Simplifying plasma balls and black holes with nonlinear diffusion
Connor Behan
Department of Physics and Astronomy, University of British Columbia, Vancouver BC, V6T 1Z1, Canada

In the Master's thesis of the author, we investigate certain aspects of gravitational physics that emerge from
stochastic toy models of holographic gauge theories. We begin by reviewing field theory thermodynamics, black hole
thermodynamics and how the AdS / CFT correspondence provides a link between the two. We then study a nonlinear evolution
equation for the energy density that was derived last year from a random walk governed by the density of states.
When one dimension is non-compact, a variety of field theories produce long lived plasma balls that are dual to black
holes. This is due to a trapping phenomenon associated with the Hagedorn density of states. With the help of numerical
and mathematical results, we show that problems arise when two or more dimensions are non-compact. A natural extension
of our model involves a system of partial differential equations for both energy and momentum. Our second model is shown
to have some desired, but also some undesired properties, such as a potential disagreement with hydrodynamics.

Firstly, I would like to thank my supervisor, Mark Van Raamsdonk, for involving me in his research and lending his
strong intuition whenever it was needed. Apart from proofreading, he works hard to ensure that students have projects
matching their interests. Gordon Semenoff agreed to proofread this thesis as well. I thank him, not only for that, but
for delivering the lectures that first taught me string theory. I wish to thank my collaborators Klaus Larjo,
Nima Lashkari and Brian Swingle who discussed many of the problems that came up in our work, and possessed the skill
and motivation to solve them. I am grateful to many fellow students, especially Michael McDermott, Fernando Nogueira and
Jared Stang, for sharing their progress and taking an interest in mine. In the first year of this work, I was supported
by the Nation Science and Engineering Research Council of Canada.

Lastly, it is a pleasure to thank my parents for all of their love and support. This
thesis is dedicated to my cousin Greg, whose wedding I missed while pursuing this degree.

§ INTRODUCTION

This thesis deals with a recently proposed toy model for the dynamics of energy distributions in thermal field
theories. These include the conformal field theories and deformations of them that have gravity duals according to the
AdS / CFT correspondence [1]. As argued in [2], our model suggests that certain important aspects of
gravitational physics emerge for thermodynamic reasons. From this perspective, it is related to the ideas of entropic
gravity in [3, 4, 5, 6].

By focusing on two adjacent sites in this energy distribution, one may check whether the system favours a
homogeneous or an inhomogeneous state. The properties of this entropic evolution are determined by the local density
of states.

Before deriving the equations of our model, it is helpful to consider Figure <ref> — energy quanta that
randomly hop between sites in a line. To each site, we ascribe a density of states $\rho(n)$ counting the number of
ways for it to have $n$ units of energy. The growth of this function lets us determine which scenario is more likely:
site 1 giving a quantum to site 2 or site 2 giving a quantum to site 1. Asking this question is equivalent to comparing
the sizes of $\rho(n_1 - 1) \rho(n_2 + 1)$ and $\rho(n_1 + 1) \rho(n_2 - 1)$. Positing a form
$\log \rho(n) \propto n^{\alpha}$, we see that site 1 is most likely to give up energy when $\alpha < 1$ and site 2
is most likely to give up energy when $\alpha > 1$. Thus, we see that this random walk leads to diffusion when the
density of states is log-concave and clustering when the density of states is log-convex. In the diffusion case
e.g., a uniform energy distribution is the inevitable final state, even when the microscopic physics are
completely reversible. Special attention is paid to the Hagedorn phase $\alpha = 1$ which is almost completely static.

Even though the essence of our model is this simple statement, it takes the form of a nonlinear partial differential
equation that accepts a $\rho(E)$ function as input. A ubiquitous density of states, which we derive using the
AdS / CFT correspondence, consists of four phases. One of the narrow phases is omitted throughout this thesis for
simplicity. The three that are left consist of a diffusive phase at high energies, a Hagedorn phase at intermediate
energies and another diffusive phase at low energies. Roughly speaking, these respectively correspond to a black hole
forming, living for a long time and ultimately evaporating away. Less ambitiously, we may say that they correspond to
balls of plasma in a purely field theoretic setting [7]. We derive rigorous bounds on the decay times for
these objects in our model and compare them to the hadronization times in [7]. We find that our times are
longer in one dimension and much shorter in higher dimensions.

To address these problems, a second model is proposed that treats momentum as another quantity that moves
stochastically through a lattice. Since the second model is much more complicated, the discussion of its properties
remains at a speculative level. Even though evolution equations for energy
and momentum sound similar to the spirit of hydrodynamics, we compare our equations to the hydro equations and only
find agreement in the crudest approximation. Despite taking the form of classical PDEs, we hasten to emphasize that our
models include quantum effects when functions like the density of states are chosen appropriately.

This thesis begins with theoretical background in Chapter 2. This chapter focuses on the tools needed to derive
thermodynamic quantities via the AdS / CFT correspondence and contains some lengthy derivations. The main model is
derived afterward in Chapter 3. In Chapter 4, various results from the mathematical literature on nonlinear diffusion
equations are applied to our PDE and used to derive the time scales for black hole evaporation. The suspicious features
of our results are discussed in this chapter as well. Chapter 5 introduces numerical methods that are suitable for our
PDE and uses them to check most of our results. The method chosen for most problems is the implicit Crank-Nicolson
approach. Chapters 6 and 7 contain the newer results that were derived after
[2] appeared. Their focus is the extension of our model that includes momentum. Just as our first model depends
on a density of states, our second model depends on a momentum restricted density of states. An expression for this
quantity is derived that allows a small amount of numerical work to be done. Code forming the basis for all of our
simulations is presented in the appendix.

§ ASPECTS OF HOLOGRAPHY

Of all the conjectures that have been made about quantum gravity, the one that has had the largest impact so far is the
AdS / CFT correspondence proposed by Juan Maldacena [1]. Known by various other names like holography or
gauge-gravity duality, it states that string theory in anti-de Sitter space is equivalent to a conformally invariant
quantum field theory living on the boundary of that space. Questions about string theory can therefore be recast in the
language of quantum field theory without gravity. Deriving the evidence for the AdS / CFT correspondence would exceed
the scope of this thesis [8]. Instead, we will explore certain dynamical processes that
can be best understood with the correspondence. The effect that will demand most of our attention is
black hole evaporation. Hawking's derivation of black hole evaporation is one of the most successful uses of quantum
field theory in curved spacetime and any eventual theory of quantum gravity is expected to account for it. Many studies
of Hawking radiation have been done using string theory and the AdS / CFT correspondence in
particular [9, 10, 11, 12].

Naturally, the first such studies focused on the original version of the correspondence in which the background is
$AdS_5 \times \mathbb{S}^5$ [1]. If one writes the six-dimensional Euclidean Dirac matrices as

\begin{equation}
\Gamma_i = \left [
\begin{tabular}{cc}
$0$ & $\bar{C}_i$ \\
$C_i$ & $0$
\end{tabular}
\right ] \; i \in \{1, \dots, 6\} \; , \nonumber
\end{equation}

the conformal field theory is specified by the Lagrangian [13]

\begin{eqnarray}
L &=& -\frac{1}{2g^2_{\mathrm{YM}}} \int_{\mathbb{R}^3} \frac{1}{2} \mathrm{Tr} \left ( F_{\mu\nu} F^{\mu\nu} + 2D_{\mu} \phi_i D^{\mu} \phi^i - [\phi_i, \phi_j][\phi^i, \phi^j] \right ) \nonumber \\
&& + \mathrm{Tr} \left ( \bar{\lambda} \sigma^{\mu} D_{\mu} \lambda - C_i \lambda [\phi^i, \lambda] - \bar{C}_i \bar{\lambda} [\phi^i, \bar{\lambda}] \right ) \textup{d}x \; . \label{sym}
\end{eqnarray}

Typically the gauge group is $U(N)$ or $SU(N)$ meaning that the scalars, spinors and vectors that show up are really
$N \times N$ matrices consisting of those types of fields. This is called the $\mathcal{N} = 4$ Super Yang-Mills theory
or sometimes the field theory of $D3$-branes. A less than encouraging fact about string theory is that
$AdS_5 \times \mathbb{S}^5$ is far from the only background we need to consider. There is really a whole landscape of
vacua whose boundary field theories may look very different. Indeeed CFT duals have been proposed for
$AdS_4 \times \mathbb{CP}^3$ [14], $AdS_3 \times \mathbb{S}^3 \times \mathbb{T}^4$ [15],
$AdS_3 \times \mathbb{S}^3 \times \mathbb{S}^3 \times \mathbb{S}^1$ [16] and many others.

Calculations involving these theories are difficult. Even showing that (<ref>) has conformal symmetry is not
trivial. Something that allows us to explore Hawking's process from the holographic viewpoint without choosing a
specific Lagrangian is the intimate connection between black holes and thermodynamics.

§.§ Thermodynamics

A number of different field theories have the same thermodynamic potentials. A useful example of this appears in a
conformal field theory. Neglecting the Casimir effect, energy and entropy are both extensive so they must be
proportional to the volume. A conformal theory has no intrinsic scale so the only dimensionful quantity that can
multiply this volume is the temperature. This leads to the expressions $E \propto VT^{d + 1}$ and $S \propto VT^d$.
Substituting them into eachother yields

\begin{equation}
S \propto V^{\frac{1}{d + 1}} E^{\frac{d}{d + 1}} \; . \label{cft-entropy}
\end{equation}

The density of states will turn out to play a fundamental role in our model so we will sometimes exponentiate this

In the calculations that follow we will see some situations in which this formula for the entropy does not hold. In
general, the rule is that (<ref>) becomes true for non-conformal theories if the energy is much larger
than any other scale. Different low energy behaviours can be introduced if one compactifies a CFT like (<ref>)
on a sphere.

§.§.§ In free field theory

An exercise done in [17] is finding the partition function of a free field theory. Starting with the fact that
$E = p$ in a massless theory, $Z^*(\textbf{p}) = 1 + e^{-\beta p}$ is the contribution of a single fermionic mode and
$Z(\textbf{p}) = 1 + e^{-\beta p} + e^{-2\beta p} + \dots = \left ( 1 - e^{-\beta p} \right )^{-1}$ is the contribution
of a single bosonic mode. Using $s^*$ and $s$ for the number of internal states, the partition function is given by

\begin{equation}
Z = \prod_{\textbf{p}} Z(\textbf{p})^s Z^*(\textbf{p})^{s^*} \; . \nonumber
\end{equation}

If we take the log, the product turns into a sum and if we take the momentum spectrum to be continuous, the sum turns
into an integral. Remembering the integration measure for momentum space, we have

\begin{eqnarray}
\log Z &\approx& \int_{\mathbb{R}^d} s^* \log \left ( 1 + e^{-\beta p} \right ) - s \log \left ( 1 - e^{-\beta p} \right ) \frac{V \textup{d}\textbf{p}}{(2\pi)^d} \nonumber \\
&=& \frac{d \omega_d V}{(2\pi)^d} \int_0^{\infty} s^* \log \left ( 1 + e^{-\beta p} \right ) - s \log \left ( 1 - e^{-\beta p} \right ) p^{d - 1} \textup{d}p \nonumber \\
&=& \frac{d! \omega_d V}{(2\pi \beta)^d} \left [ s\zeta(d + 1) + s^*\zeta^*(d + 1) \right ] \nonumber \\
&\equiv& \frac{AV}{\beta^d} \; . \label{partfunc}
\end{eqnarray}

Here, $\zeta(\sigma) = \sum_{n = 1}^{\infty} \frac{1}{n^{\sigma}}$ is the Riemann zeta function,
$\zeta^*(\sigma) = \sum_{n = 1}^{\infty} \frac{(-1)^{n - 1}}{n^{\sigma}}$ is the alternating zeta function and
$\omega_d$ is the volume of a unit ball in $\mathbb{R}^d$. We may now use
$S = \frac{\textup{d}}{\textup{d}T} (T \log Z)$ and $E = T^2 \frac{\textup{d}}{\textup{d}T} (\log Z)$ to show that
(<ref>) holds with a proportionality constant of
$\left [ \frac{(d + 1)^{d + 1}}{d^d} A \right ]^{\frac{1}{d + 1}}$.

If one is interested in the density of states, the exponential of this entropy is certainly the first term in $\rho(E)$.
However, there are an infinite number of other terms that come from the differences between the canonical and
microcanonical ensembles. The second term is a standard result that comes from treating $Z(\beta)$ as the Laplace
transform of $\rho(E)$. Performing a saddle point approximation,

\begin{eqnarray}
\rho(E) &=& \frac{1}{2\pi} \int_{-\infty}^{\infty} Z(i\beta) e^{i\beta E} \textup{d}\beta \nonumber \\
&\equiv& \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{f(i\beta)} \textup{d}\beta \nonumber \\
&\sim& \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{f(i\beta_0) - \frac{\beta^2}{2}f^{\prime\prime}(i\beta_0)} \textup{d}\beta \nonumber \\
&=& \frac{1}{\sqrt{2\pi f^{\prime\prime}(i\beta_0)}} e^{f(i\beta_0)} \; . \nonumber
\end{eqnarray}

The higher asymptotic terms cannot be found in the same way because the integral of
$e^{f(i\beta_0) - \frac{\beta^2}{2}f^{\prime\prime}(i\beta_0) - i\frac{\beta^3}{6}f^{\prime\prime\prime}(i\beta_0)}$
has no closed form solution. Instead, powers of $\beta$ after the first two must be Taylor expanded again so that
the above becomes

\begin{eqnarray}
\rho(E) &\sim& \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{f(i\beta_0) - \frac{\beta^2}{2}f^{\prime\prime}(i\beta_0)} \left ( 1 - i\frac{\beta^3}{6}f^{\prime\prime\prime}(i\beta_0) - \frac{\beta^6}{72}f^{\prime\prime\prime}(i\beta_0)^2 + \dots \right ) \nonumber \\
&& \left ( 1 + \frac{\beta^4}{24}f^{\prime\prime\prime\prime}(i\beta_0) + \frac{\beta^8}{1152}f^{\prime\prime\prime\prime}(i\beta_0)^2 + \dots \right ) \dots \textup{d}\beta \; . \nonumber
\end{eqnarray}

These calculations require us to consider an ever-growing number of ways in which a power of $\beta$ can be made.
Nevertheless, this method is still practical for finding the third term in $\rho(E)$ and the resulting expression is

\begin{eqnarray}
\rho(E) &\sim& \frac{1}{\sqrt{2\pi}} \left ( \frac{d}{(d + 1)^{d + 1}} AVE^{-d-2} \right )^{\frac{1}{2(d + 1)}} \nonumber \\
&& \exp \left [ \left ( \frac{(d + 1)^{d + 1}}{d^d} AVE^d \right )^{\frac{1}{d + 1}} - \frac{(d + 2)(2d + 1)}{24(d + 1)} \left ( dAVE^d \right )^{-\frac{1}{d + 1}} \right ] \; . \label{free-dos}
\end{eqnarray}

In [17], (<ref>) is found in a different way. The inverse Laplace transform of $Z(\beta)$ is found
exactly via a Hankel contour but as a Taylor series, not an asymptotic series. From this series

\begin{equation}
\rho(E) \sim \sum_{j = 1}^{\infty} \frac{(AV)^j E^{dj - 1}}{j!(dj - 1)!} \; , \label{free-dos-series}
\end{equation}

the first three asymptotic terms are picked off. An advantage of this is that (<ref>) can be compared
to a recent expression for the $d = 1$ density of states due to Loran, Sheikh-Jabbari and Vincon [18]:

\begin{equation}
\rho(E) \sim \frac{\pi c V}{3} \frac{I_1 \left ( \sqrt{2\pi c V E / 3} \right )}{\sqrt{2\pi c V E / 3}} \; . \label{free-dos-bessel}
\end{equation}

Neither is a generalization of the other because $d$ is arbitrary in (<ref>) and the interactions are
arbitrary in (<ref>).

In the partition function we have constructed, the $\beta$ variable is conjugate to the energy. There are also
conjugate variables associated with each momentum direction. Something special that we can do in $1 + 1$ dimensions
is combine these into a complex number. Let $p$ be a positive momentum. If there are $N$ excitations
of $p$ and $\tilde{N}$ excitations of $-p$, this state has an energy of $p(N + \tilde{N})$ and a momentum of
$p(N - \tilde{N})$. Therefore, generalized partition functions we can write down are:

\begin{eqnarray}
Z^*(p) &=& \sum_{N = 0}^1 \sum_{\tilde{N} = 0}^1 e^{-pV \tau_2 (N + \tilde{N}) + pVi \tau_1 (N - \tilde{N})} = \left ( 1 + e^{-pV (\tau_2 - i \tau_1)} \right ) \left ( 1 + e^{-pV (\tau_2 + i \tau_1)} \right ) \nonumber \\
Z(p) &=& \sum_{N = 0}^{\infty} \sum_{\tilde{N} = 0}^{\infty} e^{-pV \tau_2 (N + \tilde{N}) + pVi \tau_1 (N - \tilde{N})} = \left ( 1 - e^{-pV (\tau_2 - i \tau_1)} \right )^{-1} \left ( 1 - e^{-pV (\tau_2 + i \tau_1)} \right )^{-1} \; . \nonumber
\end{eqnarray}

Taking the product of $Z(p)^s Z^*(p)^{s^*}$ over all positive momenta, we have

\begin{eqnarray}
\log Z &\approx& \int_0^{\infty} s^* \left [ \log \left ( 1 + e^{-pV (\tau_2 - i \tau_1)}\right ) + \log \left ( 1 + e^{-pV (\tau_2 + i \tau_1)}\right ) \right ] \nonumber \\
&& - s \left [ \log \left ( 1 - e^{-pV (\tau_2 - i \tau_1)}\right ) + \log \left ( 1 - e^{-pV (\tau_2 + i \tau_1)}\right ) \right ] \frac{V \textup{d}p}{2\pi} \nonumber \\
&=& \frac{1}{2\pi} \left ( \frac{1}{\tau_2 - i\tau_1} + \frac{1}{\tau_2 + i\tau_1} \right ) \left [ s\zeta(2) + s^*\zeta^*(2) \right ] \nonumber \\
&=& -\frac{\Im \left ( \frac{1}{\tau} \right )}{\pi} \left [ s\zeta(2) + s^*\zeta^*(2) \right ] \nonumber \\
&=& -\frac{\pi \Im \left ( \frac{1}{\tau} \right )}{6} \left ( s + \frac{s^*}{2} \right ) \; . \label{gen-partfunc}
\end{eqnarray}

The dimensionless number $\tau = \tau_1 + i\tau_2$ is called the modular parameter. If
$\tau \equiv \frac{i\beta}{V}$, (<ref>) becomes the regular partition function (<ref>). The
quantity $c \equiv s + \frac{s^*}{2}$ appearing in (<ref>) is central charge that we would use in
(<ref>) if we wanted to apply it to a free theory.

