$\frac{dp}{dt}\,=\,-\gamma\,p\,,\,\,\,\,\,\text{and}\,\,\,\,\,\gamma=\frac{2\,\pi\sqrt{\lambda}T^{2}}{m}$
(4.124)

One can see that the fluctuation-dissipation theorem, which connects the drag
and diffusion of a probe particle in a thermal medium [42], is exactly
respected:

$\displaystyle D=\frac{T}{\gamma m}\,.$ (4.125)

####  4.4 Generalising to AdSd-Schwarzschild

For the off-mass-shell light quark case – as will be seen in subsection (5.3)
– generalising the diffusion coefficient to AdSd-Schwarzschild717171Remember
that a general $d$ dimensions in the bulk theory corresponds to a
($d-1$)-dimensional thermal plasma in the boundary theory. relies on the
observation that the behaviour of an arbitrary virtuality light quark at
asymptotically late times is encoded in the small virtuality light quark case.
Since $s^{2}(t)$ in the latter case can easily be calculated for any
dimensions $d\geq 3$ in AdSd-Schwarzschild, the diffusion coefficient
$D_{\text{LQ}}^{\text{AdS${}_{d}$}}$ can be determined. Unfortunately, this
method can not be applied to the on-mass-shell heavy quark case727272To talk
about the small virtuality/small mass limit of a massive, heavy probe quark,
would be nonsensical.. The avenue left is to determine, in AdSd-Schwarzschild,
the friction coefficient of the medium plasma and relate this – via the
Einstein-Sutherland equation (Eq.(3.18)) – to the diffusion coefficient. This
is precisely what is done in section (6), where the friction coefficient in
AdSd-Schwarzschild is found by first calculating the drag force on an
infinitely massive, heavy quark moving through the thermal plasma with a
constant velocity737373Implies that the probe quark is under the influence of
an external force. in the bulk theory. As will be seen, the diffusion
coefficient extracted from this method747474These calculations predominantly
follow the work of [22, 35]., agrees exactly with the the diffusion
coefficient given in Eq.(4.123) for $d=3$.

###  5 Analysing Light Quark Brownian Motion

A light quark probe in a thermal medium on the boundary starts at $t=0$ as an
off-mass-shell particle, radiates energy as it travels through the thermal
medium and finally stops radiating as it becomes an on-mass-shell particle.
Dual to this light quark in the boundary theory, a test string in the bulk can
be thought of as half the yo-yo string in $\mathbb{R}^{1,1}$. The latter is a
string which, at a time $t=0$, stretches a length of $2L$ along the x-axis; as
time progresses the string shrinks till, at a time $t=L$, it is a point; then
it begins to expand again and the process repeats. The yo-yo solution is a
well-studied classical solution of the open string [100]. There is plenty of
recent work [101, 102, 103, 104, 105, 15] modelling gluons and light quarks as
half the initial yo-yo string contracting to a point after a time $t$, in
order to calculate energy loss (see also [106, 52] for other works using this
string set-up). This idea of a falling string is precisely what is used in
order to study the off-mass-shell light quark in this article. Such a string
set-up is refered to, following the convention of [52], as the Limp Noodle.

If the test string is placed in an AdS-Schwarzschild background, the presence
of the black hole results in thermal fluctuations in the transverse $X^{I}$
directions on the string. The resultant random movement of the string endpoint
on the boundary corresponds to the light quark undergoing Brownian motion.
This set-up is depicted in figure (3).

Figure 3:  A fundamental falling open string of length $\ell_{0}$ used as a
probe in an AdS black hole background. Initially, at $t=t_{i}$, the string
starts at the boundary of the anti-de Sitter spacetime and hangs down to a
stretched horizon ($r_{s}\,=\,(1+\epsilon)\,r_{H}$ where $0<\epsilon\ll 1$)
placed just above the Schwarzschild black hole horizon. As time evolves
$t_{i}<t_{2}<t_{3}$, the string endpoint at the boundary is released and the
string collapses. Its trajectory as it falls (grey, dashed line) is predicted
by [101, 102, 103, 104, 105, 15].

An initial static set-up of the Limp Noodle such that one of its endpoints is
held fixed at the boundary for an asymptotically long time, allowing the
string to be fully populated by thermal modes emanating from the black hole
horizon, is then equivalent to the leading order static stretched string for
the heavy quark described in subsection (4.2). The two situations diverge once
the boundary endpoint of the string is released, and the endpoint starts to
fall at the local speed of light. Information, however, is also restricted to
travel at the local speed of light. Hence, all parts of the string below the
falling boundary endpoint can not know whether the boundary endpoint has
remained fixed (as in the case of the heavy quark’s test string) or been
released to fall until the endpoint crashes through it. Intuitively, it is
therefore expected that all parts of the string except the falling boundary
endpoint are described by a solution (embedding functions) identical to
Eq.(4.33) in $\mathbb{R}^{1,1}$ and Eq.(4.51) in AdS3-Schwarzschild, while the
stretched endpoint falls at the local speed of light. This will be confirmed
in the following subsection.

####  5.1 Leading Order String Behaviour

The basics of string theory discussed in subsection (4.1) – leading to the
Virasoro constraints (Eq.(4.19)) and the string equations of motion
(Eq.(4.20)) – hold for the light quark’s test string (Limp Noodle). In order
to solve Eqs.(4.19, 4.20) (with respect to the boundary conditions Eqs.(4.23,
4.24) and the relevant initial conditions) to yield the embedding functions
requires the correct partitioning of the worldsheet parameter space. This
partitioning was trivial in the heavy quark’s test string case, since its
leading order behaviour is static for all $t$. It is however more complicated
in the Limp Noodle case, considering that the boundary string endpoint is
allowed to fall at the local speed of light for $t>0$. A method in which to
partition the worldsheet in order to find the solution to the string equations
of motion was given by Itzhak Bars in 1994 [107], and expounded upon a year
later, in 1995, by himself and a collaborator Jürgen Schulze [108]. The Bars
et al. method is summarised in the following subsection.

#####  5.1.1 Parameterising the Limp Noodle: The Bars et al. method

The Bars et al. method can be used to solve the classical motion of a
relativistic string in any $(1+1)$-dimensional spacetime. Originally, the
method was used by Bars et al. to study the motion of folded strings in curved
spacetime. However the Limp Noodle can be interpreted as half a folded (or yo-
yo) string. The application of the Bars et al. method to parameterise the Limp
Noodle and find the general solution to the string equations of motion was
first proposed by Moerman et al. in [52]. The Bars et al. method can be broken
into the following steps:

  1. (i)

Define lattice-like patches on the worldsheet using the light-cone
coordinates.

Figure 4:  Lattice structure imposed on the worldsheet parameter space using
the light-cone coordinates $\sigma^{+}$ and $\sigma^{-}$. This structure is
used in the Bars et al. method [107, 108] in order to determine the general
solution for a moving open string on a manifold.

  2. (ii)

Naive solutions to the string equations of motion come in four classes,
referred to as: A, B, C, and D. They can be defined only on patches of the
worldsheet, since the full solution needs to satisfy two conditions: (a)
string solutions must be periodic in $\sigma$, and (b) the global time
coordinate must be an increasing function of $\tau$ and $\sigma$ (in order to
exclude anti-strings from appearing in the same solutions as strings). In the
static gauge the latter condition is trivially satisfied. Hence to build the
general solution from the solution classes A, B, C, D, assign the pattern of
solutions to ensure forward propagation and impose periodicity at a fixed
value of $\tau$.

  3. (iii)

Complete solutions are obtained by matching boundary conditions at the
boundary of the patches.

  4. (iv)

Define a transfer operation matrix to determine future evolution. The
propagation of the solution into the future is performed by using this matrix
to matching the boundary conditions while increasing $\tau$.

The leading order dynamics of the Limp Noodle are constrained to the
$(1+1)$-dimensional subspace of AdSd-Schwarzschild described by the set of
coordinates $(x^{0},x^{1})$, where $x^{0}$ denotes the temporal direction and
$x^{1}$ denotes the radial direction. The metric of any $(1+1)$-dimensional
subspace (where one of these dimensions is the temporal dimension) can be
transformed into a conformally flat metric [107, 108]. In order to perform
this transformation the string field in the target spacetime is redefined by
the coordinate reparametrization
$x^{\mu}(\tau,\sigma)=x^{\mu}(y^{\mu^{\prime}}(\tau,\sigma))$. This yields the
new spacetime metric in terms of isothermal coordinates
$y^{\mu^{\prime}}(\tau,\sigma)$, which can be related to the old spacetime
metric by

$G_{\mu\nu}(x)\,=\,\dfrac{\partial y^{\mu^{\prime}}}{\partial
x^{\mu}}\,\dfrac{\partial y^{\nu^{\prime}}}{\partial
x^{\nu}}\,G_{\mu^{\prime}\nu^{\prime}}(y)\,,$ (5.1)

where the isothermal coordinates in the static gauge
$y^{\mu^{\prime}}(t,\sigma)$, are given by

$y^{0^{\prime}}\,=\,\frac{1}{\sqrt{2}}\Big{(}x^{0}\,+\,x^{1}\Big{)}\,\,\,\,\text{and}\,\,\,\,y^{1^{\prime}}\,=\,\frac{1}{\sqrt{2}}\Big{(}x^{0}\,-\,x^{1}\Big{)}\,.$
(5.2)

In this isothermal coordinate frame the metric is conformally flat

$ds^{2}\,=\,G_{\mu^{\prime}\nu^{\prime}}(y)\,dy^{\mu^{\prime}}\,dy^{\nu^{\prime}}\,,$
(5.3)

and the new spacetime metric is found by inverting Eq.(5.1)

$G_{\mu^{\prime}\nu^{\prime}}(y)\,=\,\dfrac{\partial x^{\mu}}{\partial
y^{\mu^{\prime}}}\,\dfrac{\partial x^{\nu}}{\partial
y^{\nu^{\prime}}}\,G_{\mu\nu}(x)\,=\,\dfrac{\partial x^{\mu}}{\partial
y^{\mu^{\prime}}}\,\dfrac{\partial x^{\nu}}{\partial
y^{\nu^{\prime}}}\,G\,\eta_{\mu\nu}\,,$ (5.4)

where the second equality follows from choosing the conformal gauge, and
$G=G(y^{0},y^{1})\in\mathbb{R}^{+}$ is some non-zero coordinate dependent
scalar function. The metric can be calculated component-wise. For example,

$\displaystyle\begin{aligned}
G_{0^{\prime}0^{\prime}}(y)\,&=\,G\left[-\dfrac{\partial x^{0}}{\partial
y^{0^{\prime}}}\,\dfrac{\partial x^{0}}{\partial
y^{0^{\prime}}}\,+\,\dfrac{\partial x^{1}}{\partial
y^{0^{\prime}}}\,\dfrac{\partial x^{1}}{\partial
y^{0^{\prime}}}\right]\,=\,G\left[-\Big{(}\frac{1}{\sqrt{2}}\Big{)}\,\Big{(}\frac{1}{\sqrt{2}}\Big{)}\,+\,\Big{(}\frac{1}{\sqrt{2}}\Big{)}\,\Big{(}\frac{1}{\sqrt{2}}\Big{)}\right]\,=\,0\,,\\\\[5.69046pt]
\end{aligned}$ (5.5)

where the definition of the isothermal coordinates Eq.(5.2) have been used.
Similarly $G_{0^{\prime}1^{\prime}}(y)=-G$, $G_{1^{\prime}0^{\prime}}(y)=-G$,
and $G_{1^{\prime}1^{\prime}}(y)=0$. Hence $G_{\mu^{\prime}\nu^{\prime}}(y)$
has the explicit form

$G_{\mu^{\prime}\nu^{\prime}}(y)\,=\,G\,\begin{bmatrix}0&-1\\\
-1&0\end{bmatrix}\,=\,G\,\tilde{\eta}_{\mu^{\prime}\nu^{\prime}}\,,$ (5.6)

where $\tilde{\eta}_{\mu\nu}$ is the light-cone metric, and
$G:=G(y^{0^{\prime}},y^{1^{\prime}})\in\mathbb{R}^{+}$ is defined as some non-
zero coordinate dependent scalar function. Using Eq.(5.6), the metric Eq.(5.3)
becomes

$ds^{2}\,=\,-G\big{(}dy^{0^{\prime}}\,dy^{1^{\prime}}\,+\,dy^{1^{\prime}}\,dy^{0^{\prime}}\big{)}\,.$
(5.7)

The string mapping functions $Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$ are
given by

$Y^{0^{\prime}}\,=\,\frac{1}{\sqrt{2}}\Big{(}X^{0}\,+\,X^{1}\Big{)}\,\,\,\,\text{and}\,\,\,\,Y^{1^{\prime}}\,=\,\frac{1}{\sqrt{2}}\Big{(}X^{0}\,-\,X^{1}\Big{)}\,.$
(5.8)

Note that since the coordinates are isothermal, $Y^{0}\,=\,-Y_{1}$ and
$Y^{1}\,=\,-Y_{0}$. The canonically conjugate momentum densities Eq.(4.7) can
be rewritten in terms of the isothermal embedding functions,
$Y^{\mu^{\prime}}$. In the conformal gauge, the canonical momentum densities
are given by

$\displaystyle\begin{aligned}
&\Pi_{\mu}^{a}(t,\sigma)\,=\,-\frac{1}{2\pi\alpha^{\prime}}\,\eta^{ab}\,G_{\mu\nu}\,{\partial_{b}}X^{\nu}\\\\[5.69046pt]
&\Rightarrow\Pi_{\mu}^{\pm}(t,\sigma)\,=\,\frac{1}{2\pi\alpha^{\prime}}\,G\,\eta_{\mu\nu}\,{\partial_{\mp}}X^{\nu}\,,\end{aligned}$
(5.9)

if the light-cone coordinate frame is chosen (where
$\partial_{\mp}:=\frac{\partial}{\partial\sigma^{\mp}}$). Specifically,

$\Pi_{0}^{\pm}(t,\sigma)\,=\,-\frac{1}{2\pi\alpha^{\prime}}\,G\,{\partial_{\mp}}X^{0}\,\,\,\,\text{and}\,\,\,\,\Pi_{1}^{\pm}(t,\sigma)\,=\,\frac{1}{2\pi\alpha^{\prime}}\,G\,{\partial_{\mp}}X^{1}\,.$
(5.10)

Hence the conjugate momentum densities can be defined in terms of the string
mapping functions $Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$

$\displaystyle\begin{aligned}
&\Pi^{\pm}_{0^{\prime}}(t,\sigma)\,:=\,\frac{1}{\sqrt{2}}\Big{(}\Pi^{\pm}_{0}\,+\,\Pi^{\pm}_{1}\Big{)}\,\equiv\,-\frac{1}{2\,\pi\alpha^{\prime}}\,G\,{\partial_{\mp}}Y^{1^{\prime}}\,,\,\,\,\,\,\,\,\,\text{and}\\\\[5.69046pt]
&\Pi^{\pm}_{1^{\prime}}(t,\sigma)\,:=\,\frac{1}{\sqrt{2}}\Big{(}\Pi^{\pm}_{0}\,-\,\Pi^{\pm}_{1}\Big{)}\,\equiv\,-\frac{1}{2\,\pi\alpha^{\prime}}\,G\,{\partial_{\mp}}Y^{0^{\prime}}\,.\end{aligned}$
(5.11)

Combining the components in Eq.(5.11), renaming the indices
$\mu^{\prime}=\mu$, and recognising the raising and lowering property of the
isothermal coordinates, leaves

$\Pi^{\pm}_{\mu}(t,\sigma)\,=\,\frac{1}{2\,\pi\alpha^{\prime}}\,G\,{\partial_{\mp}}Y_{(1-\mu)}\,.$
(5.12)

The Virasoro constraint equations and the string equations of motion,
Eqs.(4.19, 4.20), can also be rewritten in terms the new string mapping
functions $Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$:

$\displaystyle\begin{aligned}
&\big{(}\partial_{\pm}\,Y^{0}\big{)}\,\big{(}\partial_{\pm}\,Y^{1}\big{)}\,=\,0\,,\\\\[5.69046pt]
&\partial_{+}\left(G\partial_{-}Y^{1}\right)\,+\,\partial_{-}\left(G\partial_{+}Y^{1}\right)\,=\,\left(\partial_{0}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\,,\\\\[5.69046pt]
&\partial_{+}\left(G\partial_{-}Y^{0}\right)\,+\,\partial_{-}\left(G\partial_{+}Y^{0}\right)\,=\,\left(\partial_{1}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\,,\end{aligned}$
(5.13)

where $\partial_{\mu}:=\frac{\partial}{\partial Y^{\mu}}$ for $\mu=(0,1)$.
Since, the derivation of Eq.(5.13) is somewhat tedious, it has been relegated
to appendix (B).

There are four classes of solutions to the set of equations Eqs.(5.13).
Namely,

$\displaystyle\begin{aligned}
&\textit{A}:\,\,\,\,\,\,Y^{0}(\sigma^{+},\sigma^{-})\,=\,Y^{0}_{\textit{A}}(\sigma^{+})\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Y^{1}(\sigma^{+},\sigma^{-})\,=\,Y^{1}_{\textit{A}}(\sigma^{-})\\\\[5.69046pt]
&\textit{B}:\,\,\,\,\,\,Y^{0}(\sigma^{+},\sigma^{-})\,=\,Y^{0}_{\textit{B}}(\sigma^{-})\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Y^{1}(\sigma^{+},\sigma^{-})\,=\,Y^{1}_{\textit{B}}(\sigma^{+})\\\\[5.69046pt]
&\text{C}:\,\,\,\,\,\,Y^{0}(\sigma^{+},\sigma^{-})\,=\,Y^{0}_{\textit{C}}\,=\,c_{1}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,Y^{1}(\sigma^{+},\sigma^{-})\,=\,f_{\textit{C}}\big{(}\,\alpha_{\textit{C}}(\sigma^{+})\,+\,\beta_{\textit{C}}(\sigma^{-});\,Y^{0}_{\textit{C}}\,\big{)}\\\\[5.69046pt]
&\text{D}:\,\,\,\,\,\,Y^{0}(\sigma^{+},\sigma^{-})\,=\,f_{\textit{D}}\big{(}\,\alpha_{\textit{D}}(\sigma^{+})\,+\,\beta_{\textit{D}}(\sigma^{-});\,Y^{0}_{\textit{D}}\,\big{)}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,Y^{1}(\sigma^{+},\sigma^{-})\,=\,Y^{1}_{\textit{D}}\,=\,c_{2}\,,\end{aligned}$
(5.14)

where $c_{1,2}$ are constants, and $Y^{0}_{\textit{A}}(\sigma^{+})$,
$Y^{1}_{\textit{A}}(\sigma^{-})$, $Y^{0}_{\textit{B}}(\sigma^{-})$,
$Y^{1}_{\textit{B}}(\sigma^{+})$, $\alpha_{\textit{C}}(\sigma^{+})$,
$\beta_{\textit{C}}(\sigma^{-})$, $\alpha_{\textit{D}}(\sigma^{+})$, and
$\beta_{\textit{D}}(\sigma^{-})$ are arbitrary functions [107, 108]. The
string solutions $A$, $B$, $C$ and $D$ given in Eq.(5.14) may be verified by
direct substitution into Eqs.(5.13). This is easy for classes $A$ and $B$,
since these classes exist for any metric. Taking the derivative with respect
to the light-cone coordinates $\sigma_{\pm}$, yields

$\displaystyle\begin{aligned}
&\textit{A}:\,\,\,\,\partial_{-}\,\big{(}Y^{0}_{\textit{A}}(\sigma^{+})\big{)}\,=\,0\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial_{+}\,\big{(}Y^{1}_{\textit{A}}(\sigma^{-})\big{)}\,=\,0\\\\[5.69046pt]
&\textit{B}:\,\,\,\,\partial_{+}\,\big{(}Y^{0}_{\textit{B}}(\sigma^{+})\big{)}\,=\,0\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial_{-}\,\big{(}Y^{1}_{\textit{B}}(\sigma^{-})\big{)}\,=\,0\,,\end{aligned}$
(5.15)

which is used, together with the chain rule, to easily show that the $A$ and
$B$ classes of string solutions satisfy the Virasoro constraints and the
string equations of motion Eqs.(5.13). This is harder to prove for classes $C$
and $D$. These classes are metric dependent solutions since the functions
$f_{\text{C}}$ and $f_{\text{D}}$ are metric dependent. These functions are
calculated by inverting the following relations that depend on the target
spacetime metric,

$\displaystyle\begin{aligned}
&\textit{C}:\,\,\,\,\,\,Y^{0}(\sigma^{+},\sigma^{-})\,=\,Y^{0}_{\text{C}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_{f_{\text{C}}}\,ds\,G\big{(}Y^{0}_{\text{C}},s\big{)}\,=\,\alpha_{\text{C}}(\sigma^{+})\,+\,\beta_{\text{C}}(\sigma^{-})\\\\[5.69046pt]
&\textit{D}:\,\,\,\,\,\,Y^{1}(\sigma^{+},\sigma^{-})\,=\,Y^{1}_{\text{D}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_{f_{\text{D}}}\,ds\,G\big{(}s,Y^{1}_{\text{D}}\big{)}\,=\,\alpha_{\text{D}}(\sigma^{+})\,+\,\beta_{\text{D}}(\sigma^{-})\,.\end{aligned}$
(5.16)

Taking the derivatives of Eq.(5.16) with respect to the light-cone coordinates
$\sigma_{\pm}$ and making use of Eq.(5.14),

$\displaystyle\begin{aligned}
&\textit{C}:\,\,\,\,\,\,\partial_{\pm}Y^{0}(\sigma^{+},\sigma^{-})\,=\,0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G\,\partial_{\pm}Y^{1}(\sigma^{+},\sigma^{-})\,=\,\partial_{\pm}\big{(}\alpha_{\text{C}}(\sigma^{+})\,+\,\beta_{\text{C}}(\sigma^{-})\big{)}\\\\[8.5359pt]
&\textit{D}:\,\,\,\,\,\,\partial_{\pm}Y^{1}(\sigma^{+},\sigma^{-})\,=\,0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G\,\partial_{\pm}Y^{0}(\sigma^{+},\sigma^{-})\,=\,\partial_{\pm}\big{(}\alpha_{\text{D}}(\sigma^{+})\,+\,\beta_{\text{D}}(\sigma^{-})\big{)}\,.\end{aligned}$
(5.17)

Using Eq.(5.17), the $C$ and $D$ classes of string solutions can be easily
proven to satisfy the Virasoro constraints and string equations of motion
Eqs.(5.13) by direct substitution. For the $C$ class, the Virasoro constraints
and second equation of motion are trivially satisfied. It remains to be
checked whether the first equation of motion is satisfied. To this end,
Eq.(5.17) can be substituted into the first equation of motion in Eq.(5.13)

$0\,=\,\partial_{+}\left(G\partial_{-}Y^{1}\right)\,+\,\partial_{-}\left(G\partial_{+}Y^{1}\right)=\,\partial_{+}\left(\partial_{-}\,\beta_{\text{C}}(\sigma^{-})\right)\,+\,\,\partial_{-}\left(\partial_{+}\,\alpha_{\text{C}}(\sigma^{+})\right)\,=\,0\,.$
(5.18)

Similarly for class $D$, the Virasoro constraints and the first equation of
motion are trivially satisfied, while the second equation of motion is
verified by substituting Eq.(5.17) into Eq.(5.13).

