frac{1}{c^2} \int T^{00} x^{n} x^{k} \dd[3]{x} = q^{n k} (t)
\,,
\end{align}
%
the quadrupole tensor.
We then have
%
\begin{align}
\frac{1}{2} \ddot{q}^{n k} (t) = \int T^{nk} \dd[3]{x}
\,.
\end{align}

The result is therefore \(h^{\mu 0} = 0\), and
%
\begin{align}
\overline{h}^{n k} = \frac{2G}{c^{4} r} \ddot{q}^{n k}(t)
\,.
\end{align}

\end{document}


\documentclass{article}

\usepackage{amsmath}
\usepackage[a4paper,total={6in, 8in}]{geometry}
\begin{document}

		\title{Cardano's Formula for Cubic Equations}
		\author{Dumebi Valerie Duru}
		\maketitle
	\begin{center}
		\textbf{Abstract}
	\end{center}
	\paragraph{}
		Gerolamo Cardano was born in Pavia in 1501 as the illegitimate child of a jurist. He attended the University of Padua and became a physician in the town of Sacco, after being rejected by his home town of Milan. He became one of the most famous doctors in all of Europe, having treatwd the pope. He was also an astrologer and an avid gambler, to which he wrote the Book on Games of Chance, which was the first serious treatise on the mathematics of probability. \cite{cardanoformula}

	\section{Introduction to Cardano's Formula}
		Cardano's formula for solution of cubic equations for an equation like;
		\newline
		\begin{math}
			x^{3} + a_{1}x^{2} + a_{2}x + a_{3} = 0
		\end{math}
		\newline
		the parameters Q, R, S and T can be computed thus, \newline
		\begin{align*}
			Q = \frac{3a_{2}-a_{1}^{2}}{a} &&
			R = \frac{9a_{1}a_{2}-27a_{3}-2a_{1}^{3}}{54}
		\end{align*}

		\begin{align*}
			S = 3 \sqrt{R + \sqrt{-Q^{3} + R^{2}}} && T= \sqrt{R-\sqrt{Q^{3}+R^{2}}}
		\end{align*}
			\newline
			to give the roots;
			\newline
			x$_{1} = S + T- \frac{1}{3}a_{1}$
			\newline
			x$_{2} = \frac{-(S + T)}{2}$ - $\frac{a_{1}}{3}$ + i$\frac{sqrt{3}(s-T)}{2}$
			\newline
			x$_{3} = \frac{-(S+T)}{2}$
			\newline
			Note: x$_{3}$ must not have a co-efficient.

	\subsection{Some Examples}
	\begin{itemize}
		\item $x^{3} - 3x^{2} + 4 = 0$
		\item $2x^{3} + 6x^{2} + 1 = 0$
	\end{itemize}

	\bibliography{cardano.bib}
	\bibliographystyle{ieeetr}
\end{document}


\documentclass[a4paper,11pt]{article}

\usepackage{physics}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm, mathtools}
%\usepackage{hyperref}
\usepackage{color}
\usepackage{jheppub}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\usepackage[T1]{fontenc} % if needed

% My Documents
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bes}{\begin{equation*}}
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\usetikzlibrary{decorations.pathmorphing}

\tikzset{zigzag/.style={decorate, decoration=zigzag}}
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%\linespread{1.0}
%\setlength{\parindent}{0em}
%\setlength{\parskip}{0.8em}

\title{\textbf{AdS/CFT and Other Things}}
\author{Aditya Vijaykumar}
\affiliation{International Centre for Theoretical Sciences, Bengaluru, India.}
\emailAdd{aditya.vijaykumar@icts.res.in}
\abstract{This is one's effort to understand what the AdS/CFT Correspondence actually means, with explicit calculations and intuitive explanations of whatever is covered. In the course of one's journey, one hopes to review aspects of Quantum Field Theory as well as AdS spacetimes.

One shall mainly follow the path taken by Prof. Hong Liu in his MIT OCW course String Theory and Holographic Duality, supplementing it with one's own readings and observations. One shall also try to include as many references as possible.  }
\begin{document}
\maketitle
\section{Motivations}
\subsection{Spin-2}

From a field theory perspective, it is natural to ask whether a massless spin-$2$ particle \textcolor{red}{Ex : Why is gravity massless spin $ 2 $?} can arise as the bound state of a lower spin theory. If yes, that is equivalent to saying that gravity has effectively \textit{emerged}. Gluons and quarks can indeed form massive spin-$ 2 $ bound states which are unstable. \textcolor{red}{Ex : How do spin-$2$ particles arise in QCD?}.
\subsection{Weinberg-Witten Theorem}

But can one generate massless spin-$ 2 $ particles from this by maybe changing the theory a bit? Weinberg and Witten said no. They gave the following theorems

\begin{itemize}

	\item \textbf{Theorem 1} - A theory that allows the construction of a Lorentz covariant conserved current cannot contain a massless particle of spin $  > 1/2 $ with non-vanishing charge.
	\item \textbf{Theorem 2} - A theory that allows Lorentz covariant conserved stress tensor cannot contain massless particles of spin $ >1 $.
\end{itemize}
As GR need not have a conserved stress tensor \textcolor{red}{???}, these theorems do not forbid the graviton. But the second theorem also prohibits renormalizable theories in Minkowski spacetime in having an emergent gravity description. There is, however, a hidden assumption in this theory - it only applies to particles that live in the spacetime of the original theory. This seems like a straightforward, obvious assumption. But, in AdS/CFT, gravity\footnote{gh} lives in a different spacetime!

\textbf{\textcolor{red}{{Prove the Weinberg Witten Theorem}}}

\section{Black Holes ain't so Black}

In non-gravitational physics, in principle, one can probe arbitrarily large length scales. When gravity comes in, this stops being true. When $ E \gg m_p $, there will come a point when the Schwarzchild radius $ r_s \sim GE_{c} $ will become the fundamental distance scale.

\subsection{Classical Black Holes}
The geometry outside the Schwarzschild Black Hole is given by,
\bes
ds^2 = -\qty(1-\frac{2M}{r})dt^2 + \qty(1-\frac{2M}{r})^{-1} dr^2 + r^2(d\theta^2 + \sin^2 \theta d\phi^2)
\ees
This has the following properties,
\begin{itemize}
	\item The solution is time reversal symmetric \textit{ie.} symmetric under $t \rightarrow -t$. Real black holes form through gravitational collapse, and hence this is not valid for them. Nonetheless, this can be assumed for late time black holes, which really makes the Schwarzschild solution a mathematical idealization of the actual black hole.
	\item Spacetime is non-singular at the horizon. Written in the Schwarzschild coordinates, the horizon is a coordinate singularity, and one can easily verify that it indeed is not a real singularity by \textcolor{red}{calculating curvature invariants}.
	\item $ r=r_s $ is a null hypersurface.
	\item The horizon is a surface of infinite redshift with respect to far away observer. \textcolor{red}{Show this explicitly}
	\item There are two important geometric quantities associated with the horizon,
	\bes
	\qq{Horizon Area} A_s = 4\pi r_s^2 = 16 \pi M^2
	\ees
	\bes
	\qq{\textcolor{red}{Surface Gravity}} \kappa = \frac{1}{2}f'(r_s) = \frac{1}{2r_s} = \frac{1}{4M}
	\ees
\end{itemize}

\subsection{Causal Structure and Rindler Spacetime}
Consider the region just outside the black hole horizon,
\begin{equation*}
f(r) = f(r_s) + (r-r_s)f'(r_s) + \ldots= (r-r_s)f'(r_s) + \ldots
\end{equation*}
From the metric, we can see that the proper distance $ \rho $ is just given by,
\begin{equation*}
\rho = \int_{r_s}^r \dfrac{dr}{\sqrt{f(r)}} = \int_{r_s}^r \dfrac{dr}{\sqrt{(r-r_s)f'(r_s)}} = 2\sqrt{\frac{r-r_s}{f'(r_s)}}
\end{equation*}
Hence,
\begin{equation*}
f(r) = (f'(r_s))^2\rho^2/4 = \kappa^2 \rho^2
\end{equation*}
and the metric can be written as,
\begin{align*}
ds^2 &= - \kappa^2 \rho^2 dt^2 + d\rho^2 +  r_s^2(d\theta^2 + \sin^2 \theta d\phi^2)\\
&=   \underbrace{-\rho^2 d\eta^2 + d\rho^2}_{\text{(1+1)d Minkowski Spacetime in Rindler coordinates}} +  \underbrace{r_s^2(d\Omega^2)}_{\text{2-sphere}}
\end{align*}
\textbf{\textcolor{red}{Continue this from Hartman and other sources}}
\end{document}


\documentclass[aps,pra,notitlepage,amsmath,amssymb,letterpaper,12pt]{revtex4-1}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{listings}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}

\begin{document}

\title{Classwork 13}
\author{Frank Entriken and Grady Lynch}
\affiliation{PHYS 220, Schmid College of Science and Technology, Chapman University}
\date{\today}

\maketitle

\section{Sombrero Potential}

\begin{problem}{Specifications}
Simulate the $x$ and $y$ coordinates of a ball in a double well potential (also known as the "sombrero" potential). Variables to consider are the balls mass, $m$, and the ball's friction as it rolls, $f_{\text{drag}}(\dot{x}) = -\nu \dot{x}$. The "sombrero" will be shaken back and forth repeatedly with a driving force $f_{\text{drive}}(t) = F\cos(\omega t)$.