§.§.§ In string theory

The worldsheet theory of a string can be regarded as a conformal field theory in $1 + 1$ dimensions. However,
$S \propto \sqrt{E}$ would not be correct for a macroscopic observer who has different notions of energy and
dimensionality. The worldsheet Lagrangian for a supersymmetric string theory in flat space is

\begin{eqnarray}
L &=& -\frac{1}{4\pi \alpha^{\prime}} \int_0^{2\pi} \partial_a X^{\mu} \partial^a X_{\mu} - i \bar{\Psi}^{\mu} \gamma^a \partial_a \Psi_{\mu} \textup{d}\sigma^1 \nonumber \\
&=& -\frac{1}{2\pi \alpha^{\prime}} \int_0^{2\pi} 2\partial X^{\mu} \tilde{\partial} X_{\mu} - i\psi^{\mu} \tilde{\partial} \psi_{\mu} - i\tilde{\psi}^{\mu} \partial \tilde{\psi}_{\mu} \textup{d}\sigma^1 \; . \label{string-action}
\end{eqnarray}

In the second form we have split each Dirac spinor field into two Majorana spinor fields. We have also written
derivatives with respect to $\sigma_{\pm} = \sigma^0 \pm \sigma^1$ as $\partial$ and $\tilde{\partial}$. What makes
this different from a usual quantum field theory is that the $D$ scalar fields $X^{\mu}$ can be interpreted as
positions in a $D$-dimensional target space. The worldsheet energy comes from
$\sigma^a \mapsto \sigma^a + \delta \sigma^a$ but the energy we should use for counting states is the conserved
quantity associated with $X^{\mu} \mapsto X^{\mu} + \delta X^{\mu}$. The worldsheet has $\mathrm{SO}(1, 1)$ Lorentz
symmetry regardless of how many fields there are, but the $\mathrm{SO}(D - 1, 1)$ Lorentz symmetry of the target space
is more sensitive. To survive quantization it requires that $D = 10$ [13]. If we had left the fermions out
of (<ref>) to construct bosonic string theory, the same calculation would tell us that $D = 26$.

To calculate the free energy of a gas of strings, we will begin in the same way as before.

\begin{eqnarray}
F &=& \frac{1}{\beta} \int_{\mathbb{R}^{D - 1}} \sum_m \log \left ( 1 - e^{-\beta \sqrt{p^2 + m^2}} \right ) - \sum_{m^*} \log \left ( 1 + e^{-\beta \sqrt{p^2 + m^{*2}}} \right ) \frac{V \textup{d}\textbf{p}}{(2\pi)^{D - 1}} \nonumber \\
&=& -\frac{1}{\beta} \sum_{n = 1}^{\infty} \int_{\mathbb{R}^{D - 1}} \sum_m \frac{1}{n} e^{-\beta n \sqrt{p^2 + m^2}} - \sum_{m^*} \frac{(-1)^n}{n} e^{-\beta n \sqrt{p^2 + m^{*}}} \frac{V \textup{d}\textbf{p}}{(2\pi)^{D - 1}} \nonumber
\end{eqnarray}

This expression has a sum over the masses of bosons and a sum over the masses of fermions. To arrive at
(<ref>), we set these masses to zero and replaced the sums by degeneracy factors. This was valid because the
masses became negligible in the high temperature limit. The high temperature limit of a string theory is different
because it supports arbitrarily large masses. To continue, we will use the trick

\begin{equation}
\int_0^{\infty} e^{-\frac{a^2s}{2} - \frac{b^2}{2s}} \frac{\textup{d}s}{\sqrt{s}} = \frac{\sqrt{2\pi}}{a} e^{-ab} \nonumber
\end{equation}

to rewrite the free energy density.

\begin{eqnarray}
\frac{F}{V} &=& -\sum_{n = 1}^{\infty} \int_0^{\infty} \int_{\mathbb{R}^{D - 1}} \sum_m e^{-\frac{\beta^2 n^2 s}{2} - \frac{p^2 + m^2}{2s}} - (-1)^n \sum_{m^*} e^{-\frac{\beta^2 n^2 s}{2} - \frac{p^2 + m^{*2}}{2s}} \frac{\textup{d}\textbf{p}}{(2\pi)^{D - 1}} \frac{\textup{d}s}{\sqrt{2\pi s}} \nonumber \\
&=& -\sum_{n = 1}^{\infty} \int_0^{\infty} e^{-\frac{\beta^2 n^2}{4\pi \alpha^{\prime} \tau_2}} \left [ \sum_m e^{-\pi \tau_2 \alpha^{\prime} m^2} - (-1)^n \sum_{m^*} e^{-\pi \tau_2 \alpha^{\prime} m^{*2}} \right ] (4 \pi^2 \alpha^{\prime} \tau_2)^{-\frac{D}{2}} \frac{\textup{d} \tau_2}{\tau_2} \label{partfunc-sofar}
\end{eqnarray}

Above, we have made the substitution $\tau_2 = \frac{1}{2\pi \alpha^{\prime} s}$. To proceed further, we need to know
the mass spectrum of our theory.

For concreteness we will work in Type II which is a theory of closed strings. This is natural because evidence of the
AdS / CFT correspondence was first discovered with Type IIB string theory [1]. Very little would change if
we used Type I or heterotic strings. The mode expansions for the scalar fields are identical to the ones that describe
the closed bosonic string:

\begin{equation}
X^{\mu}(\sigma^0, \sigma^1) = x^{\mu} + \alpha^{\prime} p^{\mu} \frac{\sigma_+ + \sigma_-}{2} + i \sqrt{\frac{\alpha^{\prime}}{2}} \sum_{n \neq 0} \frac{1}{n} \left ( \tilde{\alpha}^{\mu}_n e^{-in\sigma_+} + \alpha^{\mu}_n e^{-in\sigma_-} \right ) \; . \nonumber
\end{equation}

For the closed superstring, the left and right movers ($\psi$ and $\tilde{\psi}$) are independent and have the mode

\begin{eqnarray}
\psi^{\mu}(\sigma^0, \sigma^1) &=& \sqrt{\alpha^{\prime}} \sum_{r \in \mathbb{Z} + v} b^{\mu}_r e^{ir\sigma_+} \nonumber \\
\tilde{\psi}^{\mu}(\sigma^0, \sigma^1) &=& \sqrt{\alpha^{\prime}} \sum_{r \in \mathbb{Z} + v} \tilde{b}^{\mu}_r e^{-ir\sigma_-} \; . \nonumber
\end{eqnarray}

Since fermions can have two different types of boundary conditions, the parameter
$v \in \left \{ 0, \frac{1}{2} \right \}$ denotes which one we are using. For Ramond fermions, which are periodic,
$v = 0$. For Neveu-Schwarz fermions which are antiperiodic, $v = \frac{1}{2}$. The creation and anhilation
operators above obey the relations [19]

\begin{eqnarray}
[\alpha^{\mu}_m, \alpha^{\nu}_n] = &m \eta^{\mu\nu} \delta_{m + n, 0}& = [\tilde{\alpha}^{\mu}_m, \tilde{\alpha}^{\nu}_n] \nonumber \\
\{ b^{\mu}_r, b^{\nu}_s \} = &\eta^{\mu\nu} \delta_{r + s, 0}& = \{ \tilde{b}^{\mu}_r, \tilde{b}^{\nu}_s \} \; . \nonumber
\end{eqnarray}

To build up the spectrum from this, we need to consider gauge symmetries. The action (<ref>) came from
a more general action in which the worldsheet metric was dynamical. Choosing $g_{ab} = \eta_{ab}$ restricts the
physical Hilbert space to only those states which are anhilated by the Virasoro generators:

\begin{eqnarray}
L_m &=& \frac{1}{2} \sum_{n \in \mathbb{Z}} : \alpha_{m - n} \cdot \alpha_n : + \frac{1}{4} \sum_{r \in \mathbb{Z} + v} (2r - m) : b_{m - r} \cdot b_r : - a \delta_{m, 0} \nonumber \\
G_r &=& \sum_{n \in \mathbb{Z}} \alpha_n \cdot b_{r - n} \; . \nonumber
\end{eqnarray}

Analogous expressions hold for $\tilde{L}_m$ and $\tilde{G}_r$. Like $D$, the normal ordering constant $a = v$ is an
anomaly that can be fixed by demanding Lorentz invariance [19]. We may use the relativistic dispersion relation,
the mode expansions and the Virasoro generators to write down a formula for the mass operator.

\begin{eqnarray}
m^2 &=& -p_{\mu}p^{\mu} \nonumber \\
&=& -\frac{2}{\alpha^{\prime}} \alpha_0 \cdot \alpha_0 \nonumber \\
&=& -\frac{2}{\alpha^{\prime}} \left [ L_0 - \sum_{n > 0} \alpha_{-n} \cdot \alpha_n - \sum_{r > v} r b_{-r} \cdot b_r + a \right ] \nonumber \\
&=& \frac{2}{\alpha^{\prime}} \left [ \sum_{n > 0} \alpha_{-n} \cdot \alpha_n + \sum_{r > v} r b_{-r} \cdot b_r - a \right ] \label{mass-operator}
\end{eqnarray}

We must have $m^2 = \tilde{m}^2$. This translates into a condition known as level matching requiring every
state to have the same number of left and right moving excitations. Despite accounting for a gauge symmetry in this
way, the action (<ref>) still has some gauge symmetry left. A common technique for dealing with this
redundancy is fixing the lightcone gauge. This essentially means that any Lorentz index $\mu$ running from $0$ to
$D - 1$ becomes a regular index $i$ running from $1$ to $D - 2$ [19].

We now have everything we need to derive the massless spectrum of Type II string theory. In typical examples of a Fock
space, the ground state is unique. It is a singlet with respect to any symmetry group of interest and denoted most
often by $\left | 0 \right >$. This is not the case for the superstring. For Ramond fermions, the operators $b^i_0$
commute with $m^2$ meaning that many states have zero mass. This degenerate ground state in fact transforms as a
spinor in ten dimensions. Moreover it can be split into two chiralities $\left | + \right >$ and $\left | - \right >$.
This is different from four dimensions which would make the split into Weyl spinors inconsistent with the split
into Majorana spinors that we have already performed [13]. For Neveu-Schwarz fermions, the lowest lying
state has negative $m^2$. However, one of the advantages of the superstring is that it allows us to avoid this tachyon
and start at the massless states. These are also degenerate and are denoted by $b^i_{-\frac{1}{2}} \left | 0 \right >$.
Even though $b$ is an anticommuting operator for the worldsheet, the $i$ index here makes this a vector particle
in the target space. We have shown that massless R states are spacetime fermions while massless NS states are
spacetime bosons. The choice between R and NS can be made for the left and right movers separately. This means that
Type II string theories have four sectors [19].

\begin{align}
\begin{tabular}{|c|c|c|c|}
\hline
& $(\tilde{v}, v)$ & Type IIA & Type IIB \\
\hline
R-R & $(0, 0)$ & $\left | - \right > \otimes \left | + \right >$ & $\left | + \right > \otimes \left | + \right >$ \\
NS-NS & $\left ( \frac{1}{2}, \frac{1}{2} \right )$ & $\tilde{b}^i_{-\frac{1}{2}} \left | 0 \right > \otimes b^i_{-\frac{1}{2}} \left | 0 \right >$ & $\tilde{b}^i_{-\frac{1}{2}} \left | 0 \right > \otimes b^i_{-\frac{1}{2}} \left | 0 \right >$ \\
NS-R & $\left ( \frac{1}{2}, 0 \right )$ & $\tilde{b}^i_{-\frac{1}{2}} \left | 0 \right > \otimes \left | + \right > \;\;\;\;\;\;$ & $\tilde{b}^i_{-\frac{1}{2}} \left | 0 \right > \otimes \left | + \right > \;\;\;\;\;\;$ \\
R-NS & $\left ( 0, \frac{1}{2} \right )$ & $\;\;\;\;\;\; \left | - \right > \otimes b^i_{-\frac{1}{2}} \left | 0 \right >$ & $\;\;\;\;\;\; \left | - \right > \otimes b^i_{-\frac{1}{2}} \left | 0 \right >$ \\
\hline
\end{tabular}
\label{spectrum-table}
\end{align}

Each sector is 64-fold degenerate.

Our expression for the free energy density has terms like $e^{-\pi \tau_2 \alpha^{\prime} m^2}$ summed over masses.
These sums look like familiar partition functions if we substitute (<ref>) in for $m^2$. Going back to
(<ref>), it is almost correct to replace the term in square brackets with
$64 Z_{0, 0}(\tau_2) + 64 Z_{\frac{1}{2}, \frac{1}{2}}(\tau_2) - 128(-1)^n Z_{0, \frac{1}{2}}(\tau_2)$. In the notation
being used

\begin{eqnarray}
Z_{\tilde{v}, v} &=& e^{2(v + \tilde{v})\pi \tau_2} \left [ \sum_{\left \{ N^i_n, \tilde{N}^i_n \right \}} \exp \left ( -2\pi \tau_2 \sum_{i = 1}^8 \sum_{n = 1}^{\infty} n \left ( N^i_n + \tilde{N}^i_n \right ) \right ) \right ] \nonumber \\
&& \left [ \sum_{\left \{ M^i_r \right \}} \exp \left ( -2\pi \tau_2 \sum_{i = 1}^8 \sum_{r = v + 1}^{\infty} rM^i_r \right ) \right ] \left [ \sum_{\left \{ \tilde{M}^i_r \right \}} \exp \left ( -2\pi \tau_2 \sum_{i = 1}^8 \sum_{r = \tilde{v} + 1}^{\infty} r\tilde{M}^i_r \right ) \right ] \; , \label{partfunc-schematic}
\end{eqnarray}

$N^i_n$ and $\tilde{N}^i_n$ are bosonic occupation numbers while $M^i_r$ and $\tilde{M}^i_r$ are fermionic occupation
numbers. Accounting for level matching is the one correction that needs to be made. This can be done by inserting a
Kronecker delta

\begin{equation}
\delta_{L,R} = \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2\pi i \tau_1 (L - R)} \textup{d}\tau_1 \nonumber
\end{equation}

where $L = \sum_{i, n} n \tilde{N}^i_n + \sum_{i, r} r \tilde{M}^i_r$ and
$R = \sum_{i, n} n N^i_n + \sum_{i, r} r M^i_r$ are the left and right excitations respectively. Multiplying this by
(<ref>), we see that the quantity being integrated is nothing but the generalized partition function
for 8 fermions and 8 bosons. Rewriting (<ref>),

\begin{eqnarray}
\frac{F}{V} &=& -\sum_{n = 1}^{\infty} \int_0^{\infty} \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-\frac{\beta^2 n^2}{4\pi \alpha^{\prime} \tau_2}} 64 Z_{\mathrm{B}}(\tau)^8 \left [ Z_{\mathrm{R}}(\tau)^8 + Z_{\mathrm{NS}}(\tau)^8 - 2(-1)^n \sqrt{Z_{\mathrm{R}}(\tau) Z_{\mathrm{NS}}(\tau)}^8 \right ] \nonumber \\
&& (4 \pi^2 \alpha^{\prime} \tau_2)^{-5} \frac{\textup{d} \tau_1 \textup{d} \tau_2}{\tau_2} \; . \label{string-free-energy}
\end{eqnarray}

In terms of our old notation, $Z_{\mathrm{B}}(\tau) = Z(\tau)$ while $Z_{\mathrm{R}}(\tau)$ and $Z_{\mathrm{NS}}(\tau)$
approach $Z^*(\tau)$ in the small $\tau$ limit.