It is not immediately obvious that the classes of solutions presented in
Eq.(5.14) form a complete set. Completeness was shown in Bars et al. [108] by
using the curved spacetime approach based on G/H gauged WZW models. Note that
the pattern of solutions defined on the patches of the worldsheet is metric
independent and spacetime dimension independent [107]. Hence, in order to find
the general string solution for the Limp Noodle in AdSd-Schwarzschild, start
by finding the pattern of solutions defined on the patches of the worldsheet
for the Limp Noodle in flat space. This is done in the next subsection.

Prior to this, some intuition regarding the patch-parametrization of the Limp
Noodle can be developed. At a fold in a string or a string endpoint, the
determinant of the induced worldsheet metric $g_{ab}$ (Eq.(4.4)) vanishes.
Hence the string endpoint’s trajectory in the target spacetime is a null
geodesic (the point travels at the local speed of light). For classes $C$ and
$D$ – by virtue of either $Y^{0}$ or $Y^{1}$ reducing to a constant over the
patch of parameter space where the class is a solution – the induced
worldsheet metric reduces to zero, $g_{ab}=0$. From this it can be deduced
that all points in $C$ or $D$ parameter space patches map to the trajectory of
the string endpoint (or fold) in target spacetime. This is a many-to-one
mapping: the string’s endpoint in target spacetime is represented several
times on the worldsheet [107]. The string (sans the endpoints) behaves like a
massive state, hence it is expected that motion of the string is described by
classes $A$ and $B$ – ever-present, metric independent solutions.

#####  5.1.2 Test Strings in $\mathbb{R}^{1,1}$

In order to study the Limp Noodle in $\mathbb{R}^{1,1}$ the static gauge
ansatz is chosen

$X^{\mu}\,:\,[0,t_{f}]\times[0,\sigma_{f}]\,\rightarrow\,\mathbb{R}^{1,1}\,,$
(5.19)

$(t,\sigma)\,\rightarrow\,x^{\mu}(t,\sigma)\,=\,(t,x(t,\sigma))^{\mu}\,.$
(5.20)

The initial test string set-up is the same as in the heavy quark case.
Eqs.(4.26, 4.27) are chosen as the boundary conditions for the position of the
fixed endpoint at the horizon and the boundary endpoint falling in the
$x(t,\sigma)$ direction respectively. For initial conditions Eqs.(4.28, 4.29)
are chosen i.e. at $t=0$ the string is static and stretches between $x_{0}$
and $x_{0}\,+\,\ell_{0}$. Set $t_{f}$ to be the minimum time it takes for a
string to shrink from its initial length $\ell_{0}$ to a point $x_{0}$.

In flat space the total energy of the string, which represents the mass of an
initially off-mass-shell light quark, is given by Eq.(4.32)757575The
derivation of the total energy of the test string in an AdS-Schwarzschild
background is given in appendix (C).. The equation of motion for $x(t,\sigma)$
can be easily adapted from Eqs.(B.8, B.14), since the spacetime is
$\mathbb{R}^{1+1}$ (therefore $G=1$ and
$(\partial_{0}G)\,=\,(\partial_{1}G)\,=\,0$). Hence, the non-trivial string
equation of motion reduces to

$\partial_{+}\,\partial_{-}\,X^{1}\,=\,0\,\,\,\longrightarrow\,\,\,\partial_{+}\,\partial_{-}\,x(t,\sigma)\,=\,0\,\,\,\,\,\,\,\text{in}\,\,\,\,\,\,\,\,\mathbb{R}^{1+1}\,,$
(5.21)

which is simply the wave equation. The general solution to Eq.(5.21) is a sum
of continuous analytic functions of $\sigma^{+}$ and $\sigma^{-}$ (i.e. left
and right movers)

$x(t,\sigma)\,=\,\frac{1}{2}\,\left(f_{1}(\sigma^{+})\,+\,f_{2}(\sigma^{-})\right)\,.$
(5.22)

Using the boundary condition Eq.(4.26) to redefine
$f(z)\,:=\,f_{1}(z)\,=\,2\,x_{0}\,-\,f_{2}(z)$, Eq.(5.22) becomes

$x(t,\sigma)\,=x_{0}\,+\,\frac{1}{2}\,\left(f(\sigma^{+})\,-\,f(\sigma^{-})\right)\,.$
(5.23)

The Neumann boundary condition Eq.(4.27) becomes

$0\,=\frac{1}{2}\,\left[\partial_{+}\,f(\sigma^{+})\,+\,\partial_{-}\,f(\sigma^{-})\right]\big{\rvert}_{\sigma=\sigma_{f}}\,,$
(5.24)

which follows from the fact that $x_{0}\in\mathbb{R}$ and the definition
Eq.(4.15), since
$\partial_{\sigma}f(\sigma^{+})=\left(\partial_{+}-\partial_{-}\right)f(\sigma^{+})=\partial_{+}f(\sigma^{+})$.
To simplify Eq.(5.24) let $z\,:=\,t\,-\,\sigma_{f}$, then

$\sigma^{-}\rvert_{\sigma=\sigma_{f}}\,=\,t\,-\,\sigma_{f}\,=\,z\,\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,\,\sigma^{+}\rvert_{\sigma=\sigma_{f}}\,=\,t\,+\,\sigma_{f}\,=\,z\,+\,2\,\sigma_{f}\,.$
(5.25)

Using Eq.(5.25) and defining $F(z)\,:=\,\dfrac{d}{dz}f(z)$, Eq.(5.24) becomes

$F(z\,+\,2\sigma_{f})\,=\,-F(z)\,,$ (5.26)

which shows that $F(z)$ is anti-periodic with an interval of $2\,\sigma_{f}$.
The Virasoro constraint equations Eq.(4.19) can also be rewritten in terms of
$F(z)$

$\displaystyle\begin{aligned}
0\,&=\,-\,\big{(}\partial_{\pm}\,t\big{)}\,\big{(}\partial_{\pm}\,t\big{)}\,+\,\,\big{(}\partial_{\pm}\,x(t,\sigma)\big{)}\,\big{(}\partial_{\pm}\,x(t,\sigma)\big{)}\\\\[5.69046pt]
&=\,-\,\big{(}\partial_{\pm}\,t\big{)}\,\big{(}\partial_{\pm}\,t\big{)}\,+\,\,\Big{[}\partial_{\pm}\,\Big{(}x_{0}\,+\,\frac{1}{2}\,\big{(}f(\sigma^{+})\,-\,f(\sigma^{-})\big{)}\Big{)}\Big{]}^{2}\,.\end{aligned}$
(5.27)

The first line follows from Eq.(5.20), and Eq.(5.23) was used in the final
line. Eq.(5.27) breaks up into two, $(+)$ and $(-)$, constraint equations. For
the $(+)$ equation, Eq.(5.27) becomes

$\displaystyle\begin{aligned}
&0\,=\,-\,\bigg{(}\frac{1}{2}\bigg{)}\,\bigg{(}\frac{1}{2}\bigg{)}\,+\,\bigg{(}\frac{1}{2}\partial_{+}\,f(\sigma^{+})\bigg{)}^{2}\\\\[5.69046pt]
&\Rightarrow\,F(\sigma^{+})^{2}\,=\,1\,.\end{aligned}$ (5.28)

Similarly, for the $(-)$ equation $F(\sigma^{-})^{2}\,=\,1$. Hence the
Virasoro constraint equations become

$F(\sigma^{\pm})^{2}\,=\,1\,,$ (5.29)

which implies $F(\sigma^{-})^{2}\rvert_{\sigma=\sigma_{f}}\,=\,1$ or
equivalently $F(z)^{2}\,=\,1$. What remains to be determined is the sign of
$F(z)$. This is given by the anti-periodicity condition Eq.(5.26) which
results in

$F(z)\,=\,\big{(}-1\big{)}^{\big{\lfloor}\frac{z+\sigma_{f}}{2\sigma_{f}}\big{\rfloor}}\,,$
(5.30)

where $\lfloor x\rfloor$ is the floor operation which returns the largest
integer less than or equal to $x$. The function $F(z)$, plotted in figure (5),
is a step function. The definition of $F(z)$ as the derivative of $f(z)$ with
respect to $z$, implies that $f(z)$ can be found through integrating
Eq.(5.30). But $F(z)$ is a discrete function – performing this integration
involves integrating each segment of $F(z)$ separately. Since it is already
known that $F(z)$ is alternating $(+1)$ or $(-1)$ in each $2\,\sigma_{f}$
interval, and the integral of a constant is a straight monotonic function; all
that remains to be determined is if the function $f(z)$ is monotonically
increasing or decreasing on each interval. Hence $f(z)$ can be defined by

$f(z)\,=\,\big{(}-1\big{)}^{\big{\lfloor}\,\frac{z+\sigma_{f}}{2\sigma_{f}}\,\big{\rfloor}}\,\left(\left[(z\,+\,\sigma_{f})\,\,\text{mod}\,\,2\sigma_{f}\right]\,-\,\sigma_{f}\right)\,+\,\sigma_{f}\,,$
(5.31)

where
$\big{(}-1\big{)}^{\big{\lfloor}\,\frac{z+\sigma_{f}}{2\sigma_{f}}\,\big{\rfloor}}$
defines the domain of each segment to be integrated, and
$\left(\left[(z\,+\,\sigma_{f})\,\,\text{mod}\,\,2\sigma_{f}\right]\,-\,\sigma_{f}\right)$
yields $0$ if $z\in z_{1}\,\sigma_{f}$ (where $z_{1}$ is an even integer) and
$-\sigma_{f}$ if $z\in z_{2}\,\sigma_{f}$ (where $z_{2}$ is an odd integer).
The function $f(z)$ is a triangular wave function and is plotted in figure
(5).

Figure 5: The functions (i) $F(z)$ and (ii) $f(z)$ over the range
$[-6\sigma_{f},\,6\sigma_{f}]$, given by Eq.(5.30) and Eq.(5.31) respectively.

Now that a specific form of the function $f(z)$ has been obtained,
$x(t,\sigma)$ (Eq.(5.23)) is completely known. For example, at
$t\,=\,\sigma_{f}$, Eq.(5.23) becomes

$x(\sigma_{f},\sigma)\,=\,x_{0}\,+\,\frac{1}{2}\,\left(f(\sigma_{f}\,+\,\sigma)\,-\,f(\sigma_{f}\,-\,\sigma)\right)=\,x_{0}\,,$
(5.32)

where the second equality follows from using the definition Eq.(5.31). In
studying the Limp Noodle, the primary interest is in the behaviour of the
string as it shrinks from a length of $\ell_{0}$ to a single point $x_{0}$.
From Eq.(5.32) this clearly is achieved in a $t=\sigma_{f}$ amount of time;
hence the choice $t_{f}\,=\,\sigma_{f}$ is made. The worldsheet parameter
space is therefore given by
$\mathcal{M}\,=\,[0,\sigma_{f}]\times[0,\sigma_{f}]$, i.e. the parameter space
is square.

Given the insight of [107, 108, 106] the parameter space $\mathcal{M}$ can be
minimally partitioned into two exclusive triangular regions

$\displaystyle\begin{aligned}
&\mathcal{M}_{1}\,:=\,\big{\\{}(t,\sigma)\in\mathcal{M}\rvert_{\sigma}\,\in\,[0,\sigma_{f}-t]\big{\\}}\\\\[5.69046pt]
&\mathcal{M}_{2}\,:=\,\big{\\{}(t,\sigma)\in\mathcal{M}\rvert_{\sigma}\,\in\,(\sigma_{f}-t,\sigma_{f}]\big{\\}}\,.\end{aligned}$
(5.33)

In order to satisfy the boundary and initial conditions two classes of
solutions exist, one in each of the two parameter space regions: the upper
triangular region $(\mathcal{M}_{2})$ and the lower triangular region
$(\mathcal{M}_{1})$. The embedding functions of the Limp Noodle in
$\mathbb{R}^{1+1}$ is therefore given by

$X^{\mu}_{\text{Mink}}(t,\sigma)\,=\,\left(t,\,x_{0}\,+\,\begin{Bmatrix}\sigma,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
\sigma_{f}-t,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\right)^{\mu}\,.$ (5.34)

The Limp Noodle embedding functions satisfy the given boundary and initial
conditions. The boundary conditions can be checked first: for $\sigma=0$,
$(t,\sigma)\in\mathcal{M}_{1}$ and the Dirichlet condition Eq.(4.26) is
satisfied; while for $\sigma=\sigma_{f}$, $(t,\sigma)\in\mathcal{M}_{2}$ and
the Neumann condition Eq.(4.27) is satisfied. The initial conditions are also
fulfilled. For $t=0$, $(t,\sigma)\in\mathcal{M}_{1}$ and the initial condition
Eq.(4.28) is satisfied. Further the initial condition Eq.(4.29) is satisfied,
since $\sigma_{f}\,=\,\ell_{0}$ is a requirement of the Virasoro constraints
in flat space.

From Eq.(5.34), observe that the functions $X^{\mu}_{\text{Mink}}(t,\sigma)$
map the parameter space $\mathcal{M}_{1}$ region to an extended, space-like
region in the target spacetime; while the parameter space $\mathcal{M}_{2}$
region is mapped to a null geodesic in the target spacetime. Null geodesics
describe the trajectories of massless ‘point-like’ objects, and as such the
worldsheet is considered to be wrapped up to a point along this geodesic.
Hence, the string solution on $\mathcal{M}_{1}$ describes the entire string
sans the falling endpoint, while the string solution on $\mathcal{M}_{2}$
describes the falling endpoint. This is evident from figure (6)767676Details
regarding figures (6)-(8) can be found in Mathematica Notebook [a]
(MappingWorldSheetToTarget.nb) – see appendix (G) for access..

Figure 6: Left: The worldsheet parameter space, where $\mathcal{M}_{1}$ is the
blue region and $\mathcal{M}_{2}$ is the red region (Eq.(5.33)). Right: The
embedding functions $X^{\mu}_{\text{Mink}}(t,\sigma)$, Eq.(5.34), where
$\mathcal{M}_{1}$ is mapped to a space-like region (blue region) and
$\mathcal{M}_{2}$ is mapped to a null geodesic (red, dashed line) in the
target spacetime.

$X^{\mu}_{\text{Mink}}(\mathcal{M}_{2})$ lies on the boundary of
$X^{\mu}_{\text{Mink}}(\mathcal{M}_{1})$, and momentum flows between these two
regions in the target spacetime. In order to continue to satisfy the Neumann
boundary condition Eq.(4.27), momentum flows out of the $\mathcal{M}_{1}$
region at $\sigma\,=\,\sigma_{f}-t$, but immediately stops at
$\mathcal{M}_{2}$ since any momentum flowing out of the string endpoint would
change the physics in the boundary theory777777An infinite force is required
to stop the flow of momentum abruptly at the $\mathcal{M}_{1}/\mathcal{M}_{2}$
divide, and yet the string has a finite energy density (since the string has a
finite tension). This discrepancy stems from the fact that strings are
fundamentally one-dimensional objects and are not made up of $d=0$ constituent
objects..

If Eq.(5.34) is converted into isothermal coordinates in terms of light-cone
parameter space ($\sigma^{+},\sigma^{-}$),
$Y^{\mu}_{\text{Mink}}(\sigma^{+},\sigma^{-})$ defined on the region
$\mathcal{M}_{1}$ is a Bars et al. [107, 108] class $A$ solution787878See
subsection (5.1.1) for details. and the solution defined on the region
$\mathcal{M}_{2}$ is a class $C$ solution, as expected. To see this, use the
definition Eq.(5.8) to define the isothermal embedding functions on
$\mathcal{M}_{1}$,

$\displaystyle\begin{aligned}
Y^{0^{\prime}}&\,=\,\frac{1}{\sqrt{2}}\Big{(}t\,+\,\left(x_{0}+\sigma\right)\Big{)}\\\\[5.69046pt]
&\,=\,\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}\left(\sigma^{+}\,+\,\sigma^{-}\right)\,+\,x_{0}\,+\,\frac{1}{\sqrt{2}}\left(\sigma^{+}\,-\,\sigma^{-}\right)\right)\\\\[5.69046pt]
&\,=\sigma^{+}\,+\,\frac{1}{\sqrt{2}}\,x_{0}\,=:\,Y^{0^{\prime}}\left(\sigma^{+}\right)\,,\end{aligned}$
(5.35)

where the definition of the light-cone coordinates (Eq.(4.15)) is used in the
second line. Similarly,

$\displaystyle\begin{aligned}
Y^{1^{\prime}}\,=\,\sigma^{-}\,-\,\frac{1}{\sqrt{2}}\,x_{0}\,=:Y^{1^{\prime}}\left(\sigma^{-}\right)\,.\end{aligned}$
(5.36)

Comparing Eqs.(5.35, 5.36) against the four classes of Bars et al. solutions
(Eq.(5.14)), it is obvious that the isothermal embedding functions
$Y^{\mu}_{\text{Mink}}(\sigma^{+},\sigma^{-})$ defined on the region
$\mathcal{M}_{1}$ are a class $A$ solution. This is depicted in figure (7).

The isothermal embedding functions on $\mathcal{M}_{2}$ are given by

$\displaystyle\begin{aligned}
Y^{0^{\prime}}\,=\,\frac{1}{\sqrt{2}}\Big{(}t\,+\,\left(x_{0}+\sigma_{f}-t\right)\Big{)}\,=\,\frac{1}{\sqrt{2}}\left(x_{0}+\sigma_{f}\right)\,=:\,Y^{0^{\prime}}_{C}\,,\end{aligned}$
(5.37)

and

$\displaystyle\begin{aligned}
Y^{1^{\prime}}\,=\,\frac{1}{\sqrt{2}}\Big{(}t\,-\,\left(x_{0}+\sigma_{f}-t\right)\Big{)}\,=\,-\left(x_{0}+\sigma_{f}-\left(\sigma^{+}+\sigma^{-}\right)\right)\,=:\,Y^{1^{\prime}}\left(\sigma^{+},\sigma^{-}\right)\,.\end{aligned}$
(5.38)

Matching Eqs.(5.37, 5.38) with four classes of Bars et al. solutions
(Eq.(5.14)) it is easily apparent that the embedding functions
$Y^{\mu}_{\text{Mink}}(\sigma^{+},\sigma^{-})$ defined on the region
$\mathcal{M}_{2}$ are a class $C$ solution. Figure (7) displays this
graphically.