According to Newton's Second Law, the ball must satisfy the equation of motion: $$m\ddot{x} = f_{\text{hat}}(x) + f_{\text{drag}}(\dot{x}) + f_{\text{drive}}(t) = x - x^3 - \nu \dot{x} + F\cos(\omega t)$$
\end{problem}

\section{The Solution}

\begin{solution}
In order to compute the coordinates of the ball in motion we used the Runge-Kutta 4th Method. This method accurately predicts each subsequent $x$ and $x$ value by considering the approximations before and after the desired value. By using this method we were able to graphically represent the ball's $x$ and $x$ values over the course of its motion. The graphs below represent the ball at different starting values: $x0$ is the starting $x$ position of the ball, $y0$ is the starting $y$ position of the ball, and $F$ which is the force of the shake.

\subsection{Our Graphs}

\begin{figure}[h!]
  \includegraphics[width=0.4\textwidth]{p1.png}
  \caption{$x0=-0.9$, $y0=0$, $x0=0.18$}
  \label{fig:figlabel}
\end{figure}

\begin{figure}[h!]
  \includegraphics[width=0.4\textwidth]{p2.png}
  \caption{$x0=0.2$, $y0=0.1$, $x0=0.25$}
  \label{fig:figlabel}
\end{figure}

\begin{figure}[h!]
  \includegraphics[width=0.4\textwidth]{p3.png}
  \caption{$x0=0$, $y0=0$, $x0=0.4$ \\ The movement of the this graph uses a slightly modified version of the code from $RK4_2$, where our N value is multiplied by a value of 1000 up from 50.}
  \label{fig:figlabel}
\end{figure}

\end{solution}
\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                 %
%  GEANT manual in LaTeX form                              %
%                                                                 %
%  Michel Goossens (for translation into LaTeX)                   %
%  Version 1.00                                                   %
%  Last Mod. Jan 24 1991  1300   MG + IB                          %
%                                                                 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Origin{G.N.Patrick, L.Urb\'{a}n}
\Submitted{26.09.83}\Revised{16.12.93}
\Version{Geant 3.16}\Routid{PHYS221}
\Makehead{Simulation of Compton scattering}
\section{Subroutines}
\Shubr{GCOMP}{}
\Rind{\tt GCOMP} generates the Compton scattering of a photon on an
atomic electron. It uses the random number techniques of Butcher and
Messell \cite{bib-BUTC} to sample the scattered photon energy according
to the Klein-Nishina formula \cite{bib-KLEI}.

The interaction produces one electron, which is put in the \FCind{/GCKING/}
common block for further tracking. Tracking of the scattered photon will
continue, with direction and energy changed by the interaction.
All input/output information is through {\tt GEANT} common blocks.

\begin{tabular}{ll}
Input: & via COMMON \FCind{/GCTRACK/} \\
Output: & via COMMONs \FCind{/GCTRAK/} and \FCind{/GCKING/}
\end{tabular}

Compton scattering is selected in {\tt GEANT} by the input data
record {\tt COMP}. When Compton scattering is selected, \Rind{GCOMP}
is called automatically from the {\tt GEANT} photon tracking
routine \Rind{GTGAMA}.

\section {Method}

For a complete account of the Monte Carlo methods used the
interested
user is referred to the publications of Butcher and Messel
\cite{bib-BUTC}, Messel
and Crawford \cite{bib-MESS} and Ford and Nelson \cite{bib-EGS3}.
Only the basic formalism is outlined here.

The quantum  mechanical Klein-Nishina differential cross-section
is:
\[
\Phi(E,E') =\frac{X_0 n \pi r_0^2 m_{\rm e}}{E^2}
     \left[\frac{1}{\epsilon}+\epsilon\right]
     \left[1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2}\right]
\]
where,\quad
\begin{tabular}[t]{l@{\ = \ }l}
$E$         & energy of the incident photon   \\
$E'$        & energy of the scattered photon  \\
$\epsilon$  & $E'/E$                          \\
$m_{e}$     & electron mass                   \\
$n$         & electron density                \\
$r_0$       & classical electron radius       \\
$X_0$       & radiation length
\end{tabular}

Assuming an elastic collision, the scattering angle $\theta$ is
defined by the Compton formula:

\[
E'   = E \frac{m_{\rm e}}{ m_{\rm e} + E(1-\cos\theta )}
\]

Using the combined ``composition and rejection'' Monte Carlo methods
described in chapter {\tt PHYS211}, we may set:

\[
\begin{array}{LcLLcL}
f(\epsilon)   & = & \left[\frac{1}{\epsilon}+\epsilon\right] =
                    \sum^{2}_{i=1} \alpha_i f_i(E)
                    &  \multicolumn{2}{l}{\mbox{for}}
                    & \epsilon_0 > \epsilon > 1     \\
g(\epsilon)   & = & \left[ 1 - \frac{\epsilon\sin^2\theta}{1+\epsilon^2}
                    \right] & \multicolumn{3}{l}{\mbox{rejection function}} \\
\alpha_1      & = & \frac{1}{\ln(1/\epsilon_0)}   &
\alpha_2      & = & \frac{1}{2} (1-\epsilon_0^2)                             \\
f_1(\epsilon) & = & \frac{1}{\epsilon\ln(1/\epsilon_0)} &
f_2(\epsilon) & = & \frac{2\epsilon}{1-\epsilon^2}
\end{array}
\]

The value of $\epsilon$ corresponding to the minimum
photon energy (backward scattering) is given by:
\begin{eqnarray*}
\epsilon_0 & = & \frac{1}{1+2E/m_{\rm e}}
\end{eqnarray*}

Given a set of random numbers $r_i$ uniformly distributed in [0,1],
the sampling procedure for $\epsilon$ is the following:
\begin{enumerate}
\item
decide which element of the $f(\epsilon)$ distribution to sample from.
Let $\alpha_T = (\alpha_1+\alpha_2)r_0$. If $\alpha_1\geq\alpha_T$
select $f_1(\epsilon)$, otherwise select $f_2(\epsilon)$;

\item  sample $\epsilon$ from the distributions
corresponding to $f_1$ or $f_2$. For $f_1$ this is simply achieved by:
\[
\epsilon = \epsilon_0 e^\alpha_1 r_1
\]
For $f_2$, we change variables and use:
\[
\epsilon' = \left\{ \begin{array}{lll}
\max(r_2,r_3) & \mbox{for } & E/m \geq (E/m+1)r_4 \\
r_5           & \multicolumn{2}{l}{\mbox{for all other cases}}
 \end{array} \right.
\]
Then, $\epsilon = \epsilon_0+(1-\epsilon_0)\epsilon'$;

\item calculate $\sin^2\theta = \max(0,t(2-t))$
where $t=m_{\rm e}(1-\epsilon)/E'$

\item test the rejection function, if $r_6 \leq g(\epsilon)$ accept
$\epsilon$, otherwise return to step 1.
\end{enumerate}

After the successful sampling of $\epsilon$, \Rind{GCOMP} generates the
polar angles of the scattered photon with respect to the direction of
the parent photon. The azimuthal angle, $\phi$, is generated isotropically and
$\theta$ is as defined above. The momentum vector of the scattered
photon is then calculated according to kinematic considerations. Both
vectors are then transformed into the {\tt GEANT} coordinate system.

\section{Restriction}

The differential cross-section is only valid for those
collisions in which the energy of the recoil electron is large compared
with its binding energy (which is ignored). However, as pointed out by
Rossi \cite{bib-ROSS}, this has a negligible effect
because of the small number of
recoil electrons produced at very low energies.


\documentclass{article}  %Need this.