Our goal is to investigate the high temperature limit of (<ref>). Since this corresponds to
$\beta \rightarrow 0$, the integral is dominated by the $n = 1$ term of the sum and the small $\tau$ limits of the
worldsheet partition functions. A curious fact about string theories is that at a high enough temperature, called the
Hagedorn temperature $T_{\mathrm{H}}$, the free energy density diverges. We will solve for $\beta_{\mathrm{H}}$. This
can be done by looking at any one of the four terms in the integrand of (<ref>). Substituting the
generalized partition function (<ref>), the function we are integrating is

\begin{equation}
64 e^{-\frac{\beta^2}{4\pi \alpha^{\prime} \tau_2}} e^{-2\pi \Im \left ( \frac{1}{\tau} \right )} \frac{(4 \pi^2 \alpha^{\prime} \tau_2)^{-5}}{\tau_2} \; . \nonumber
\end{equation}

The value of $\beta_{\mathrm{H}}$ is reached when the overall exponent is zero. This means

\begin{eqnarray}
\beta_{\mathrm{H}}^2 &=& 8\pi^2 \alpha^{\prime} \frac{\tau_2^2}{\tau_1^2 + \tau_2^2} \nonumber \\
&\approx& 8\pi^2 \alpha^{\prime} \; . \label{hagedorn-temperature}
\end{eqnarray}

The partition function for a system first diverges when the density of states becomes exponential and the
decay of the Boltzmann factor can no longer overpower such growth. This is equivalent to saying
that $S \propto E$. The proportionality constant can be read off from (<ref>) because
$\beta = \frac{\textup{d} S}{\textup{d} E}$ must give $\beta_{\mathrm{H}}$. The result of this, in contrast to
(<ref>) is:

\begin{equation}
S = 2\pi \sqrt{2 \alpha^{\prime}} E \; . \label{string-entropy}
\end{equation}

§.§ Black holes

It is clear how entropy arises in the field theories we have discussed. If we only know the energy $E$ of a field,
the corresponding ensemble of particles can be in any one of $\rho(E)$ microstates contributing to our lack of
knowledge about the system. Microstates of this form do not appear to be present for black holes. Classically, one can
learn everyting about a black hole from just three numbers: mass, charge and angular momentum. The discovery that black
holes have entropy as well, has led to some of the deepest results in theoretical physics [20].

§.§.§ Useful metrics

Black hole metrics in arbitrary dimension have seen increasing interest since the discovery of the AdS / CFT
correspondence [21]. Most authors take the Einstein equations to be fundamental so that they read

\begin{equation}
\mathcal{R}_{\mu\nu} - \frac{1}{2} \mathcal{R} g_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu} \label{einstein}
\end{equation}

regardless of how many dimensions there are. The same cannot be said of the Newtonian limit. If (<ref>)
describes the full theory of gravity in $d + 1$ dimensions, one can show that dimension dependent prefactors
necessarily appear in the Poisson equation [21]:

\begin{eqnarray}
\Delta \Phi &=& 8\pi G \frac{d - 2}{d - 1} \rho \nonumber \\
&=& 8\pi G \frac{d - 2}{d - 1} M \delta(0) \; . \label{newtonian-limit}
\end{eqnarray}

The second form above specializes to a point mass of $M$.

The neutral, irrotational black hole in arbitrary dimension is called the Schwarzschild-Tangherlini solution.

\begin{equation}
ds^2 = -\left ( 1 - \frac{\mu}{r^{d - 2}} \right ) dt^2 + \left ( 1 - \frac{\mu}{r^{d - 2}} \right )^{-1} dr^2 + r^2 d\Omega_{d - 1}^2 \label{schwarzschild}
\end{equation}

We will take this opportunity to review some of the basic properties that black hole metrics should have.

* It can be checked that (<ref>) solves Einstein's equation with no cosmological constant and no
stress-energy tensor. In fact, it is the unique spherically symmetric and time independent solution. The requirement
that it be time independent is redundant if $d = 3$.

* It is clear that (<ref>) is asymptotically flat. Therefore, the notion of “escaping” from a potential
well in this metric is well defined.

* It is also clear that beyond a certain radius, one can no longer escape. Far away from the origin, $t$ is timelike and
$r$ is spacelike but this reverses when $r$ falls below $\mu^{\frac{1}{d - 2}}$. On the inside of this
event horizon, the flow of time in (<ref>) is such that an object is inexorably drawn toward
the centre.

* This implies that the mass generating the event horizon has been compressed to a point. This, along with the fact that
metrics for equal point masses should be indistinguishable, tells us that (<ref>)
represents the most efficient packing of said mass into a sphere of radius $\mu^{\frac{1}{d - 2}}$.

* The mass may be computed by taking the Newtonian limit. For those unfamiliar with the ADM procedure [22], we will
consider a test particle far away from the origin, moving radially outward. If the motion is non-relativistic, the
timelike geodesic condition becomes $g_{tt} \dot{t}^2 \approx -1$ or
$\dot{t}^2 \approx \left ( 1 - \frac{\mu}{r^{d - 2}} \right )^{-1}$. Substituting this into the geodesic equation,

\begin{eqnarray}
\ddot{r} &\approx& -\Gamma^r_{tt} \dot{t}^2 \nonumber \\
&\approx& \frac{1}{2} \partial_r g_{tt} \frac{g^{rr}}{1 - \frac{\mu}{r^{d - 2}}} \nonumber \\
&=& -\nabla \left ( \frac{\mu}{2r^{d - 2}} \right ) \; . \nonumber
\end{eqnarray}

The function inside the gradient should be a solution to (<ref>). Recalling the Green's function for
the $d$-dimensional Laplacian, this is only true if $\mu = \frac{16 \pi GM}{d(d - 1) \omega_d}$.

A further generalization of interest to us is the metric for a black hole that is asymptotically AdS. Anti-de Sitter
space is the maximally symmetric solution to Einstein's equations when they have a negative cosmological constant. A
cosmological constant defines a length scale for the spacetime with

\begin{equation}
\Lambda \equiv -\frac{d(d - 1)}{2L^2} \label{cosmological-constant}
\end{equation}

by convention. Because of this, AdS posesses a conformal boundary. Its radial co-ordinate is infinite but an observer
is able to reach it in a finite amount of proper time. The solution is

\begin{equation}
ds^2 = -\left ( 1 + \frac{r^2}{L^2} \right ) dt^2 + \left ( 1 + \frac{r^2}{L^2} \right )^{-1} dr^2 + r^2 d\Omega_{d - 1}^2 \label{ads}
\end{equation}

and a black hole metric asymptotic to this is

\begin{equation}
ds^2 = -\left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right ) dt^2 + \left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right )^{-1} dr^2 + r^2 d\Omega_{d - 1}^2 \; . \label{schwarzschild-ads}
\end{equation}

Uniqueness of (<ref>) in the same sense as (<ref>) is suspected but not known.
The horizon radius is a solution to $\frac{\mu}{r_0^{d - 2}} = 1 + \frac{r_0^2}{L^2}$ with the outermost one being
the point of no return. The mass is again given by $\mu = \frac{16 \pi GM}{d(d - 1) \omega_d}$. To see this,
a non-relativistic radial trajectory satisfies

\begin{eqnarray}
\ddot{r} &\approx& -\Gamma^r_{tt} \dot{t}^2 \nonumber \\
&\approx& \frac{1}{2} \partial_r g_{tt} \nonumber \\
&=& -\nabla \left ( \frac{\mu}{2r^{d - 2}} \right ) - \frac{r}{L^2} \; . \nonumber
\end{eqnarray}

While this lacks the sophistocation of methods like [23], we obtain the right answer if we simply subtract
the acceleration that a particle would have in pure AdS.

It will be convenient to rewrite this metric in Eddington-Finkelstein co-ordinates. This can be done using either
the retarded time or advanced time, which take the form

\begin{eqnarray}
u &=& t - r^* \nonumber \\
v &=& t + r^* \nonumber
\end{eqnarray}

respectively. Differentiating these to arrive at

\begin{eqnarray}
\dot{u} &=& \dot{t} - \frac{\textup{d}r^*}{\textup{d}r} \dot{r} \nonumber \\
\dot{v} &=& \dot{t} + \frac{\textup{d}r^*}{\textup{d}r} \dot{r} \; , \nonumber
\end{eqnarray}

we see that $r^*$ should be chosen so that its radial derivative is the factor relating $\dot{t}$ and
$\dot{r}$. For a null geodesic,

\begin{equation}
\dot{t} = \pm \left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right )^{-1} \dot{r} \nonumber
\end{equation}

where the positive sign corresponds to an outgoing particle and the negative sign corresponds to an ingoing particle.
We now have

\begin{equation}
\frac{\textup{d}r^*}{\textup{d}r} = \left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right )^{-1} \; . \nonumber
\end{equation}

In certain cases, this can be integrated to give $r^*$ explicitly. However, this is not needed for replacing $dt$.
Performing the change of variables,

\begin{eqnarray}
ds^2 &=& -\left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right ) du^2 - 2 du dr + r^2 d\Omega_{d - 1}^2 \nonumber \\
ds^2 &=& -\left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right ) dv^2 + 2 dv dr + r^2 d\Omega_{d - 1}^2 \label{ed-fink-ads}
\end{eqnarray}

are the desired metrics.

§.§.§ Hawking radiation

To derive the relation between entropy and the area of a black hole, we will follow Hawking's original paper
[24] as well as the clarifications in [25, 26]. What we will see is that two observers — one
observing spacetime before a black hole has formed, the other after — will have different definitions of the quantum
vacuum. The Klein-Gordon equation for a field in curved spacetime is

\begin{eqnarray}
\nabla_{\mu} \nabla^{\mu} \phi - m^2 \phi = 0 \nonumber \\
\frac{1}{\sqrt{-g}} \partial_{\mu} \left ( \sqrt{-g} g^{\mu\nu} \partial_{\nu} \phi \right ) - m^2 \phi = 0 \; . \label{kg-equation}
\end{eqnarray}

For any two solutions $\phi_1$ and $\phi_2$, the Klein-Gordon inner product

\begin{equation}
\left < \phi_1, \phi_2 \right > = \int_S \left [ \phi^*_1 \nabla^{\mu} \phi_2 - \phi_2 \nabla^{\mu} \phi^*_1 \right ] \textup{d}\Sigma_{\mu} \label{kg-inner-product}
\end{equation}

will be conserved. The notation above suggests a scalar field, but this does not have to be the case. Fields with
multiple components like vectors and spinors satisfy the Klein-Gordon equation componentwise. To indicate that Hawking
radiation is a mixture of all types of particles, we will write creation and anhilation operators as
$a^{\mathcal{I}\dagger}$ and $a^{\mathcal{I}}$ where $\mathcal{I}$ is an index set. One simplification we will make,
however, is that the fields are massless.

We may write a basis of solutions to (<ref>) as $f_i$ and choose them to be orthonormal with respect to
(<ref>). If we do this, the field operator takes the form

\begin{equation}
\phi^{\mathcal{I}} = \sum_i f_i a^{\mathcal{I}}_i + f^*_i a^{\mathcal{I}\dagger}_i \; . \nonumber
\end{equation}

In other words, positive frequency modes multiply anhilation operators while negative frequency modes multiply creation
operators. We will let these $f_i$ represent any particles that can be seen before a black hole forms. Since there are
no such particles, the past observer will see the vacuum state $\left | 0 \right >_a$ defined as the state that is
anhilated by all $a^{\mathcal{I}}_i$. The future observer sees a different metric and in particular a different time
component of the metric. This means he will have a different definition of positive and negative frequency. Writing

\begin{equation}
\phi^{\mathcal{I}} = \sum_i g_i b^{\mathcal{I}}_i + g^*_i b^{\mathcal{I}\dagger}_i \; , \nonumber
\end{equation}

each $g_i$ representing a particle in the black hole spacetime should be expressible as a linear combination
of the $f_i$. A positive frequency $g_i$ may therefore include a contribution from a negative frequency $f_i$ and
vice versa. If so, the $b^{\mathcal{I}}_i$ will not anhilate the $a^{\mathcal{I}}_i$ vacuum and the
$a^{\mathcal{I}}_i$ will not anhilate the $b^{\mathcal{I}}_i$ vacuum. This discrepancy between
$\left | 0 \right >_a$ and $\left | 0 \right >_b$ means that the future observer will see radiation precisely because
the past observer did not.

It is not correct to say that the only modes of $\phi^{\mathcal{I}}$ are $f_i$ waves that the past observer can see
and $g_i$ waves that the future observer can see. There are also $h_i$ waves in the future that cannot be seen because
they are behind the event horizon of the black hole. We would have to consider these if we wanted to write the past
modes as linear combinations of the future modes. As it happens, we will only need to write the future modes as linear
combinations of the past modes. Converting

\begin{eqnarray}
g_i &=& \sum_j \alpha_{ij} f_j + \beta_{ij} f^*_j \nonumber \\
h_i &=& \sum_j \sigma_{ij} f_j + \tau_{ij} f^*_j \nonumber
\end{eqnarray}

into a set of relations between operators, we arrive at the so-called Bogoliubov transformation:

\begin{eqnarray}
b^{\mathcal{I}}_i &=& \sum_j \alpha^*_{ij} a^{\mathcal{I}}_j - \beta^*_{ij} a^{\mathcal{I}\dagger}_j \nonumber \\
c^{\mathcal{I}}_i &=& \sum_j \sigma^*_{ij} a^{\mathcal{I}}_j - \tau^*_{ij} a^{\mathcal{I}\dagger}_j \; . \nonumber
\end{eqnarray}

This tells us that the number of particles detected as belonging to $g_i$ in the future is given by

\begin{equation}
n_i = {}_a \left < 0 \right | b^{\mathcal{I}\dagger}_i b^{\mathcal{I}}_i \left | 0 \right >_a = \sum_j \left | \beta_{ij} \right |^2 \; . \label{emission}
\end{equation}

Assuming that we are dealing with bosons, we also have

\begin{equation}
1 = {}_a \left < 0 \right | \left [ b^{\mathcal{I}}_i, b^{\mathcal{I}\dagger}_i \right ] \left | 0 \right >_a = \sum_j \left | \alpha_{ij} \right |^2 - \left | \beta_{ij} \right |^2 \; . \label{absorbtion}
\end{equation}

We will now be more explicit about what the modes are so that we may plug them into the inner product and find the
$\alpha_{ij}$ and $\beta_{ij}$ coefficients.

Spherical waves are convenient choices, but it is important not to use the expressions for flat space spherical waves
when we are really in a curved space. By construction, outgoing null geodesics are lines of constant $u$ while ingoing
null geodesics are lines of constant $v$. Therefore, the advanced and retarded times should be used in place of
$t \pm r$ giving us

\begin{eqnarray}
f_{\omega, l_1, \dots, l_{d-1}} (v, \theta_1, \dots, \theta_{d - 1}) = \frac{e^{i \omega v}}{\sqrt{(\omega r)^{d - 1} d \omega_d}} Y_{l_1, \dots, l_{d - 1}} (\theta_1, \dots, \theta_{d - 1}) \nonumber \\
g_{\omega, l_1, \dots, l_{d-1}} (u, \theta_1, \dots, \theta_{d - 1}) = \frac{e^{i \omega u}}{\sqrt{(\omega r)^{d - 1} d \omega_d}} Y_{l_1, \dots, l_{d - 1}} (\theta_1, \dots, \theta_{d - 1}) \label{spherical-waves}
\end{eqnarray}

as approximate solutions for large $r$. We could similarly consider
$f_{\omega, l_1, \dots, l_{d-1}} (u, \theta_1, \dots, \theta_{d - 1})$ and
$g_{\omega, l_1, \dots, l_{d-1}} (v, \theta_1, \dots, \theta_{d - 1})$ but these would affect the result very little.
The interesting effects come from waves that switch from ingoing to outgoing while the black hole is forming. By this,
we mean that waves of constant $v$ travel toward the collapsing mass at $r = 0$. As long as an event horizon has not
formed yet, such waves may emerge from the other side and start moving away with constant $u$. A natural question to
ask is which constant $u$? That is, what will $u$ be in terms of the $v$ that the wave used to have? This is
the key question that must be answered before we can take an inner product and derive Hawking's result. The difficulty
in relating these is explained in Figure <ref>.

[][Before]     [][After]
Photon A leaves a light source and starts heading toward $r = 0$. After a certain time interval, photon B does
the same. If the spacetime is Minkowski, the distance between the two photons will not change. Conversely, if a
black hole at $r = 0$ forms at just the right time, photon A will escape but photon B will stay trapped at the event
horizon forever.

A better way to compare objects A and B is to imagine that A throws a ball C backwards until it is caught by B. The
proper time for C to travel should be equal along all stages of the journey. Instead of proper time, we will
use a difference of affine parameters which is appropriate for signals travelling at the speed of light.