Figure 7: Left: The worldsheet parameter space in light-cone coordinates
($\sigma^{+},\sigma^{-}$), where $\mathcal{M}_{1}$ is the blue region and
$\mathcal{M}_{2}$ is the red region. The parameter space regions
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are plotted from Eq.(5.33) which is
converted to ($\sigma^{+},\sigma^{-}$) coordinates using Eq.(4.15). Right: The
isothermal embedding functions $Y^{\mu}_{\text{Mink}}(\sigma^{+},\sigma^{-})$,
Eqs.(5.35, 5.36, 5.37, 5.38), where $\mathcal{M}_{1}$ is mapped to a space-
like region (blue region) and $\mathcal{M}_{2}$ is mapped to a null geodesic
(red, dashed line) in the target spacetime.

#####  5.1.3 Test Strings in AdS3-Schwarzschild

In the previous subsection, the general embedding functions for the Limp
Noodle in $\mathbb{R}^{1+1}$, Eq.(5.34), were discovered. From this the
embedding functions for the Limp Noodle in AdS3-Schwarzschild can be found
using the same minimal partitioning of the parameter space and the determined
pattern of solution classes on the patches of the parameter space.

In order to trivially write down the general solution for the Limp Noodle in
AdS3-Schwarzschild a set of isothermal coordinates is needed. As was shown in
subsection (4.2.2), the tortoise coordinate $r_{*}$ together with the temporal
coordinate $t$ forms such a set of coordinates in AdSd-Schwarzschild.

The Dirichlet and Neumann conditions, Eqs.(4.45, 4.46), are chosen as boundary
conditions for the Limp Noodle in AdS3-Schwarzschild. The boundary conditions
can be rewritten in terms of the isothermal $(t,r_{*})$ coordinate set. In
subsection (4.2.2), it was noticed that these boundary conditions in the
conformally flat description of AdS3-Schwarzschild, Eqs.(4.47, 4.49), are
respectively analogous to the boundary conditions for the Limp Noodle (and the
heavy quark test string) in $\mathbb{R}^{1+1}$, Eqs.(4.26, 4.27). Hence, the
embedding functions of the Limp Noodle in the conformally flat description of
AdS3-Schwarzschild are the direct analogue of Eq.(5.34). In $(t,r_{*})$
coordinates, the embedding functions are

$X^{\mu}_{\text{\text{AdS${}_{3}$-Sch}}}(t,\sigma)\,=\,\left(t,\,r_{s*}\,+\,\begin{Bmatrix}\sigma,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
\sigma_{f}-t,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\,,\,0\right)^{\mu}\,,$ (5.39)

where the $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ parameter space regions are
given by Eq.(5.33). Using the inverse tortoise transformation Eq.(4.41), the
embedding functions for the AdS3-Schwarzschild Limp Noodle (Eq.(5.39)) can be
rewritten in $(t,r)$ coordinates

$X^{\mu}_{\text{\text{AdS${}_{3}$-Sch}}}(t,\sigma)\,=\,\left(t,\,\begin{Bmatrix}-r_{H}\,\coth\big{(}\frac{r_{H}}{l^{2}}\,\left(r_{s*}\,+\,\sigma\right)\big{)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
-r_{H}\,\coth\big{(}\frac{r_{H}}{l^{2}}\,\left(r_{s*}\,+\,\sigma_{f}-t\right)\big{)},\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\,,\,0\right)^{\mu}\,,$ (5.40)

where the position of the fixed string endpoint attached to the stretched
horizon $r_{s*}$ is given by Eq.(4.48) and the length of the string in
tortoise coordinates is given by Eq.(4.52). Notice that the identification
Eq.(4.53) still holds. The embedding functions are plotted in figure (8).

Figure 8: Left: The worldsheet parameter space, where $\mathcal{M}_{1}$ is the
blue region and $\mathcal{M}_{2}$ is the red region (Eq.(5.33)). Right: The
embedding functions $X^{\mu}_{\text{AdS3-Sch}}(t,\sigma)$, Eq.(5.40), where
$\mathcal{M}_{1}$ is mapped to a space-like region (blue region) and
$\mathcal{M}_{2}$ is mapped to a null geodesic (red, dashed line) in the
target spacetime.

####  5.2 Fluctuating Test String Dynamics

As discussed previously, the boundary endpoint of the Limp Noodle falls at the
local speed of light – the same speed at which information propagates. Hence,
any part of the string below the falling endpoint does not know if the string
is stretched like an on-mass-shell quark or its endpoint is falling like an
off-mass-shell quark. Therefore, the transverse fluctuations of all parts of
the Limp Noodle except the falling endpoint are described by the same
transverse equations of motion as those of the on-mass-shell heavy quark’s
test string and its solution (discussed in subsections (4.3.1) and (4.3.2)).

The leading order solution to the string equations of motion – renamed
$X^{\mu}_{0}(t,\sigma)$ – is given by Eq.(5.40) in AdS3-Schwarzschild. Adding
non-zero fluctuations in the transverse $x$-direction, the Limp Noodle
solution becomes

$X^{\mu}_{\text{\text{AdS${}_{3}$-Sch}}}(t,\sigma)\Big{\rvert}_{\mathcal{M}_{1}}=\,X^{\mu}_{0}(t,\sigma)\,+\,\big{(}0,\,0,\,X(t,\sigma)\,\big{)}^{\mu}\,=\,\left(t,\,-r_{H}\coth\left(\frac{r_{H}}{l^{2}}\,\left(r_{s*}+\sigma\right)\right),\,X(t,\sigma)\right)^{\mu}\,.$
(5.41)

The transverse string equations of motion are supplied in Eq.(4.76); and its
solution is given by Eq.(4.93) in AdS3-Schwarzschild. Continuity of the string
solution in spacetime means that it is unnecessary to find the explicit string
solution on the $\mathcal{M}_{2}$ parameter space region which maps onto the
falling string endpoint.

#####  5.2.1 Calculating the Light Quark’s Mean-Squared Displacement
$s^{2}(t)$ in AdS3-Schwarzschild

Unlike in the case of the heavy quark’s test string where the boundary
endpoint is fixed in the radial direction (subsection (4.3.2)); for the Limp
Noodle the position of the boundary endpoint is falling at the local speed of
light and is defined by the operator

$\hat{X}_{\text{end}}(t)\,:=\,\hat{X}(t,\,\sigma_{f}-t)\,.$ (5.42)

As in subsection (4.3.2), in order to determine $s^{2}(t)$ – the falling
string endpoint’s mean-squared transverse displacement defined in Eq.(4.104) –
begin by calculating the expectation value of the position of the free
endpoint at two different times. Using Eq.(4.93),

$\displaystyle\begin{aligned}
\langle:\mathrel{\hat{X}_{\text{end}}(t_{1})\hat{X}_{\text{end}}(t_{2})}:\rangle&=\frac{\beta^{2}}{4\pi\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega\,d\omega^{\prime}}{(2\pi)^{2}}\frac{1}{\sqrt{\omega\omega^{\prime}}}\Big{[}f_{\omega}(\sigma_{f}-t_{1})f^{\,*}_{\omega^{\prime}}(\sigma_{f}-t_{2})e^{-i\omega
t_{1}+i{\omega^{\prime}}t_{2}}\langle:\mathrel{\hat{a}_{\omega}\hat{a}^{\,\dagger}_{\omega^{\prime}}}:\rangle\,\\\\[5.69046pt]
&\,\,\,\,\,+f^{\,*}_{\omega}(\sigma_{f}-t_{1})f_{\omega^{\prime}}(\sigma_{f}-t_{2})e^{i\omega
t_{1}-i{\omega^{\prime}}t_{2}}\langle:\mathrel{\hat{a}^{\,\dagger}_{\omega}\hat{a}_{\omega^{\prime}}}:\rangle\Big{]}\\\\[5.69046pt]
&=\frac{\beta^{2}}{4\pi\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega\,d\omega^{\prime}}{2\pi}\,\frac{1}{\sqrt{\omega\,\omega^{\prime}}}\bigg{[}\delta(\omega^{\prime}-\omega)f_{\omega}(\sigma_{f}-t_{1})f^{\,*}_{\omega^{\prime}}(\sigma_{f}-t_{2})\frac{e^{-i\omega
t_{1}+i{\omega^{\prime}}t_{2}}}{e^{\beta\omega^{\prime}}-1}\,\\\\[5.69046pt]
&\,\,\,\,\,+\delta(\omega-\omega^{\prime})f^{\,*}_{\omega}(\sigma_{f}-t_{1})f_{\omega^{\prime}}(\sigma_{f}-t_{2})\frac{e^{i\omega
t_{1}-i{\omega^{\prime}}t_{2}}}{e^{\beta\omega}-1}\bigg{]}\\\\[5.69046pt]
&=\frac{\beta^{2}}{4\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\,\text{Re}\left(f_{\omega}(\sigma_{f}-t_{1})f^{\,*}_{\omega}(\sigma_{f}-t_{2})e^{-i\omega(t_{1}-t_{2})}\right)\,,\end{aligned}$
(5.43)

where
$:\mathrel{\hat{a}^{\,\dagger}_{\omega}\hat{a}_{\omega^{\prime}}}:\,=\,:\mathrel{\hat{a}_{\omega^{\prime}}\hat{a}^{\,\dagger}_{\omega}}:\,=\,\hat{a}^{\,\dagger}_{\omega}\hat{a}_{\omega^{\prime}}$
is the normal ordering operator797979Normal ordering of the annihilation and
creation operators $\hat{a}_{\omega},\hat{a}^{\,\dagger}_{\omega}$ is used to
remove logarithmic UV divergences., and the definition of the Bose-Einstein
distribution (Eq.(4.103)) is used in the third equality. The final equality
follows from the property of the Dirac delta function
$\int_{0}^{\infty}dk\,e^{-kx}\,\delta(k-a)\,=\,e^{-ax}$; and the complex
conjugate property $\text{Re}(z)\,=\,(z+\bar{z})/2$ has also been used. At
specific times, the regularized correlator Eq.(5.43) becomes

$\langle:\mathrel{\hat{X}_{\text{end}}(t)\hat{X}_{\text{end}}(0)}:\rangle\,=\,\frac{\beta^{2}}{4\,\pi^{2}\,\sqrt{\lambda}}\,\int_{0}^{\infty}\frac{d\omega}{\omega}\,\frac{1}{e^{\beta\omega}-1}\,\text{Re}\big{(}f_{\omega}(\sigma_{f}-t)f^{\,*}_{\omega}(\sigma_{f})e^{-i\omega
t}\big{)}\,,$ (5.44)

$\langle:\mathrel{\hat{X}_{\text{end}}(0)\hat{X}_{\text{end}}(0)}:\rangle\,=\,\frac{\beta^{2}}{4\,\pi^{2}\,\sqrt{\lambda}}\,\int_{0}^{\infty}\frac{d\omega}{\omega}\,\frac{1}{e^{\beta\omega}-1}\,\text{Re}\big{(}f_{\omega}(\sigma_{f})f^{\,*}_{\omega}(\sigma_{f})\big{)}\,,$
(5.45)

$\langle:\mathrel{\hat{X}_{\text{end}}(t)\hat{X}_{\text{end}}(t)}:\rangle\,=\,\frac{\beta^{2}}{4\,\pi^{2}\,\sqrt{\lambda}}\,\int_{0}^{\infty}\frac{d\omega}{\omega}\,\frac{1}{e^{\beta\omega}-1}\,\text{Re}\big{(}f_{\omega}(\sigma_{f}-t)f^{\,*}_{\omega}(\sigma_{f}-t)\big{)}\,.$
(5.46)

Inputting Eqs.(5.44, 5.45, 5.46) into the definition Eq.(4.104), the string
falling endpoint’s mean-squared transverse displacement can be calculated as

$\displaystyle\begin{aligned}
s^{2}(t)\,\,&=\frac{\beta^{2}}{4\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\text{Re}\left(f_{\omega}(\sigma_{f}-t)f^{\,*}_{\omega}(\sigma_{f}-t)+f_{\omega}(\sigma_{f})f^{\,*}_{\omega}(\sigma_{f})-2f_{\omega}(\sigma_{f}-t)f^{\,*}_{\omega}(\sigma_{f})e^{-i\omega
t}\right)\\\\[5.69046pt]
&=\frac{\beta^{2}}{4\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\big{\rvert}f_{\omega}(\sigma_{f}-t)-f_{\omega}(\sigma_{f})e^{i\omega
t}\big{\rvert}^{2}\,,\end{aligned}$ (5.47)

where in the second line the complex conjugate property
$\text{Re}(z)\,=\,(z+\bar{z})/2$ is again used, and in the final line another
complex conjugate property $\rvert z\rvert^{2}\,=\,z\,\bar{z}$ is used.

#####  5.2.2 The Limiting Cases of $s^{2}(t)$

The mass (or virtuality) of the probe, off-mass-shell light quark808080An off-
mass-shell quark does not satisfy Einstein’s energy and momentum relation
$E^{2}=(pc)^{2}+(2mc)^{2}$. In the case of the light quark it starts at $t=0$
as an off-mass-shell particle (corresponding to the initial static string),
radiates energy as it travels through the thermal medium (string contracts as
the boundary endpoint falls at the local speed of light), and finally stops
radiating as it becomes an on-mass-shell particle. in the boundary theory
determines the length of the test string $\ell_{0}$ in the bulk.
Specifically818181See table (1), page 1, for a comprehensive dictionary
between quantities in the boundary theory and the bulk theory.,

$\tilde{r}_{0}r_{H}\,\equiv\,r_{c}\,:=\,(r_{s}+\ell_{0})\,\,\iff\,\,2\,\pi\,\alpha^{\prime}(M_{\text{rest}}+\Delta
m)\,,$ (5.48)

where $M_{\text{rest}}(T)$ is the static thermal mass of external
particle828282In the AdS5/$\mathcal{N}=4$ SYM case, this is the free energy of
a quark at rest in $\mathcal{N}=4$ SYM plasma. In the limit of zero
temperature it is equal to the QCD Lagrangian quark mass, $m_{q}$ [68]. and
$\Delta m(T)$ is the thermal rest mass shift. In the boundary theory this
corresponds to the virtuality of the probe light quark being much less than
the temperature of the thermal plasma, i.e. $Q^{2}\ll T^{2}$. Equivalently in
the dual gravitational picture,

$Q^{2}\,\ll\,T^{2}\,\,\,\,\implies\,\,\,\,\frac{\lambda\,\ell_{0}^{2}}{4\,\pi^{2}\,l^{4}}\,\ll\,\frac{1}{\beta^{2}}\,\,\,\,\implies\,\,\,\,\frac{4\lambda}{(d-1)^{2}}\,\frac{\ell_{0}^{2}}{r_{H}^{2}}\,\ll\,1\,\,\,\,\implies\,\,\,\,\ell_{0}\,\ll\,r_{H}\,,$
(5.49)

for fixed $\lambda$. In the second equality, the Hawking temperature Eq.(4.36)
and the definition of the quark mass (or virtuality) Eq.(4.32) is used; and in
the third equality, Eq.(4.36) is used again. In the small virtuality limit,
the minimum radius of the space filling D7-brane is small, i.e.
$\tilde{r}_{0}\rightarrow 1+\epsilon$, where $0<\epsilon\ll 1$ and
$\tilde{r}_{0}$ is defined in Eq.(4.89). In juxtaposition, the large
virtuality limit refers $\tilde{r}_{0}\rightarrow\infty$.

In the current subsection the string falling endpoint’s mean-squared
transverse displacement, $s^{2}(t)$, is examined in a number of different
limiting cases. Videlicet, the small virtuality case and the arbitrary
virtuality case. In both of these cases, the asymptotically early time and the
asymptotically late time limit is explored. For the arbitrary virtuality case,
a further limit of small or large virtuality can be considered. It is expected
that the large virtuality, early time limit will correspond exactly with the
early time limit of the on-mass-shell heavy quark – since, at early times the
light quark’s test string is static (i.e. the boundary endpoint has not yet
begun to fall).

Small Virtuality

In this subsection the behaviour of $s^{2}(t)$, Eq.(5.47), in the small
virtuality limit is considered; and after obtaining a suitable expression for
$s_{\text{small}}^{2}(t)$, the asymptotic early and late time dynamics are
explored. The defining condition for the small virtuality limit is given in
Eq.(5.49). This equation implies838383 Explicitly, for the case
$\sigma=\sigma_{f}$,
$\coth^{2}\left(\frac{r_{H}\,(r_{s*}+\sigma)}{l^{2}}\right)=\coth^{2}\left(\frac{r_{H}\,(r_{s*}+\sigma_{f})}{l^{2}}\right)=\coth^{2}\left(\coth^{-1}\left(-\frac{r_{s}\,+\,\ell_{0}}{r_{H}}\right)\right)\approx\,\coth^{2}\left(\coth^{-1}\left(-\frac{r_{s}}{r_{H}}\right)\right)\stackrel{{\scriptstyle(\epsilon\rightarrow
0)}}{{\longrightarrow}}\,1$ where the second equality follows from using the
definition for $\sigma_{f}$, Eq.(4.52); the approximation follows since
$\ell_{0}\,\ll\,r_{H}$; and in taking the limit the definition of the
stretched horizon $r_{s}\,=\,(1+\epsilon)\,r_{H}$ (where $0<\epsilon\ll 1$) is
used. The limit follows since, $\coth^{-1}(-(1+\epsilon))\rightarrow-\infty$
as $\epsilon\rightarrow 0$; and $\coth^{2}(x)\rightarrow 1$ as
$x\rightarrow-\infty$. Further, for the case $\sigma=0$,
$\coth^{2}\left(\frac{r_{H}\,(r_{s*}+\sigma)}{l^{2}}\right)=\coth^{2}\left(\frac{r_{H}\,r_{s*}}{l^{2}}\right)=\coth^{2}\left(\coth^{-1}\left(-\frac{r_{s}}{r_{H}}\right)\right)\stackrel{{\scriptstyle(\epsilon\rightarrow
0)}}{{\longrightarrow}}\,1$ where the first equality follows from the
definition for $r_{s*}$, Eq.(4.48).