\usepackage{amsmath,amsthm,amssymb}

\usepackage[margin=1in]{geometry}

\newtheorem*{thm}{Theorem}
\newtheorem*{cnj}{Conjecture}
\newtheorem*{lem}{Lemma}
\newtheorem*{cor}{Corollary}
\newtheorem*{prop}{Proposition}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}

\title{Proof Portfolio Problem 8}
\author{}
\date{}

\begin{document}
\maketitle

As a reminder, you should pick only one of the following problems. Remember to start ASAP and see me if you need help.

\emph{The initial deadline for Problems 5-8 is Monday, March 23 (11:59PM). The final deadline is Friday, March 30 (11:59PM).}

%Sets
\noindent\textbf{Conjecture 8A.}\footnote{The symbol $\times$ is defined on page 256 of your text.  For two sets $X$ and $Y$, $X \times Y = \{(x,y) \mid x \in X \text{ and } y\in Y.\}$ Think of ordered pairs, like you're graphing on the Cartesian plane.} If $A,B,$ and $C$ are subsets of some universal set $U$ then
 \[A\times (B\cup C) = (A\times B) \cup(A\times C).\]\\

\noindent\textbf{Conjecture 8B.} Let $X = \{x \in\mathbb{Z} : x \equiv 2 \pmod{6} \}$ and $Y = \{y\in\mathbb{Z} : 3 \mid y-5\}$.  Prove that one of these sets is a proper subset of the other (stating your result as a theorem).\\

\section*{Some LaTeX Notes:}
For 8A:
\begin{verbatim}
$A\times (B-C) = (A\times B) - (A\times C)$
\end{verbatim}

\noindent For 8B:  \begin{verbatim}
$X = \{x \in\mathbb{Z} : x \equiv 2 \pmod{6} \}$ \end{verbatim}
and

\begin{verbatim}$Y = \{y\in\mathbb{Z} : 3 \mid y-5\}$\end{verbatim}
(The \begin{verbatim} \ \end{verbatim} makes the set braces appear.) You could also use \begin{verbatim} \Z \end{verbatim} if you are using my LaTeX file (instead of \begin{verbatim}\mathbb{Z} \end{verbatim}) .)

\end{document}


\section{Simulation Results}

The results presented in this section were generated by running Monte Carlo simulations of the search procedure, since finding a closed-form solution to the mean \textbf{T}ime \textbf{T}o \textbf{D}ecision (TTD) is not readily available in the general case \cite{Chung2012AnalysisStrategies}. We simulated the grid, the agents and the targets in order to evaluate the performance of the system.
%and to analyse how modifying search parameters affects the outcome.
We present statistics related to the Monte Carlo simulations, which reveal how modifying parameters of the search procedure affect the outcome. For each set of parameters in tables \ref{table:VaryingPriorDistribution}, \ref{table:VaryingInitialBelief}, \ref{table:MiscalibratedSensor}, \ref{table:MultipleTargetEGSweep}, \ref{table:MultipleTargetSaccadicRandom} and \ref{table:VaryingNumberOfAgents} we ran 5000 simulations, which finish when the agent (or agents) terminates the search. The parameters that we vary in the simulations are shown in Table \ref{table:SimRunParameters}:

\begin{table}[H]
    \centering
    \begin{tabular}{|c|c|c|c|}
    \hline
         Parameter& Configuration 1 & Configuration 2 & Configuration 3 \\
         \hline
         Initial Distribution & Gaussian & Uniform & - \\

         \hline
         Initial Cumulative Probability & 0.25 & 0.5 & 0.75 \\

         \hline
         Sensor Model Parameters & $\alpha$=0.05, $\beta$ = 0.02 & $\alpha$=0.2, $\beta$ = 0.15 &
         $\alpha$=0.4, $\beta$ = 0.4 \\

         \hline
         \# of Targets Present & 1 & 2 & 3 \\

         \hline
         \# of Agents Used & 1 & 2 & 3 \\

         \hline
    \end{tabular}
    \caption{Parameters varied for each simulation run}
    \label{table:SimRunParameters}
\end{table}

The meaning of each of these parameters is outlined as follows:
\begin{enumerate}
    \item The initial belief distribution of each agent describes the distribution of $p(x_0 | e_0, u_0)$, which corresponds to the probability distribution describing its initial belief of the location of the target, prior to gathering any evidence. This may come from prior information about the scene. A Gaussian distribution is peaked, meaning that prior information suggests that some grid cells are more likely to contain the target than others. This can be seen in Figure \ref{fig:InitialGaussian}. A Uniform distribution is not peaked, reflecting no prior information related to the location of the target. This can be seen in Figure \ref{fig:InitialUniform}.
    \item The initial cumulative belief that the target is present in the region is the agent's belief in whether the target is present in the region, prior to gathering any evidence. It is the cumulative sum of the probabilities of the target presence in each of the grid cells. An initial cumulative belief of 0.25 means that initially, the agent is 25\% sure that the target is present in the region.
    \item The sensor model false positive rate and false negative rate and the parameters $\alpha$ and $\beta$ are set out in Section \ref{subsec:stochasticEnvModel}. These can differ from the true rate at which the sensor will observe positive and negative observations. For example, given that the sensor actually outputs false positives and false negatives at a rate of 0.2 and 0.15 respectively, if the sensor model is calibrated with a false positive rate and false negative rate of 0.05 and 0.02 respectively, it will drastically under-estimate the rate at which false readings are recorded, which is reflected in the way it updates its estimated state.
    \item \# of Targets Present gives the number of distinct targets present in the search region. These are assumed to exist in distinct locations.
    \item \# of Agents Used is the number of agents participating in the search.
\end{enumerate}
The results of the simulations show how varying these parameters and suggest how to set them to achieve a desired result. We focus on the most commonly reported metrics in the literature, which are related to the distribution of time to decision \cite{Chung2012AnalysisStrategies}, \cite{Waharte2010ProbabilisticUAVs}, \cite{Waharte2010SupportingUAVs} and \cite{Lau2007OptimalEnvironments}. The time to decision is measured as the number of discrete time-steps before the search concludes. We also report on the rate at which incorrect target locations are returned and the rate at which the agents incorrectly conclude that the target is not present.

\par For each of the simulations, we arbitrarily chose to use the SPRT cut-off criteria with the upper Type \Romannum{1} error probability set to 0.1 and the upper Type \Romannum{2} error probability set to 0.15. In practice, this meant the agent would terminate the search if its cumulative belief that the target was present exceeded 0.895 or fell below 0.143. We generated simulated sensor readings using arbitrarily chosen values of the false positive rate = 0.2 and a false negative rate = 0.15. For each simulation run, we generated random starting locations for the agents and targets in a uniformly spaced 10 $\times$ 10 grid.
\par Unless specified otherwise, we use the following default parameters: sensor model false negative rate = 0.15, sensor model false positive rate = 0.2, initial belief distribution = uniform, initial cumulative belief target is present = 0.5, number of targets present = 1, number of active agents = 1, the $\epsilon$-greedy search has $\epsilon$=0.2 and a neighbourhood radius of 4. Histograms showing the results of running the simulation with varying parameters are shown in Appendix \ref{chap:AppendixTwo}.

\input{Chapters/MultiAgentTargetDetection/Results/SingleAgentResults.tex}

\input{Chapters/MultiAgentTargetDetection/Results/MultipleAgentResults.tex}


%!TEX root = ../thesis.tex
%*******************************************************************************
%****************************** Chapter 3: intro2 *********************************
%*******************************************************************************

\chapter{The projective to Einstein correspondence $N\rightarrow M$}
\label{chap:intro2}

The purpose of this chapter is to introduce the preliminaries that are required to understand the remainder of the thesis. We will first review projective geometry, including the Cartan and tractor bundles associated with a projective structure, before moving on to the projective to Einstein correspondence of \cite{DM}. We begin with some notation and conventions.