Waves hoping to escape the black hole must start off with a $v$ smaller than the one posessed by photon B in Figure
<ref>. We will call this largest advanced time $v_0$. If the spacetime is Minkowski, long before the
black hole has formed, $v = t + r$ and affine parametrizations are

\begin{eqnarray}
t(\lambda) &=& t(0) + \frac{\lambda}{2} \nonumber \\
r_{\mathrm{A}}(\lambda) &=& r_{\mathrm{A}}(0) - \frac{\lambda}{2} \nonumber \\
r_{\mathrm{B}}(\lambda) &=& r_{\mathrm{B}}(0) - \frac{\lambda}{2} \nonumber \\
r_{\mathrm{C}}(\lambda) &=& r_{\mathrm{A}}(0) + \frac{\lambda}{2} \; . \nonumber
\end{eqnarray}

By the time C makes it to B, it will have the same position and time co-ordinate as B so it must have the same $v$ as
B. Therefore subtracting the advanced times corresponding to A and C, we have

\begin{equation}
v - v_0 = r_{\mathrm{A}}(\lambda) - r_{\mathrm{C}}(\lambda) = -\lambda \; . \nonumber
\end{equation}

We may therefore call $v_0 - v$ an affine parameter that vanishes for the wave that stays at the event horizon. This
means that after the horizon at $r_0$ has formed, a wave's radial co-ordinate must look like $r = r_0 - \lambda$. We
will substitute this into the retarded time for C noting that C is not a wave of constant $u$ because it travels
backwards from A back to B.

\begin{eqnarray}
\dot{u} &=& \dot{t} - \frac{\textup{d}r^*}{\textup{d}r} \dot{r} \nonumber \\
&=& 2 \left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right )^{-1} \nonumber \\
&=& 2 \left ( 1 + \frac{r^2}{L^2} - \left ( \frac{r_0}{r} \right )^{d - 2} \left ( 1 + \frac{r_0^2}{L^2} \right ) \right )^{-1} \nonumber \\
&=& 2 \left ( 1 + \frac{(r_0 - \lambda)^2}{L^2} - \left ( \frac{r_0}{r_0 - \lambda} \right )^{d - 2} \left ( 1 + \frac{r_0^2}{L^2} \right ) \right )^{-1} \nonumber \\
\end{eqnarray}

This does not have a closed form integral, but the interesting effects come from waves that are close to the horizon.
Keeping only the lowest order in $\lambda$,

\begin{eqnarray}
\dot{u} &\approx& -\frac{2r_0}{\lambda} \frac{L^2}{(d - 2)L^2 + dr_0^2} \nonumber \\
u &\approx& -2r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{\lambda}{C} \right ) \nonumber \\
&=& -2r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} \right ) \; . \nonumber
\end{eqnarray}

The equation relating $u$ to $v$ has now been found, so we may substitute (<ref>) into
(<ref>) for a surface whose normal derivative is $\partial^r$. We will abbreviate the
$l_1, \dots, l_{d - 1}$ dependence as $l$ and the $\theta_1, \dots, \theta_{d - 1}$ dependence as $\theta$.

\begin{eqnarray}
\alpha_{\omega, l}^{\omega^{\prime}, l^{\prime}} &=& \left < g_{\omega, l}, f_{\omega^{\prime}, l^{\prime}} \right > \nonumber \\
&=& i \int_{\mathbb{S}^{d - 1}} \int_{-\infty}^{v_0} f^*_{\omega^{\prime}, l^{\prime}}(v, \theta) \partial^r g_{\omega, l}(v, \theta) - g_{\omega, l}(v, \theta) \partial^r f^*_{\omega^{\prime}, l^{\prime}}(v, \theta) \textup{d}v \textup{d}\Omega \nonumber \\
&=& i \int_{\mathbb{S}^{d - 1}} \int_{-\infty}^{v_0} f^*_{\omega^{\prime}, l^{\prime}}(v, \theta) \left [ \partial_v + \left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right ) \partial_r \right ] g_{\omega, l}(v, \theta) \nonumber \\
&& - g_{\omega, l}(v, \theta) \left [ \partial_v + \left ( 1 + \frac{r^2}{L^2} - \frac{\mu}{r^{d - 2}} \right ) \partial_r \right ] f^*_{\omega^{\prime}, l^{\prime}}(v, \theta) \textup{d}v \textup{d}\Omega \nonumber \\
&=& \frac{i}{d \omega_{d - 1} \sqrt{\omega \omega^{\prime}}^{d - 1}} \int_{\mathbb{S}^{d - 1}} Y_{l}(\theta) Y^*_{l^{\prime}}(\theta) \textup{d}\Omega \nonumber \\
&& e^{i\omega^{\prime}v} \partial_v e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} \right )} - e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} \right )} \partial_v e^{i \omega^{\prime} v} \textup{d}v \nonumber \\
&=& \frac{\delta_{l_1}^{l_1^{\prime}} \delta_{l_2}^{l_2^{\prime}} \dots \delta_{l_{d - 1}}^{l_{d - 1}^{\prime}}}{d \omega_{d - 1} \sqrt{\omega \omega^{\prime}}^{d - 1}} \int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} \right )} e^{i \omega^{\prime} v} \textup{d}v \nonumber
\end{eqnarray}

The other Bogoliubov coefficient is found similarly. The only difference is that when integrating two spherical
harmonics without a complex conjugate, we have to use the identity
$Y_{l_1, l_2, \dots, l_{d - 1}} = (-1)^{l_1} Y^*_{-l_1, l_2, \dots, l_{d - 1}}$.

\begin{eqnarray}
\beta_{\omega, l}^{\omega^{\prime}, l^{\prime}} &=& \left < g_{\omega, l}, f^*_{\omega^{\prime}, l^{\prime}} \right > \nonumber \\
&=& \frac{(-1)^{l_1}\delta_{-l_1}^{l_1^{\prime}} \delta_{l_2}^{l_2^{\prime}} \dots \delta_{l_{d - 1}}^{l_{d - 1}^{\prime}}}{d \omega_{d - 1} \sqrt{\omega \omega^{\prime}}^{d - 1}} \int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} - \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} \right )} e^{-i \omega^{\prime} v} \textup{d}v \nonumber
\end{eqnarray}

The integrands above have a branch cut on the real axis because of the $\log \left ( \frac{v_0 - v}{C} \right )$. In
order to manipulate them with complex analysis, it is convenient to displace them with $\pm i \epsilon$. The
$\alpha$ and $\beta$ integrals become

\begin{eqnarray}
\int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} + i \epsilon \right )} e^{i \omega^{\prime} v} \textup{d}v \label{alpha-integral} \\
\int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} - \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} - i \epsilon \right )} e^{-i \omega^{\prime} v} \textup{d}v \label{beta-integral}
\end{eqnarray}

respectively. The signs for $\pm i \epsilon$ above are dictated by our requirement that $e^{\pm i \omega^{\prime} v}$
vanish at infinity.

For the function with $+i \epsilon$, it is not difficult to show that the semi-circular arc has no
contribution to the integral. Therefore the integral between $-\infty$ and $\infty$ is zero.

The integral in (<ref>) vanishes if the domain is the contour in Figure <ref>. We may
therefore split it up as follows.

\begin{eqnarray}
&& \int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} + i \epsilon \right )} e^{i \omega^{\prime} v} \textup{d}v \nonumber \\
&=& -\int_{v_0}^{\infty} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} + i \epsilon \right )} e^{i \omega^{\prime} v} \textup{d}v \nonumber \\
&=& -e^{i \omega^{\prime} v_0} \int_{-\infty}^0 \left ( \frac{2 \omega r_0}{v^{\prime}} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( - \left ( \frac{-v^{\prime}}{C} - i \epsilon \right ) \right )} e^{i \omega^{\prime} v^{\prime}} \textup{d}v^{\prime} \nonumber
\end{eqnarray}

In the second step, we made the substitution $v^{\prime} = v_0 - v$. This integral is written with the understanding
that it should be evaluated with a contour in the lower half plane. We must therefore write
$\log(-A) = \log(A) - i\pi$ instead of $\log(-A) = \log(A) + i\pi$ to avoid crossing the branch cut. We will perform
this step and then make another substitution $v^{\prime} = v - v_0$.

\begin{eqnarray}
&& \int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} + i \epsilon \right )} e^{i \omega^{\prime} v} \textup{d}v \nonumber \\
&=& -e^{i \omega^{\prime} v_0 + 2\pi \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2}} \int_{-\infty}^0 \left ( \frac{2 \omega r_0}{v^{\prime}} \frac{L^2}{(d - 2)L^2 + dr_0^2} + \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{-v^{\prime}}{C} - i \epsilon \right )} e^{i \omega^{\prime} v^{\prime}} \textup{d}v^{\prime} \nonumber \\
&=& e^{2i \omega^{\prime} v_0 + 2\pi \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2}} \int_{-\infty}^{v_0} \left ( \frac{2 \omega r_0}{v_0 - v} \frac{L^2}{(d - 2)L^2 + dr_0^2} - \omega^{\prime} \right ) e^{2i \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2} \log \left ( \frac{v_0 - v}{C} - i \epsilon \right )} e^{i \omega^{\prime} v} \textup{d}v \nonumber
\end{eqnarray}

By manipulating (<ref>), we have turned it into a multiple of (<ref>). This implies the

\begin{equation}
\left | \alpha_{\omega, l_1 \dots, l_{d - 1}}^{\omega^{\prime}, l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \right | = e^{2\pi \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2}} \left | \beta_{\omega, l_1 \dots, l_{d - 1}}^{\omega^{\prime}, l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \right | \; . \label{alpha-beta-ratio}
\end{equation}

Going back to (<ref>) and (<ref>), the integral
$\sum_{l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \int_0^{\infty} \left | \alpha_{\omega, l_1 \dots, l_{d - 1}}^{\omega^{\prime}, l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \right |^2 - \left | \beta_{\omega, l_1 \dots, l_{d - 1}}^{\omega^{\prime}, l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \right |^2 \textup{d}\omega^{\prime}$
describes what the black hole will absorb. The integral
$\sum_{l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \int_0^{\infty} \left | \beta_{\omega, l_1 \dots, l_{d - 1}}^{\omega^{\prime}, l_1^{\prime}, \dots, l_{d - 1}^{\prime}} \right |^2 \textup{d}\omega^{\prime}$
describes what the black hole will emit. Without worrying about normalization, (<ref>) tells us that
the ratio between a mode's absorbtion and emission cross sections is

\begin{equation}
n_{\omega} = \left ( e^{4\pi \omega r_0 \frac{L^2}{(d - 2)L^2 + dr_0^2}} - 1 \right )^{-1} \; . \label{bose-einstein-factor}
\end{equation}

This is precisely the Bose-Einstein thermal factor for a blackbody at temperature

\begin{equation}
T_{\mathrm{BH}} = \frac{1}{4\pi r_0} \left ( d - 2 + d \frac{r_0^2}{L^2} \right ) \label{bh-temperature}
\end{equation}

Had we used an anticommutator in (<ref>), we would have seen the Fermi-Dirac factor for the same
temperature. The famous Bekenstein-Hawking entropy, $S = \frac{A}{4G}$, clearly follows from this if $L$ is large.
For a general $L$, we will use the fact that $\mu = r_0^{d - 2} + \frac{r_0^d}{L^2}$ to write

\begin{equation}
\textup{d}M = \frac{d(d-1)\omega_d}{16\pi G} \textup{d}\mu = \frac{d(d-1)\omega_d}{16\pi G} \left [ (d - 2) r_0^{d - 3} + \frac{d}{L^2} r_0^{d - 1} \right ] \textup{d}r_0 \; . \nonumber
\end{equation}

Then integrating,

\begin{eqnarray}
S_{\mathrm{BH}} &=& \int \frac{\textup{d}E}{T_{\mathrm{BH}}} \nonumber \\
&=& \int \frac{\textup{d}M}{T_{\mathrm{BH}}} \nonumber \\
&=& \frac{d(d-1)\omega_d}{4G} \int r_0^{d - 2} \textup{d}r_0 \nonumber \\
&=& \frac{d\omega_d}{4G} r_0^{d - 1} \nonumber \\
&=& \frac{A}{4G} \label{bh-entropy}
\end{eqnarray}

and we see that the entropy formula is exactly the same in $AdS_{d + 1}$.

Hints that the area of a black hole somehow describes an entropy were already known in 1973 when Bekenstein proposed
the proportionality with a coefficient “close to” $\frac{\log 2}{8\pi}$ [27]. Apart from improving the
coefficient to $\frac{1}{4}$, Hawking's 1975 paper established that (<ref>) is the genuine entropy of a
thermal spectrum [24]. Modern techniques can derive (<ref>) much more quickly but at the cost of
once again obscuring the nature of this entropy [28, 29]. In 1981, Bekenstein noticed that
(<ref>) is more than just the entropy of a black hole. It is an upper bound on the entropy that any
system occupying the same volume can have [30]. The argument, which was strong motivation for the
AdS / CFT correspondence [8], is remarkably simple. Suppose that a non-black hole system fills a ball of
radius $r_0$ and has more entropy than $\frac{A}{4G}$. Its mass must be less than that of a black hole with horizon
radius $r_0$ and therefore, it can be turned into said black hole through the addition of mass. Such a procedure would
give the system an entropy of $\frac{A}{4G}$ later on, violating the second law of thermodynamics. Incidentally, two
major open problems in physics are related to the evaporation of black holes. A featureless object described uniquely
by mass, charge and angular momentum should not have entropy and yet we have seen that it contains more entropy than
anything else. While their exact nature remains unknown, some methods for elucidating black hole microstates are
provided by string theory [31]. A more serious problem is the black hole information paradox.
This is concerned with the fact that a bath of radiation cannot contain information about the formation of a black
hole. If a thermal state is all that a black hole leaves behind after it evaporates, one effectively has a pure state
evolving into a mixed state which is a violation of unitarity. String theoretic resolutions to this have been proposed
as well but are, at the time of writing, much more speculative [32, 33].

§.§ Strong coupling

The AdS / CFT correspondence delivers on a 1974 promise to make strongly coupled $U(N)$ and $SU(N)$ gauge theories
more tractable when $N$ is large [34]. More precisely, the quantum gravity theory in the bulk that is dual to
a CFT becomes increasingly classical as we take $N \rightarrow \infty$ with $\lambda = g^2_{\mathrm{YM}} N$ fixed. This
is known as the large $N$, planar or `tHooft limit. The `tHooft coupling $\lambda$ which only needs to be fixed, is the
parameter that would have to be small for the usual Feynman diagram expansion to be valid. When a series of Feynman
diagrams is written down using powers of $\frac{1}{N}$ instead of $g_{\mathrm{YM}}$, the expansion looks very similar
to that of a closed string theory with coupling $g_{\mathrm{s}}$. For this reason, the identification

\begin{equation}
g_{\mathrm{s}} = \frac{\lambda}{4\pi N} \label{gs-identification}
\end{equation}

appears in the duality [8]. This small string coupling allows a perturbative calculation to be done in AdS
when the field theory on the boundary is strongly coupled. Some of the coupling strengths not covered by this limit
(e.g. large $g_{\mathrm{YM}}$ and $N$) can be explored with the help of string dualities. For instance, a
weak-strong symmetry known as S-duality is often associated with Type IIB string theory [35]. If IIB in AdS is
equivalent to SYM on the boundary, this statement implies that (<ref>) is invariant under
$g_{\mathrm{YM}} \mapsto \frac{1}{g_{\mathrm{YM}}}$. Although such a result could have been discovered through
holography, it was discovered earlier using some of the same evidence that led to the correspondence
[36, 37]. In addition, we should note that even if one believes AdS / CFT, the idea of S-duality holding
for Type IIB string theory is also a conjecture [35].

Using classical gravity to approximate field theories in the strong coupling regime has become the most widely explored
aspect of the AdS / CFT correspondence [38, 39]. For this application, questions about whether string
theory is realized in nature, are irrelevant. We will go through an example of this duality, whereby the interacting
spectrum of (<ref>) can be understood through our seemingly unrelated calculations regarding free field theories
and black holes.

§.§.§ Gauge theory phases

When compactified on a sphere, there are at least four interesting phases posessed by Super Yang-Mills. The
transitions in and out of these phases are gradual, as they must be for a theory with finitely many fields. However,
the transitions may become sharp in the strict $N \rightarrow \infty$ limit. Following our pattern above, we will
give an expression for the entropy of each phase in order of increasing energy.

The first thing we need to know is that in the original version of the correspondence, the bulk geometry is
$AdS_5 \times \mathbb{S}^5$ where the length scale of $AdS_5$ and the radius of $\mathbb{S}^5$ are equal
[1]. This radius, which we will call $L$ is given by the duality prescription as

\begin{equation}
L^4 = 4\pi g_{\mathrm{s}} \alpha^{\prime 2} N \; . \label{l4-identification}
\end{equation}

There is also a radius for the $\mathbb{S}^3$ of the the field theory, which we will call $R$. It is natural to
compare the dimensionless energies of the string theory $EL$ to “some multiple” of the dimensionless energies of the
field theory $ER$. By studying the Klein-Gordon equation (<ref>) for a graviton propagating in
$AdS_5 \times \mathbb{S}^5$, one may show that the second-lowest energy it can have is

\begin{equation}
E_1 = \frac{1}{L} \; . \label{first-excited-graviton}
\end{equation}

While this would not be the case for a general gauge theory, a highly supersymmetric theory like SYM has some excited
state energies that can be computed without the correspondence. This is a harder calculation but an analysis of
chiral primary operators [8, 40] tells us that $E_1 = \frac{1}{R}$ on the field theory side. The multiple
in question is therefore 1 and we will be able to replace $EL$ with $ER$ in what follows.