$\coth^{2}\left(\frac{r_{H}\,(r_{s*}+\sigma)}{l^{2}}\right)\,\rightarrow\,1,\,\,\,\,\,\,\text{since}\,\,\,\,\,\,\ell_{0}\,\ll\,r_{H}\,,$
(5.50)

where $\sigma\,\in\,[0,\sigma_{f}]$. Using Eq.(5.50) the equations of motion
in AdS3-Schwarzschild, Eq.(4.76), simplifies to the wave equation

$-\,\partial_{t}^{2}\,X(t,\sigma)\,+\,\partial_{\sigma}^{2}\,X(t,\sigma)\,=\,0\,.$
(5.51)

The plane wave solutions to Eq.(5.51) are the near-horizon solutions described
in Eq.(4.85). Hence in the small virtuality limit the in-falling $(+)$ and
out-going $(-)$ modes of the general solution $f_{\omega}(\sigma)$ are given
by,

$f^{(\pm)}_{\omega}(\sigma)\,=\,e^{\pm i\omega\,(r_{s*}\,+\,\sigma)}\,.$
(5.52)

The general solution $f_{\omega}(\sigma)$ is a linear combination of the in-
falling and out-going modes (defined in Eq.(4.86)). The coefficient
$B_{\omega}$ is fixed in the same way as in subsection (4.3.2): by imposing a
Neumann boundary condition at the falling string endpoint in the radial
direction at $t=0$,

$\partial_{\sigma}\,f_{\omega}(\sigma)\big{\rvert}_{\sigma=\sigma_{f}}\,=\,0\,.$
(5.53)

Using Eqs.(4.86, 5.52), Eq.(5.53) can be easily solved in Mathematica848484See
Mathematica Notebook [b]: BrownianMotion.nb. to yield

$B_{\omega}\,=\,e^{i2\omega\,(r_{s*}\,+\,\sigma_{f})}\,.$ (5.54)

Hence, the general solution $f_{\omega}(\sigma)$ is given by,

$\displaystyle\begin{aligned}
f_{\omega}(\sigma)\,&=\,f_{\omega}^{(+)}(\sigma)\,+\,B_{\omega}\,f_{\omega}^{(-)}(\sigma)\\\\[5.69046pt]
&=\,2e^{i\omega\,(r_{s*}\,+\,\sigma_{f})}\,\cos\big{(}\omega\,(\sigma-\sigma_{f})\big{)}\,,\end{aligned}$
(5.55)

where Eqs.(5.52, 5.54) and the trigonometric identity
$\cos(x)\,=\,(e^{ix}+e^{-ix})/2$ are used in the second line. In order to
calculate the falling string endpoint’s mean-squared transverse displacement
in the small virtuality limit, Eq.(5.55) is substituted into Eq.(5.47):

$\displaystyle\begin{aligned}
s^{2}_{\text{small}}(t):=s^{2}(t)\big{\rvert}_{Q^{2}\ll
T^{2}}\,&=\frac{\beta^{2}}{4\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\Big{\rvert}2e^{i\omega(r_{s*}+\sigma_{f})}\cos\big{(}\omega
t\big{)}-2e^{i\omega\,(r_{s*}+\sigma_{f})}e^{i\omega
t}\Big{\rvert}^{2}\\\\[5.69046pt]
&=\frac{\beta^{2}}{4\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\big{(}2-e^{-2i\omega
t}-e^{2i\omega t}\big{)}\\\\[5.69046pt]
&=\frac{\beta^{2}}{\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\sin^{2}(\omega
t)\,,\end{aligned}$ (5.56)

where the complex conjugate property $\rvert z\rvert^{2}\,=\,z\,\bar{z}$ is
used in the second line, and the trigonometric identities
$\cos(x)\,=\,(e^{ix}+e^{-ix})/2$ and $\sin^{2}(x)\,=\,(1-\cos(2x))/2$ are used
in the final line. The integral in Eq.(5.56) can be solved analytically.
Introducing a change of variables

$x\,:=\,\beta\omega\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,k\,:=\,\frac{t}{\beta}\,,$
(5.57)

Eq.(5.56) becomes

$\displaystyle\begin{aligned}
s^{2}_{\text{small}}(t)&=\frac{\beta^{2}}{\pi^{2}\sqrt{\lambda}}\,\sum_{n=1}^{\infty}\,\int_{0}^{\infty}\frac{dx}{x}\,\sin^{2}(kx)\,e^{-nx}\\\\[5.69046pt]
&=\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\,\ln\left(\prod_{n=1}^{\infty}\left(1+\frac{4k^{2}}{n^{2}}\right)\right)\\\\[5.69046pt]
&=\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\,\ln\left(\frac{\beta}{2\,\pi\,t}\sinh\left(\frac{2\,\pi\,t}{\beta}\right)\right)\,,\end{aligned}$
(5.58)

where in the second line the geometric series
$1/(1-x)\,=\,\sum_{n=0}^{\infty}x^{n}$ is used; and in the third line the
integral was performed using Mathematica858585See Mathematica Notebook [b]:
BrownianMotion.nb.. In the final line, Mathematica was again used, followed by
a change of variables using Eq.(5.57). Notice from Eq.(5.58) that $\beta$
naturally defines a cross-over time scale between the dynamics of early and
late times868686See footnote 67, page 67., and that the cross-over time is
independent of the length of the string $\ell_{0}$.

(I) Asymptotic Early Time Dynamics

Since, in the early time limit $t\ll\beta$, Eq.(5.58) can be expanded in
powers of $k=t/\beta$. For small $k$ a Taylor expansion (about $k=0$) can be
performed. Explicitly,
$\ln(\sinh(x)/x)=x^{2}/6-\,x^{4}/180+\,x^{6}/2835+\,\mathcal{O}(x^{8})$ (where
$x=2k\pi$). Therefore, Eq.(5.58) becomes

$s^{2}_{\text{small}}(t)\,\stackrel{{\scriptstyle(t\ll\beta)}}{{\longrightarrow}}\,\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\,\left[\frac{2\pi^{2}\,t^{2}}{3\,\beta^{2}}\,+\,\mathcal{O}\left(\frac{t}{\beta}\right)^{4}\right]\,=\,\frac{t^{2}}{6\sqrt{\lambda}}\,+\,\mathcal{O}\left(\frac{t}{\beta}\right)^{4}\,.$
(5.59)

Since $s^{2}_{\text{small}}(t)\sim t^{2}$, the early time dynamics exhibit
ballistic behaviour878787The early time limit of the on-mass-shell heavy quark
also displays ballistic behaviour (see Eq.(4.119)). (see Eq.(3.20)).

(II) Asymptotic Late Time Dynamics

At asymptotically late times ($t\gg\beta$) the integral in Eq.(5.58) can be
expanded in powers of $1/k=\beta/t$. Specifically,

$\displaystyle\begin{aligned}
s^{2}_{\text{small}}(t)&\,\,\,=\,\,\,\,\,\,\,\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\,\Big{[}\ln\big{(}\sinh\big{(}2\,\pi\,k\big{)}\big{)}\,+\,\ln\Big{(}\frac{1}{2\,\pi\,k}\Big{)}\Big{]}\\\\[5.69046pt]
&\,\,\,=\,\,\,\,\,\,\,\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\,\bigg{[}\frac{2\,\pi\,t}{\beta}\,+\,\ln\big{(}1\,-\,e^{-4\,\pi\,k}\big{)}\,+\,\ln\bigg{(}\frac{\beta}{4\,\pi\,t}\bigg{)}\bigg{]}\\\\[5.69046pt]
&\,\stackrel{{\scriptstyle(\beta\ll
t)}}{{\longrightarrow}}\,\,\frac{\beta\,t}{2\,\pi\sqrt{\lambda}}\,+\,\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\ln\bigg{(}\frac{\beta}{4\,\pi\,t}\bigg{)}\,+\,\mathcal{O}\Big{(}\beta^{2}\,e^{\frac{-4\,\pi\,t}{\beta}}\Big{)}\,,\end{aligned}$
(5.60)

where the second line follows from trigonometric identity
$\sinh(x)=(e^{x}-e^{-x})/2$; and, in the final line, the expansion
$\ln{(1-x)}=-x-\,x^{2}/2-\,x^{3}/3\,+\mathcal{O}(x^{4})$ (where
$x=e^{-4\,\pi\,k}$) is performed, since $e^{-4\,\pi\,k}\rightarrow 0$ in the
late time regime ($k\gg 1$).

Since $s^{2}_{\text{small}}(t)\sim t$, the late time dynamics exhibit
diffusive behaviour888888The late time limit of the on-mass-shell heavy quark
also displays diffusive behaviour (see Eq.(4.122)).. From Eq.(3.21) it is
expected that $s^{2}_{\text{small}}(t)\,=\,2\,D\,t$ at late times, where $D$
is the diffusion coefficient. Comparing this to Eq.(5.60), it is easy to see
that the diffusion coefficient is given by

$D_{\text{LQ}}^{\text{AdS${}_{3}$}}\,=\,\frac{\beta}{4\,\pi\sqrt{\lambda}}\,.$
(5.61)

The diffusion coefficient for the on-mass-shell heavy quark is given by
Eq.(4.123). By comparing Eq.(5.61) with Eq.(4.123), notice that an off-mass-
shell small virtuality light quark which is initially at rest in a strongly-
coupled thermal plasma has a diffusion coefficient related to that of a
massive on-mass-shell heavy quark by

$D_{\text{LQ}}^{\text{AdS${}_{3}$}}\,=\,\frac{1}{2}\,D_{\text{HQ}}^{\text{AdS${}_{3}$}}\,.$
(5.62)

The factor of $1/2$ may arise through the differences in partitioning the
worldsheet for the heavy and light quark test strings. This is explored
further in subsection (6.2).

Arbitrary Virtuality

(I) Asymptotic Early Time Dynamics

Now consider a probe light quark with arbitrary virtuality. The behaviour of
$s^{2}(t)$, Eq.(5.47), at asymptotically early times is considered. Since
$t\ll\beta$, Eq.(5.47) can be expanded in powers of $k=t/\beta$,

$\displaystyle\begin{aligned}
\big{\rvert}f_{\omega}(\sigma_{f}-t)-f_{\omega}(\sigma_{f})e^{i\omega
t}\big{\rvert}^{2}&\stackrel{{\scriptstyle(t\ll\beta)}}{{\longrightarrow}}\,\left(f_{\omega}(\sigma_{f})-f_{\omega}(\sigma_{f})\,e^{i\omega
t}\right)\left(f_{\omega}^{*}(\sigma_{f})-f_{\omega}^{*}(\sigma_{f})\,e^{-i\omega
t}\right)\\\\[5.69046pt]
&=\big{\rvert}f_{\omega}(\sigma_{f})\big{\rvert}^{2}\left(2\,-\,e^{-i\omega
t}\,-\,e^{i\omega t}\right)\\\\[5.69046pt]
&=\big{\rvert}f_{\omega}(\sigma_{f})\big{\rvert}^{2}\left(\omega^{2}t^{2}+\mathcal{O}(\omega
t)^{4}\right)\\\\[5.69046pt]
&=\,4\,\frac{1+\nu^{2}}{1+\tilde{r}_{0}\nu^{2}}\left(\omega^{2}t^{2}+\mathcal{O}(\omega
t)^{4}\right)\,,\end{aligned}$ (5.63)

where the first line follows since in the early time regime $t\ll\beta$
implies $f_{\omega}(\sigma_{f}-t)\approx f_{\omega}(\sigma_{f})$; and in the
third line $e^{i\omega t}$ and $e^{-i\omega t}$ are Taylor expanded. The final
line follows898989Mathematica is used to simplify the algebra. This
computation (and others in this section) are explicitly shown in Mathematica
Notebook [b] (BrownianMotion.nb). For access, see appendix (G). from using
Eqs.(4.84, 4.86, 4.91) to calculate
$\big{\rvert}f_{\omega}(\sigma_{f})\big{\rvert}^{2}=f_{\omega}(\sigma_{f})f_{\omega}^{\,*}(\sigma_{f})$.
Inputting Eq.(5.63) into Eq.(5.47), yields

$s^{2}(t)\big{\rvert}_{t\ll\beta}\,=\,\frac{4\,t^{2}}{\sqrt{\lambda}}\,\int_{0}^{\infty}\,d\nu\,\frac{\nu}{e^{2\pi\nu}-1}\,\frac{1+\nu^{2}}{1+\tilde{r}_{0}\nu^{2}}\,,$
(5.64)

where a change of variables was performed using Eqs.(4.36, 4.79). Eq.(5.64) is
evaluated by breaking the integral into two parts and recognising that these
terms are the series expansion to $\mathcal{O}(k)^{0}$ (around $k=0$, where
$k=t/\beta$) of the second derivatives with respect to $k$ of the integrals
$I_{1}$ and $I_{2}$ given in Eq.(4.113) and Eq.(4.114) respectively. To
clarify, the second derivative of $I_{1}$ and $I_{2}$ are given by

$\displaystyle\begin{aligned}
&I_{1}\,=\,4\int_{0}^{\infty}\,\frac{dx}{x(1+a^{2}x^{2})}\,\frac{\sin^{2}\big{(}\frac{kx}{2}\big{)}}{e^{x}-1}\\\\[5.69046pt]
&\Rightarrow\frac{\partial^{2}I_{1}}{\partial
k^{2}}\bigg{\rvert}_{k=0}=\,4\pi\int_{0}^{\infty}\,d\nu\,\frac{\nu}{e^{2\pi\nu}-1}\,\frac{1}{1+\tilde{r}_{0}^{2}\,\nu^{2}}\,,\end{aligned}$
(5.65)

and,

$\displaystyle\begin{aligned}
&I_{2}=\,4\int_{0}^{\infty}\,\frac{dx}{x}\,\frac{\sin^{2}\big{(}\frac{kx}{2}\big{)}}{e^{x}-1}\\\\[5.69046pt]
&\Rightarrow\frac{\partial^{2}I_{2}}{\partial
k^{2}}\bigg{\rvert}_{k=0}=\,4\pi\int_{0}^{\infty}\,d\nu\,\frac{\nu}{e^{2\pi\nu}-1}\,,\end{aligned}$
(5.66)

where $x=2\pi\nu$, $a=\tilde{r}_{0}/(2\pi)$ and $k=t/\beta$. Explicitly
rewriting Eq.(5.64) in terms of $\partial_{k}^{2}I_{1}$ and
$\partial_{k}^{2}I_{2}$ yields

$\displaystyle\begin{aligned}
s^{2}(t)\big{\rvert}_{t\ll\beta}&=\,\frac{4\,t^{2}}{\sqrt{\lambda}}\int_{0}^{\infty}\,d\nu\left(\frac{\tilde{r}_{0}^{2}-1}{\tilde{r}_{0}^{2}}\frac{\nu}{e^{2\pi\nu}-1}\,\frac{1}{1+\tilde{r}_{0}^{2}\nu^{2}}\,+\,\frac{1}{\tilde{r}_{0}^{2}}\frac{\nu}{e^{2\pi\nu}-1}\right)\\\\[5.69046pt]
&=\,\frac{t^{2}}{\pi\sqrt{\lambda}}\left(\frac{\tilde{r}_{0}^{2}-1}{\tilde{r}_{0}^{2}}\,\frac{\partial^{2}I_{1}}{\partial
k^{2}}\bigg{\rvert}_{k=0}+\,\frac{1}{\tilde{r}_{0}^{2}}\,\frac{\partial^{2}I_{2}}{\partial
k^{2}}\bigg{\rvert}_{k=0}\right)\,.\end{aligned}$ (5.67)

The solution to the integrals $I_{1}$ and $I_{2}$ are given by Eqs.(4.116,
4.117) respectively. Taking the second derivative with respect to $k$ followed
by a series expansion to $\mathcal{O}(k)^{0}$ (around $k=0$) for each of these
solutions, yields

$\frac{\partial^{2}I_{1}}{\partial
k^{2}}\bigg{\rvert}_{k=0}\,=\,\frac{2\left(-\pi^{2}\ln\left(\frac{\tilde{r}_{0}}{2\pi}\right)+\pi^{2}\ln\left(\frac{2\pi}{\tilde{r}_{0}}\right)+\pi^{3}\cot\left(\frac{\pi}{\tilde{r}_{0}}\right)-\pi^{2}\psi\left(1+\frac{1}{\tilde{r}_{0}}\right)-\pi^{2}\psi\left(1-\frac{1}{\tilde{r}_{0}}\right)-2\pi^{2}\ln(2\pi)\right)}{\tilde{r}_{0}^{2}}\,,$
(5.68)

where it was recognised in the early time case909090For the physically
relevant solution $t\geq 0$, the early time limit $t\ll\beta$ implies
$\beta>0$. Hence, $k=t/\beta\geq 0$, and $\rvert k\rvert=k$. that $\rvert
k\rvert=k$, and $\psi(z)$ is the digamma function. In addition,

$\frac{\partial^{2}I_{2}}{\partial
k^{2}}\bigg{\rvert}_{k=0}\,=\,\frac{\pi^{2}}{3}\,.$ (5.69)

Inputting Eq.(5.68) and Eq.(5.69) into Eq.(5.67), yields919191Eq.(5.70) agrees
with Eq.(3.42) in Moerman et al.’s [52] if the the digamma function is
rewritten in terms of the harmonic numbers. Specifically,
$\psi(n)=H_{(n-1)}-\gamma_{E}$, where $\gamma_{E}$ is the Euler-Mascheroni
constant and $n$ is a positive integer. It is worth remarking that this form
was avoid here since $\big{(}1+\frac{1}{\tilde{r}_{0}}\big{)}$ and
$\big{(}1-\frac{1}{\tilde{r}_{0}}\big{)}$ are not necessarily positive
integers.

$s^{2}(t)\big{\rvert}_{t\ll\beta}\,=\,\frac{t^{2}}{6\tilde{r}_{0}^{4}\sqrt{\lambda}}\,\left[\tilde{r}_{0}^{2}+6\left(1-\tilde{r}_{0}^{2}\right)\left(2\ln(\tilde{r}_{0})-\pi\cot\left(\frac{\pi}{\tilde{r}_{0}}\right)+\psi\left(1+\frac{1}{\tilde{r}_{0}}\right)+\psi\left(1-\frac{1}{\tilde{r}_{0}}\right)\right)\right]\,.$
(5.70)

The limit of small or large virtuality can now be considered.

  1. (i)

Small Virtuality Limit ($\tilde{r}_{0}\rightarrow 1+\epsilon$)
Using Mathematica to take the limit of Eq.(5.70) as $\tilde{r}_{0}\rightarrow
1+\epsilon$, yields

$s^{2}(t)\big{\rvert}_{t\ll\beta}\,\stackrel{{\scriptstyle(\tilde{r}_{0}\rightarrow
1+\epsilon)}}{{\longrightarrow}}\,\frac{t^{2}}{6\sqrt{\lambda}}\,+\,\mathcal{O}(\epsilon)\,.$
(5.71)

This agrees exactly with Eq.(5.59), i.e. taking the early time limit followed
by the small virtuality limit is equivalent to taking small virtuality limit
followed by the early time limit – a necessary consistency check.

  2. (ii)

Large Virtuality Limit ($\tilde{r}_{0}\rightarrow\infty$)
Since $\tilde{r}_{0}\gg 0$, define a variable $y=1/\tilde{r}_{0}$ (where $y\ll
0$) so that a Taylor expansion in $y$ can be performed. Rewriting Eq.(5.70) in
terms of $y$ and performing a series expansion (around $y=0$) in Mathematica,
yields

$s^{2}(t)\big{\rvert}_{t\ll\beta}\,\stackrel{{\scriptstyle(\tilde{r}_{0}\rightarrow\infty)}}{{\longrightarrow}}\,\frac{t^{2}}{\tilde{r}_{0}\sqrt{\lambda}}\,+\,\mathcal{O}\left(\frac{1}{\tilde{r}_{0}}\right)^{2}\,.$
(5.72)

As expected, the large virtuality early time behaviour corresponds exactly
with the early time limit of the on-mass-shell heavy quark (i.e. the static
string solution at early times, Eq.(4.122)).

(II) Asymptotic Late Time Dynamics

The behaviour of $s^{2}(t)$, Eq.(5.47), at asymptotically late times is
considered. Defining two dimensionless quantities

$u\,:=\,\frac{\beta}{t}\,=\,\frac{1}{k}\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,z\,:=\,\omega
t\,,$ (5.73)

the late time dynamics of $s^{2}(t)$ can be examined. At asymptotically late
times $t\gg\beta$, or equivalently $u\ll 1$. The integral Eq.(5.47) can then
be rewritten

$s^{2}(t)=\frac{\beta^{2}}{4\,\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{dz}{z}\frac{1}{e^{zu}-1}\big{\rvert}f_{z/t}\left(\sigma_{f}-\beta/u\right)-f_{z/t}(\sigma_{f})e^{iz}\big{\rvert}^{2}\,.$
(5.74)

Using Eqs.(4.84, 4.86, 4.91) to simplify the integrand, yields

$\displaystyle\begin{aligned}
\big{\rvert}f_{z/t}\left(\sigma_{f}-\beta/u\right)-f_{z/t}(\sigma_{f})e^{iz}\big{\rvert}^{2}&=\left(f_{z/t}\left(\sigma_{f}-\beta/u\right)-f_{z/t}(\sigma_{f})e^{iz}\right)\,\left(f_{z/t}^{\,*}\left(\sigma_{f}-\beta/u\right)-f_{z/t}^{\,*}(\sigma_{f})e^{iz}\right)\\\\[5.69046pt]
&\stackrel{{\scriptstyle(u\ll 1)}}{{\longrightarrow}}\,4\sin^{2}(\omega
t)\,+\,\mathcal{O}\left(\beta/t\right)\,,\end{aligned}$ (5.75)

where the final line is written in terms of $\omega$, $\beta$ and $t$ using
Eq.(5.73). Further, it has also been recognised that at asymptotically late
times the test string’s falling endpoint is in the near-horizon region,
therefore $r\approx r_{H}$. Substituting Eq.(5.75) into Eq.(5.74), yields

$s^{2}(t)\big{\rvert}_{t\gg\beta}=\frac{\beta^{2}}{\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\sin^{2}(\omega
t)\,\equiv\,s^{2}_{\text{small}}(t)\,.$ (5.76)

Therefore, the late time dynamics of an arbitrary virtuality light quark in a
thermal plasma is diffusive $s^{2}_{\text{small}}(t)\sim t$, and the diffusion
coefficient is given by Eq.(5.61). This statement can be made due to a key
insight of Moerman et al. [52] – that the integral in Eq.(5.76) is identical
to $s^{2}_{\text{small}}(t)$, the string falling endpoint’s mean-squared
transverse displacement in the limit of small virtualities (see subsection
(5.2.2)). This allows them to conclude that the behaviour of arbitrary
virtuality quarks at asymptotically late times (i.e. in the near-horizon
region) is encoded in the small virtuality case. This is intuitive, since the
length of the Limp Noodle is arbitrarily short at asymptotically late
times929292Comparing Eq.(5.50) with Eq.(E.6) the limit of small virtualities
appears to be equivalent to the near-horizon limit.. Since
$s^{2}_{\text{small}}(t;d)$ can be solved for any dimensions $d\geq 3$ in
AdSd-Schwarzschild, the late time behaviour (and therefore the diffusion
coefficient) of Limp Noodles with arbitrary length (equivalently, light quarks
of arbitrary virtuality) in any number of transverse spatial dimensions are
able to be determined. This generalisation to AdSd-Schwarzschild is presented
in the following subsection.