\subsubsection{Notation and conventions}
\begin{itemize}
\item We use $\R^n$ to mean the real vector space of dimension $n$, and $\R_n$ to mean its dual. We think of vectors in $\R^n$ as column vectors, and vectors in $\R_n$ as row vectors.
\item We will projectivise vector spaces by \textit{line projectivisation}, that is, we will take the space of \textit{unoriented} lines through the origin, unless stated otherwise.
\item When we refer to the projective to Einstein correspondence, the projective manifold will be called $N$ and will have dimension $n$, whilst the Einstein manifold will be called $M$ and will have dimension $2n$.
\item We use the letter $\pi$ for maps to $N$, and the letter $\kappa$ for maps to $M$. We will attach subscripts to $\pi$ and $\kappa$ to give information about the preimage or the significance of the map.
\item We use lower case Latin indices $i,j,k,\dots=1,\dots,n$ for tensorial objects on $N$ and $a,b,c,\dots=1,\dots,2n$ for tensorial objects on $M$.
\item We use $\odot$ and $\wedge$ to denote the symmetrised and antisymmetrised tensor product respectively. That is,
\[
A\odot B = \frac{1}{2}(A\otimes B + B\otimes A),\qquad A\wedge B = \frac{1}{2}(A\otimes B - B\otimes A).
\]
We also occasionally write $A^2$ for $A\odot A$, and in Chapter \ref{chap:EW_and_toda} we will sometimes omit the $\odot$ altogether.
\item Where indices have been symmetrised or antisymmetrised over, we will enclose them in round or square brackets respectively.
\item Our conventions for differential forms are
\[
(d\omega)_{ab\dots c}=\partial_{[a}\omega_{b\dots c]},\qquad (\eta\wedge\omega)_{a\dots d}=\eta_{[a\dots b}\omega_{c\dots d]},\]
\[
\omega=\omega_{a\dots b}\,dx^{a}\wedge\dots\wedge dx^{b}, \qquad F_{ab}\,{d}x^{a}\wedge{d}x^{b}=F_{[ab]}\,{d}x^{a}\otimes{d}x^{b}.
\]
\item We use $\hook$ to denote a contraction between a vector and a form.
\item The Riemann curvature tensor $R_{abc}^{\quad d}$ of a connection $\nabla_a$ is defined by
\[
(\nabla_a\nabla_b - \nabla_b\nabla_a)X^d = R_{abc}^{\quad d}X^c,
\]
where $X$ is any vector field.
\item The Ricci tensor is defined by the contraction $R_{bc} = R_{abc}^{\quad a}$.
\end{itemize}

\section{Projective Geometry}\label{sec:projgeom}

Our discussion follows Eastwood \cite{Eastwood}.

\begin{defi}\label{def:projstruct} A projective structure $(N,[\nabla])$
on a manifold $N$ is an equivalence class $[\nabla]$ of torsion--free affine connections on $N$ which have the same geodesics as unparametrised curves.
\end{defi}

The following proposition converts definition \ref{def:projstruct} to a more operational form.

\begin{prop} Two torsion--free connections $\nabla$ and $\ov{\nabla}$ belong to the same projective class if and only if their components $\Gamma^i_{jk}$ and $\ov{\Gamma}^i_{jk}$ are related by
\begin{equation}
\ov{\Gamma}^i_{jk} - \Gamma^i_{jk} = \delta_{j}^{i}\Upsilon_{k}+\delta_{k}^{i}\Upsilon_{j}\label{eq:proj_change}
\end{equation}
for some one--form $\Upsilon.$
\end{prop}

{\bf Proof.} We denote by $\mathcal{V}$ the vertical sub--bundle of $T(TN)$, where $\pi_T:TN\rightarrow N$ is the tangent bundle to $N$. A connection defines a splitting of the exact sequence
\be \label{eq:TTMsequence}
0 \longrightarrow \mathcal{V} \longrightarrow T(TN) \longrightarrow \pi_T^*TN \longrightarrow 0
\ee
so that each $\xi\in T_xN$ has a unique pull--back in the horizontal sub--bundle complementary to $\mathcal{V}_x$. The integral curves of these pull--backs, when projected down to $N$, then define the geodesics of the connection.

Any two connections are related by some $\delta\Gamma^k_{ij}$, which satisfies $\delta\Gamma^k_{ij}=\delta\Gamma^k_{(ij)}$ as long as both connections are torsion--free. A change of connection is equivalent to a change in the splitting of (\ref{eq:TTMsequence}). At $\xi\in T_xN$, the change is given by the homomorphism from $T_xN$ to $T_xN=\mathcal{V}_x$ defined by the contraction $\xi^i\Gamma^k_{ij}$. Thus the two connections define the same geodesics if and only if $\xi^i\xi^j\Gamma^k_{ij}$ is a multiple of $\xi^k$ for all $\xi^i$. This is true if and only if there is a one--form $\Upsilon_i$ such that (\ref{eq:proj_change}) is satisfied.\footnote{To see this, take some one--form $\omega_i$ and note that $2\xi^i\xi^j\delta^k_{(i}\Upsilon_{j)}\omega_k$ vanishes if and only if $\xi^k\omega_k$ does.}
\koniec

One can show that the curvature of a connection $\nabla$ in the projective class can be uniquely decomposed as
\be \label{eq:projcurvdecomp}
R_{ijk}^{\ \ \ l} = W_{ijk}^{\ \ \ l} + 2\delta^l_{[i}\Rho_{j]k} -2\Rho_{[ij]}\delta^l_k,
\ee
where the Weyl projective curvature tensor, $W_{ijk}^{\ \ \ l}$, is trace free, and the Schouten tensor, $\Rho_{ij}$, is given in terms of the Ricci tensor by
\[
\Rho_{ij}=\frac{1}{n-1}R_{(ij)}+\frac{1}{n+1}R_{[ij]}.
\]
The objects $W_{ijk}^{\ \ \ l}$ and $\Rho_{ij}$ transform as
\be \label{eq:schout_change}
\ov{W}_{ijk}^{\ \ \ l} = W_{ijk}^{\ \ \ l}, \qquad \ov{\Rho}_{ij} = \Rho_{ij} - \nabla_i\Upsilon_j + \Upsilon_i\Upsilon_j
\ee
under a change of representative connection (\ref{eq:proj_change}). Note that for $n=2$ the Weyl tensor always vanishes.

A projective structure in dimension $n$ is said to be flat if it is diffeomorphic to the real projective space $\RP^n$ with its standard flat projective structure.
\begin{defi} \label{def:RPn}
The real projective space $\RP^n$ of dimension $n$ is the space of unoriented lines through the origin in $\R^{n+1}$, thought of as $\RP^n=(\R^{n+1}\backslash\{0\})/\R^*$, where the quotient identifies points $P\in\R^{n+1}$ under the equivalence relation
\[
(P^0,\dots,P^n)\sim (cP^0,\dots,cP^n)\ \forall\ c\in\R^*.
\]
The geodesics on $\RP^n$ are given by planes through the origin in $\R^{n+1}$ under the projection $\pi_\PP:\R^{n+1}\rightarrow\RP^n$.
\end{defi}

\begin{rmk}
Let $P$ denote a non--zero point in $\R^{n+1}$ with coordinates $(P^0,\dots,P^n)^T$, and let $[P]$ denote the corresponding point in $\RP^n$, labelled by homogeneous coordinates. In a patch $\mathcal{U}_0$ where $P^0\neq 0$, we can write $[P]=[1,P^1/P^0,\dots,P^n/P^0]^T$ and define inhomogeneous coordinates on $\RP^n$ by
\[
(x^1,\dots,x^n) = (P^1/P^0,\dots,P^n/P^0).
\]
If we combine this with coordinate patches $\mathcal{U}_i$ where $P^i\neq 0,\ i=1,\dots,n$, we can build an atlas for $\RP^n$.
\end{rmk}

\begin{rmk}
The flat projective structure on $\RP^n$ has a special duality property which we now discuss. Consider the set of hyperplanes through the origin in $\R^{n+1}$. These can be specified by their normal vector, which is defined only up to multiplication by $\R^*$. Let us denote such a hyperplane by a non--zero row vector $L\in\R_{n+1}$. A point $P\in\R^{n+1}$ lies in the hyperplane defined by $L$ if and only if $L\cdot P=0$.

When we projectivise the $\R^{n+1}$, any $P\neq 0$ descends to a point $[P]\in\RP^n$, and any hyperplane descends to a hypersurface $[L]\subset\RP^n$. The incidence relation $L\cdot P=0$ is now equivalent to the point $[P]$ lying in the hypersurface $[L]$. The homogeneous coordinates $[L]$ parametrise a second projective space which we think of as the dual to the $\RP^n$ parametrised by $[P]$, and denote $\RP_n$.
\end{rmk}

\begin{rmk}
Real projective space can be viewed as homogeneous space as follows.
The group $SL(n+1,\mathbb{R})$ acts from the left via the fundamental representation on coordinates $(P^0,\dots,P^n)^T$ in $\R^{n+1}$, and this descends to a transitive action on $\RP^n$. By the orbit stabiliser theorem, $\RP^n=SL(n+1,\R)/S$, where $S$ is a subgroup stabilising a point. If we choose the point $[1,0,\dots,0]^T$, the elements of $S$ are matrices of the general form
\[
\begin{pmatrix}\mathrm{det}a^{-1} & b\\
0 & a
\end{pmatrix}
\]
for some $a\in GL(n,\mathbb{R})$ and $b\in\mathbb{R}_{n}$.
\end{rmk}

\begin{rmk}
The necessary and sufficient condition for flatness of a projective structure depends on the dimension of the manifold on which it is defined. In dimension $n>2$, a projective structure is flat if and only if its Weyl projective curvature tensor vanishes. However, for $n=2$ the projective Weyl tensor is always vanishing. It can be shown that a projective structure on a \textit{surface} is flat if and only if the Cotton tensor $\nabla_{[i}\Rho_{j]k}$ vanishes for any choice of representative connection.
\end{rmk}

\subsection{The Cartan bundle}

One way of understanding the construction in \cite{DM}
is via the Cartan bundle \cite{Cartan} of the projective structure $(N,[\nabla])$ (see also \cite{KobNag,Sharpe}). Cartan geometries generalise Klein's Erlangen programme \cite{Klein}, a study of homogeneous spaces $G/S$, to the curved case, in which the total space $G$ is replaced by a principal right $S$-bundle over a manifold $N$ such that the tangent space to $N$ at every point is isomorphic to the Lie algebra quotient $\mathfrak{g}/\mathfrak{s}$. Since projective structures are modelled on $\mathbb{RP}^{n}$, which can be viewed as a homogeneous space, they constitute a type of Cartan geometry.