The low energy behaviour of the bulk is described by a free gas of strings in their worldsheet ground states.
Refering to (<ref>), this is a gas of 128 bosons and 128 fermions — essentially
gravitons and their superpartners. The free field theory result (<ref>) includes a volume $V$, which is
only well defined if there is a clear separation between space and time, i.e.

\begin{equation}
ds^2 = -dt^2 + g_{ij} dx^i dx^j \nonumber
\end{equation}

where $g_{ij}$ is a Riemanian metric. Since the line element

\begin{equation}
ds^2 = -\left ( 1 + \frac{r^2}{L^2} \right ) dt^2 + \left ( 1 + \frac{r^2}{L^2} \right )^{-1} dr^2 + r^2 d\Omega_3^2 + L^2 d\Omega_5^2 \label{ads5-s5}
\end{equation}

is not in this form, the brute force calculation of the partition function would have to start with the Klein-Gordon
equation. Solving the relevant Klein-Gordon equation is certainly a useful exercise. In addition to energy eigenvalues
like (<ref>), it would allow us to derive a bound on the mass that any particle in $AdS$ must
satisfy [41, 42]. However, there is another method that can tell us the appropriate $V$ more quickly.
Performing a conformal transformation on (<ref>), we may turn it into

\begin{equation}
ds^2 = -dt^2 + \left ( 1 + \frac{r^2}{L^2} \right )^{-2} dr^2 + \left ( 1 + \frac{r^2}{L^2} \right )^{-1} \left ( r^2 d\Omega_3^2 + L^2 d\Omega_5^2 \right ) \; . \label{rescaled-ads5-s5}
\end{equation}

The massless version of (<ref>) (called the minimally coupled Klein-Gordon equation) is not
invariant under such a rescaling, but the conformally coupled Klein-Gordon equation is. This equation is

\begin{equation}
\nabla_{\mu} \nabla^{\mu} \phi + \frac{d - 1}{4d} \mathcal{R} \phi = 0 \label{kgr-equation}
\end{equation}

where $\mathcal{R}$ is the Ricci scalar. The volume associated with the (<ref>) metric would
therefore appear in the partition function for this graviton gas in the conformally coupled case.

\begin{eqnarray}
V &=& \int_0^{\infty} \iint_{\mathbb{S}^3 \times \mathbb{S}^5} \left ( 1 + \frac{r^2}{L^2} \right )^{-5} L^5 r^3 \textup{d}\Omega_3 \textup{d}\Omega_5 \textup{d}r \nonumber \\
&=& 15 \omega_3 \omega _5 L^5 \int_0^{\infty} \left ( 1 + \frac{r^2}{L^2} \right )^{-5} r^3 \textup{d}r \nonumber \\
&=& \frac{15}{24} \omega_3 \omega_5 L^9 \nonumber
\end{eqnarray}

Because the difference between minimal and conformal coupling only appears beyond the leading order thermodynamics
[17, 8], it is sufficient to substitute $d = 9$, $s = s^* = 128$ and the $V$ above. This yields

\begin{eqnarray}
S_1 &=& \left ( \frac{10^{10}}{9^9} \frac{15}{24} \frac{9! \omega_3 \omega_5 \omega_9}{(2\pi)^9} 128 \left ( \zeta(10) + \zeta^*(10) \right ) E^9 L^9 \right )^{\frac{1}{10}} \nonumber \\
&=& 10 \left ( 9! \omega_3 \omega_5 \omega_9 \frac{15}{24} \frac{1023}{4} \zeta(10) \right )^{\frac{1}{10}} \left ( \frac{EL}{18\pi} \right )^{\frac{9}{10}} \nonumber \\
&=& 10 \left ( \frac{2728}{9355} \pi^8 \right )^{\frac{1}{10}} \left ( \frac{ER}{9} \right )^{\frac{9}{10}} \label{sym-entropy1}
\end{eqnarray}

as the entropy of the lowest energy phase.

As energy increases, the infinite tower of worldsheet vibrations becomes important and the entropy enters the
Hagedorn regime. Starting with (<ref>), we just need to use the (<ref>) and
(<ref>) identifications to get the entropy for the second phase in terms of gauge theory parameters:

\begin{eqnarray}
S_2 &=& 2\pi \sqrt{2 \alpha^{\prime}} E \nonumber \\
&=& 2\pi \left ( \frac{L^4}{\pi g_s N} \right )^{\frac{1}{4}} E \nonumber \\
&=& 2\pi \left ( \frac{4}{\lambda} \right )^{\frac{1}{4}} ER \; . \label{sym-entropy2}
\end{eqnarray}

As with any proper string theory, the $AdS_5 \times \mathbb{S}^5$ background is not static. It receives a
backreaction from stringy states that becomes more significant as the energy increases. The highest energy phases of
Super-Yang Mills will therefore involve Newton's constant $G$. The formula

\begin{equation}
G_5 L^5 = 8\pi^3 g_{\mathrm{s}}^2 \alpha^{\prime 4} \label{g5-identification}
\end{equation}

is the last piece of the correspondence that we need [43]. We mentioned previously that Einstein's
equations (and Newton's constant in particular) should be the same in all dimensions. Thus, it may seem strange to
refer to a five-dimensional gravitational constant $G_5$. The explanation is that $G_5$ is not the gravitational
constant at all, but rather an illusion created by the presence of compact dimensions. The true $G$ appears as a
prefactor in the Einstein-Hilbert action

\begin{equation}
\mathcal{S} = \frac{1}{16\pi G} \int_{AdS_5 \times \mathbb{S}^5} (\mathcal{R} - 2\Lambda) \sqrt{-g} \textup{d}x \; . \nonumber
\end{equation}

If the radius of the sphere is small enough, a macroscopic observer only sees $AdS_5$. Integrating out the
$\mathbb{S}^5$ and looking at the prefactor once again will tell us the relation between $G_5$ and $G$. Using the fact
that Ricci scalars add for direct product manifolds

\begin{eqnarray}
\mathcal{S} &=& \frac{1}{16\pi G} \int_{AdS_5 \times \mathbb{S}^5} (\mathcal{R}_{AdS_5} - 2\Lambda) \sqrt{-g} \textup{d}x + \frac{1}{16\pi G} \int_{AdS_5 \times \mathbb{S}^5} \mathcal{R}_{\mathbb{S}^5} \sqrt{-g} \textup{d}x \nonumber \\
\mathcal{S}_{\mathrm{eff}} &=& \frac{1}{16\pi G} \int_{\mathbb{S}^5} \sqrt{g_{\mathbb{S}^5}} \textup{d}x \int_{AdS_5} (\mathcal{R}_{AdS_5} - 2\Lambda) \sqrt{-g_{AdS_5}} \textup{d}x + \mathcal{S}_{\mathrm{shift}} \nonumber \\
&=& \frac{6 \omega_6 L^5}{16 \pi G} \int_{AdS_5} (\mathcal{R}_{AdS_5} - 2\Lambda) \sqrt{-g_{AdS_5}} \textup{d}x + \mathcal{S}_{\mathrm{shift}} \; . \nonumber
\end{eqnarray}

We have turned the action into an effective action by evaluating part of it. This makes it clear that
$G_5 = \frac{G}{6 \omega_6 L^5}$. The fact that these dimensionally reduced Newton constants are generally much
smaller than $G$ has led to the hypothesis that the apparent strength of gravity increases when the distance is very
small. Indeed, proponents of extra-dimension phenomenology have discussed the possibility of forming black holes
at the LHC [44, 45, 46].

With these constants in hand, we need to calculate the entropy associated with the geometry that develops in the
third phase. Since entropy increases with energy, it is only logical that our spacetime should eventually achieve the
geometry that has a monopoly on entropy — that of a black hole. The mysterious microstates of this black hole can
be put in a one-to-one correspondence with the well defined microstates of the CFT. When the event horizon $r_0$ first
forms, it is smaller than the radius $L$. It is therefore a good approximation to describe it with the Schwarzschild
solution (<ref>) involving all ten spacetime dimensions. Using the entropy formula (<ref>),

\begin{eqnarray}
S_3 &=& \frac{9 \omega_9}{4G} r_0^8 \nonumber \\
&=& \frac{9 \omega_9}{4G} \mu^{\frac{8}{7}} \nonumber \\
&=& \frac{9}{4} \left ( \frac{G}{\omega_9} \right )^{\frac{1}{7}} \left ( \frac{2\pi E}{9} \right )^{\frac{8}{7}} \nonumber \\
&=& \frac{9}{4} \left ( 6 L^5 G_5 \frac{\omega_6}{\omega_9} \right )^{\frac{1}{7}} \left ( \frac{2\pi E}{9} \right )^{\frac{8}{7}} \nonumber \\
&=& \frac{9}{4} \left ( \frac{3\pi \omega_6}{N^2 \omega_9} \right )^{\frac{1}{7}} \left ( \frac{2\pi EL}{9} \right )^{\frac{8}{7}} \nonumber \\
&=& \frac{9}{4} \left ( \frac{1890}{N^2} \right )^{\frac{1}{7}} \left ( \frac{\pi ER}{9} \right )^{\frac{8}{7}} \; . \label{sym-entropy3}
\end{eqnarray}

As the black hole grows to a radius $r_0 \gg L$, the five small dimensions become negligible allowing us to use the
asymptotically AdS black hole (<ref>). This also makes it a good approximation to say
$\mu = r_0^2 \left ( 1 + \frac{r_0^2}{L^2} \right ) \approx \frac{r_0^4}{L^2}$. Inserting this into (<ref>),

\begin{eqnarray}
S_4 &=& \frac{\omega_4}{G_5} r_0^3 \nonumber \\
&=& \frac{\omega_4}{G_5} \mu^{\frac{3}{4}} \nonumber \\
&=& \left ( \frac{\omega_4}{G_5} \right )^{\frac{1}{4}} \left ( \frac{4\pi EL^2}{3} \right )^{\frac{3}{4}} \nonumber \\
&=& \left ( \frac{2N^2 \omega_4}{\pi L^3} \right )^{\frac{1}{4}} \left ( \frac{4\pi EL^2}{3} \right )^{\frac{3}{4}} \nonumber \\
&=& \pi \sqrt{N} \left ( \frac{4}{3} ER \right )^{\frac{3}{4}} \; . \label{sym-entropy4}
\end{eqnarray}

This matches the behaviour that a conformal theory must have at high energies (<ref>).

We have yet to give estimates for the energy ranges where these phases are valid. Prefactors for these energies would
be suspicious due to the gradual nature of the phase transitions. We will therefore only keep factors that may be
comparable to $N^2$. To determine when the Hagedorn phase becomes important, we should set $E$ to the mass of an
excited string. From (<ref>), we see that this is of order
$\frac{1}{\sqrt{\alpha^{\prime}}} \propto \frac{\lambda^{\frac{1}{4}}}{R}$.

Strings have a characteristic length and a black hole with this length as its horizon radius has a characteristic
energy. When the energy of a string gas exceeds this, it is expected to collapse to the small black hole that we
discussed before. Of course there are some non-black hole geometries having energies of this magnitude
(e.g. a giant graviton [47]) but these are “rare”. This is consistent with the
“heat death” proposal in which a black hole is the inevitable final state of a system that evolves via thermal
fluctuations. An equivalent statement on the CFT side is that as the dimensions of gauge invariant operators increase,
the fraction of them that describe black holes approaches unity [48]. The transition for this black hole
“probably forming” can be found by checking when the Hagedorn entropy becomes comparable to the small black hole
entropy. Setting (<ref>) equal to (<ref>), this energy is of order
$\frac{N^2}{\lambda^{\frac{7}{4}} R}$.

Finally, the midpoint between the small black hole and the large black hole occurs when $r_0 = R$. Expressing the
event horizon radius in terms of the mass, $E \propto \frac{R^7}{G} \propto \frac{R^2}{G_5} \propto \frac{N^2}{R}$.
Putting this together we see that the entropy for strongly coupled SYM is given by

\begin{equation}
S(E) =
\begin{cases}
10 \left ( \frac{2728}{9355} \pi^8 \right )^{\frac{1}{10}} \left ( \frac{ER}{9} \right )^{\frac{9}{10}} & ER \ll \lambda^{\frac{1}{4}} \\
2\pi \left ( \frac{4}{\lambda} \right )^{\frac{1}{4}} ER & \lambda^{\frac{1}{4}} \ll ER \ll \lambda^{-\frac{7}{4}} N^2 \\
\frac{9}{4} \left ( \frac{1890}{N^2} \right )^{\frac{1}{7}} \left ( \frac{\pi ER}{9} \right )^{\frac{8}{7}} & \lambda^{-\frac{7}{4}} N^2 \ll ER \ll N^2 \\
\pi \sqrt{N} \left ( \frac{4}{3} ER \right )^{\frac{3}{4}} & N^2 \ll ER
\end{cases}
\; . \label{sym-entropy}
\end{equation}

Notice that if we were to find the entropy of free Super-Yang Mills by substituting $d = 3$ and $s = s^* = 8N^2$ in
(<ref>), the result would be $\frac{4}{3} \pi \sqrt{N} \left ( ER \right )^{\frac{3}{4}}$. The entropies
differ by a factor of $\left ( \frac{4}{3} \right )^{\frac{1}{4}}$ or equivalently, the free energies differ by a
factor of $\frac{4}{3}$. Writing

\begin{equation}
F = -\frac{1}{6} h(\lambda) \pi^2 N^2 V T^4 \label{sym-free-energy4}
\end{equation}

with $h(0) = 1$ and $\lim_{\lambda \rightarrow \infty} h(\lambda) = \frac{3}{4}$, various authors have studied how
$h$ interpolates between these limits using curvature corrections on the string theory side [49] and loop
diagrams on the field theory side [50, 51]. It was later found that interpolating between weakly coupled
and strongly coupled free energy is not as simple as multiplying by $h$. Corrections to (<ref>)
involving $T^2$ need to be multiplied by different functions of the `tHooft coupling [43].

§.§.§ Plasma balls

The microcanonical entropy of strongly coupled SYM on $\mathbb{S}^3$ (<ref>) is a formula that we will use
repeatedly. Part of its derivation relied on the fact that the theory's dual description involved black holes radiating
a thermal spectrum. The goal of this thesis is to argue for the converse: an arbitrary field theory with an entropy
sufficiently similar to (<ref>) exhibits dynamics that are indicative of black hole formation and
evaporation. There is a large class of field theory solutions, called plasma balls, that have been shown to be of this
type [7]. Most studies of them are numerical [7, 52, 53] but at least one has been
constructed analytically [54].

Consider the canonical phases of Super Yang-Mills found by fixing the temperature instead of the energy. We may
differentiate the entropy in (<ref>) to plot $\beta$ as a function of $E$.

For an inverse temperature like the one shown, the gravity side must choose the geometry that minimizes
the free energy.

At low temperatures, the system must be in the graviton gas phase. One could raise the temperature (lower the
dotted line) all the way to the Hagedorn temperature at which the canonical ensemble ceases to exist but the
most interesting situation occurs for an intermediate value where there is competition between three phases. From
(<ref>), a straightforward calculation of the graviton gas free energy yields

\begin{equation}
F = -\frac{2728}{9355} \pi^8 R^9 T^{10} \label{sym-free-energy1}
\end{equation}

and we have already written the large black hole free energy (<ref>). If we were to calculate the
small black hole free energy in the same way, we would find that it is positive, so (<ref>) and
(<ref>) are the only ones we need. Setting them equal, we find a first order phase transition at

\begin{equation}
T_{\mathrm{D}} = \frac{1}{R} \left ( \frac{9355 N^2}{16368 \pi^5} \right )^{\frac{1}{6}} \; . \label{dec-temperature}
\end{equation}

Were it not for the complication of the internal manifold $\mathbb{S}^5$, this would be the Hawking-Page transition
[28] showing that a sufficiently large black hole in AdS can come to equilibrium with the radiation it emits.
Since the energy, entropy and temperature of a black hole are all known in terms of its event horizon radius $r_0$,

\begin{eqnarray}
F &=& E - TS \nonumber \\
&=& \frac{d \omega_d}{16 \pi G} r_0^{d - 2} \left ( 1 - \frac{r_0^2}{L^2} \right ) \; . \nonumber
\end{eqnarray}

When $r_0 < L$, this is minimized for an $r_0$ that rolls to zero. When $r_0 > L$, this is minimized for as large an
$r_0$ as possible. Substituting $r_0 = L$ into (<ref>), we find

\begin{equation}
T_{\mathrm{HP}} = \frac{d - 1}{2\pi L} \; . \label{hp-temperature}
\end{equation}

This phenomenon on the field theory side has the interpretation of a deconfinement phase transition related to the
scale $R$. As $R \rightarrow \infty$, the temperature (<ref>) vanishes and there is no confinement as
expected for a CFT in Minkowski space. Since the confining theory of greatest physical interest
(quantum chromodynamics) lives in infinite volume, it has little in common with Super Yang-Mills on $\mathbb{S}^3$.
A holographic study of QCD requires one to introduce a scale to SYM in a more drastic way.