####  5.3 Generalising to AdSd-Schwarzschild

To determine $s^{2}_{\text{small}}(t;d)$ the AdSd-Schwarzschild metric
(Eq.(4.34)) is examined in the near-horizon limit939393As discussed in
subsection (2.1), ‘dropping the 1’ from the function $H(r)$ in the
AdS${}_{d}\times S^{d}$ metric corresponds to being in the near-horizon region
(or in the ‘deep-throat’) of AdS${}_{d}\times S^{d}$ spacetime – see figure
(1), page 1. Here, a further limit is considered – letting
$r=(1+\tilde{\epsilon})r_{H}$ and studying the near-horizon geometry of the
black-brane metric – which is the near-horizon limit refered to in this
subsection. , which surmounts to letting $r=(1+\tilde{\epsilon})r_{H}$, where
($0<\epsilon\leq\tilde{\epsilon}\ll 1$) and series expanding each term
individually around $\tilde{\epsilon}=0$.

Setting $r=(1+\tilde{\epsilon})r_{H}$, the AdSd-Schwarzschild metric Eq.(4.34)
becomes

$ds_{d}^{2}=\frac{r_{H}^{2}}{l^{2}}(1+\tilde{\epsilon})^{2}\,d\vec{X}_{I}^{2}\,-\,\frac{r_{H}^{2}}{l^{2}}(1+\tilde{\epsilon})^{2}\left(1-\Big{(}\frac{1}{1+\tilde{\epsilon}}\Big{)}^{d-1}\right)\,dt^{2}\,+\,\frac{l^{2}}{(1+\tilde{\epsilon})^{2}}\frac{1}{1-\left(1/(1+\tilde{\epsilon})\right)^{d-1}}\,d\tilde{\epsilon}^{2}\,,$
(5.77)

where each term is now able to be expanded in $\tilde{\epsilon}$ (around
$\tilde{\epsilon}=0$). Specifically949494In Eq.(5.79), all terms greater than
$\mathcal{O}\left(\tilde{\epsilon}\right)^{2}$ disappear when $d=3$.,

$\frac{r_{H}^{2}}{l^{2}}(1+\tilde{\epsilon})^{2}\,d\vec{X}_{I}^{2}\,=\,\left(\frac{r_{H}^{2}}{l^{2}}\,+\,\frac{2r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,+\,\frac{r_{H}^{2}\tilde{\epsilon}^{2}}{l^{2}}\right)d\vec{X}_{I}^{2}\,,$
(5.78)

$\frac{r_{H}^{2}}{l^{2}}(1+\tilde{\epsilon})^{2}\left(1-\Big{(}\frac{1}{1+\tilde{\epsilon}}\Big{)}^{d-1}\right)\,dt^{2}\,\longrightarrow\,\left(-\frac{(d-1)r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,+\,\frac{\left(d^{2}-5d+4\right)r_{H}^{2}\tilde{\epsilon}^{2}}{2l^{2}}\,+\,\mathcal{O}\left(\tilde{\epsilon}\right)^{3}\right)dt^{2}\,,$
(5.79)

$\frac{l^{2}}{(1+\tilde{\epsilon})^{2}}\frac{1}{1-\left(1/(1+\tilde{\epsilon})\right)^{d-1}}\,d\tilde{\epsilon}^{2}\,\longrightarrow\,\left(\frac{l^{2}}{(d-1)\tilde{\epsilon}}\,+\,\frac{(d-4)l^{2}}{2(d-1)}\,+\,\frac{\left(d^{2}-14d+36\right)l^{2}\,\tilde{\epsilon}}{12(d-1)}\,+\,\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\right)d\tilde{\epsilon}^{2}\,.$
(5.80)

The question now arises as how to truncate the series expansions
(Eqs.(5.78)-(5.80)), such that the $dt^{2},d\vec{X}_{I}^{2}$ and
$d\tilde{\epsilon}^{2}$ terms in the metric are all to the same
$\mathcal{O}(\tilde{\epsilon})$. To help clarify this task of consistent
truncation, there are actually only two options available:

  1. (i)

assume $dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$d\vec{X}_{I}^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$, such that the AdSd-
Schwarzschild metric becomes

$ds_{d}^{2}=\left(\frac{r_{H}^{2}}{l^{2}}\,+\,\frac{2r_{H}^{2}\tilde{\epsilon}}{l^{2}}\right)d\vec{X}_{I}^{2}\,-\,\frac{(d-1)r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,dt^{2}\,+\,\frac{l^{2}}{(d-1)\tilde{\epsilon}}\,d\tilde{\epsilon}^{2}\,+\,\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\,,$
(5.81)

  2. (ii)

or, assume $dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$d\vec{X}_{I}^{2}\sim\mathcal{O}(\tilde{\epsilon})$, such that the AdSd-
Schwarzschild metric becomes

$ds_{d}^{2}=\frac{r_{H}^{2}}{l^{2}}\,d\vec{X}_{I}^{2}\,-\,\frac{(d-1)r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,dt^{2}\,+\,\frac{l^{2}}{(d-1)\tilde{\epsilon}}\,d\tilde{\epsilon}^{2}\,+\,\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\,.$
(5.82)

These are the only options, since if higher order terms in the $dt^{2}$ or
$d\tilde{\epsilon}^{2}$ series are included, the different signs in Eqs.(5.79,
5.80) will result in a metric which is not conformally flat in ($t,r_{*}$)
coordinates. The leading order metric includes only the first term in the
$d\tilde{\epsilon}^{2}$ series expansion (Eq.(5.80)), therefore – to be
consistent – all terms in the leading order metric must be of
$\mathcal{O}(\tilde{\epsilon})$. One of the ramifications of this is that it
can not be assumed that $d\vec{X}_{I}\sim\mathcal{O}(\tilde{\epsilon})$ (or
higher) at leading order, since the $d\vec{X}_{I}^{2}$ term in the metric
would then disappear.

To decide between options (i) or (ii), establish which option ensures that
$t,\,\vec{X}_{I},\,\tilde{\epsilon}$ are on equal footing, and that the
$dt,\,d\vec{X}_{I},\,d\tilde{\epsilon}$ terms scale the same way as
$\tilde{\epsilon}\rightarrow\lambda\,\tilde{\epsilon}$. This will ensure the
resulting near-horizon AdSd-Schwarzschild metric has directions which encode
transverse fluctuations959595It is clear that the near-horizon AdSd-
Schwarzschild metric needs to have transverse directions since these are
present in the near-horizon AdS3-Schwarzschild metric (which was found
directly from inverting the tortoise coordinate in $d=3$, and calculating the
transverse string equations of motion without series expanding the parameters
– see subsection (5.2)).. For option (ii), considering the scaling
$\tilde{\epsilon}\rightarrow\lambda\,\tilde{\epsilon}$, the
$d\tilde{\epsilon}^{2}$ term in Eq.(5.82) scales as

$\frac{l^{2}}{(d-1)\tilde{\epsilon}}\,d\tilde{\epsilon}^{2}\rightarrow\lambda\,\frac{l^{2}}{(d-1)\tilde{\epsilon}}\,d\tilde{\epsilon}^{2}\,.$

Under the same transformation, in order for the $dt^{2}$ term in Eq.(5.82) to
scale as

$\frac{-(d-1)r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,dt^{2}\rightarrow\lambda\,\frac{-(d-1)r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,dt^{2}\,,$

the temporal variable needs to scale like $t\rightarrow t$. Similarly, in
order for the $d\vec{X}_{I}^{2}$ term in Eq.(5.82) to scale as

$\frac{r_{H}^{2}}{l^{2}}\,d\vec{X}_{I}^{2}\rightarrow\lambda\,\frac{r_{H}^{2}}{l^{2}}\,d\vec{X}_{I}^{2}\,,$

the transverse variable needs to scale like
$\vec{X}_{I}\rightarrow\sqrt{\lambda}\,\vec{X}_{I}$969696Accepting that $t$
and $\vec{X}_{I}$ scale differently isn’t far-fetched considering this is the
near-horizon region of the black-brane.. A consistent way to transform $t$ and
$\vec{X}_{I}$, such that the $dt,\,d\vec{X}_{I},\,d\tilde{\epsilon}$ terms in
the AdSd-Schwarzschild metric scale the same way under the transformation
$\tilde{\epsilon}\rightarrow\lambda\,\tilde{\epsilon}$, can not be found for
option (i). This strongly suggests that option (ii) is the correct way to
truncate the near-horizon AdSd-Schwarzschild metric.

Intuitive arguments aside, a rigorous mathematical approach to proving
$dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$d\vec{X}_{I}^{2}\sim\mathcal{O}(\tilde{\epsilon})$ can be applied in the
$d=3$ case. Since in AdS3-Schwarzschild the tortoise coordinate is invertible,
the transverse string equations of motion and its solution can be exactly
found. The equations of motion and its solution can be series expanded in
$\tilde{\epsilon}$ in the near-horizon region. Order by order these should
match with the equations of motion and its solution derived from the near-
horizon AdS3-Schwarzschild (found by expanding the metric in
$\tilde{\epsilon}$ and assuming $dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$
and $dx^{2}\sim\mathcal{O}(\tilde{\epsilon})$)979797This is done in detail
Mathematica Notebook [c] (NearHorizonAdSd.nb) – see appendix (G) for access..
If this method of matching terms proves to be consistent, the order of
$\tilde{\epsilon}$ of the temporal and transverse directions will be known,
and this remains true for a general number of dimensions, $d$. The near-
horizon AdSd-Schwarzschild metric will, therefore, also be correctly known.
The remainder of the subsection will follow the standard method of finding the
equations of motion for the transverse fluctuations, its solution, the string
falling endpoint’s mean-squared transverse displacement $s^{2}(t)$, and –
finally – the diffusion coefficient for the light quark in AdSd-Schwarzschild.

#####  5.3.1 Expanding the AdS3-Schwarzschild Transverse Equations of Motion
and its Solution in the Near-Horizon Region

The AdS3-Schwarzschild metric, the resulting transverse equation of motion,
and its solution are given by Eqs.(4.42, 4.75, 4.77) respectively989898The
full solution to the transverse equation of motion is given by Eq.(4.86),
where the constant $B_{\omega}$ is defined in Eq.(4.91) and the modes
$f^{(\pm)}_{\omega}(r)$ are given in Eq.(4.81). The linearly independent
solutions $f^{(+)}_{\omega}(r)$ or $f^{(-)}_{\omega}(r)$ are sufficient to
work with; since, if an ODE is solved by $f^{(+)}_{\omega}(r)$ or
$f^{(-)}_{\omega}(r)$, it will also be solved by the superposition of those
solutions, $f_{\omega}(r)$. Therefore, in order to simplify the algebra, take
$f_{\omega}(r)=f^{(+)}_{\omega}(r)$ (equivalently
$f_{\omega}(r)=f^{(-)}_{\omega}(r)$ could have been chosen).. In the near-
horizon region (which corresponds to setting $r=(1+\tilde{\epsilon})r_{H}$),
Eq.(4.42) becomes

$ds_{3}^{2}=-\frac{r_{H}^{2}}{l^{2}}\,\tilde{\epsilon}\,(2+\tilde{\epsilon})\,dt^{2}\,+\,\frac{r_{H}^{2}}{l^{2}}\,(1+\tilde{\epsilon}^{2})\,dx^{2}\,+\,\frac{l^{2}}{\tilde{\epsilon}\,(2+\tilde{\epsilon})}\,d\tilde{\epsilon}^{2}\,,$
(5.83)

which is equivalent to Eq.(5.77) when $d=3$; Eq.(4.75) becomes

$\displaystyle\begin{aligned}
0\,&=\,-\,\partial_{t}^{2}X(t,\tilde{\epsilon})\,+\,\frac{r_{H}^{2}}{l^{4}}\,\frac{\tilde{\epsilon}\,(2+\tilde{\epsilon})}{(1+\tilde{\epsilon})^{2}}\,\partial_{\tilde{\epsilon}}\left(\tilde{\epsilon}\,(1+\tilde{\epsilon})^{2}(2+\tilde{\epsilon})\,{\partial_{\tilde{\epsilon}}}X(t,\tilde{\epsilon})\right)\\\\[5.69046pt]
&=-\,\partial_{t}^{2}X_{\text{reg}}(t,\tilde{\epsilon})+\frac{(\tilde{\epsilon}+2)}{4l^{4}(\tilde{\epsilon}+1)}\Big{[}-l^{2}\omega\left(l^{2}\omega(\tilde{\epsilon}+1)(\tilde{\epsilon}+2)-2ir_{H}\tilde{\epsilon}(3\tilde{\epsilon}+5)\right)X_{\text{reg}}(t,\tilde{\epsilon})\\\\[5.69046pt]
&\,\,\,\,\,\,\,+\,4r_{H}\tilde{\epsilon}\left(il^{2}\omega(\tilde{\epsilon}+1)(\tilde{\epsilon}+2)+4r_{H}\tilde{\epsilon}(\tilde{\epsilon}+2)+2r_{H}\right)\partial_{\tilde{\epsilon}}X_{\text{reg}}(t,\tilde{\epsilon})\,+\,4r_{H}^{2}\tilde{\epsilon}^{2}(\tilde{\epsilon}+1)(\tilde{\epsilon}+2)\partial^{2}_{\tilde{\epsilon}}X_{\text{reg}}(t,\tilde{\epsilon})\Big{]}\,,\end{aligned}$
(5.84)

where $r=(1+\tilde{\epsilon})r_{H}$ and
$\partial_{\tilde{\epsilon}}=r_{H}\partial_{r}$ are used in the first line;
and Eq.(4.77) becomes

$X(t,\tilde{\epsilon})\,=\,\left(\frac{\tilde{\epsilon}}{\tilde{\epsilon}+2}\right)^{\frac{il^{2}\omega}{2rH}}\frac{i\,l^{2}\omega+r_{H}\tilde{\epsilon}+r_{H}}{(\tilde{\epsilon}+1)\left(r_{H}+i\,l^{2}\omega\right)}\,e^{-i\omega
t}\,=:\,\tilde{\epsilon}^{-\frac{il^{2}\omega}{2rH}}\,X_{\text{reg}}(t,\tilde{\epsilon})\,,$
(5.85)

where $f_{\omega}(r)=f^{(+)}_{\omega}(r)$ (which is defined in Eq.(4.81)), and
the regular part of the solution is given by
$X_{\text{reg}}(t,\tilde{\epsilon})$. The regular equation of motion
(Eq.(5.84)) can now be series expanded in $\tilde{\epsilon}$ (around
$\tilde{\epsilon}=0$). However, in order to perform this expansion, an ansatz
for the expansion of $X_{\text{reg}}(t,\tilde{\epsilon})$ is required. Up to
$\mathcal{O}(\tilde{\epsilon})^{2}$, the simple ansatz
$X_{\text{reg}}(t,\tilde{\epsilon})=x_{0}(t)+\tilde{\epsilon}\,x_{1}(t)+\tilde{\epsilon}^{2}\,x_{2}(t)$
can be used. The regular equation of motion then becomes

$\displaystyle\begin{aligned}
0\,&=\left(-x_{0}^{\prime\prime}(t)-\omega^{2}x_{0}(t)\right)+\tilde{\epsilon}\left(\frac{\omega\left(-l^{2}\omega+5ir_{H}\right)}{l^{2}}x_{0}(t)+\frac{\left(2r_{H}+il^{2}\omega\right)^{2}}{l^{4}}x_{1}(t)-x_{1}^{\prime\prime}(t)\right)\\\\[5.69046pt]
&+\tilde{\epsilon}^{2}\left(-\frac{\omega\left(l^{2}\omega-2ir_{H}\right)}{4l^{2}}x_{0}(t)+\frac{\left(-l^{4}\omega^{2}+9il^{2}r_{H}\omega+14r_{H}^{2}\right)}{l^{4}}x_{1}(t)+\frac{\left(4r_{H}+il^{2}\omega\right)^{2}}{l^{4}}x_{2}(t)-x_{2}^{\prime\prime}(t)\right)+\mathcal{O}(\tilde{\epsilon})^{3},\end{aligned}$
(5.86)

where $x^{\prime\prime}(t)=d^{2}x/dt^{2}$. Series expanding the regular part
of the solution (Eq.(5.85)) in $\tilde{\epsilon}$ (around
$\tilde{\epsilon}=0$), yields

$\displaystyle\begin{aligned}
X_{\text{reg}}(t,\tilde{\epsilon})&=e^{-it\omega}\,2^{-\frac{il^{2}\omega}{2rH}}\left(1-\frac{il^{2}\omega\left(l^{2}\omega-5ir_{H}\right)}{4\,r_{H}\left(l^{2}\omega-
ir_{H}\right)}\tilde{\epsilon}-\frac{l^{2}\omega\left(l^{4}\omega^{2}-11il^{2}r_{H}\omega-34r_{H}^{2}\right)}{32\,r_{H}^{2}\left(l^{2}\omega-
ir_{H}\right)}\tilde{\epsilon}^{2}+\mathcal{O}(\tilde{\epsilon})^{3}\right)\\\\[5.69046pt]
&=:x_{0}(t)+\tilde{\epsilon}\,x_{1}(t)+\tilde{\epsilon}^{2}\,x_{2}(t)+\mathcal{O}(\tilde{\epsilon})^{3}\,.\end{aligned}$
(5.87)

As a consistency check, notice that the solution Eq.(5.87) solves the
transverse equation of motion Eq.(5.86) at each respective order of
$\tilde{\epsilon}$.

#####  5.3.2 Expanding the AdS3-Schwarzschild Metric in the Near-Horizon
Region

The transverse equation of motion and its solution which have been derived
from the AdS3-Schwarzschild metric and then series expanded to yield the near-
horizon limit (Eqs.(5.86, 5.87)) should match exactly with the transverse
equation of motion and its solution derived from the series expanded, near-
horizon AdS3-Schwarzschild metric. Going on physical intuition, it seems like
a good starting point would be to assume
$dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$dx^{2}\sim\mathcal{O}(\tilde{\epsilon})$ and check if the transverse equation
of motion and its solution derived from expanding this metric in the near-
horizon limit match with those found in the previous subsection. To this end,
assuming $dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$dx^{2}\sim\mathcal{O}(\tilde{\epsilon})$ results in an AdS3-Schwarzschild
metric

$ds_{3}^{2}=\frac{r_{H}^{2}}{l^{2}}\,dx^{2}\,-\,\frac{2r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,dt^{2}\,+\,\frac{l^{2}}{2\tilde{\epsilon}}\,d\tilde{\epsilon}^{2}\,+\,\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\,,$
(5.88)

where Eq.(5.82) for $d=3$ has been used. The near-horizon tortoise coordinate,
defined by setting $r=(1+\tilde{\epsilon})r_{H}$ in the definition of the
tortoise coordinate Eq.(4.39), can also be expanded and truncated around
$\tilde{\epsilon}=0$. To leading order it is sufficient to take999999It’s
proven that the leading order tortoise coordinate is given by Eq.(5.89) in
appendix (F).