In the Riemannian case, the model space is $\mathbb{R}^{n}\cong\mathrm{Euc}(n)/SO(n)$. The corresponding Cartan geometry is a general, curved Riemannian manifold. One has an obvious subclass of frames which are ``adapted'' to the metric, i.e. those which are orthonormal. We can thus think of a curved Riemannian manifold as a principal $SO(n)$ bundle whose tangent spaces are modelled on $\mathbb{R}^{n}\cong\mathfrak{Euc}(n)/\mathfrak{so}(n)$. We say that Riemannian manifolds are Cartan geometries of type $(\mathrm{Euc}(n),SO(n))$.

The theory of Cartan geometries was developed as part of Cartan's
\textit{method of moving frames}. The idea is to pick out some adapted frames for manifolds equipped with some non-metric structure. The bundle of such frames over a manifold is then a principal bundle $\pi_\mathcal{G}:\mathcal{G}\rightarrow N$ with structure group $S$.

The bundle $\mathcal{G}$ is equipped with a $\mathfrak{g}$-valued one-form
$\theta$ called the Cartan connection. It defines an isomorphism $\theta:T_{u}\mathcal{G}\rightarrow\mathfrak{g}$ at every point $u\in \mathcal{G}$ such that the vertical subspace $\mathcal{V}_{u}\mathcal{G}\subset T_{u}\mathcal{G}$ is mapped to $\mathfrak{s}$ and the horizontal subpace $\mathcal{H}_{u}\mathcal{G}\subset T_{u}\mathcal{G}$ is defined as the inverse image of $\mathfrak{g}/\mathfrak{s}$. Note that it is not a connection in the usual sense of a principal bundle connection, since it takes value in a Lie algebra larger than that of the structure group. Further details can be found in \cite{Sharpe}.

%The importance of the Cartan connection is that it satisfies a number of properties, in particular equivariance, i.e. $R_{h}^{*}\theta=\mathrm{Ad}(h^{-1})\theta$ for all $h\in H$.

In the projective case, if we choose the point which is stabilised by $S$ to be $[1,0,\dots,0]$, the Cartan connection can be written as a matrix
\be \label{eq:cartan_connection}
\theta=\begin{pmatrix}-\mathrm{tr}\phi & \eta\\
\omega & \phi
\end{pmatrix},
\ee
where $\omega$, $\eta$ and $\phi$ are one-forms valued in $\mathbb{R}^{n}$, $\mathbb{R}_{n}$ and $\mathfrak{gl}(n,\mathbb{R})$ respectively.
We will refer to the components of $\omega$ and $\eta$ with respect
to the natural basis of $\mathfrak{sl}(n+1,\mathbb{R})$ as $\{\omega^{(i)}\}$ and $\{\eta_{(i)}\}$, so that $\omega^{(i)}$ and $\eta_{(i)}$ are both one-forms taking values in $\R$.

\begin{defi}
The Cartan geometry of a projective structure $(N,[\nabla])$ consists of a principal right $S$--bundle $\pi_\mathcal{G}:\mathcal{G}\rightarrow N$, where the right--action of some $s\in S$ on $\mathcal{G}$ is denoted by $R_s$, and a one--form $\theta$ on $\mathcal{G}$ called the Cartan connection, which takes values in $\mathfrak{sl}(n+1,\mathbb{R})$. The Cartan connection can be written in the form (\ref{eq:cartan_connection}) and has the following properties:
\begin{enumerate}
\item $\theta_u:T_u\mathcal{G}\rightarrow\mathfrak{sl}(n+1,\R)$ is an isomorphism for all $u\in \mathcal{G}$;
\item $\theta(\xi_\mathfrak{v})=\mathfrak{v}$ for all fundamental vector fields $\xi_\mathfrak{v}$ on $\mathcal{G}$;
\item $R^*_s\theta = \mathrm{Ad}(s^{-1})\theta=s^{-1}\theta s$ for all $s\in S$.
\item If $\xi$ is a vector field on $\mathcal{G}$ with the property that $\eta(\xi)=\phi(\xi)=0$ and $\omega(\xi)\in\R^n\backslash\{0\}$, then the integral curve of $\xi$ projects down to a geodesic on $N$ and conversely every geodesic of $[\nabla]$ arises in this way.
\item The $\mathfrak{sl}(n+1,\mathbb{R})$-valued
curvature two-form $\Theta$ satisfies
\be \label{eq:curvature_2-form}
\Theta=d\theta+\theta\wedge\theta=\begin{pmatrix}0 & L(\omega\wedge\omega)\\
0 & W(\omega\wedge\omega)
\end{pmatrix},
\ee
where $L$ and $W$ are smooth curvature functions valued in $\mathrm{Hom}(\R^n\wedge\R^n,\R_n)$ and $\mathrm{Hom}(\R^n\wedge\R^n,\R_n\otimes\R^n)$ respectively. The function $W$ represents the Weyl projective curvature tensor appearing in (\ref{eq:projcurvdecomp}).
\end{enumerate}
\end{defi}

\begin{rmk}The Cartan geometry of a projective structure is unique in the sense that for any two Cartan geometries $(\widehat{\pi}_\mathcal{G}:\widehat{\mathcal{G}}\rightarrow N,\widehat{\theta})$ and $(\pi_\mathcal{G}:\mathcal{G}\rightarrow N,\theta)$ of type $(SL(n+1,\R),S)$ satisfying the above properties there is a $S$--bundle isomorphism $\nu:\mathcal{G}\rightarrow\widehat{\mathcal{G}}$ such that $\nu^*\widehat{\theta}=\theta$. %This means that although we cannot choose a unique connection on the tangent bundle to $N$, we can choose a unique connection on the Cartan bundle.
\end{rmk}

\begin{rmk} \label{rmk:theta_symmetry}
For every open set $\mathcal{U}\subset N$, projective vector fields on $\mathcal{U}$ are in one-to-one correspondence with vector fields on $\pi_\mathcal{G}^{-1}(\mathcal{U})$
which preserve $\theta$ and are equivariant under the principal $S$--action.
\end{rmk}

\subsection{Tractor bundles}
The Cartan connection also gives us a unique connection on any bundle associated to $\mathcal{G}$ via some $S$--module. In particular, let $\mathcal{B}$ be a vector space and $\rho_\mathcal{B}:S\rightarrow GL(\mathcal{B})$ a representation of $S$ acting on $\mathcal{B}$. We can construct an \textit{associated bundle}
\[\pi_\mathcal{B}:\mathcal{G}\times_{\rho_\mathcal{B}} \mathcal{B}\rightarrow N \]
where points in $\mathcal{G}\times_{\rho_\mathcal{B}} \mathcal{B}$ are equivalence classes of pairs $[u,v]$, where $u\in \mathcal{G}$ and $v\in \mathcal{B}$, up to the equivalence relation
\[
(u_1,v_1)\sim (u_2,v_2) \quad \Leftrightarrow \quad \exists\ s\ \mbox{such that}\  u_2=u_1 s,\  v_2 = \rho_\mathcal{B}(s^{-1}) v_1.
\]

We thus obtain a vector bundle over $N$ whose fibres are diffeomorphic to $\mathcal{B}$. A section $\tilde{\sigma}:N\rightarrow \mathcal{G}\times_{\rho_\mathcal{B}} \mathcal{B}$ is represented by a map ${\sigma}:\mathcal{G}\rightarrow \mathcal{B}$ which is equivariant in the sense that ${\sigma}(us)=\rho_\mathcal{B}(s^{-1}){\sigma}(u)$ for all $s\in S$. Importantly, any such bundle inherits a connection from the Cartan connection $\theta$ on $\mathcal{G}$. The concept of an associated bundle applies to any principal bundle, but we call vector bundles which are associated to a Cartan bundle \textit{tractor} bundles, and the connections that they inherit from the Cartan connection are called tractor connections.