This can be done by compactifying some but not all of the directions in a Minkowski CFT. Witten's model [29]
e.g. compactifies SYM on a Scherk-Schwarz circle — $\mathbb{S}^1$ with antiperiodic fermions. Since the
other directions are extended, it is helpful to rewrite (<ref>) so that they manifestly appear as Minkowski space.
Global AdS is large enough for this to be done several times yielding disjoint patches separated by co-ordinate
singularities. A given patch has the following metric known as Poincaré AdS

\begin{equation}
ds^2 = \frac{L^2}{z^2} \left [ -d\tau^2 + dz^2 + dx_i dx^i \right ] \label{poincare}
\end{equation}

where the boundary is located at $z = 0$. The transformation

\begin{eqnarray}
t &=& \arctan \left ( \frac{2L\tau}{L^2 + z^2 + x^2 - \tau^2} \right ) \nonumber \\
r &=& \frac{L}{z} \sqrt{z^2 + \tau^2 + \frac{1}{4L^2} \left ( L^2 + z^2 + x^2 - \tau^2 \right )^2} \label{global-to-poincare} \\
\sin \theta_i \dots \sin \theta_{i - 1} \cos \theta_i &=& \frac{x_i}{\sqrt{z^2 + \tau^2 + \frac{1}{4L^2} \left ( L^2 + z^2 + x^2 - \tau^2 \right )^2}} \nonumber \\
\end{eqnarray}

converts between the global and Poincaré metrics [8]. Some sources assume $r \gg L$ before deriving
(<ref>) in order to write the simpler transformation $z = \frac{L^2}{r}$ [55]. This gives the
false impression that (<ref>) is only approximately equal to a patch of AdS. Applying
(<ref>) to (<ref>) gives another form of the AdS black hole:

\begin{equation}
ds^2 = \frac{L^2}{z^2} \left [ -\left ( 1 - \frac{z^d}{z_0^d} \right ) d\tau^2 + \left ( 1 - \frac{z^d}{z_0^d} \right )^{-1} dz^2 + dx_i dx^i \right ] \; . \label{schwarzschild-poincare}
\end{equation}

If we compactify a spatial direction and let $j$ take on fewer values than $i$, the resulting metric is:

\begin{equation}
ds^2 = \frac{L^2}{z^2} \left [ -\left ( 1 - \frac{z^d}{z_0^d} \right ) d\tau^2 + \left ( 1 - \frac{z^d}{z_0^d} \right )^{-1} dz^2 + dx_j dx^j + d\theta^2 \right ] \; . \label{witten-model}
\end{equation}

An interesting procedure, that would not have worked for any of the previous metrics, is available to be used on
(<ref>). We may Euclideanize, exchange the $\theta$ circle with the $\tau$ circle and switch back to a
Minkowskian signature. This yields a spacetime known as the AdS soliton without us having to solve Einstein's equations

\begin{equation}
ds^2 = \frac{L^2}{z^2} \left [ -d\tau^2 + \left ( 1 - \frac{z^d}{z_0^d} \right )^{-1} dz^2 + dx_j dx^j + \left ( 1 - \frac{z^d}{z_0^d} \right ) d\theta^2 \right ] \label{ads-soliton}
\end{equation}

The deconfinement we saw earlier was a transition between two spacetimes that shared the same boundary: empty AdS and
the AdS black hole. We now have (<ref>) and (<ref>) competing for the same boundary. However,
the horizon position $z_0$ has a very different interpretation in (<ref>) because it causes
the $\theta$ circle to shrink to zero size. A horizon that observers can safely cross changes the signature according
to $(-, +, +, \dots, +, +) \mapsto (+, -, +, \dots, +, +)$. On the other hand, if we allowed $z > z_0$, we would see
$(-, +, +, \dots, +, +) \mapsto (-, -, +, \dots, +, -)$. Because this is a Lorentzian theory, $z_0$ is simply a point
where the spacetime ends. This enduring scale, called the infrared wall, is what leads to a mass gap [56].

[][Separate solutions]     [][Interpolation]
The green sheet is the boundary at $z = 0$. While Poincaré AdS looks like empty space ending on this
boundary, the black hole and soliton solutions are more interesting. Instead of enclosing a region of finite volume,
the horizon is a sheet at $z = z_0$ like that of a black brane. To remind us that there is a
Scherk-Schwarz circle at each point, we have drawn the horizon as a set of black circles inside cylinders that extend
infinitely far to the left. These cylinders become cigar shaped regions in the soliton solution that has the infrared
wall. On the right, we show an interpolation between these behaviours. For $y = \infty$, the IR wall does not exist.
For finite positive $y$ it is behind the horizon and for negative $y$ it is in front.

The Witten model with these two backgrounds is the typical arena for seeing plasma balls. These were conjectured
[7] based on the observation that stable domain walls should exist between solutions like
(<ref>) and (<ref>). Roughly, such a domain wall is constructed by choosing a special
direction $y$ and making $z_0$ a function of $y$. Choosing this function appropriately, the bulk metric can be
made to look like the black hole at $y = \infty$ and the soliton at $y = -\infty$. This solution, which cannot be
found analytically, may look like the one in Figure <ref>. In order for it to be stable, the pressure of
the deconfined phase must be small enough to balance the domain wall tension at some temperature. Intuitive arguments
for this are given in [7] with the final confirmation being numerical.

[][Domain wall]     [][Plasma ball]
On the field theory side, the domain wall shows up as a sudden jump in the energy density. The minimum is
close to zero while the maximum is the expected energy at the deconfinement temperature. The existence of a plasma
ball is the additional assumption that we can make this domain wall “wrap around”. The decay of the resulting
object might look like diffusion governed by the heat equation.

This process can be repeated to further localize the domain wall. Instead of going from the confining vacuum at
$y = -\infty$ to the deconfined plasma at $y = \infty$, one may change the solution so that it goes from confined to
deconfined and back [56]. This can also be done using directions other than $y$ to make the area of the black
hole horizon finite. The black hole made in this way decays in a process that looks like some combination of shrinking
in $y$ and hitting the infrared wall in $z$. The field theory state dual to this black hole near the IR is called a
plasma ball. The dual decay process consists of hadrons leaving the ball and travelling outwards. Because they are
travelling into a confining vacuum, they must be color singlets, leading to a lifetime proportional to $N^2$

When deriving (<ref>), the black holes we discussed were dual to energy eigenstates of SYM on
$\mathbb{S}^3$. In analogy with free theories (whose momentum eigenstates are completely delocalized), these states
have uniform energy density of order $\frac{N^2}{R^4}$ on the whole sphere. The situation is very different for plasma
balls. They have non-uniform energy densities like those in Figure <ref> because their dual black
holes come from an interpolation of gravity saddle points. When we construct plasma balls (in a completely different
manner), we should keep in mind that small compact directions and infrared walls are likely to appear in the
corresponding geometries. As a check, it is interesting to see what goes wrong when trying to construct a plasma ball
for Super-Yang Mills on $\mathbb{S}^3$. In principle, one could prepare a state in the CFT that has Figure
<ref>'s energy density at $t = 0$. Rather than a thermalization process dual to Hawking radiation,
this state's future is governed by the phenomenon of collective oscillations [57]. When quantized on a
sphere, the generators of conformal transformations obey the same commutation relations as traditional raising and
lowering operators. Rewriting them as $L^i_+$ and $L^i_-$, [57] constructed undamped oscillating states
by applying a function of them to a density matrix:
$\rho \mapsto \mathcal{N} g \left ( L^1_{\pm}, \dots, L^d_{\pm} \right ) \rho g^{\dagger} \left ( L^1_{\pm}, \dots, L^d_{\pm} \right )$.
In analogy with coherent states of the harmonic oscillator, explicit functions were given such as the simplest one:

\begin{equation}
g \left ( L^i_+, L^i_- \right ) = e^{\alpha L^i_+ + \beta L^i_-} \; . \label{coherent-state}
\end{equation}

When $\alpha = -\beta^*$, (<ref>) is unitary and $\mathcal{N} = 1$, but [57] gave
normalization constants for other $\alpha$ and $\beta$ as well. Crucially, the AdS isometry dual to
(<ref>) is no more complicated than a boost. This allows its effect on strongly coupled states to be
found with the AdS / CFT correspondence. The example considered in [57] starts with a three-dimensional
spacetime known as the BTZ black hole [58]:

\begin{equation}
ds^2 = -\left ( \frac{r^2}{R^2} - \frac{r_0^2}{R^2} \right ) dt^2 + \left ( \frac{r^2}{R^2} - \frac{r_0^2}{R^2} \right )^{-1} dr^2 + \frac{r^2}{R^2} dx^2 \; . \label{btz-ads}
\end{equation}

As before, the boundary theory dual to this has a uniform stress-energy tensor when compactified on the
$\frac{x}{R} \sim \frac{x}{R} + 2\pi$ sphere:

\begin{eqnarray}
T_{tt} &=& T_{xx} = \frac{r_0^2}{16\pi GR^3} \nonumber \\
T_{tx} &=& T_{xt} = 0 \; . \label{unboosted-tensor}
\end{eqnarray}

This localized lump of energy is certainly something like the $t = 0$ slice of a plasma ball. We are claiming
that if it forms on a sphere, its evolution is too dynamic to be considered that of a plasma ball. Its motion cannot
be undone by a boost because this turns it back into the uniform profile of (<ref>).

We could use $AdS_3$ / $CFT_2$ analogues of (<ref>) and (<ref>) to replace $r_0$
and $G$ with gauge theory expressions above. Because AdS has a boundary, boosting (<ref>) to a velocity of $v$
yields a black hole that oscillates about the origin indefinitely. The CFT state dual to this bouncing black hole has
the stress-energy tensor:

\begin{eqnarray}
T_{tt} &=& T_{xx} = \frac{1}{32\pi G R^3} \left [ \frac{(1 - v^2)(R^2 + r_0^2)}{\left ( v \cos \left ( \frac{t - x}{R} \right ) - 1 \right )^2} + \frac{(1 - v^2)(R^2 + r_0^2)}{\left ( v \cos \left ( \frac{t + x}{R} \right ) - 1 \right )^2} - 2R^2\right ] \nonumber \\
T_{tx} &=& T_{xt} = \frac{v(1 - v^2)(R^2 + r_0^2)}{8\pi G R^3} \left [ \frac{\sin \left ( \frac{t}{R} \right ) \sin \left ( \frac{x}{R} \right ) \left ( v \cos \left ( \frac{t}{R} \right ) \cos \left ( \frac{x}{R} \right ) - 1\right )}{\left ( v \cos \left ( \frac{t - x}{R} \right ) - 1 \right )^2 \left ( v \cos \left ( \frac{t + x}{R} \right ) - 1 \right )^2} \right ] \; . \label{boosted-tensor}
\end{eqnarray}

Thus we see that a valid CFT solution having Figure <ref> as its $t = 0$ energy density is not a
meta-stable ball at a fixed position, but a stable flow with an oscillating position [57].

§ TREATING ENERGY STOCHASTICALLY

Our goal is to model certain features of an interacting field theory, without recourse to what the specific interaction
is. One way to accomplish this is to construct a model that is based on the system's density of states. A thermodynamic
quantity like this is easier to understand than the Hamiltonian because it is only an indication of the
spectrum of the Hamiltonian. The result of our derivation will be an evolution equation for the energy density
at each point in space.

§.§ Main equations

To begin our analysis, we consider a cubic lattice of identical sites in $d$ dimensions and keep track of the number of
units of energy that can be found on each site. Following [2], we will write $\{ n_r \}$ to mean
$n_1$ units on site 1, $n_2$ units on site 2, $n_3$ units on site 3, etc. Each configuration has a certain
probability of being realized. This probability is $P \left ( \{ N_r \}(t) = \{ n_r \} \right )$. Uppercase letters
have been used for random variables with the lowercase versions denoting specific values. However, we will often
shorten this to $P \left ( \{ n_r \} \right )$. In a stochastic process with continuous time, the probabilities as a
function of time obey the master equation [59]:

\begin{equation}
\frac{\partial P \left ( \{ n_r \} \right )}{\partial t} = \sum_{\{ n_r^{\prime} \}} \left [ P \left ( \{ n_r^{\prime} \} \right ) W_{\{ n_r^{\prime} \} \rightarrow \{ n_r \}} - P \left ( \{ n_r \} \right ) W_{\{ n_r \} \rightarrow \{ n_r^{\prime} \}} \right ] \; . \label{master}
\end{equation}

The $W$ quantities which determine the process are called the transition rates and are defined by:

\begin{equation}
W_{\{ n_r \} \rightarrow \{ n_r^{\prime} \}} = \lim_{\tau \rightarrow 0} \frac{P \left ( \{ N_r \}(t + \tau) = \{ n_r^{\prime} \} | \{ N_r \}(t) = \{ n_r \} \right )}{\tau} \; . \nonumber
\end{equation}

To convert (<ref>) into something more concrete, we will make three physical assumptions: local energy
conservation, detailed balance and entropic dominance.

§.§.§ Physical assumptions

Inline with our first assumption, we declare that any transition which is nonlocal or does not conserve energy has a
$W$ value of zero. In the transition rates that are left, energy is transferred between two sites and those sites must
be nearest neighbours. Instead of listing all $\{ n_r^{\prime} \}$ configurations that can be reached from $\{ n_r \}$,
we may simply choose a pair of sites $\left < a, b \right >$ and a number $k$ to transfer between them. The master
equation therefore becomes

\begin{eqnarray}
\frac{\partial P \left ( \{ n_r \} \right )}{\partial t} &=& \sum_{\left < a, b \right >} \sum_{k \neq 0} \left [ P (\dots, n_a + k, n_b - k, \dots) W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)} \right. \nonumber \\
&& \left. - P (\dots, n_a, n_b, \dots) W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)} \right ] \; . \label{physical-master}
\end{eqnarray}

We will not work directly with probabilities, but rather the expectation of a particular site's energy:

\begin{equation}
\bar{n}_c \equiv \left < N_c \right > = \sum_{\{ n_r \}} P \left ( \{ n_r \} \right ) n_c \; . \label{expected-n}
\end{equation}

The next step is to differentiate (<ref>) and substitute (<ref>):

\begin{eqnarray}
\frac{\partial \bar{n}_c}{\partial t} &=& \sum_{\{ n_r \}} n_c \frac{\partial P \left ( \{ n_r \} \right )}{\partial t} \nonumber \\
&=& \sum_{\{ n_r \}} n_c \sum_{\left < a, b \right >} \sum_{k \neq 0} \left [ P (\dots, n_a + k, n_b - k, \dots) W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)} \right. \nonumber \\
&& \left. - P (\dots, n_a, n_b, \dots) W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)} \right ] \; . \nonumber
\end{eqnarray}

Every $P (\dots, n_a + k, n_b - k, \dots) W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)}$ in the sum is a
$P (\dots, n_a, n_b, \dots) W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)}$ for some other $\{ n_r \}$ and the
negative $k$ value. As long as $n_c$ does not appear, these cancel with the same coefficient. We may therefore
let $a = c$ and reindex.

\begin{eqnarray}
\frac{\partial \bar{n}_c}{\partial t} &=& \sum_{\left < b, c \right >} \sum_{k \neq 0} \sum_{\{ n_r \}} \left [ n_c P (\dots, n_b + k, n_c - k, \dots) W_{(n_b + k, n_c - k) \rightarrow (n_b, n_c)} \right. \nonumber \\
&& \left. - n_c P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \right ] \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} \sum_{\{ n_r \}} \left [ (n_c + k) P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \right. \nonumber \\
&& \left. - n_c P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \right ] \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} \sum_{\{ n_r \}} k P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \nonumber \\
&=& \left < \sum_{\left < b, c \right >} \sum_{k \neq 0} k W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \right > \nonumber \\
&\approx& \sum_{\left < b, c \right >} \sum_{k \neq 0} k W_{(\bar{n}_b, \bar{n}_c) \rightarrow (\bar{n}_b - k, \bar{n}_c + k)} \label{equation-for-general-w}
\end{eqnarray}

We have yet to show that it is safe to replace random variables by their expectations in the last step. Since
(<ref>) is still quite general, further work is required to narrow down our choices for $W$.

The configurations $\{ n_r \}$ represent collections of several microstates. Introducing the function $\rho(n_r)$
giving the number of ways for site $r$ to have energy $n_r$, it is easy to count the number of ways in which our
configurations can be realized. There are $\prod_r \rho(n_r)$ microstates with the distribution $\{ n_r \}$. The
most familiar situation in statistical mechanics is that of thermal equilibrium. State $\mu$ in equilibrium is
achieved with probability $P(\mu) = \frac{1}{Z} e^{-\beta E}$. Since this is just one microstate, we should add up a
sufficient number of them to get

\begin{equation}
P \left ( \{ n_r \} \right ) = \frac{1}{Z} e^{-\beta E} \prod_r \rho(n_r) \; . \label{equilibrium-p}
\end{equation}

These are equilibrium probabilities so the master equation should vanish when they are inserted. This condition
provides a constraint on the possible transition rates but we will make use of a stronger condition; the principle of
detailed balance. Detailed balance, which holds for reversible Markov chains, states that all terms in
(<ref>) should separately vanish in equilibrium instead of just the entire sum:

\begin{equation}
P (\dots, n_a + k, n_b - k, \dots) W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)} = P (\dots, n_a, n_b, \dots) W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)} \; . \label{detailed-balance}
\end{equation}

This principle was famously used by Einstein to predict spontaneous emission rates before quantum field theory had
been developed [60]. Substituting (<ref>) into (<ref>), our condition becomes

\begin{equation}
\rho(n_a + k) \rho(n_b - k) W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)} = \rho(n_a) \rho(n_b) W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)} \; . \label{w-constraint}
\end{equation}

There are many solutions to this system of equations but some make more sense than others.