$\tilde{\epsilon}_{*}:=\frac{l^{2}}{2\,r_{H}^{2}}\ln(\tilde{\epsilon})\,,$
(5.89)

where $r_{*}=r_{H}\tilde{\epsilon}_{*}$, and the number of dimensions $d=3$
has been set posthumously. Barring an integration constant of
$\mathcal{O}(\tilde{\epsilon})^{0}$ (specifically $-l^{2}/(2r_{H})\ln(2)$),
Eq.(5.89) agrees with expanding and truncating Eq.(4.40) after setting
$r=(1+\tilde{\epsilon})r_{H}$100100100See Mathematica Notebook [c]
(NearHorizonAdSd.nb) for details: access in appendix (G).. Inverting and
calculating the differential yields

$\tilde{\epsilon}=e^{\frac{2\,r_{H}^{2}}{l^{2}}\,\tilde{\epsilon}_{*}}\,,\,\,\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,\,\,\,d\tilde{\epsilon}=\frac{2\,r_{H}^{2}}{l^{2}}\,e^{\frac{2\,r_{H}^{2}}{l^{2}}\,\tilde{\epsilon}_{*}}\,d\tilde{\epsilon}_{*}\,.$
(5.90)

Eqs.(5.89, 5.90) are used to transform the expanded and truncated metric into
($t,\tilde{\epsilon}_{*}$) coordinates. The metric Eq.(5.88) becomes

$\displaystyle\begin{aligned}
ds_{3}^{2}&=2\,\frac{r_{H}^{2}}{l^{2}}\,\,e^{\frac{2\,r_{H}^{2}}{l^{2}}\,\tilde{\epsilon}_{*}}\left(-dt^{2}\,+\,r_{H}^{2}\,d\tilde{\epsilon}_{*}^{2}\right)\,+\,\frac{r_{H}^{2}}{l^{2}}\,dx^{2}\\\\[5.69046pt]
&=2\,\frac{r_{H}^{2}}{l^{2}}\,e^{\frac{2\,r_{H}\,r_{*}}{l^{2}}}\left(-dt^{2}\,+\,dr_{*}^{2}\right)\,+\,\frac{r_{H}^{2}}{l^{2}}\,dx^{2}\,,\end{aligned}$
(5.91)

where the final line is a conformally flat description of near-horizon
AdS3-Schwarzschild in $(t,r_{*})$ coordinates. It follows from the near-
horizon definition $r=(1+\tilde{\epsilon})r_{H}$, and Eqs.(5.89, 5.90) that

$r=r_{H}\,\left(1\,+\,e^{\frac{2\,r_{H}\,r_{*}}{l^{2}}}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,\,\,\,r_{*}=\frac{l^{2}}{2\,r_{H}}\,\ln\left(\frac{r}{r_{H}}-1\right)\,.$
(5.92)

As discussed in subsection (5.1.3), in the conformally flat $(t,r_{*})$
coordinate system the boundary conditions for the Limp Noodle in
AdS3-Schwarzschild are given by Eqs.(4.47, 4.49), which are respectively
analogous to the boundary conditions for the Limp Noodle in
$\mathbb{R}^{1+1}$, Eqs.(4.26, 4.27). Hence, the leading order solution for
the Limp Noodle in AdS3-Schwarzschild in conformal ($t,r_{*}$) coordinates can
be written down (Eq.(5.39)). Converting then to $(t,r)$ coordinates, the
string solution for the $d=3$ Limp Noodle in the near-horizon limit at leading
order is given by

$X^{\mu}_{\text{\text{AdS${}_{3}$-Sch}}}(t,\sigma)\,=\,\left(t,\,\begin{Bmatrix}r_{H}\Big{(}1\,+\,e^{\frac{2\,r_{H}}{l^{2}}\,\left(r_{s*}\,+\,\sigma\right)}\Big{)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
r_{H}\Big{(}1\,+\,e^{\frac{2\,r_{H}}{l^{2}}\,\left(r_{s*}\,+\,\sigma_{f}\,-\,t\right)}\Big{)},\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\,,\,0\right)^{\mu}\,,$ (5.93)

where Eq.(5.92) is used. The position of the fixed string endpoint attached to
the stretched horizon is given by

$r_{s*}\,=\,\frac{l^{2}}{2\,r_{H}}\,\ln\Big{(}\frac{r_{s}}{r_{H}}\,-\,1\Big{)}\,\,\,\,\,\,\,\,\,\,\text{or,
equivalently}\,\,\,\,\,\,\,\,\,\,\tilde{\epsilon}_{s*}\,=\,\frac{l^{2}}{2\,r_{H}^{2}}\,\ln(\epsilon)\,,$
(5.94)

since the stretched horizon is defined as $r_{s}=(1+\epsilon)r_{H}$, where
$0<\epsilon\leq\tilde{\epsilon}\ll 1$. The length of the string in tortoise
coordinates is given by

$\sigma_{f}\,=\,\frac{l^{2}}{2\,r_{H}}\,\ln\Big{(}\frac{r_{s}\,+\,\ell_{0}}{r_{H}}\,-\,1\Big{)}\,-\,r_{s*}\,\,\,\,\,\,\,\,\,\,\text{or,
equivalently}\,\,\,\,\,\,\,\,\,\,\sigma_{f}\,=\,\frac{l^{2}}{2\,r_{H}}\,\ln\left(\epsilon+\frac{\ell_{0}}{r_{H}}\right)\,-\,r_{H}\,\tilde{\epsilon}_{s*}\,,$
(5.95)

and the $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ parameter space regions are
still given by Eq.(5.33). Using the definition of the near-horizon limit
($r=(1+\tilde{\epsilon})r_{H}$) and $r_{*}=r_{H}\tilde{\epsilon}_{*}$, the
leading order string solution Eq.(5.93) can be converted into
$(t,\tilde{\epsilon})$ coordinates. Specifically,

$X^{\mu}_{\text{\text{AdS${}_{3}$-Sch}}}(t,\sigma)\,=\,\left(t,\,\begin{Bmatrix}e^{\frac{2\,r_{H}}{l^{2}}\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
e^{\frac{2\,r_{H}}{l^{2}}\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma_{f}\,-\,t\right)},\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\,,\,0\right)^{\mu}\,.$ (5.96)

Therefore, the identification
$\tilde{\epsilon}\,=\,e^{\frac{2\,r_{H}}{l^{2}}\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)}$
is made. The equation of motion for the transverse fluctuations in the near-
horizon limit of AdS3-Schwarzschild can be derived from Eq.(4.70),

$\displaystyle\begin{aligned}
0\,&=\,\partial_{a}\left(-\frac{1}{2\pi\alpha^{\prime}}\,\left(\sqrt{-g}\,g^{ab}\,G_{IJ}\right)\big{\rvert}_{X^{\mu}_{0}}{\partial_{b}}X^{J}\right)\\\\[5.69046pt]
&=\,\partial_{a}\left(\left(\sqrt{-g}\,g^{ab}\,\frac{r_{H}^{2}}{l^{2}}\right)\bigg{\rvert}_{X^{\mu}_{0}}{\partial_{b}}X(t,\tilde{\epsilon})\right)\,,\end{aligned}$
(5.97)

where the spacetime metric in the near-horizon limit $G_{\mu\nu}$ is given
explicitly by

$G_{\mu\nu}\,=\,\begin{bmatrix}-\frac{2\,r_{H}^{2}\,\tilde{\epsilon}}{l^{2}}&0&0\\\
0&\frac{l^{2}}{2\,\tilde{\epsilon}}&0\\\
0&0&\frac{r_{H}^{2}}{l^{2}}\end{bmatrix}\,,$ (5.98)

where $\mu,\nu$ index over the directions $(t,\tilde{\epsilon},x)$. The
explicit entries of the induced worldsheet metric are calculated from
Eq.(4.4)101101101This calculation is similar to Eq.(C.4) where the induced
metric is calculated explicitly in ($t,r$) coordinates (see appendix (C)).. In
the near-horizon limit, for $d=3$, the leading order induced metric in
($t,\tilde{\epsilon}$) coordinates is given by

$g_{ab}\big{\rvert}_{X^{\mu}_{0}}\,:=\,\begin{bmatrix}g_{tt}&g_{t\sigma}\\\
g_{\sigma
t}&g_{\sigma\sigma}\end{bmatrix}\Bigg{\rvert}_{X^{\mu}_{0}}\,=\,\begin{bmatrix}G_{tt}&0\\\
0&\tilde{\epsilon}^{\prime
2}\,G_{\tilde{\epsilon}\tilde{\epsilon}}\end{bmatrix}\,=\,\begin{bmatrix}-\frac{2\,r_{H}^{2}\,\tilde{\epsilon}}{l^{2}}&0\\\
0&\tilde{\epsilon}^{\prime
2}\,\frac{l^{2}}{2\,\tilde{\epsilon}}\end{bmatrix}\,,$ (5.99)

where Eq.(5.98) is used, and
$\tilde{\epsilon}^{\prime}=\partial_{\sigma}\tilde{\epsilon}$. Hence, its
inverse is given by

$g^{ab}\big{\rvert}_{X^{\mu}_{0}}\,:=\,\begin{bmatrix}g^{tt}&g^{t\sigma}\\\
g^{\sigma
t}&g^{\sigma\sigma}\end{bmatrix}\Bigg{\rvert}_{X^{\mu}_{0}}\,=\,\frac{1}{\det\big{(}g_{ab}\rvert_{X^{\mu}_{0}}\big{)}}\,\begin{bmatrix}\tilde{\epsilon}^{\prime
2}\,G_{\tilde{\epsilon}\tilde{\epsilon}}&0\\\
0&G_{tt}\end{bmatrix}\,=\,\begin{bmatrix}-\frac{l^{2}}{2\,r_{H}^{2}\,\tilde{\epsilon}}&0\\\
0&\frac{2\,\tilde{\epsilon}}{l^{2}\,\tilde{\epsilon}^{\prime
2}}\end{bmatrix}\,,$ (5.100)

where $\det\big{(}g_{ab}\rvert_{X^{\mu}_{0}}\big{)}$ is calculated from
Eq.(5.99),

$g\rvert_{X^{\mu}_{0}}\,\equiv\,\det\big{(}g_{ab}\rvert_{X^{\mu}_{0}}\big{)}\,=\,\tilde{\epsilon}^{\prime
2}\,G_{\tilde{\epsilon}\tilde{\epsilon}}\,G_{tt}\,=\,\bigg{(}\tilde{\epsilon}^{\prime
2}\,\frac{l^{2}}{2\,\tilde{\epsilon}}\bigg{)}\,\bigg{(}-\frac{2\,r_{H}^{2}}{l^{2}}\,\tilde{\epsilon}\bigg{)}\,=\,-r_{H}^{2}\,\tilde{\epsilon}^{\prime
2}\,.$ (5.101)

Expanding the indices of Eq.(5.97) using Eqs.(5.99)-(5.101), the transverse
equation of motion become

$\displaystyle\begin{aligned}
0\,&=\,\partial_{\sigma}\left(\bigg{(}\sqrt{-g}\,g^{\sigma\sigma}\,\frac{r_{H}^{2}}{l^{2}}\bigg{)}\bigg{\rvert}_{X^{\mu}_{0}}\,\,{\partial_{\sigma}}X(t,\sigma)\right)\,+\,\partial_{t}\left(\bigg{(}\sqrt{-g}\,g^{tt}\,\frac{r_{H}^{2}}{l^{2}}\bigg{)}\bigg{\rvert}_{X^{\mu}_{0}}\,\,{\partial_{t}}X(t,\sigma)\right)\\\\[5.69046pt]
&=\,{\partial_{\sigma}}\left(r_{H}\tilde{\epsilon}^{\prime}\left(\frac{2\,\tilde{\epsilon}}{l^{2}\,\tilde{\epsilon}^{\prime
2}}\right)\,\frac{r_{H}^{2}}{l^{2}}\,{\partial_{\sigma}}X(t,\sigma)\right)\,+\,\partial_{t}\left(r_{H}\tilde{\epsilon}^{\prime}\left(-\frac{l^{2}}{2\,r_{H}^{2}\,\tilde{\epsilon}}\right)\,\frac{r_{H}^{2}}{l^{2}}\,{\partial_{t}}X(t,\sigma)\right)\,\\\\[5.69046pt]
&=\,-\,\partial_{t}^{2}X(t,\sigma)\,+\,\frac{4\,r_{H}^{2}}{l^{4}}\,\tilde{\epsilon}\,\frac{1}{\partial_{\sigma}\tilde{\epsilon}}\,\partial_{\sigma}\left(\frac{1}{\partial_{\sigma}\tilde{\epsilon}}\,\tilde{\epsilon}\,{\partial_{\sigma}}X(t,\sigma)\right)\,\\\\[5.69046pt]
&=\,-\,\partial_{t}^{2}X(t,\tilde{\epsilon})\,+\,\frac{4\,r_{H}^{2}}{l^{4}}\,\tilde{\epsilon}\,\partial_{\tilde{\epsilon}}\big{(}\tilde{\epsilon}\,{\partial_{\tilde{\epsilon}}}X(t,\tilde{\epsilon})\big{)}\,.\end{aligned}$
(5.102)

where the last line is written completely in terms of near-horizon spacetime
coordinates ($t,\tilde{\epsilon}$) by differentiating the identification
$\tilde{\epsilon}\,=\,e^{\frac{2\,r_{H}}{l^{2}}\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)}$,

$\partial_{\sigma}\,=\,2\,\frac{r_{H}}{l^{2}}\,e^{2\frac{r_{H}}{l^{2}}\,\big{(}r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\big{)}}\,\partial_{\tilde{\epsilon}}\,=\,\,2\,\frac{r_{H}}{l^{2}}\,\tilde{\epsilon}\,\partial_{\tilde{\epsilon}}\,.$
(5.103)

In terms of $X_{\text{reg}}(t,\tilde{\epsilon})$ the equation of motion is
given by

$0\,=\,-\partial_{t}^{2}X_{\text{reg}}(t,\tilde{\epsilon})+\frac{4r_{H}\tilde{\epsilon}\left(r_{H}\tilde{\epsilon}\,\partial_{\tilde{\epsilon}}^{2}X_{\text{reg}}(t,\tilde{\epsilon})+\left(r_{H}+il^{2}\omega\right)\partial_{\tilde{\epsilon}}X_{\text{reg}}(t,\tilde{\epsilon})\right)}{l^{4}}-\omega^{2}X_{\text{reg}}(t,\tilde{\epsilon})\,,$
(5.104)

where the definition of the regular solution in Eq.(5.85) has been used. In
order to check whether, at leading order, the regular equation of motion
agrees with the series expanded regular equation of motion (Eq.(5.86)), the
leading order ansatz $X_{\text{reg}}(t,\tilde{\epsilon})=x_{0}(t)$ should be
used. The leading order contribution to Eq.(5.104) then becomes

$0\,=\,-x_{0}^{\prime\prime}(t)-\omega^{2}x_{0}(t)\,,$ (5.105)

which agrees exactly with the leading order contribution of Eq.(5.86). This
proves that the expanded near-horizon metric is equivalent to the near-horizon
metric at leading order in AdS3-Schwarzschild. The expanded near-horizon
metric given in Eq.(5.88) is therefore correct, and
$dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$dx^{2}\sim\mathcal{O}(\tilde{\epsilon})$. As a final check, the calculations
in this subsection are repeated assuming
$dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$dx^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$. The equation of motion found
is,

$\displaystyle\begin{aligned}
0\,&\,\,\,=\,\,\,-\,\partial_{t}^{2}\,X(t,\tilde{\epsilon})\,+\,\frac{4\,r_{H}^{2}}{l^{4}}\,\frac{\tilde{\epsilon}}{\left(1+\,2\,\tilde{\epsilon}\right)}\,\partial_{\tilde{\epsilon}}\big{(}\tilde{\epsilon}\,\left(1+\,2\,\tilde{\epsilon}\right)\,{\partial_{\tilde{\epsilon}}}X(t,\tilde{\epsilon})\big{)}\\\\[5.69046pt]
&\,\,\,=\,\,\,-\,\partial_{t}^{2}X_{\text{reg}}(t,\tilde{\epsilon})+\frac{4r_{H}^{2}}{l^{4}}\tilde{\epsilon}^{2}\partial^{2}_{\tilde{\epsilon}}X_{\text{reg}}(t,\tilde{\epsilon})+\frac{4r_{H}\tilde{\epsilon}\,\partial_{\tilde{\epsilon}}X_{\text{reg}}(t,\tilde{\epsilon})\left(il^{2}\omega(2\tilde{\epsilon}+1)+4r_{H}\tilde{\epsilon}+r_{H}\right)}{l^{4}(2\tilde{\epsilon}+1)}\\\\[5.69046pt]
&\,\,\,\,\,\,\,\,\,\,\,\,\,-\,\frac{\omega
X_{\text{reg}}(t,\tilde{\epsilon})\left(l^{2}\omega(2\tilde{\epsilon}+1)-4ir_{H}\tilde{\epsilon}\right)}{l^{2}(2\tilde{\epsilon}+1)}\\\\[5.69046pt]
&\stackrel{{\scriptstyle
LO}}{{\longrightarrow}}\,-\,x_{0}^{\prime\prime}(t)+\left(-\omega+\frac{4ir_{H}}{l^{2}}\frac{\tilde{\epsilon}}{(2\tilde{\epsilon}+1)}\right)\omega
x_{0}(t)\,,\end{aligned}$ (5.106)

which does not match the leading order contribution of Eq.(5.86).

#####  5.3.3 The Diffusion Constant in AdSd-Schwarzschild

In the previous two subsections it was proven that in order for a near-horizon
expansion of the metric to be consistent, the leading order metric is taken to
$\mathcal{O}(\tilde{\epsilon})$ – where
$dt^{2}\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$dx^{2}\sim\mathcal{O}(\tilde{\epsilon})$ (equivalently
$t\sim\mathcal{O}(\tilde{\epsilon})^{0}$ and
$x\sim\mathcal{O}(\sqrt{\tilde{\epsilon}})$). Generalising to $d$ dimensions
this holds true, where $X_{I}$ (instead of $x$) denotes the transverse
directions in AdSd-Schwarzschild. Hence the leading order, near-horizon AdSd-
Schwarzschild metric is given by Eq.(5.82).

From the near-horizon metric, which is conformal in $(t,r_{*})$ coordinates,
the leading order string solution followed by the transverse equations of
motion can be found. A similar method to subsection (5.3.2) is followed. The
near-horizon tortoise coordinate, defined by setting
$r=(1+\tilde{\epsilon})r_{H}$ in the definition of the tortoise coordinate
Eq.(4.39), can be expanded and truncated around $\tilde{\epsilon}=0$. To
leading order it is sufficient to take102102102It’s proven that the leading
order tortoise coordinate is given by Eq.(5.107) in appendix (F).

$\tilde{\epsilon}_{*}:=\frac{l^{2}}{r_{H}^{2}}\,\frac{\ln(\tilde{\epsilon})}{(d-1)}\,,$
(5.107)

where $r_{*}=r_{H}\tilde{\epsilon}_{*}$. Inverting and calculating the
differential yields

$\tilde{\epsilon}=e^{\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,\tilde{\epsilon}_{*}}\,,\,\,\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,\,\,\,d\tilde{\epsilon}=\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,e^{\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,\tilde{\epsilon}_{*}}\,d\tilde{\epsilon}_{*}\,.$
(5.108)

The definition of the near-horizon tortoise coordinate, its inverse and
differential (Eqs.(5.107, 5.108)) can be used to convert the metric Eq.(5.82)
into $(t,\tilde{\epsilon}_{*})$ coordinates

$\displaystyle\begin{aligned}
ds_{d}^{2}&=\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,e^{\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,\tilde{\epsilon}_{*}}\left(-dt^{2}\,+\,r_{H}^{2}\,d\tilde{\epsilon}_{*}^{2}\right)\,+\,\frac{r_{H}^{2}}{l^{2}}\,d\vec{X}_{I}^{2}\\\\[5.69046pt]
&=\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,e^{\frac{r_{H}\,r_{*}}{l^{2}}\,(d-1)}\left(-dt^{2}\,+\,dr_{*}^{2}\right)\,+\,\frac{r_{H}^{2}}{l^{2}}\,d\vec{X}_{I}^{2}\,.\end{aligned}$
(5.109)

From the final line in Eq.(5.109) notice that the metric – like the near-
horizon AdS3-Schwarzschild metric (Eq.(5.91)) – is conformal in $(t,r_{*})$
coordinates. It follows from the near-horizon definition
$r=(1+\tilde{\epsilon})r_{H}$, and Eqs.(5.107, 5.108) that

$r=r_{H}\,\left(1\,+\,e^{\frac{r_{H}\,r_{*}}{l^{2}}\,(d-1)}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,\,\,\,r_{*}=\frac{l^{2}}{(d-1)\,r_{H}}\,\ln\left(\frac{r}{r_{H}}\,-\,1\right)\,,$
(5.110)

which reduces down to Eq.(5.92) for $d=3$. From the metric Eq.(5.109) the
method laid out in subsection (5.1.3) can be followed in order to write down
the leading order solution for the Limp Noodle in AdSd-Schwarzschild in
conformal ($t,r_{*}$) coordinates, which is given by Eq.(5.39)103103103This
equation – derived from the worldsheet parameter space partitioning of the
test string in $\mathbb{R}^{1,1}$, Eq.(5.34) – is independent of $d$, the
number of spacetime dimensions.. Converting then to $(t,r)$ coordinates, the
string solution for the Limp Noodle in the near-horizon limit at leading order
is given by

$X^{\mu}_{\text{\text{AdS${}_{d}$-Sch}}}(t,\sigma)\,=\,\left(t,\,\begin{Bmatrix}r_{H}\Big{(}1\,+\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{s*}\,+\,\sigma\right)}\Big{)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
r_{H}\Big{(}1\,+\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{s*}\,+\,\sigma_{f}\,-\,t\right)}\Big{)},\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\,,\,0\right)^{\mu}\,,$ (5.111)

where Eq.(5.110) is used104104104It will be much harder to find next-to-
leading order corrections to the string solution in the near-horizon limit,
since if the near-horizon metric includes higher order terms
($\mathcal{O}(\tilde{\epsilon})^{2}$ terms) it will no longer be conformal in
($t,r_{*}$) coordinates.. The position of the fixed string endpoint attached
to the stretched horizon is given by