A particularly important example of a vector bundle associated to $\mathcal{G}$ is the \textit{cotractor bundle}, which defined by the canonical action of $S$ on $\R_{n+1}$ given by $(s,L)\mapsto Ls^{-1}$. We call this bundle $\pi_\mathcal{T}:\cT^*\rightarrow N$. In order to describe its connection, we consider a section represented by $\sigma:\mathcal{G}\rightarrow\R_{n+1}$ and define the one--form
\be \label{eq:df-ftheta}
d\sigma - \sigma\theta.
\ee
This turns out to be a \textit{semi--basic}\footnote{Recall that a semi--basic form on a fibre bundle $\mathcal{G}\rightarrow N$ is a form which is a linear combination, with coefficients parametrised by the fibres, of basic forms on $\mathcal{G}$ (i.e. forms which are the pull-backs of forms on $N$).} one--form satisfying
\[
R_s^*(d\sigma-\sigma\theta) = (d\sigma - \sigma\theta)s,
\]
making $\sigma\mapsto d\sigma-\sigma\theta$ an equivariant connection on $\cT^*$.

Although this construction of $\mathcal{T}^*$ relies on the Cartan bundle, it is possible to construct it independently. In order to do so we need the notion of a \textit{projective density}.%To do so, we consider the transformation of the derivative of a volume form under a projective change (\ref{eq:proj_change}). This approach can be motivated by the fact that the special linear group $SL(n,\R)$, which we have already seen to be important in projective geometry, is the group of volume preserving transformations of $\R^n$.
\subsubsection{Projective densities}
From the projective change of connection (\ref{eq:proj_change}) we can derive the corresponding change in $\nabla\chi$ for some $m$--form $\chi$ on $N$:
\be \label{eq:p-form_change}
\ov{\nabla}_i\chi_{jk\dots l} = \nabla_i\chi_{jk\dots l} - (m+1)\Upsilon_i\chi_{jk\dots l} - (m+1)\Upsilon_{[i}\chi_{jk\dots l]}.
\ee
In particular, for a volume form ($m=n$) we find
\[
\ov{\nabla}_i\chi_{jk\dots l} = \nabla_i\chi_{jk\dots l} - (n+1)\Upsilon_i\chi_{jk\dots l},
\]
where the final term in (\ref{eq:p-form_change}) has vanished because it contains a symmetrisation over $n+1$ indices. We can write this in a more compact way as
\[
\ov{\nabla}_i\chi = \nabla_i\chi - (n+1)\Upsilon_i\chi.
\]

Note that for sections $\tau$ of the bundle $\mathcal{E}(w):=(\Lambda^n)^{-w/(n+1)}$ we have
\be \label{eq:ddensity_change}
\ov{\nabla}_i\tau = \nabla_i\tau + w\Upsilon_i\tau.
\ee
We called such sections \textit{projective densities of weight $w$}, and for any vector bundle $\mathcal{B}\rightarrow N$ we write $\mathcal{B}(w)$ for the tensor product of $\mathcal{B}$ with $\mathcal{E}(w)$. For example, $T^*N(w)$ is the bundle of one--forms with projective weight $w$, and for sections $\mu_i$ of $T^*N(w)$ we have
\be \label{eq:dweighted_form_change}
\ov{\nabla}_i\mu_j = \nabla_i\mu_j + (w-1)\Upsilon_i\mu_j - \Upsilon_j\mu_i.
\ee

\subsubsection{The cotractor bundle}
We can now define the cotractor bundle $\pi_\cT:\mathcal{T}^*\rightarrow N$. For a choice of connection in the projective class we identify
\be \label{eq:T*splitting}
\mathcal{T}^* = \mathcal{E}(1)\oplus T^*N(1),
\ee
so that a section can be represented by a pair
\be \label{eq:T*coords}
\begin{pmatrix}
{\tau} \\ {\mu}_i
\end{pmatrix}.
\ee
Under a change of projective connection (\ref{eq:proj_change}), this splitting changes according to
\be \label{eq:chi_mu_change}
\ov{\begin{pmatrix}
{\tau} \\ {\mu}_i
\end{pmatrix}} =
\begin{pmatrix}
\tau \\ \mu_i + \Upsilon_i\tau
\end{pmatrix}.
\ee
Note the exact sequence
\be \label{eq:T*sequence}
0\longrightarrow T^*N(1)\longrightarrow \mathcal{T}^* \overset{V}\longrightarrow \mathcal{E}(1)\longrightarrow 0,
\ee
where we call the map $V$ the projective \textit{canonical tractor}\footnote{In fact $V$ is a section of a bundle $\mathcal{T}(1)$, where $\mathcal{T}$ can be identified with a direct sum $\mathcal{E}(1)\oplus TN(1)$ given a choice of connection in the projective class. The natural pairing between $\mathcal{T}$ and $\mathcal{T}^*$ defines the map $V:\mathcal{T}^*\rightarrow\mathcal{E}(1)$.}. A choice of connection in the projective class defines a splitting (\ref{eq:T*splitting}) of (\ref{eq:T*sequence}).

The bundle $\mathcal{T}^*$ admits a projectively invariant \textit{tractor connection} given by
\be \label{eq:tractor_connection}
\nabla^\mathcal{T}_i \begin{pmatrix}
\tau \\ \mu_j
\end{pmatrix}
= \begin{pmatrix}
\nabla_i \tau - \mu_i \\
\nabla_i\mu_j + \Rho_{ij}\tau
\end{pmatrix},
\ee
where $\nabla$ is the choice of projective connection and $\Rho_{ij}$ is its Schouten tensor. This turns out to agree with (\ref{eq:df-ftheta}). Under a change of projective connection (\ref{eq:proj_change}), we find
\begin{align*}
\ov{\nabla}^\mathcal{T}_i\ov{\begin{pmatrix}
\tau \\ \mu_j
\end{pmatrix}}
&= \ov{\nabla}^\mathcal{T}_i\begin{pmatrix}
\tau \\ \mu_j + \Upsilon_j\tau \end{pmatrix} \\
&= \begin{pmatrix}
\ov{\nabla}_i\tau - (\mu_i + \Upsilon_i\tau) \\
\ov{\nabla}_i(\mu_j+\Upsilon_j\tau) + \ov{\Rho}_{ij}\tau
\end{pmatrix} \\
&= \begin{pmatrix}
\nabla_i\tau + \Upsilon_i\tau - (\mu_i + \Upsilon_i\tau) \\
\nabla_i(\mu_j + \Upsilon_j\tau) - \Upsilon_j(\mu_i + \Upsilon_i\tau)
+ (\Rho_{ij} - \nabla_i\Upsilon_j + \Upsilon_i\Upsilon_j) \tau
\end{pmatrix},
\end{align*}
where we have used (\ref{eq:chi_mu_change}) in the first line, (\ref{eq:tractor_connection}) in the second and (\ref{eq:ddensity_change}), (\ref{eq:dweighted_form_change}) and (\ref{eq:schout_change}) in the third. After some cancellation, we identify
\begin{align*}
\ov{\nabla}^\mathcal{T}_i\ov{\begin{pmatrix}
\tau \\ \mu_j
\end{pmatrix}}
&= \begin{pmatrix}
\nabla_i\tau - \mu_i \\
\nabla_i\mu_j + \Upsilon_j\nabla_i\tau - \Upsilon_j\mu_i + \Rho_{ij}\tau
\end{pmatrix} \\
&= \ov{\begin{pmatrix}
\nabla_i\tau - \mu_i \\
\nabla_i\mu_j + \Rho_{ij}\tau
\end{pmatrix}}
= \ov{\nabla^\mathcal{T}_i \begin{pmatrix}
\tau \\ \mu_j
\end{pmatrix}}
\end{align*}
using (\ref{eq:tractor_connection}) and the change of splitting (\ref{eq:chi_mu_change}) adapted to the tensor product of $T^*N$ and $ \cT^*$ of which the derivative is a section.

Any tensor product of $\cT,\cT^*$ and $\cE(1)$ is equipped with a tractor connection which is inherited from the connection on the standard cotractor bundle. Equivalently, any such tensor product can be thought of as an associated vector bundle to the Cartan bundle $\mathcal{G}$, with its connection inherited from the Cartan connection via the corresponding representation of $S$. It is this connection, with its special equivariance property, that allows us to construct an Einstein metric as an invariant of the projective structure. This construction is the subject of the following section.