One solution to (<ref>) has $W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)}$ proportional to the number of
final states $\rho(n_a) \rho(n_b)$. This type of transition rate is the one most compatible with the ergodic principle.
When fluctuations are completely thermal, a higher number of final states should be the only thing favouring one
transition over another. We write

\begin{equation}
W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)} \propto C(n_a, n_b) \rho(n_a + k) \rho(n_b - k) \nonumber
\end{equation}

where there can be some additional dependence on $k$. Substituting this into (<ref>), we find

\begin{equation}
C(n_a, n_b) = C(n_a + k, n_b - k) \; . \nonumber
\end{equation}

Using this relation repeatedly, we may set $k$ equal to $n_a$ or $n_b$, telling us that $C$ is only a function of the
total energy. For this to be valid, any configuration must be reachable from the configuration obtained by having one
site shift all of its energy to a neighbour. This is the same as saying that there are no superselection sectors.
We now have

\begin{equation}
W_{(n_a, n_b) \rightarrow (n_a + k, n_b - k)} \propto C \left ( \frac{n_a + n_b}{2} \right ) \rho(n_a + k) \rho(n_b - k) \label{w-solution}
\end{equation}

where we have included the factor of $\frac{1}{2}$ for later convenience. We should note that
$W_{(n_a + k, n_b - k) \rightarrow (n_a, n_b)}$ is not proportional to the number of configurations that have $n_a$
on site $a$ and $n_b$ on site $b$. This would be $\rho (n_a) \rho (n_b) \prod_{r \notin \{a, b\}} \rho(n_r)$. A
transition rate involving all of these factors would be inconsistent with the nearest neighbour logic we have been
using. The only transition we have allowed is one in which site $b$ sends $k$ units of energy to site $a$. If a
transition rate compatible with the ergodic principle also depends linearly on $\rho(n_c)$ for some other site $c$,
this is not a transition rate for $(n_a, n_b) \rightarrow (n_a + k, n_b - k)$ but rather
$(n_a, n_b, n_c) \rightarrow (n_a + k, n_b - k, n_c)$. In other words, site $c$ has itself undergone a transition to
some other internal state that keeps the same energy $n_c$. This amounts to two transition happening in the same
timestep. Moreover, there is no way to tell that this is indeed a nearest neighbour transition paired with a
transition between internal states far away. It could have been site $b$ sending $k$ units of energy to site $c$
followed immediately by $c$ sending $k$ units of energy to site $a$, thus violating locality.

§.§.§ The continuum limit

The equation we wish to build on is

\begin{equation}
\frac{\partial n_c}{\partial t} = \sum_{\left < b, c \right >} \sum_{k \neq 0} k W_{(n_c, n_b) \rightarrow (n_c + k, n_b - k)} \label{non-continuum}
\end{equation}

where the transition rates are given by (<ref>). So far, we have been assuming that the energy and the
spatial co-ordinate both vary by discrete amounts. One way to write this is to have site $c$ labelled by $\textbf{x}$
which means that site $b$ is $\textbf{x} \pm a\textbf{e}$ for some unit vector $\textbf{e}$. To consider a
continuous version of (<ref>), the lattice constant $a$ must approach zero. Additionally, the sum over
$k \neq 0$ must become a sum over $\pm \epsilon$ with $\epsilon$ also going to zero. Our formula (<ref>)

\begin{equation}
W_{(E(\textbf{x}), E(\textbf{x} + a\textbf{e})) \rightarrow (E(\textbf{x}) + \epsilon, E(\textbf{x} + a\textbf{e}) - \epsilon)} = C\left ( \frac{E(\textbf{x}) + E(\textbf{x} + a\textbf{e})}{2} \right ) \rho(E(\textbf{x}) + \epsilon) \rho(E(\textbf{x} + a\textbf{e}) - \epsilon) \label{transition-rates}
\end{equation}

where we use $E$ instead of $n$ to make it clear that we are talking about energy densities that are being incremented
continuously. One should keep in mind that $C$ has units of inverse time in order for $W$ to be a rate. Since our
differential equation for the energy density is now a function of the small parameters $\epsilon$ and $a$, a useful
approximation to it can be derived with a Taylor expansion.

Using (<ref>) in the continuous version of (<ref>), we have

\begin{eqnarray}
\frac{\partial E(\textbf{x})}{\partial t} &=& \epsilon \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} \left [ W_{(E(\textbf{x}), E(\textbf{x} + a\textbf{e})) \rightarrow (E(\textbf{x}) + \epsilon, E(\textbf{x} + a\textbf{e}) - \epsilon)} - W_{(E(\textbf{x}), E(\textbf{x} + a\textbf{e})) \rightarrow (E(\textbf{x}) - \epsilon, E(\textbf{x} + a\textbf{e}) + \epsilon)}\right ] \nonumber \\
&=& \epsilon \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} C\left ( \frac{E(\textbf{x}) + E(\textbf{x} + a\textbf{e})}{2} \right ) \left [ \rho(E(\textbf{x}) + \epsilon) \rho(E(\textbf{x} + a\textbf{e}) - \epsilon) \right. \nonumber \\
&& \left. - \rho(E(\textbf{x}) - \epsilon) \rho(E(\textbf{x} + a\textbf{e}) + \epsilon) \right ] \; . \label{continuum}
\end{eqnarray}

We will define

\begin{equation}
\frac{\partial E}{\partial t} = X(\epsilon, a) = \epsilon \tilde{X}(\epsilon, a) \nonumber
\end{equation}

in which case the relevant Taylor series becomes

\begin{equation}
\frac{\partial E}{\partial t} = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \frac{\epsilon^m a^n}{m! n!} \left. \frac{\partial^{m + n} X}{\partial \epsilon^m \partial a^n} \right |_{\substack{a = 0 \\ \epsilon = 0}} \; . \label{2d-taylor}
\end{equation}

We can see from (<ref>) that $\tilde{X}(0, a) = 0 = \tilde{X}(\epsilon, 0)$, so any term in (<ref>)
that survives, must involve at least three derivatives of $X$: two with respect to $\epsilon$ and one with respect to
$a$. In fact, the number of derivatives we need to take is even higher. Differentiating something like
$\rho(E(\textbf{x} + a\textbf{e}))$ with respect to $a$ would contribute a
$\frac{\partial \rho}{\partial E} \partial_i E e_i$ term inside the sum. If we add up the $i^{\mathrm{th}}$ components
of $\textbf{e}$ where $\textbf{e}$ runs over all positive and negative standard basis vectors, the result is zero.
This means we need at least one more derivative with respect to $a$ and the approximation we seek is:

\begin{equation}
\frac{\partial E}{\partial t} = \frac{1}{4} \epsilon^2 a^2 \left. \frac{\partial^4 X}{\partial \epsilon^2 \partial a^2} \right |_{\substack{a = 0 \\ \epsilon = 0}} \; . \label{e-derivative}
\end{equation}

We will use the abbreviated notation $\rho_{+} = \rho(E(\textbf{x} + a\textbf{e}))$ and
$C_{+} = C \left ( \frac{E(\textbf{x}) + E(\textbf{x} + a\textbf{e})}{2} \right )$ which satisfy

\begin{eqnarray}
\frac{\partial \rho_{+}}{\partial a} &=& e_j \partial_j \rho_{+} \nonumber \\
\frac{\partial C_{+}}{\partial a} &=& \frac{1}{2} e_j \partial_j C_{+} \; . \label{convenient-relations}
\end{eqnarray}

Also, the derivatives with respect to $\epsilon$ are not calculated here but in the appendix. Picking up from where
the appendix leaves off,

\begin{eqnarray}
\left. \frac{\partial^4 X}{\partial \epsilon^2 \partial a^2} \right |_{\substack{a = 0 \\ \epsilon = 0}} &=& 2 \left. \frac{\partial^3 \tilde{X}}{\partial \epsilon \partial a^2} \right |_{\substack{a = 0 \\ \epsilon = 0}} \nonumber \\
&=& 4 \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} \frac{\partial^2}{\partial a^2} \left. \left [ C_{+} \left ( \rho_{+} \frac{\textup{d} \rho}{\textup{d} E} - \rho \frac{\textup{d} \rho_{+}}{\textup{d} E} \right ) \right ] \right |_{a = 0} \nonumber \\
&=& 4 \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} e_i \frac{\partial}{\partial a} \left. \left [ \frac{1}{2} \partial_i C_{+} \left ( \rho_{+} \frac{\textup{d} \rho}{\textup{d} E} - \rho \frac{\textup{d} \rho_{+}}{\textup{d} E} \right ) + C_{+} \left ( \partial_i \rho_{+} \frac{\textup{d} \rho}{\textup{d} E} - \rho \partial_i \frac{\textup{d} \rho_{+}}{\textup{d} E} \right ) \right ] \right |_{a = 0} \nonumber \\
&=& 4 \left [ C \left ( \partial_i \partial_j \rho \frac{\textup{d} \rho}{\textup{d} E} - \rho \partial_i \partial_j \frac{\textup{d} \rho}{\textup{d} E} \right ) -\partial_i C \left ( \rho \partial_j \frac{\textup{d} \rho}{\textup{d} E} - \partial_j \rho \frac{\textup{d} \rho}{\textup{d} E} \right ) \right ] \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} e_i e_j \nonumber \\
&=& 8 \left [ C \left ( \partial_i \partial_i \rho \frac{\textup{d} \rho}{\textup{d} E} - \rho \partial_i \partial_i \frac{\textup{d} \rho}{\textup{d} E} \right ) -\partial_i C \left ( \rho \partial_i \frac{\textup{d} \rho}{\textup{d} E} - \partial_i \rho \frac{\textup{d} \rho}{\textup{d} E} \right ) \right ] \; . \label{expression-before-log}
\end{eqnarray}

It is not immediately obvious but (<ref>) simplifies to a more compact expression involing a
logarithm. If we expand

\begin{eqnarray}
-C \rho^2 \partial_i \frac{\textup{d} \log \rho}{\textup{d} E} &=& -C \rho^2 \partial_i \frac{\frac{\textup{d} \rho}{\textup{d} E}}{\rho} \nonumber \\
&=& C \rho^2 \frac{\partial_i \rho \frac{\textup{d} \rho}{\textup{d} E} - \rho \partial_i \frac{\textup{d} \rho}{\textup{d} E}}{\rho^2} \nonumber \\
&=& C \left ( \partial_i \rho \frac{\textup{d} \rho}{\textup{d} E} - \rho \partial_i \frac{\textup{d} \rho}{\textup{d} E} \right ) \; , \nonumber
\end{eqnarray}

we get something whose $\partial_i$ derivative is (<ref>). This shows that

\begin{equation}
\frac{\partial E}{\partial t} = -2 \epsilon^2 a^2 \partial_i \left ( C(E) \rho^2(E) \partial_i \frac{\textup{d} \log \rho(E)}{\textup{d} E} \right ) \; . \label{hypercorrect-pde}
\end{equation}

Checking the dimensions of (<ref>), the left hand side is an energy density over a time. On the right
hand side, we have an energy density in the form of $\epsilon$ because the other $\epsilon$ cancels with the
$\textup{d}E$. We also have an inverse time because the function $C$ had inverse time units. The $a^2$ cancels with
the two spatial derivatives. From now on, we will drop unknown dimensionful parameters by absorbing them into the time.
The main differential equation of our model is

\begin{equation}
\frac{\partial E}{\partial t} = - \partial_i \left ( C \rho^2 \partial_i \frac{\textup{d} \log \rho}{\textup{d} E} \right ) \label{main-pde}
\end{equation}

in which $C$ is assumed to be a dimensionless function. Common choices for it will be $1$ and $\rho^{-2}$.

§.§ Interesting features

Our continuum limit equation has particularly nice things to say about a system with microcanonical phases like
(<ref>). At least two of these phases only appear at energies that are large compared to the spatial volume.
After considering some insights in [2] concerned with the case $\frac{\partial E}{\partial t} = 0$, we
will see that large energies are required to even trust the model at a basic level.

§.§.§ Static situations

A special role is played by the density of states whose logarithm is linear in $E$. This is the Hagedorn density of
states that we saw appearing in string theory and Super Yang-Mills: $\rho(E) \propto e^{\beta_{\mathrm{H}} E}$. In this
case $\frac{\textup{d} \log \rho(E)}{\textup{d}E} = \beta_{\mathrm{H}}$, a constant. The
$\partial_i$ acting on this constant will set the left hand side of (<ref>) to zero. Under Hagedorn behaviour,
the energy distribution $E(\textbf{x}, t)$ does not change with time.

The gauge theories with holographic duals have a Hagedorn regime as well as other phases. As stated before, we expect
$\log \rho(E) \propto E^{\alpha}$ where we could have $\alpha < 1$, $\alpha > 1$ or $\alpha = 1$ depending on the
energy. Since the dynamics are frozen with a purely Hagedorn density of states, we expect changes in the energy to
take place very slowly if $\alpha = 1$ is the widest phase. The $\alpha = 1$ phase can equivalently be described as the
energy range for which the inverse temperature $\beta(E)$ is flat.

Using the fact that $\log \rho(E)$ is the microcanonical entropy, we can rewrite our main equation in terms of
$\beta(E)$ as well:

\begin{equation}
\frac{\textup{d} \log \rho(E)}{\textup{d} E} = \frac{\textup{d} S(E)}{\textup{d} E} = \beta(E) \nonumber
\end{equation}

and our equation becomes

\begin{equation}
\frac{\partial E}{\partial t} = - \partial_i \left ( C \rho^2 \partial_i \beta \right ) \; . \label{beta-pde}
\end{equation}

The phases can be characterized by whether $\beta$ is decreasing ($\alpha < 1$), increasing ($\alpha > 1$) or
neither ($\alpha = 1$).

With a Hagedorn density of states, $\frac{\partial E}{\partial t}$ vanishes for any energy distribution. Conversely
with a uniform energy distribution, $\frac{\partial E}{\partial t}$ vanishes for any density of states. This
equilibrium distribution may be stable or unstable depending on the phase we are in. We will decompose the energy as

\begin{equation}
E(\textbf{x}, t) = E_0 + \tilde{E}(\textbf{x}, t) \nonumber
\end{equation}

where $\tilde{E}$ is small, allowing us to keep only one power of it in the PDE (<ref>). First,

\begin{eqnarray}
\partial_i \beta(E) &\approx& \partial_i \left ( \beta(E_0) + \tilde{E} \frac{\textup{d} \beta(E_0)}{\textup{d}E} \right ) \nonumber \\
&=& \frac{\textup{d} \beta(E_0)}{\textup{d}E} \partial_i \tilde{E} \; . \nonumber
\end{eqnarray}

This expression with one power of $\tilde{E}$ is multiplied by $C(E) \rho^2(E)$. A first order expansion of
$C(E) \rho^2(E)$ would give an overall result that is second order in $\tilde{E}$ so we only expand it to zeroth order:

\begin{equation}
\frac{\partial \tilde{E}}{\partial t} \approx - C(E_0) \rho^2(E_0) \frac{\textup{d} \beta(E_0)}{\textup{d}E} \partial_i \partial_i \tilde{E} \; . \label{linear-beta-pde}
\end{equation}

This is either the heat equation or the reverse heat equation depending on whether the overall coefficient is negative
or positive. The sign of $\frac{\textup{d} \beta_0}{\textup{d}E}$ is what matters because $C$ and $\rho$ are positive
functions. Agreeing with our earlier intuition about the entropic dynamics of energy, we have the following three

* $\frac{\textup{d}\beta}{\textup{d}E} < 0$ is a decreasing inverse temperature, a concave entropy and a log-concave
density of states. It leads to diffusion or inhomogeneities that decrease with time due to the heat equation.

* $\frac{\textup{d}\beta}{\textup{d}E} > 0$ is an increasing inverse temperature, a convex entropy and a log-convex
density of states. It leads to clustering or inhomogeneities that increase with time due to the reverse heat equation.

* $\frac{\textup{d}\beta}{\textup{d}E} = 0$ is a constant inverse temperature, a linear entropy and a Hagedorn density
of states. It leads to static behaviour.

Understanding the detailed properties of the diffusion and to a lesser extent the clustering caused by this PDE will be
the focus of the next chapter.

§.§.§ Mean-field variances

A loose end in this chapter has been the assumption that we may work only with expected values in
(<ref>). In general, mean-field approximations may be used on quantities that have a small
variance. An energy with a small variance is also one of the desired features of our model. After all, the model is
an attempt at connecting the excitations of field theory degrees of freedom to Einstein gravity, something that is
completely deterministic.