$r_{s*}\,=\,\frac{l^{2}}{(d-1)\,r_{H}}\,\ln\Big{(}\frac{r_{s}}{r_{H}}\,-\,1\Big{)}\,\,\,\,\,\,\,\,\,\,\text{or,
equivalently}\,\,\,\,\,\,\,\,\,\,\tilde{\epsilon}_{s*}\,=\,\frac{l^{2}}{(d-1)\,r_{H}}\,\ln(\epsilon)\,,$
(5.112)

since the stretched horizon is defined as $r_{s}=(1+\epsilon)r_{H}$, where
$0<\epsilon\leq\tilde{\epsilon}\ll 1$. The length of the string in tortoise
coordinates is given by

$\sigma_{f}\,=\,\frac{l^{2}}{(d-1)\,r_{H}}\,\ln\Big{(}\frac{r_{s}\,+\,\ell_{0}}{r_{H}}\,-\,1\Big{)}\,-\,r_{s*}\,\,\,\,\,\,\,\,\text{or,
equivalently}\,\,\,\,\,\,\,\,\sigma_{f}\,=\,\frac{l^{2}}{(d-1)\,r_{H}}\,\ln\left(\epsilon+\frac{\ell_{0}}{r_{H}}\right)\,-\,r_{H}\,\tilde{\epsilon}_{s*}\,,$
(5.113)

and the $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ parameter space regions are
still given by Eq.(5.33). Using the definition of the near-horizon limit
($r=(1+\tilde{\epsilon})r_{H}$) and $r_{*}=r_{H}\tilde{\epsilon}_{*}$, the
leading order string solution Eq.(5.111) can be converted into
$(t,\tilde{\epsilon})$ coordinates. Specifically,

$X^{\mu}_{\text{\text{AdS${}_{d}$-Sch}}}(t,\sigma)\,=\,\left(t,\,\begin{Bmatrix}e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{1}\\\
e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma_{f}\,-\,t\right)},\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(t,\sigma)\,\in\,\mathcal{M}_{2}\end{Bmatrix}\,,\,0\right)^{\mu}\,,$ (5.114)

where $\tilde{\epsilon}_{s*}$ and $\sigma_{f}$ are defined in Eqs.(5.112,
5.113). Therefore, the identification
$\tilde{\epsilon}\,=\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)}$
is made. The equations of motion for the transverse fluctuations in the near-
horizon limit of AdSd-Schwarzschild can be derived from Eq.(4.67), and are
equivalent to Eq.(5.97) with the generalised spacetime metric in the near-
horizon limit ($G_{\mu\nu}$) given by

$G_{\mu\nu}\,=\,\begin{bmatrix}-\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,\tilde{\epsilon}&0&0&0&0&0\\\
0&\frac{l^{2}}{(d-1)}\,\frac{1}{\tilde{\epsilon}}&0&0&0&0\\\
0&0&\frac{r_{H}^{2}}{l^{2}}&0&0&0\\\ 0&0&0&\frac{r_{H}^{2}}{l^{2}}\
&\ddots&0\\\ 0&0&0&\ddots&\ddots&0\\\
0&0&0&0&0&\frac{r_{H}^{2}}{l^{2}}\end{bmatrix}\,,$ (5.115)

where $\mu,\nu$ index over $\left(t,\,r,\,2,\,...\,,\,d-1\right)$. The
explicit entries of the induced worldsheet metric are calculated from
Eq.(4.4). In the near-horizon limit, the leading order induced metric in
($t,\tilde{\epsilon}$) coordinates is given by

$g_{ab}\big{\rvert}_{X^{\mu}_{0}}\,:=\,\begin{bmatrix}g_{tt}&g_{t\sigma}\\\
g_{\sigma
t}&g_{\sigma\sigma}\end{bmatrix}\Bigg{\rvert}_{X^{\mu}_{0}}\,=\,\begin{bmatrix}G_{tt}&0\\\
0&\tilde{\epsilon}^{\prime
2}\,G_{\tilde{\epsilon}\tilde{\epsilon}}\end{bmatrix}\,=\,\begin{bmatrix}-\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,\tilde{\epsilon}&0\\\
0&\tilde{\epsilon}^{\prime
2}\,\frac{l^{2}}{(d-1)}\,\frac{1}{\tilde{\epsilon}}\end{bmatrix}\,,$ (5.116)

where Eq.(5.115) is used, and
$\tilde{\epsilon}^{\prime}=\partial_{\sigma}\tilde{\epsilon}$. Hence, its
inverse is given by

$g^{ab}\big{\rvert}_{X^{\mu}_{0}}\,:=\,\begin{bmatrix}g^{tt}&g^{t\sigma}\\\
g^{\sigma
t}&g^{\sigma\sigma}\end{bmatrix}\Bigg{\rvert}_{X^{\mu}_{0}}\,=\,\frac{1}{\det\big{(}g_{ab}\rvert_{X^{\mu}_{0}}\big{)}}\,\begin{bmatrix}\tilde{\epsilon}^{\prime
2}\,G_{\tilde{\epsilon}\tilde{\epsilon}}&0\\\
0&G_{tt}\end{bmatrix}\,=\,\begin{bmatrix}-\frac{l^{2}}{(d-1)r_{H}^{2}\,\tilde{\epsilon}}&0\\\
0&\frac{(d-1)\,\tilde{\epsilon}}{l^{2}\,\tilde{\epsilon}^{\prime
2}}\end{bmatrix}\,,$ (5.117)

where $\det\big{(}g_{ab}\rvert_{X^{\mu}_{0}}\big{)}$ is calculated from
Eq.(5.116),

$g\rvert_{X^{\mu}_{0}}\,\equiv\,\det\big{(}g_{ab}\rvert_{X^{\mu}_{0}}\big{)}\,=\,\tilde{\epsilon}^{\prime
2}\,G_{\tilde{\epsilon}\tilde{\epsilon}}\,G_{tt}\,=\,\bigg{(}\tilde{\epsilon}^{\prime
2}\frac{l^{2}}{(d-1)}\,\frac{1}{\tilde{\epsilon}}\bigg{)}\,\bigg{(}-\frac{r_{H}^{2}}{l^{2}}\,(d-1)\,\tilde{\epsilon}\bigg{)}\,=\,-r_{H}^{2}\,\tilde{\epsilon}^{\prime
2}\,.$ (5.118)

Expanding the indices of Eq.(5.97) using Eqs.(5.116)-(5.118), the transverse
equation of motion become

$\displaystyle\begin{aligned}
0\,&=\,\partial_{\sigma}\left(\bigg{(}\sqrt{-g}\,g^{\sigma\sigma}\,\frac{r_{H}^{2}}{l^{2}}\bigg{)}\bigg{\rvert}_{X^{\mu}_{0}}\,\,{\partial_{\sigma}}X^{I}(t,\sigma)\right)\,+\,\partial_{t}\left(\bigg{(}\sqrt{-g}\,g^{tt}\,\frac{r_{H}^{2}}{l^{2}}\bigg{)}\bigg{\rvert}_{X^{\mu}_{0}}\,\,{\partial_{t}}X^{I}(t,\sigma)\right)\\\\[5.69046pt]
&=\,\partial_{\sigma}\left(r_{H}\tilde{\epsilon}^{\prime}\left(\frac{(d-1)\,\tilde{\epsilon}}{l^{2}\,\tilde{\epsilon}^{\prime
2}}\right)\,\frac{r_{H}^{2}}{l^{2}}\,{\partial_{\sigma}}X^{I}(t,\sigma)\right)\,+\,\partial_{t}\left(r_{H}\tilde{\epsilon}^{\prime}\left(-\frac{l^{2}}{(d-1)r_{H}^{2}}\,\frac{1}{\tilde{\epsilon}}\right)\,\frac{r_{H}^{2}}{l^{2}}\,{\partial_{t}}X^{I}(t,\sigma)\right)\,\\\\[5.69046pt]
&=\,-\,\partial_{t}^{2}X^{I}(t,\tilde{\epsilon})\,+\,\frac{r_{H}^{2}}{l^{4}}\,(d-1)^{2}\,\tilde{\epsilon}\,\partial_{\tilde{\epsilon}}\big{(}\tilde{\epsilon}\,{\partial_{\tilde{\epsilon}}}X^{I}(t,\tilde{\epsilon})\big{)}\,,\end{aligned}$
(5.119)

where $X^{I}$ denotes the transverse direction ($I\in(2,\,3,\,...\,,\,d-1)$);
and the last line is written completely in terms of near-horizon spacetime
coordinates ($t,\tilde{\epsilon}$) by differentiating the identification
$\tilde{\epsilon}\,=\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)}$,

$\displaystyle\begin{aligned}
&d\tilde{\epsilon}\,=\,(d-1)\,\frac{r_{H}}{l^{2}}\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\big{(}r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\big{)}}\,d\sigma\\\\[5.69046pt]
&\Rightarrow\,\partial_{\sigma}\,=\,(d-1)\,\frac{r_{H}}{l^{2}}\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\big{(}r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\big{)}}\,\partial_{\tilde{\epsilon}}\\\\[5.69046pt]
&\Rightarrow\,\partial_{\sigma}\,=\,\,(d-1)\,\frac{r_{H}}{l^{2}}\,\tilde{\epsilon}\,\partial_{\tilde{\epsilon}}\,.\end{aligned}$
(5.120)

Eq.(5.120) can also be used to convert the equations of motion Eq.(5.119) to
parameter space coordinates $(t,\sigma)$,

$\displaystyle\begin{aligned}
0\,&=\,-\,\partial_{t}^{2}\,X^{I}(t,\sigma)\,+\,\frac{r_{H}^{2}}{l^{4}}\,(d-1)^{2}\,\tilde{\epsilon}\,\frac{l^{2}}{(d-1)\,r_{H}}\,\frac{1}{\tilde{\epsilon}}\,\partial_{\sigma}\left(\tilde{\epsilon}\,\frac{l^{2}}{(d-1)\,r_{H}}\,\frac{1}{\tilde{\epsilon}}\,\partial_{\sigma}X^{I}(t,\sigma)\right)\\\\[5.69046pt]
&=\,-\,\partial_{t}^{2}\,X^{I}(t,\sigma)\,+\,\partial_{\sigma}^{2}\,X^{I}(t,\sigma)\,.\end{aligned}$
(5.121)

The near-horizon string equations of motion for the transverse fluctuations on
the Limp Noodle is given by wave equation. Remember that the small virtuality
limit of the $d=3$ Limp Noodle also yielded the wave equation as the
transverse equation of motion (Eq.(5.51))105105105The solution to Eq.(5.121)
is therefore given by Eq.(5.52), or alternatively by
$f_{\omega}^{(\pm)}(\tilde{\epsilon})=\tilde{\epsilon}^{\pm\frac{1}{(d-1)}\frac{il^{2}\omega}{r_{H}}}$,
where the identification
$\tilde{\epsilon}\,=\,e^{\frac{r_{H}}{l^{2}}\,(d-1)\,\left(r_{H}\,\tilde{\epsilon}_{s*}\,+\,\sigma\right)}$
is used to convert from parameter space coordinates ($t,\sigma$) to near-
horizon spacetime coordinates ($t,\tilde{\epsilon}$). Notice when
$f_{\omega}^{(+)}(\tilde{\epsilon})$ is used as the solution (as in subsection
(5.3.1)), $f_{\omega}^{(-)}(\tilde{\epsilon})$ is used in the definition of
$X_{\text{reg}}(t,\tilde{\epsilon})$, Eq.(5.85).. Since this is true,
repeating the calculations laid out in subsection (5.2.2) is sufficient to
calculate $s^{2}(t;d)$. The only change is that $\beta$ scales with the number
of spacetime dimensions, so Eq.(5.56) needs to be adapted by adding in the
appropriate dimensionally dependent factor. In order to calculate this, notice
that $\beta$ enters the calculation of $s^{2}_{\text{small}}(t)$ in subsection
(5.2.2) through the use of the normalization constant $A_{\omega}$. For $d\geq
3$, Eq.(4.101) is generalised

$A_{\omega}(d)\,:=\,\frac{l}{r_{H}}\,\sqrt{\frac{\pi\alpha^{\prime}}{\omega}}\,=\,\frac{(d-1)\beta}{4\,\sqrt{\pi\omega}\,\lambda^{1/4}}\,\equiv\,\frac{(d-1)}{2}\,A_{\omega}\,,$
(5.122)

where the second equality follows from using the definition of the AdS radius
of curvature $l$ (Eq.(4.65)) and the Hawking temperature (Eq.(4.36)). Using
the generalised normalization constant $A_{\omega}(d)$, the expectation value
of the position of the boundary endpoint at two different times (Eq.(5.43))
becomes106106106Note that the Bose-Einstein distribution Eq.(4.103) does not
carry any dimensional dependence.

$\displaystyle\begin{aligned}
\langle:\mathrel{\hat{X}_{\text{end}}(t_{1};d)\hat{X}_{\text{end}}(t_{2};d)}:\rangle&=\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\text{Re}\left(f_{\omega}(\sigma_{f}-t_{1})f^{\,*}_{\omega}(\sigma_{f}-t_{2})e^{-i\omega(t_{1}-t_{2})}\right)\\\\[5.69046pt]
&\equiv\frac{(d-1)^{2}}{4}\langle:\mathrel{\hat{X}_{\text{end}}(t_{1})\hat{X}_{\text{end}}(t_{2})}:\rangle\,,\end{aligned}$
(5.123)

while the string falling endpoint’s mean-squared transverse displacement,
Eq.(5.47), becomes

$\displaystyle\begin{aligned}
s^{2}(t;d)&=\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\big{\rvert}f_{\omega}(\sigma_{f}-t)-f_{\omega}(\sigma_{f})e^{i\omega
t}\big{\rvert}^{2}\\\\[5.69046pt]
&\equiv\frac{(d-1)^{2}}{4}s^{2}(t)\,,\end{aligned}$ (5.124)

due to the factors of $A_{\omega}A_{\omega^{\prime}}$ appearing in the
calculations of both
$\langle:\mathrel{\hat{X}_{\text{end}}(t_{1};d)\hat{X}_{\text{end}}(t_{2};d)}:\rangle$
and $s^{2}(t;d)$.

In subsection (5.2.2), the general solution to the wave equation,
$f_{\omega}(\sigma)$, is given by Eqs.(5.52)-(5.55). In order to calculate the
falling string endpoint’s mean-squared transverse displacement in the small
virtuality limit, Eq.(5.55) is substituted into Eq.(5.124). Specifically,

$\displaystyle\begin{aligned}
s^{2}(t;d)\,&=\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\Big{\rvert}2e^{i\omega(r_{s*}+\sigma_{f})}\cos\big{(}\omega
t\big{)}-2e^{i\omega\,(r_{s*}+\sigma_{f})}e^{i\omega
t}\Big{\rvert}^{2}\\\\[5.69046pt]
&=\frac{(d-1)^{2}\beta^{2}}{4\pi^{2}\sqrt{\lambda}}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{1}{e^{\beta\omega}-1}\sin^{2}(\omega
t)\\\\[5.69046pt]
&\equiv\frac{(d-1)^{2}}{4}s^{2}_{\text{small}}(t)\\\\[5.69046pt]
&=\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\,\ln\left(\frac{\beta}{2\pi
t}\sinh\left(\frac{2\pi t}{\beta}\right)\right)\,,\end{aligned}$ (5.125)

where Eq.(5.56) is used in the third line, and Eq.(5.58) in the final line. In
order to calculate the diffusion coefficient, the late time dynamics of
Eq.(5.125) must be explored. Expanding in powers of $1/k=\beta/t$ yields

$\displaystyle\begin{aligned}
s^{2}(t;d)&\,\,\,=\,\,\,\,\,\,\,\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\,\Big{[}\ln\big{(}\sinh\big{(}2\pi
k\big{)}\big{)}\,+\,\ln\Big{(}\frac{1}{2\pi k}\Big{)}\Big{]}\\\\[5.69046pt]
&\,\stackrel{{\scriptstyle(\beta\ll
t)}}{{\longrightarrow}}\,\,\frac{(d-1)^{2}\beta
t}{8\pi\sqrt{\lambda}}\,+\,\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\ln\bigg{(}\frac{\beta}{4\,\pi\,t}\bigg{)}\,+\,\mathcal{O}\Big{(}\beta^{2}\,e^{\frac{-4\pi
t}{\beta}}\Big{)}\,,\end{aligned}$ (5.126)

where the calculation steps in Eq.(5.60) have been followed closely. Since
$s^{2}(t;d)\sim t$, the late time dynamics exhibit diffusive behaviour. From
Eq.(3.21) it is expected that $s^{2}_{\text{small}}(t)\,=\,2\,D\,t$ at late
times, where $D$ is the diffusion coefficient. Comparing this to Eq.(5.126) it
is easy to see that the diffusion coefficient is given by

$D_{\text{LQ}}^{\text{AdS$d$}}(d)\,=\,\frac{(d-1)^{2}\beta}{16\pi\sqrt{\lambda}}\,.$
(5.127)

For $d=3$, this reduces down to Eq.(5.61) as expected.

Moerman et al. in [52] present an expression for the light quark diffusion
coefficient in $d$ dimensions which agrees with Eq.(5.127). However, their
method of expanding the near-horizon AdSd-Schwarzschild metric (described in
subsection (3.3) of [52]) is inconsistent. The near-horizon AdSd-Schwarzschild
metric given in Eq.(3.48) of [52] is dimensionally incorrect107107107Since
$r_{H}^{2}$, $l^{2}$ $dx_{d-2}^{2}$ and $dt^{2}$ all go like $Energy^{-1}$,
and $\tilde{\epsilon}$ (as well as $d\tilde{\epsilon}^{2}$) are defined as
dimensionless quantities; the first term in the metric Eq.(3.48) of [52] goes
like $Energy^{-2}$, the second term also goes like $Energy^{-2}$, but the
third term is dimensionless. In juxtaposition, all terms in the correct,
leading order near-horizon AdSd-Schwarzschild metric given in this subsection
(Eq.(5.82)) consistently go like $Energy^{-2}$.108108108A note on terminology:
$\epsilon$ in [52] corresponds to $\tilde{\epsilon}$ in this article.. Using
the near-horizon tortoise coordinate Eq.(5.107) (no attempt is made to explain
how the order of truncation is consistent with the leading order near-horizon
metric109109109The paper [52] simply states ”Expanding (and truncating) each
term in the AdSd metric, Eq.(2.32), to lowest non-vanishing order in
$\epsilon$”. There does not appear to have been an attempt to examine if each
term is consistently taken to the same order of $\epsilon$ ($\tilde{\epsilon}$
– see footnote 108).), the metric in Eq.(3.48) of [52] is converted to
($t,\tilde{\epsilon}_{*}$) coordinates, and found to be conformal in this
system110110110The correct near-horizon AdSd-Schwarzschild metric (Eq.(5.82))
is actually conformal in ($t,r_{*}$) coordinates (as shown in Eq.(5.109))..
Hence, the leading order string solution can be written down in
($t,\tilde{\epsilon}_{*}$) coordinates, from which Moerman et al. proceeded to
finding the transverse equations of motion – this, coincidentally, being the
wave equation. Therefore, although there is agreement between the final
results, Eq.(5.127) and the diffusion coefficient in $d$ dimensions derived by
Moerman et al., the authors conclude that the calculations in this subsection
arriving at Eq.(5.127), present the first complete, consistent derivation of
the light quark diffusion coefficient in $d$ dimensions.

###  6 Drag Force in AdS/CFT

Besides the test string set-ups depicted in figures (2) and (3) (see section
(4), page 2, and section (5), page 3, respectively), there are many other
string set-ups – illuminating various aspects or properties of the thermal
plasma in the boundary theory – that could be considered. One such set-up is a
trailing string in AdS5-Schwarzschild which models an infinitely massive probe
quark111111111Due to its infinite mass such a quark would not undergo Brownian
motion in thermal plasma. moving with a constant velocity $v$ in a
$\mathcal{N}=4$ SYM thermal plasma112112112The constant velocity is measured
with respect to the rest frame of the plasma.. This was considered by Gubser
[22] and Herzog et al. [35] independently in 2006, who aimed to approximately
explain jet-quenching – the phenomenon whereby energy loss is experienced by
high-energy quarks travelling through the quark-gluon plasma113113113Around
the same time, studies examining trailing strings in the AdS spacetime in
order to investigate the dissipative and diffusive behaviour of a massive
quark moving through a field theory plasma were common (see [36, 37, 38, 39,
40, 41]).. The succeeding subsection follows the calculations of [22, 35] in
using the AdS/CFT correspondence to calculate the drag force experienced by a
massive probe quark in the $\mathcal{N}=4$ SYM thermal medium. The string set-
up used is depicted in figure (9). Further, in subsection (6.2), the set-up is
generalised to AdSd-Schwarzschild.