\section{The projective to Einstein correspondence}
Consider a quotient of the total space $\mathcal{G}$ of the
Cartan bundle by $GL(n,\mathbb{R})$, which is embedded in $S$
in the obvious way:
\be \label{eq:GL(n)_embedding}
GL(n,\mathbb{R})\ni a\longmapsto\begin{pmatrix}\mathrm{det}a^{-1} & 0\\
0 & a
\end{pmatrix}\in S.
\ee
It is easily verified that
\[
\begin{pmatrix}\mathrm{det}a^{-1} & 0\\
0 & a
\end{pmatrix}\begin{pmatrix}0 & \eta\\
\omega & 0
\end{pmatrix}\begin{pmatrix}\mathrm{det}a^{-1} & 0\\
0 & a
\end{pmatrix}^{-1}=\begin{pmatrix}0 & \eta a^{-1}\mathrm{det}a^{-1}\\
(\mathrm{det}a)a\omega & 0
\end{pmatrix},
\]
for any $a\in GL(n,\mathbb{R})$, meaning that due to the equivariance
property of the Cartan connection, the natural contraction $\eta\omega:=\sum_{i}\eta_{(i)}\otimes\omega^{(i)}$
defined by $\theta$ is preserved by the adjoint action of this $GL(n,\mathbb{R})$
subgroup. It thus descends to a naturally defined object on the quotient
$M=\mathcal{G}/GL(n,\mathbb{R})$.

\begin{theo}{\cite{DM}}\label{thm:DM}
There exist a metric and two--form
$(g,\Omega)$ on $\mbox{\ensuremath{M=\mathcal{G}/GL(n,\mathbb{R})}}$ such
that the quotient map $\kappa_q:P\rightarrow M$ gives
\begin{eqnarray}
\kappa_q^{*}g & = & \mathrm{Sym}(\eta\omega) \label{eq:g_cartan} \\
\kappa_q^{*}\Omega & = & \mathrm{Ant}(\eta\omega), \label{eq:Omega_cartan}
\end{eqnarray}
where $\mathrm{Sym}$ and $\mathrm{Ant}$ denote the symmetric and
anti-symmetric parts of the $(0,2)$ tensor $\eta\omega$. Moreover,
$\Omega$ is closed as a consequence of the Bianchi identity satisfied by the curvature two--form (\ref{eq:curvature_2-form}), $g$
is Einstein with non-zero scalar curvature, and the two are related
by an endomorphism $J$ satisfying $J^{2}=Id$. Hence $(g,\Omega)$
is an almost para--K\"ahler structure on $M$.
\end{theo}

\begin{rmk}
The full proof of Theorem \ref{thm:DM} only appears explicitly in \cite{DM} in the case $n=2$, although it can be generalised to $n>2$. This generalisation is discussed in their appendix. They show that the Ricci scalar of $g$ is $24$ in the case $n=2$. In Chapter \ref{chap:KK_lift} we will need the Ricci scalar for general $n$. We will calculate this under the assumption (stated without proof in \cite{DM}) that $g$ is Einstein.
\end{rmk}

\begin{rmk}
The quotient $M$ turns out to be an affine bundle over $N$ with
structure group $S$, i.e. $S$ acts affinely on the fibres of $\pi_M:M\rightarrow N$,
and sections of this bundle are in one-to-one correspondence with
representative connections $\nabla\in[\nabla]$. This means that given
some choice of connection $\nabla\in[\nabla]$ we have a diffeomorphism
$\kappa_A:T^{*}N\rightarrow M$ with which we can pull back the pair
$(g,\Omega)$. In canonical local coordinates $(x^{i},\zeta_{i})$ on
the cotangent bundle, we find
\begin{eqnarray}
\kappa_A^{*}g & = &  d\zeta_{i}\odot dx^{i}-(\Gamma_{ij}^{k}\zeta_{k}-\zeta_{i}\zeta_{j}-\Rho_{ij}) dx^{i}\odot dx^{j},\label{eq:coord_form}\\
\kappa_A^{*}\Omega & = &  d\zeta_{i}\wedge dx^{i}+\Rho_{ij} dx^{i}\wedge dx^{j},\qquad i,j=1,\dots ,n.\nonumber
\end{eqnarray}
Here $\Gamma_{jk}^{i}$ are the connection components of the representative
connection $\nabla$ that we chose, and its Schouten tensor is denoted $\Rho_{ij}$. This can be shown to be projectively invariant
in the sense that a different choice of $\nabla\in[\nabla]$ corresponds
to shifting the fibre coordinates $\zeta_{i}$, i.e. metrics on $T^{*}N$
resulting from pulling back $g$ using different representative connections
are isometric. Explicitly, a projective transformation (\ref{eq:proj_change})
corresponds to a change
\begin{equation}
\zeta_{i}\longrightarrow \zeta_{i}+\Upsilon_{i}.\label{eq:zeta_change}
\end{equation}
\end{rmk}

\begin{rmk}
In fact, the metric and symplectic form (\ref{eq:coord_form}) turn
out to belong to a one-parameter family $\{(g_\Lambda,\Omega_\Lambda)\,;\,\Lambda\neq 0\}$, which
can be written in local coordinates as
\begin{eqnarray}
g_{\Lambda} & = &  d\zeta_{i}\odot dx^{i}-(\Gamma_{ij}^{k}\zeta_{k}-\Lambda \zeta_{i}\zeta_{j}-\Lambda^{-1}\Rho_{ij}) dx^{i}\odot dx^{j}\label{eq:general_g}\\
\Omega_{\Lambda} & = &  d\zeta_{i}\wedge dx^{i}+\frac{1}{\Lambda}\Rho_{ij} dx^{i}\wedge dx^{j},\qquad i,j=1,\dots ,n.\label{eq:general_omega}
\end{eqnarray}
Metrics of the form (\ref{eq:general_g}) are a subclass of so-called
Osserman metrics. More details can be found in \cite{osserman}. They are all Einstein with non--zero scalar curvature $24\Lambda$, but for $\Lambda\neq1$ the relation to projective geometry is lost. For the remainder of the thesis we will write $g$ for $g_{\Lambda=1}$ unless stated otherwise. Note that $\{g_\Lambda\}$ will be the subject of Chapter \ref{chap:KK_lift}, whilst in Chapters  \ref{chap:EW_and_toda} and \ref{chap:c-proj} we will restrict our attention to $g$ because the projective geometry is a key aspect of the content of these chapters.
\end{rmk}

\begin{rmk}
One could also consider taking a quotient of $\mathcal{G}$ by a different subgroup of $S$. The $\R^*$ bundle over $M$ which we will discuss in Chapter \ref{chap:KK_lift} will turn out to be a quotient of $\mathcal{G}$ by $SL(n,\R)$.
\end{rmk}

\begin{rmk}
As mentioned above, in the special case where the $(N,[\nabla])$ is a projective surface, $M$ has dimension four, and so anti--self--duality is defined. It turns out that both the symplectic form $\Omega$ and the conformal curvature of $g$ are ASD. Both of these facts will play an important role in Chapter \ref{chap:EW_and_toda}.
\end{rmk}

\begin{rmk}
Note that an endomorphism $J$ which squares to the identity defines two $n$--dimensional sub--bundles of the tangent bundle $TM$ defined at each $m\in M$ as the vector subspaces of $T_mM$ which have eigenvalues $\pm 1$ with respect to $J$. These sub--bundles form a pair of smooth distributions $D_\pm$ in $TM$. Further discussion of the endomorphism $J$ will appear in Chapter \ref{chap:c-proj}.
\end{rmk}

\subsection{Symmetries of $M$}

Recall that a projective vector field on any manifold with a connection
generates a one--parameter family of transformations which preserve the
geodesics of that connection up to parametrisation. Projective vectors
fields thus naturally arise as the symmetries of a projective structure.
Explicitly, a vector field $\widehat{K}$ is projective if it satisfies
\begin{equation}
\mathcal{L}_{\widehat{K}}\Gamma_{ij}^{k}=\delta_{i}^{k}\Upsilon_{j}+\delta_{j}^{k}\Upsilon_{i}\label{eq:proj_transf}
\end{equation}
for some 1-form $\Upsilon$, where $\Gamma_{ij}^{k}$ are the connection
components, and their Lie derivative is defined by\footnote{Despite the fact that connection components are not tensorial objects, one can still define their Lie derivative with respect to a vector $\widehat{K}$ by considering how they transform when one moves infinitesimally along the curve defined by $\widehat{K}$. See \cite{yano} for further details.}
\begin{equation}
\mathcal{L}_{\widehat{K}}\Gamma_{ij}^{k}\equiv\frac{\partial^{2}\widehat{K}^{k}}{\partial x^{i}\partial x^{j}}+\widehat{K}^{m}\frac{\partial\Gamma_{ij}^{k}}{\partial x^{m}}-\Gamma_{ij}^{m}\frac{\partial \widehat{K}^{k}}{\partial x^{m}}+\Gamma_{im}^{k}\frac{\partial \widehat{K}^{m}}{\partial x^{j}}+\Gamma_{mj}^{k}\frac{\partial \widehat{K}^{m}}{\partial x^{i}}.\label{eq:liederivGamma}
\end{equation}