If variances are initially small, we want to make sure that they grow slowly so that our model stays valid for a long
time. Just as we derived an expression for $\frac{\partial \bar{n}_c}{\partial t}$ from the master equation, we can
repeat the calculation for $\frac{\partial \bar{n^2}_c}{\partial t}$.

\begin{eqnarray}
\frac{\partial \bar{n^2}_c}{\partial t} &=& \sum_{\{ n_r \}} n^2_c \frac{\partial P \left ( \{ n_r \} \right )}{\partial t} \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} \sum_{\{ n_r \}} \left [ n^2_c P (\dots, n_b + k, n_c - k, \dots) W_{(n_b + k, n_c - k) \rightarrow (n_b, n_c)} \right. \nonumber \\
&& \left. - n^2_c P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \right ] \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} \sum_{\{ n_r \}} \left [ ((n_c + k)^2 - n^2_c) P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \right ] \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} \sum_{\{ n_r \}} (2kn_c + k^2) P (\dots, n_b, n_c, \dots) W_{(n_b, n_c) \rightarrow (n_b - k, n_c + k)} \nonumber \\
&\approx& \sum_{\left < b, c \right >} \sum_{k \neq 0} (2kn_c + k^2) W_{(\bar{n}_b, \bar{n}_c) \rightarrow (\bar{n}_b - k, \bar{n}_c + k)} \label{v-equation-for-general-w}
\end{eqnarray}

If we combine (<ref>) with (<ref>),

\begin{eqnarray}
\sigma^2_c &=& \bar{n^2}_c - \bar{n}_c^2 \nonumber \\
\frac{\partial \sigma^2_c}{\partial t} &=& \frac{\partial \bar{n^2}_c}{\partial t} - 2\bar{n}_c \frac{\partial \bar{n}_c}{\partial t} \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} k^2 W_{(\bar{n}_b, \bar{n}_c) \rightarrow (\bar{n}_b - k, \bar{n}_c + k)} \nonumber \\
&=& \sum_{\left < b, c \right >} \sum_{k \neq 0} k^2 C \left ( \frac{\bar{n}_b + \bar{n}_c}{2} \right ) \rho(\bar{n}_b - k) \rho(\bar{n}_c + k) \nonumber
\end{eqnarray}

where we have substituted the $W$ solution (<ref>). This equation for the variance can be examined in the
continuum limit and the key is that we do not need as many orders as $\epsilon^2 a^2$ in the subsequent Taylor
expansion. The continuum limit is

\begin{eqnarray}
\frac{\partial \sigma^2(\textbf{x})}{\partial t} &=& \epsilon^2 \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} \left [ W_{(E(\textbf{x}), E(\textbf{x} + a\textbf{e})) \rightarrow (E(\textbf{x}) + \epsilon, E(\textbf{x} + a\textbf{e}) - \epsilon)} + W_{(E(\textbf{x}), E(\textbf{x} + a\textbf{e})) \rightarrow (E(\textbf{x}) - \epsilon, E(\textbf{x} + a\textbf{e}) + \epsilon)}\right ] \nonumber \\
&=& \epsilon^2 \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} C\left ( \frac{E(\textbf{x}) + E(\textbf{x} + a\textbf{e})}{2} \right ) \left [ \rho(E(\textbf{x}) + \epsilon) \rho(E(\textbf{x} + a\textbf{e}) - \epsilon) \right. \nonumber \\
&& \left. + \rho(E(\textbf{x}) - \epsilon) \rho(E(\textbf{x} + a\textbf{e}) + \epsilon) \right ] \label{v-continuum}
\end{eqnarray}

where the positive sign is due to the fact that we have $k^2$ instead of $k$. Again, define

\begin{equation}
\frac{\partial \sigma^2}{\partial t} = X(\epsilon, a) = \epsilon^2 \tilde{X}(\epsilon, a) \; . \nonumber
\end{equation}

When Taylor expanding $X$, we need at least two derivatives with respect to $\epsilon$ because of the
$\epsilon^2$ prefactor. This is all we need because $\tilde{X}(0, 0)$ is nonzero.

\begin{eqnarray}
\frac{\partial \sigma^2}{\partial t} &=& \frac{1}{2} \epsilon^2 \left. \frac{\partial^2 X}{\partial \epsilon^2} \right |_{\substack{a = 0 \\ \epsilon = 0}} \nonumber \\
&=& \epsilon^2 \tilde{X}(0, 0) \nonumber \\
&=& 2 \epsilon^2 \sum_{\textbf{e} \in \{ \pm \textbf{e}_1, \dots, \pm \textbf{e}_d \}} C \rho^2 \nonumber \\
&=& 4d \epsilon^2 C \rho^2 \label{variance-pde}
\end{eqnarray}

Unlike (<ref>), it is not the same unknown function that appears on the left and right hand sides of
(<ref>). One must first solve for $E(\textbf{x}, t)$ in order to solve for $\sigma(\textbf{x}, t)$.

Since the expression for $\frac{\partial E}{\partial t}$ has a small prefactor of $a^2$ while that for
$\frac{\partial \sigma^2}{\partial t}$ does not, it would seem that the rate of change of the variance is
parametrically larger, something we wished to avoid. However, it does not make sense to compare these quantities
directly. The energy has the same units as its standard deviation so we should compare
$\frac{\partial \sigma}{\partial t}$ to $\frac{\partial E}{\partial t}$ or more conveniently,
$\frac{\partial \sigma^2}{\partial t}$ to $\frac{\partial E^2}{\partial t}$.

\begin{equation}
\frac{\frac{\partial \sigma^2}{\partial t}}{\frac{\partial E^2}{\partial t}} = \frac{\frac{\partial \sigma^2}{\partial t}}{2E \frac{\partial E}{\partial t}} \label{need-high-energy}
\end{equation}

Factors of $\epsilon^2$ in the numerator and denominator cancel leaving $E a^2$ in the denominator. This tells us that
such a ratio of derivatives can indeed be small if the energy is large enough. In other words, a PDE like
(<ref>) can be trusted to model high energy phenomena. Thinking about gravity, this includes the extreme
environments of black holes but not the everyday motion of test particles around them. On a more practical level, it
would be difficult to even write down the equation (<ref>) if we were concerned with it holding for low
energies. For the field theories we are interested in, only asymptotic expressions are known for the density of states.
Even for situations in which the exact number of states is known for all energies, this $\rho(E)$ is not continuous.

§ NONLINEAR DIFFUSION

We now cover some of the properties of equations like (<ref>) that are known analytically. The main
assumption we will use throughout this chapter is that $\beta$ is weakly decreasing, i.e.
$\frac{\textup{d}\beta}{\textup{d}E} \leq 0$. This is a slight departure from the strongly coupled gauge theory result
as shown in Figure <ref>.

[][Sharp phases]
[][Smooth phases]
[][Purely diffusive]
To emphasize the four phases expected in holographic gauge theories, we have drawn a piecewise
$\beta(E)$ function. It is more realistic to expect an approximation to this function that is differentiable
whenever $E > 0$. The function plotted in the middle figure is still difficult to work with because there is a small
range of energies for which it is increasing. Assuming that this phase is negligible is the best way to predict
the behaviour of our nonlinear PDE.

If $C(E) = \rho^{-2}(E)$,

\begin{equation}
\frac{\partial E}{\partial t} = - \partial_i \left ( C(E) \rho^2(E) \partial_i \beta(E) \right ) \label{non-filtration-form}
\end{equation}

\begin{equation}
\frac{\partial E}{\partial t} = - \partial_i \partial_i \tilde{\beta}(E) \label{filtration-form}
\end{equation}

where $\tilde{\beta}(E) = \beta(E)$. We will in fact consider (<ref>) regardless of $C$. This is
because (<ref>) is always (<ref>) for some other $\tilde{\beta}$. Simply
define $\tilde{\beta}^{\prime} = C \rho^2 \beta^{\prime}$. Because $C$ and $\rho$ are positive functions,
$\tilde{\beta}$ is decreasing if and only if $\beta$ is. Therefore we will drop the tilde and write

\begin{equation}
\frac{\partial E}{\partial t} = - \Delta \beta(E) \label{filtration-equation}
\end{equation}

from now on. Of course the $\beta(E)$ in (<ref>) no longer has to be of the form plotted in
Figure <ref> but this will be unimportant for most of the results that follow.

§.§ Basic properties on a bounded domain

We will start by assuming that $x \in \Omega$ where $\Omega$ is an open, bounded domain in $\mathbb{R}^d$. This allows
the initial energy density $E(x, 0)$ to be integrable without decaying to zero. As shown in Figure
<ref>, a potential problem with (<ref>) is thus avoided because the low energies for
which $\beta$ diverges are not realized.

[][Energy distribution]
[][Realized range of energies]
Resricting the size of the domain allows us to avoid the vanishingly small energies for which our PDE
no longer applies.

The initial condition will be denoted $E_0(x)$ reflecting the fact that we choose an energy distribution to evolve
forward in time, i.e. $E_0$ is an input to the Cauchy problem.

§.§.§ Conservation of energy

Energy conservation was one of the properties we demanded from the start. As with any Cauchy problem, whether or not
energy is conserved depends on the boundary conditions. As with the heat equation, Neumann boundary conditions
are the appropriate ones to consider. When discussing these mathematical results, we will use “mass” to refer to
the integral of $E(x, t)$ over space rather than a gap in the spectrum of a field theory.

If $E$ solves

\begin{equation}
\begin{dcases}
\frac{\partial E}{\partial t} (x, t) = - \Delta \beta(E(x, t)) & (x, t) \in \Omega \times (0, \infty) \\
E(x, 0) = E_0(x) & x \in \Omega \\
\nabla E(x, t) \cdot n = 0 & (x, t) \in \partial \Omega \times (0, \infty)
\end{dcases} \nonumber
\end{equation}

then $M(t) = \int_{\Omega} E(x, t) \textup{d}x$ is constant.

We can show that the derivative of $M$ is zero using Green's first identity.

\begin{eqnarray}
\frac{\textup{d} M}{\textup{d} t} &=& \int_{\Omega} \frac{\partial E}{\partial t} \textup{d}x \nonumber \\
&=& \int_{\Omega} \Delta \beta(E) \textup{d}x \nonumber \\
&=& \int_{\Omega} \nabla \cdot \left ( \beta^{\prime}(E) \nabla E \right ) \textup{d}x \nonumber \\
&=& \int_{\Omega} \beta^{\prime}(E) \Delta E + \nabla \left ( \beta^{\prime}(E) \right ) \cdot \nabla E \textup{d}x \nonumber \\
&=& \int_{\partial \Omega} \beta^{\prime}(E) \left ( \nabla E \cdot n \right ) \textup{d}S_x \nonumber \\
&=& 0 \; . \nonumber
\end{eqnarray}

Dirichlet boundary conditions would not lead to conserved energy unless we finely tuned $\beta^{\prime}$ to be zero
on the boundary.

§.§.§ The maximum principle

Perhaps the most ubiquitous tool in the study of elliptic and parabolic equations is the maximum principle. Although
it is sometimes introduced as a tool for studying linear equations, many nonlinear versions of it have appeared over
the years [61]. Our proof of a suitable maximum principle will be very similar to the one in [62].

Suppose that $\frac{\partial u}{\partial t} \geq - \Delta \beta(u)$ and
$\frac{\partial v}{\partial t} \leq - \Delta \beta(v)$ for a monotonically decreasing $\beta$. If
$u(x, 0) > v(x, 0)$ for all $x \in \Omega$, there does not exist a spacetime point
$(x_0, t_0) \in \Omega \times (0, \infty)$ for which $u(x_0, t_0) < v(x_0, t_0)$.

First suppose that $w \in C^2(\Omega \times (0, \infty))$ is initially non-negative but not always non-negative. Then
there must exist some point $(x_0, t_0) \in \Omega \times (0, \infty)$ such that $w(x_0, t_0) < 0$. We can let $x_0$
be the position of the minimum of $w(\cdot, t_0)$ so that $\nabla w(x_0, t_0) = 0$ and $\Delta w(x_0, t_0) \geq 0$
are also satisfied. If $t_0$ is the first time such a point occurs, $\frac{\partial w}{\partial t}(x_0, t_0) \leq 0$.

Now let $w(x, t) = e^{-At} \left [ \beta(v(x, t)) - \beta(u(x, t)) \right ]$ where $A$ is some positive constant.
Saying that $w$ is non-negative is the same as saying that $u \geq v$ because $\beta$ is monotone. Therefore if
$u \geq v$ initially but not always, the above says that there must be a point
$(x_0, t_0) \in \Omega \times (0, \infty)$ such that:

* $\beta(u(x_0, t_0)) > \beta(v(x_0, t_0))$

* $\beta^{\prime}(u(x_0, t_0)) \nabla u(x_0, t_0) = \beta^{\prime}(v(x_0, t_0)) \nabla v(x_0, t_0)$

* $-\Delta \beta(u(x_0, t_0)) \geq -\Delta \beta(v(x_0, t_0))$

* $\beta^{\prime}(u(x_0, t_0)) \frac{\partial u}{\partial t}(x_0, t_0) - \beta^{\prime}(v(x_0, t_0)) \frac{\partial v}{\partial t}(x_0, t_0) \geq A \left [ \beta(u(x_0, t_0)) - \beta(v(x_0, t_0)) \right ]$

Since $u$ is a supersolution and $v$ is a subsolution, $\frac{\partial u}{\partial t} \geq -\Delta \beta(u)$ and
$\frac{\partial v}{\partial t} \leq -\Delta \beta(v)$ can be combined to give
$\frac{\partial}{\partial t}(u - v) + \Delta \left [ \beta(u) - \beta(v) \right ] \geq 0$. On the other hand, our four
conditions above can be combined into an inequality that contradicts this telling us that $u$ cannot drop below $v$ if
$u(x, 0) \geq v(x, 0)$.

First we use the fact that $\Delta \beta(u(x_0, t_0)) \leq \Delta \beta(v(x_0, t_0))$ to write
$\frac{\partial}{\partial t}(u(x_0, t_0) - v(x_0, t_0)) + \Delta \left [ \beta(u(x_0, t_0)) - \beta(v(x_0, t_0)) \right ] \leq \frac{\partial}{\partial t}(u(x_0, t_0) - v(x_0, t_0))$.
We must now show that the difference of the time derivatives is less than zero. To do this, we note that at least
one of the following must be true:

\begin{eqnarray}
\beta^{\prime}(u(x_0, t_0)) \frac{\partial u}{\partial t}(x_0, t_0) - \beta^{\prime}(v(x_0, t_0)) \frac{\partial v}{\partial t}(x_0, t_0) \leq \beta^{\prime}(u(x_0, t_0)) \frac{\partial}{\partial t}(u(x_0, t_0) - v(x_0, t_0)) \nonumber \\
\beta^{\prime}(u(x_0, t_0)) \frac{\partial u}{\partial t}(x_0, t_0) - \beta^{\prime}(v(x_0, t_0)) \frac{\partial v}{\partial t}(x_0, t_0) \leq \beta^{\prime}(v(x_0, t_0)) \frac{\partial}{\partial t}(u(x_0, t_0) - v(x_0, t_0)) \; . \nonumber
\end{eqnarray}

The first one is true if $\frac{\partial v}{\partial t}(x_0, t_0)$ and
$\beta^{\prime}(v(x_0, t_0)) - \beta^{\prime}(u(x_0, t_0))$ have the same sign. The second one is true if they have
different signs. Use $\beta^{\prime}_0$ to denote whatever prefactor appears in the correct statement, either
$\beta^{\prime}(u(x_0, t_0))$ or $\beta^{\prime}(v(x_0, t_0))$. The key is that this is a negative constant. The
last inequality in our list of four can turn this into

\begin{eqnarray}
\beta^{\prime}_0 \frac{\partial}{\partial t} (u(x_0, t_0) - v(x_0, t_0)) &\geq& \beta^{\prime}(u(x_0, t_0)) \frac{\partial u}{\partial t}(x_0, t_0) - \beta^{\prime}(v(x_0, t_0)) \frac{\partial v}{\partial t}(x_0, t_0) \nonumber \\
&\geq& A \left [ \beta(u(x_0, t_0)) - \beta(v(x_0, t_0)) \right ] \; . \nonumber
\end{eqnarray}

Dividing through by $\beta^{\prime}_0$, we get

\begin{equation}
\frac{\partial}{\partial t} (u(x_0, t_0) - v(x_0, t_0)) \leq \frac{A}{\beta^{\prime}_0} \left [ \beta(u(x_0, t_0)) - \beta(v(x_0, t_0)) \right ] \; . \nonumber
\end{equation}

Using the first inequality in the list of four, we see that this is less than zero, completing the proof.

Even though this could still be generalized further [63], it is already more general than the maximum
principle. To get the maximum principle, let $v$ be a solution (a special case of a subsolution) and $u$ be
$\max \{ v(x, 0) : x \in \bar{\Omega} \}$. Since $u$ is a constant, it is also a solution and therefore a supersolution.
The theorem above now tells us that $u$ must continue to upper bound $v$ at all later times which means the maximum
of $v$ decreases with time. As always, this is equivalent to the minimum principle which states that the minimum
increases with time. We have already seen in (<ref>) that our PDE turns into the heat equation when
the energy distribution is close to uniform. The maximum principle tells us that our PDE is similar to the heat
equation in some respects for general energy distributions as well.

In addition to talking about the value of the maximum of $E$, we can gain some information about its position.
If our initial condition $E_0$ is spherically symmetric, our equation gives

\begin{eqnarray}
\frac{\partial E}{\partial t} &=& -r^{1-d} \frac{\partial}{\partial r} \left ( r^{d-1} \frac{\partial \beta(E)}{\partial r} \right ) \nonumber \\
\frac{\partial}{\partial t} \frac{\partial E}{\partial r} &=& -\frac{\partial}{\partial r} \left [ r^{1-d} \frac{\partial}{\partial r} \left ( r^{d-1} \frac{\partial \beta(E)}{\partial r} \right ) \right ] \nonumber \\
&=& -\frac{\partial}{\partial r} \left [ \frac{d-1}{r} \beta^{\prime}(E) \frac{\partial E}{\partial r} + \beta^{\prime\prime}(E) \left ( \frac{\partial E}{\partial r} \right )^2 + \beta^{\prime}(E) \frac{\partial^2 E}{\partial r^2} \right ] \nonumber \\
&=& \frac{d-1}{r^2} \beta^{\prime}(E) \frac{\partial E}{\partial r} - \frac{d-1}{r} \beta^{\prime\prime}(E) \left ( \frac{\partial E}{\partial r} \right )^2 - \frac{d-1}{r} \beta^{\prime}(E) \frac{\partial^2 E}{\partial r^2} - \beta^{\prime\prime\prime}(E) \left ( \frac{\partial E}{\partial r} \right )^3 \nonumber \\
&& - 3\beta^{\prime\prime}(E) \frac{\partial E}{\partial r} \frac{\partial^2 E}{\partial r^2} - \beta^{\prime}(E) \frac{\partial^3 E}{\partial r^3} \; . \nonumber
\end{eqnarray}

Now suppose that $E_0$ achieves its maximum at the origin and has no other local minima or maxima. In other words
$\frac{\partial E_0}{\partial r} < 0$ for $r \neq 0$. If $\frac{\partial E}{\partial r}$ were to become positive away