Figure 9:  A fundamental open string used as a probe in an AdS black hole
background to model an infinitely massive quark moving through a thermal
plasma at a constant velocity $v$ on the boundary. One endpoint of the test
string is attached to the boundary of anti-de Sitter spacetime at infinity
($r=\infty$), from which the string hangs down to a stretched horizon
($r_{s}\,=\,(1+\epsilon)\,r_{H}$ where $0<\epsilon\ll 1$) placed just above
the Schwarzschild black hole horizon. Left: A string which trails behind its
boundary endpoint moving with velocity $v$ in the $X^{2}$ direction. This is
the physical solution. Energy flows from the boundary, down the string, and
towards the horizon. Right: A string which trails in front of its boundary
endpoint moving with velocity $v$ in the $X^{2}$ direction. This solution is
unphysical. Energy would need to be flowing away from the horizon, up the
string, and towards the boundary.

####  6.1 Test Strings in AdS5-Schwarzschild

Consider a static open string in $AdS_{5}$-Schwarzschild, with one endpoint
attached to the boundary at $r=\infty$ and the other allowed to hang down
towards the horizon. Since the string is infinitely long (which models an
infinitely massive quark in the boundary theory), the transverse fluctuations
that the string experiences due to the presence of the black hole horizon
never reach the boundary, i.e. the infinitely massive quark does not
experience Brownian motion in the thermal medium. Regardless of its length,
this test string is still a classical relativistic string and, as such, is
described by the standard Polyakov action (Eq.(4.3)) or Nambu-Goto action
(Eq.(4.54)). It is convenient to use the Nambu-Goto action here114114114The
Nambu-Goto action given in Eq.(4.54) is defined in the String Frame. In the
Einstein Frame, the Nambu-Goto action is constructed with an added dilaton
factor $e^{\phi/2}$,
$S_{NG}^{E}\,:=\,-\frac{1}{2\pi\alpha^{\prime}}\,\int_{\mathcal{M}}d^{2}\sigma\,e^{\phi/2}\,\sqrt{-g}\,,$
where $g\,:=\,\text{det}\,(g_{ab})$ and the induced worldsheet metric $g_{ab}$
is given by Eq.(4.4). In [22] the Einstein Frame is chosen for the proceeding
calculations. The phenomenologically relevant case of AdS5-Schwarzschild is
considered. In this spacetime the dilaton factor can be ignored, since
$e^{\phi/2}=H^{(5-d)/4}$ (where $H$ is the warping factor and $e^{\phi/2}$
disappears when $d=5$). It is worth a footnote to remark, however, that the
Einstein Frame is related to the String Frame by a conformal rescaling of the
metric, and that the frames are considered interchangeable when describing the
physics of massless modes on the string [109]. The calculations in this
article are in the String Frame..

From Eq.(4.34), the AdS5-Schwarzschild metric is given by

$ds_{5}^{2}=\frac{r^{2}}{l^{2}}\,\left(-h(r)\,dt^{2}\,+\,d\vec{X}_{I}^{2}\,\right)\,+\,\frac{l^{2}}{r^{2}}\frac{dr^{2}}{h(r)}\,,\,\,\,\,\,\,\,\,\text{where}\,\,\,\,\,\,\,\,h(r)\,=\,1-\left(\frac{r_{H}}{r}\right)^{4}\,\in[0,1]$
(6.1)

is the blackening factor of the Schwarzschild black hole situated at the
horizon; $t\in[0,\infty)$ is the temporal coordinate; $r\in[0,\infty)$ is the
radial coordinate; and the transverse spatial directions along which the
D3-brane is extended are denoted by
$\vec{X}_{I}=\left(X^{2},X^{3},X^{4}\right)\in\mathbb{R}^{3}$. Further,
$l\in\mathbb{R}^{+}$ is the curvature radius of AdSd and $S^{d}$.

Following [22, 35], a gauge choice is made where $\tau=t$ and $\sigma=r$ are
the worldsheet coordinates. The target spacetime metric $G_{\mu\nu}$ – where
$\mu,\nu$ index over the spacetime directions $\left(t,\,r,\,2,\,3,\,4\right)$
– is given explicitly in Eq.(C.2); and the induced worldsheet metric $g_{ab}$,
defined in Eq.(4.4), takes the form Eq.(C.3). The explicit entries of the
metric $g_{ab}$, its inverse $g^{ab}$, and its determinant
$g:=\text{det}(g_{ab})$ are given by Eqs.(C.4, C.5, C.6) respectively. Using
the target spacetime metric $G_{\mu\nu}$ and remembering the gauge choice
$\tau=t$ and $\sigma=r$, the determinant of the induced metric becomes

$g=-\left(1\,-\,\frac{1}{h(r)}\dot{X}_{I}^{2}+\,\frac{r^{4}}{l^{4}}h(r)\,X_{I}^{\prime
2}\right)\,,$ (6.2)

where $\dot{X}\equiv\partial_{\tau}X=\partial_{t}X$ and
$X^{\prime}\equiv\partial_{\sigma}X=\partial_{r}X$ (in both definitions the
second equality follows from the worldsheet gauge choice). Inputting Eq.(6.2),
into the definition of the Nambu-Goto action Eq.(4.54) yields

$S_{NG}=-\frac{1}{2\pi\alpha^{\prime}}\int_{\mathcal{M}}d^{2}\sigma\,\sqrt{1\,-\,\frac{1}{h}\dot{X}_{I}^{2}+\,\frac{r^{4}}{l^{4}}h\,X_{I}^{\prime
2}}\,,$ (6.3)

where the notational adjustment $h(r)\equiv h$ is used for brevity. The
Lagrangian density, also defined in Eq.(4.54), is therefore given by

$\mathcal{L}=-\,\sqrt{1\,-\,\frac{1}{h}\dot{X}_{I}^{2}+\,\frac{r^{4}}{l^{4}}h\,X_{I}^{\prime
2}}\,.$ (6.4)

To proceed, consider movement of the test string with a constant velocity $v$
in one of the transverse directions, the $X^{2}$ direction115115115Due to a
symmetry of anti-de Sitter spacetime all transverse directions are identical.
Therefore any of the transverse directions can be chosen – considering motion
in either the $X^{2}$, $X^{3}$ or $X^{4}$ directions would be equivalent..
Following [22], an ansatz to describe the late-time behaviour of the string is
made. Specifically116116116To clarify the notation: the mapping function for
the second transverse direction will be denoted $X^{2}$, while squaring this
mapping function (if necessary) will be denoted $(X^{2})^{2}$.,

$X^{2}(t,r)\,=\,vt\,+\,\xi(r)\,+\,o(t)\,,$ (6.5)

where, at late times, all other motions are damped out and the $o(t)$ term
disappears. The ansatz Eq.(6.5) relies on the assumption that steady state
behaviour of the string’s motion is achieved at late times. Substituting this
ansatz into the Lagrangian density Eq.(6.4) – remembering that $X^{2}$ is the
only direction in which the string is considered to move – yields

$\mathcal{L}\,=\,-\,\sqrt{1-\frac{v^{2}}{h}+\frac{r^{4}}{l^{4}}h\,\xi^{\prime
2}}\,,$ (6.6)

where $\xi(r)\equiv\xi$ is implied, and $\xi^{\prime}=\partial_{r}\xi$. In
order to determine $\xi(r)$, the Euler-Lagrange equations are derived from the
Lagrangian density (Eq.(6.6)), and are given by

$\frac{{\partial}}{{\partial
t}}\,\frac{{\partial\mathcal{L}}}{\partial\xi^{\prime}}-\frac{{\partial\mathcal{L}}}{{\partial\,\xi}}\,=\,0\,,$
(6.7)

where the quantity $\pi_{\xi}$ can be defined as

$\pi_{\xi}\,:=\,\frac{{\partial\mathcal{L}}}{{\partial\big{(}{\partial_{r}}\,\xi\big{)}}}\,.$
(6.8)

Since the Lagrangian density (Eq.(6.6)) does not depend on $\xi$ (i.e.
${\partial\mathcal{L}}/{\partial\,\xi}=0$), the Euler-Lagrange equations
become

$\displaystyle\begin{aligned}
\pi_{\xi}\,&:=\,\frac{{\partial\mathcal{L}}}{{\partial\,\xi^{\prime}}}\,=\,\mathcal{C}\\\\[5.69046pt]
&=\,-\,\frac{r^{4}}{l^{4}}h\,\xi^{\prime}\left(1-\frac{v^{2}}{h}+\frac{r^{4}}{l^{4}}h\,\xi^{\prime
2}\right)^{-\frac{1}{2}}\,,\end{aligned}$ (6.9)

where Eq.(6.6) is used in the second line. Hence, the equation of motion of
the string is $\pi_{\xi}$ being a constant of motion. Solving Eq.(6.9) yields
a defining relation for $\xi^{\prime}(r)$. Mathematica’s $\tt{Solve}$ function
is used117117117Mathematica is used to help simplify the algebra. See
Mathematica Notebook [d] (DragForce.nb) for details on the calculations
presented in section (6): access in appendix (G)., to find

$\xi^{\prime}(r)\,=\,\pm\,\pi_{\xi}\,\frac{l^{4}}{r^{4}\,h}\,\sqrt{\frac{h\,-v^{2}}{h\,-\,\frac{l^{4}}{r^{4}}\pi_{\xi}^{2}}}\,.$
(6.10)

Finding $\xi(r)$ will involve integrating Eq.(6.10) with respect to $r$.
However, $\xi(r)$ is required to be real everywhere, and the square root in
Eq.(6.10) is not necessarily always real. The only variable which can be
adjusted is $\pi_{\xi}$. Defining $\pi_{\xi}$ such that
$(h\,-v^{2})/(h\,-\,\frac{l^{4}}{r^{4}}\pi_{\xi}^{2})$ is always positive
(thereby ensuring the square root is always real) is equivalent to setting the
appropriate boundary conditions. Since $h(r)\in[0,1]$, at some intermediate
radius $h\,=\,h^{*}$ the numerator $(h\,-\,v^{2})$ changes sign. In order for
the square root to be real everywhere the denominator must change sign at the
same point, i.e. at $h\,=\,h^{*}$,

$h\,-\,\frac{l^{4}}{r^{4}}\pi_{\xi}^{2}\,=\,h\,-\,v^{2}\,=\,0\,.$ (6.11)

Rearranging the first equality leaves

$\pi_{\xi}^{2}\,=\,\frac{r^{4}}{l^{4}}\,v^{2}\,.$ (6.12)

While the second equality in Eq.(6.11) is rearranged to

$r^{4}\,=\,\frac{r_{H}^{4}}{1-v^{2}}\,.$ (6.13)

where the definition of $h(r)$ (Eq.(6.1)) is used. Inputting Eq.(6.13) into
Eq.(6.12) results in an appropriate expression for $\pi_{\xi}$,

$\pi_{\xi}\,=\,\pm\,\frac{v}{\sqrt{1-v^{2}}}\,\frac{r_{H}^{2}}{l^{2}}\,.$
(6.14)

Plugging this defining relation for $\pi_{\xi}$ into Eq.(6.10) guarantees that
the square root in Eq.(6.10) is always positive, and $\xi(r)$ is everywhere
real. The $(\pm)$ in Eq.(6.14) determines if the quark is moving in the $(+)$
or $(-)$ $X^{2}$ direction. Consider the case where $\pi_{\xi}>0$ for the
remainder of the calculation. Eq.(6.10) becomes,

$\xi^{\prime}(r)\,=\,\pm\,v\,\frac{r_{H}^{2}\,l^{2}}{r^{4}\,-\,r_{H}^{4}}\,.$
(6.15)

An integrable expression for $\xi^{\prime}$ has finally been found.
Integrating Eq.(6.15) with respect to $r$ yields118118118Mathematica is used
to easily integrate (see Mathematica Notebook [d]: DragForce.nb).

$\xi(r)\,=\,\mp\,\frac{l^{2}\,v}{2\,r_{H}}\,\left(\tan^{-1}\left(\frac{r}{r_{H}}\right)\,+\,\ln\sqrt{\frac{r\,+\,r_{H}}{r\,-\,r_{H}}}\right)\,,$
(6.16)

where $\ln(-x)\,=\,\ln(x)\,+\,i\,\pi$ for $x>0$ is used and it is recognised
that $i\,\pi$ can be absorbed into the integration constant, which (if the
calculation was entirely rigorous) would be present in Eq.(6.16). The function
$\xi(r)$ can take on two values: (i) for $(\xi^{\prime}(r)>0,\,\xi(r)<0)$ the
string trails behind its boundary endpoint which is moving with a velocity $v$
in the $(+)$ $X^{2}$ direction, and (ii) for $(\xi^{\prime}(r)<0,\,\xi(r)>0)$
the string trails in front of its boundary endpoint which is moving with a
velocity $v$ in the $(+)$ $X^{2}$ direction. The ‘tail wagging the dog’
scenario depicted in (ii) is an unphysical solution [35]; however both of
these cases are depicted in figure (9), page 9.

Using the expression for $\xi^{\prime}(r)$ given in Eq.(6.15) the Lagrangian
density (Eq.(6.6)) becomes

$\mathcal{L}\,=\,-\,\sqrt{1-v^{2}}\,.$ (6.17)

Since the definition of the Nambu-Goto action (Eq.(4.54)) implies
$\mathcal{L}=-\sqrt{-g}$, the determinant of the induced metric becomes

$g\,=\,-\,\left(1-v^{2}\right)\,.$ (6.18)

To calculate the flow of momentum down the string $dp_{2}/dt$, the canonical
momentum densities $\Pi^{a}_{\mu}(t,\sigma)$ (Eq.(4.7)), and the string
equations of motion (Eq.(4.20)) are needed. The conserved charges associated
with the momentum densities $\Pi^{a}_{\mu}$ are defined [96, 15] for a general
curve $\gamma$ on the worldsheet, as

$p^{\gamma}_{\mu}\,:=\,\int_{\gamma}\,d\sigma^{b}\,\tilde{\epsilon}_{ab}\,\Pi^{a}_{\mu}\,,$
(6.19)

where $\tilde{\epsilon}_{ab}$ is the Levi-Civita symbol119119119The Levi-
Civita symbol is defined in two dimensions as
$\tilde{\epsilon}_{ab}\,=\,\begin{Bmatrix}+1\,,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
(a,b)=(1,2)\\\ -1\,,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\ (a,b)=(2,1)\\\
0\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
a=b\end{Bmatrix}\,.$ , and $g_{ab}$ is the induced metric on the worldsheet in
curved spacetime (Eq.(4.4)). Since the string moves only in the $X^{2}$
direction by construction, the string’s transverse momentum densities are only
non-zero in this direction. Using Eq.(6.19) the $X^{2}$ component of the
spacetime momentum flowing over some general time interval $\mathcal{I}$ of
length $\Delta t$ is

$\frac{dp_{2}}{dt}\,\Delta
t\,=\,p^{\mathcal{I}}_{2}\,=\,\int_{\mathcal{I}}dt\,\Pi^{r}_{2}\,,$ (6.20)

where the gauge choice $\tau=t$ and $\sigma=r$ is applied, and
$\tilde{\epsilon}_{rt}=+1$ from the definition of the Levi-Civita symbol. The
radius at which the integral in Eq.(6.20) is evaluated does not matter, since
the momentum densities $\Pi^{a}_{\mu}(t,\sigma)$ are conserved (Eq.(4.20)).
For the physical solution where the string trails behind its boundary endpoint
$(\xi^{\prime}(r)>0,\,\xi(r)<0)$, the quantity $dp_{2}/dt$ is identified as
the drag force and the orientation of the integral is chosen such that the
drag force is negative (i.e. acts to oppose the motion of the string).
Therefore, $dp_{2}/dt=\Pi^{r}_{2}$. According to the definition of the
momentum densities Eq.(4.7),

$\displaystyle\begin{aligned}
\frac{dp_{2}}{dt}\,:=\,\Pi^{r}_{2}\,&=\,-\frac{1}{2\pi\alpha^{\prime}}\,\sqrt{-\gamma}\,\gamma^{rb}\,G_{2\,\nu}\,{\partial_{b}}X^{\nu}\\\\[5.69046pt]
&=\,-\frac{1}{2\pi\alpha^{\prime}}\,\sqrt{1-v^{2}}\,g^{rb}\,G_{2\,\nu}\,{\partial_{b}}X^{\nu}\,,\end{aligned}$
(6.21)

where the constraint equation, Eq.(4.12), and Eq.(6.18) is used in the second
line. Expanding the implied summation over repeated indices in Eq.(6.21)
yields

$\displaystyle\begin{aligned}
\frac{dp_{2}}{dt}\,&=\,-\frac{1}{2\pi\alpha^{\prime}}\sqrt{1-v^{2}}\,G_{2\,\nu}\,\left(g^{rr}\,\partial_{r}X^{\nu}\,+\,g^{rt}\,\partial_{t}X^{\nu}\right)\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\frac{1}{\sqrt{1-v^{2}}}\,G_{2\,2}\,\left(\left(G_{tt}\,+\,G_{2\,2}\,\partial_{t}X^{2}\partial_{t}X^{2}\right)\,\partial_{r}X^{2}\,+\,\left(-G_{2\,2}\,\partial_{r}X^{2}\,\partial_{t}X^{2}\right)\,\partial_{t}X^{2}\right)\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\frac{1}{\sqrt{1-v^{2}}}\,G_{2\,2}\,G_{tt}\,\partial_{r}\xi\\\\[5.69046pt]
&=\,-\frac{1}{2\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\frac{v}{\sqrt{1-v^{2}}}\,,\end{aligned}$
(6.22)

where, in the second line, it was recognised that $G_{2\,\nu}=G_{2\,2}$ is the
only non-zero possibility since $G_{\mu\nu}$ is a diagonal matrix, and
Eq.(C.5) was used to find the explicit entries of the inverse induced metric
$g^{ab}$ (remembering that the gauge choice $\tau=t$ and $\sigma=r$ must be
made). Further, Eq.(6.18) was used and it was noted that, by construction, the
only non-zero transverse direction was the $X^{2}$ direction. In the third
line the commutative property of partial derivatives and the ansatz Eq.(6.5)
was used. The final line follows from inputting the target spacetime metric
entries (Eq.(C.2)) and the defining relation for $\xi^{\prime}(r)$, Eq.(6.15)
(where $\xi^{\prime}(r)>0$).

Comparing Eq.(6.14) (where $\pi_{\xi}<0$) and Eq.(6.22), notice that
$\pi_{\xi}$ is understood to be a momentum density, since

$\Pi_{2}^{r}\,=\,\frac{1}{2\pi\alpha^{\prime}}\pi_{\xi}\,.$ (6.23)

Eq.(6.23) follows from Eq.(4.7) which gives
$\Pi_{2}^{\sigma}:=\frac{1}{2\pi\alpha^{\prime}}\partial_{({\partial_{\sigma}}X^{2})}\mathcal{L}$;
and from the worldsheet gauge choice $\sigma=r$, from which the momentum
density becomes
$\Pi_{2}^{r}\,=\,\frac{1}{2\pi\alpha^{\prime}}\partial_{({\partial_{r}}\,\xi)}\mathcal{L}\,=\,\frac{1}{2\pi\alpha^{\prime}}\pi_{\xi}$
(where the ansatz Eq.(6.5) is used in the first equality and Eq.(6.8) in the
second equality).

For the phenomenologically relevant AdS5/$\mathcal{N}=4$ SYM case, the ‘t
Hooft coupling is $\lambda\,\equiv\,g_{\text{YM}}^{2}\,N_{c}$, where $N_{c}$
is the number of colours and the Yang-Mills coupling is related to the string
coupling by: $g_{\text{YM}}\,=\,2\sqrt{\pi\,g_{s}}$. The fundamental string
length scale Eq.(4.65) becomes

$\alpha^{\prime}\,=\,\frac{l^{2}}{\sqrt{g_{\text{YM}}^{2}\,N_{c}}}\,.$ (6.24)

Using Eq.(6.24) and the definition of Hawking temperature (Eq.(4.36)) with
$d=5$, the drag force (Eq.(6.22)) can be expressed in terms of
phenomenologically relevant gauge theory variables,

$\frac{dp_{2}}{dt}\,=\,-\frac{\pi\,\sqrt{g_{YM}^{2}N_{c}}}{2}\,T^{2}\,\frac{v}{\sqrt{1-v^{2}}}\,.$
(6.25)

Following the formal frameworki of Gubser [22], in the gauge theory the
momentum $p_{2}$ of the probe quark can be theoretically120120120Both the
momentum $p_{2}$ and the mass $m$ of the external quark are, in fact,
infinite. related to its mass $m$ through

$p_{2}(t)\ =\,\frac{v}{\sqrt{1\,-\,v^{2}}}\,m\,,$ (6.26)