One consequence of the symmetry property of the Cartan connection discussed in remark \ref{rmk:theta_symmetry} is that for every open set $\mathcal{U}\subset N$ we have an isomorphism between the Lie algebra of projective vector fields on $\mathcal{U}$ and the Lie algebra of vector fields on $\pi_\mathcal{G}^{-1}(\mathcal{U})$ preserving the natural contraction $\eta\omega$. Such vector fields must descend to vector fields on $\pi_M^{-1}(\mathcal{U})$ preserving $(g,\Omega)$. In fact, it can be shown that every Killing vector field of $(M,g_\Lambda)$ is also symplectic with respect to $\Omega_\Lambda$ and is therefore the lift of a projective vector field on $(N,[\nabla])$.

Explicitly, for every projective vector field $\widehat{K}$ of $(N,[\nabla])$
there is a corresponding symmetry $K$ of $(M,g_{\Lambda},\Omega_\Lambda)$
given in local coordinates by
\begin{equation}
{K}=\widehat{K}-\zeta_{i}\frac{\partial \widehat{K}^{j}}{\partial x^{i}}\frac{\partial}{\partial \zeta_{j}}+\frac{1}{\Lambda}\Upsilon_{i}\frac{\partial}{\partial \zeta_{i}},\label{eq:kvf_from_pvf}
\end{equation}
where $\Upsilon_{i}$ is defined by (\ref{eq:proj_transf}).

\subsection{Tractor perspective} \label{sec:trac_construction}

From the tractor perspective, the space $M$ will turn out to be the projectivised cotractor bundle of $N$ with an $\RP^{n-1}$ sub--bundle removed from each fibre. We can understand what this $\RP^{n-1}$ sub--bundle is as follows.

On the total space of $\cT^*$ we pull back $\pi_\mathcal{T}:\cT^*\to N$ along $\pi_\mathcal{T}$ to get $\pi_\mathcal{T}^*(\cT^*)\to \cT^*$ as a vector bundle over the total space $\cT^*$. By construction this bundle has a tautological section $W\in \Gamma (\pi_\cT^*(\cT^*))$.  We also have $\pi_\cT^*(\cT(w))$ for any weight $w$, and we shall write simply $V\in \Gamma(\pi_\cT^*(\cT(1)))$ for the pull back to $\cT^*$ of the canonical tractor $V$ on $N$.

Now define
\be
\label{projection_map}
\kappa_\PP: \cT^*\longrightarrow \mathcal{M}:=\mathbb{P}(\cT^*)
\ee
by the fibre--wise projectivisation, and use $\pi_\cM$ for the map
$$
\pi_\cM:\mathcal{M}\to N.
$$
We denote by $\cE_{\cT^*}(w')$, for $w'\in \mathbb{R}$, the line
bundle on $\mathbb{P}(\cT^*)$ whose sections correspond to functions
$f: \pi_\cT^*\cT^* \to\mathbb{R} $ that are homogeneous of degree $w^\prime$ in
the fibres of $\pi_\cT^*\cT^*\to \mathbb{P}(\cT^*)$. For any weight $w$ we also have $\cE(w)$ on $N$ and its pull back to the bundle $\pi_\cM^*\cE(w)\to \mathbb{P}(\cT^*)$.
We define the product of these two density bundles on $\cM$ as
$$
\ce(w,w'):= \pi_\cT^*\cE(w) \otimes \cE_{\cT^*}(w').
$$

On $\cT^*$ there is  a canonical density $\tau\in \Gamma(\pi_\cT^*\cE(1))$ given by
$$
\tau:= V\hook W,
$$
%% and this is homogeneous of degree 1 up the fibres of $\pi$ in the total space $\cT^*$.
Note that $\tau$ is homogeneous of degree 1 up the fibres of the
map $\kappa_\PP:\cT^*\to \mathcal{M}$. Thus $\tau$ determines, and is equivalent
to, a section (that we also denote) $\tau$ of the density bundle $\cE(1,1)$. So $\mathcal{M}$ is stratified according to
whether or not $\tau$ is vanishing, and we write $\mathcal{Z}(\tau)$
to denote, in particular, the zero locus of $\tau$. We will show in Chapter \ref{chap:c-proj} that $M$ can be identified with $\mathcal{M}\setminus \mathcal{Z}(\tau)$.

%\rs{Note the
%  second density bundle is not necessarily orientable.}

\subsection{The model case} \label{sec:intro_model}
When $(N,[\nabla])$ is the flat projective structure $\RP^n$, the metric and symplectic form (\ref{eq:coord_form}) reduce to
\be \label{eq:intro_model_g}
g = d\zeta_i\odot dx^i + \zeta_i\zeta_j\,dx^i\odot dx^j, \quad \Omega = d\zeta_i\wedge dx^i,\quad i,j=1,\dots,n.
\ee
In this case the Cartan bundle of $N$ is just $SL(n+1,\R)$, and $M$ is simply the Lie group quotient $SL(n+1,\R)/GL(n,\R)$. The cotractor bundle has zero curvature, and although it is not trivial, the restriction of $\PP(\cT^*)$ to the set $\mathcal{Z}(\tau)\neq 0$ is. %in fact if we instead take $N$ to be the sphere $S^n$ with its standard flat projective structure where the geodesics are great circles, the tractor bundle is trivial. Note that $S^n$ is a double cover of $\RP^n$ which is orientable in all dimensions, and consists of the set of \textit{oriented} lines in $\R^{n+1}$. We obtain it by taking an analogous quotient of $\R^{n+1}$ where points are considered equivalent only up to multiplication by a \textit{positive} number. Replacing $\RP^n$ with $S^n$ allows us to write $cT=S^n\times\R^3$, so
We can thus write $M$ as
\[
M=\{([P],[L])\in\RP^n\times\RP_n\ |\ P\cdot L\neq 0\}.
\]

As discussed in Section \ref{sec:projgeom}, a point $[L]\in\RP_n$ represents a line $[L]\subset\RP^n$ which passes through $P\in\RP^n$ if and only if $L\cdot P=0$. In Chapter \ref{chap:EW_and_toda} we will show that for $n=2$, the conformal structure on $M$ can be obtained by demanding that two pairs $([P], [L])$ and $([\tP], [\tL])$ are null--separated if there exists a line which contains the three points $([P], [\tP], [L]\cap [\tL])$.

We also find in the model case that the symplectic form $\Omega$ is parallel with respect to the Levi--Civita connection $^{\bf g}\nabla$ of $g$, meaning that the endomorphism $J$ of $TM$ which relates $g$ and $\Omega$ is also parallel. As a result of this, the distributions $D_\pm$ defined by the two $n$--dimensional eigen--bundles of $J$ are parallel in the sense that $^{\bf g}\nabla_{\xi_1}\xi_2\in\Gamma(D_\pm)$ for all $\xi_1\in \Gamma(TM),\ \xi_2\in\Gamma(D_\pm)$. This makes the distributions \textit{Frobenius integrable}, meaning that the Lie bracket of any two sections of $D_\pm$ is also a section of $D_\pm$, or equivalently (as shown by Frobenius) that each of the two distributions is tangent to a foliation by sub--manifolds of dimension $n$ at every point.

To see that a parallel distribution is necessarily Frobenius integrable, note that the Lie bracket can be written
\[
[\xi_1,\xi_2]= \nabla_{\xi_1}\xi_2- \nabla_{\xi_2}\xi_1\in\Gamma(D)\quad\mbox{for all}\quad \xi_1,\xi_2\in\Gamma(D),
\]
where $D$ is a distribution which is parallel with respect to a connection $\nabla$. The Frobenius integrability of $D_\pm$ makes $(M,g,\Omega)$ not only almost para--K\"ahler but also para--K\"ahler. Further discussion about this distinction can be found in Chapter \ref{chap:c-proj}.