# Dark bubbles: decorating the wall

Souvik Banerjee  Ulf Danielsson  Suvendu Giri

###### Abstract

Motivated by the difficulty of constructing de Sitter vacua in string theory,
a new approach was proposed in arXiv:1807.01570 and arXiv:1907.04268, where
four dimensional de Sitter space was realized as the effective cosmology, with
matter and radiation, on an expanding spherical bubble that mediates the decay
of non supersymmetric AdS5 to a more stable AdS5 in string theory. In this
third installment, we further expand on this scenario by considering the
backreaction of matter in the bulk and on the brane in terms of how the brane
bends. We compute the back reacted metric on the bent brane as well as in the
five dimensional bulk. To further illuminate the effect of brane-bending, we
compare our results with an explicit computation of the five dimensional
graviton propagator using a holographic prescription. Finally we comment on a
possible localization of four dimensional gravity in our model using two
colliding branes.

##  1 Introduction and overview

The extreme difficulty of constructing a model of the observed positive
cosmological constant in string theory, has led to the recent proposal of de
Sitter (dS) swampland conjectures [1, 2, 3], which forbid the existence of
such vacua in string theory (see [4] for a review). To get around this
difficulty, a possible new approach was proposed in [5, 6], where four
dimensional dS is realized as a time-dependent geometry in an unstable five
dimensional AdS vacuum of string theory. This scenario is inspired by the
Randall-Sundrum braneworld [7, 8] construction, but differs from it in a
crucial way. Instead of a flat brane with a $\mathbb{Z}_{2}$ symmetry across
it, our shellworld is a spherical brane that mediates the decay of an unstable
five dimensional AdS to a more stable one. This leads to important differences
in the way that four dimensional matter and radiation is realized on the
shellworld. For instance, a cloud of strings stretching out from the shell
leads to matter, while a black hole in the bulk give rise to radiation on the
four dimensional shellworld. In fact, it was shown in [9] that the presence of
these ingredients in the five dimensional bulk facilitate the decay of the
unstable AdS and favor the formation of a bubble with a small cosmological
constant.

Let us begin by summarizing the shellworld construction of [5, 6]. Consider a
theory that has two AdS5 vacua with cosmological constants $\Lambda_{+}$
$(=-6k_{+}^{2}=-6/L_{+}^{2})$ and $\Lambda_{-}$ $(=-6k_{-}^{2}=-6/L_{-}^{2})$
with $k_{-}>k_{+}$. The vacuum with higher energy $\Lambda_{+}$ can decay into
the vacuum with lower energy $\Lambda_{-}$ non-perturbatively via the
nucleation of a Brown-Teitelboim instanton. 111In string theory, such decays
are supported by the conjecture that all non supersymmetric AdS vacua must
decay as a consequence of the weak gravity conjecture [10, 11, 12]. The decay
AdS vacua supported by flux through the nucleation of charged membranes, and
its relation to the Weak Gravity Conjecture [13] in the context of
shellworlds, was discussed in [6]. Let us consider these AdS vacua in global
coordinates (where the subscripts $+$ and $-$ denote quantities outside and
inside 222Inside (outside) refers to the direction away from the brane in
which the volume of radial slices decreases (increases). the bubble
respectively)

$\mathrm{d}s^{2}=-f_{\pm}(r)\mathrm{d}t^{2}+f_{\pm}(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{3}^{2},$
(1.1)

where $f_{\pm}(r)=1+k_{\pm}^{2}r^{2}$. For the metric inside the bubble with
$f_{-}(r)$, the radial coordinate $r$ goes from the centre of AdS to the
position of the bubble i.e. $r\in\left(0,a(t)\right)$, while for the metric
outside the bubble with $f_{+}(r)$, $r\in\left(a(t),\infty\right)$. The
complete metric for all $r$ i.e. $r\in\left(0,\infty\right)$ in the five
dimensional bulk is then given by

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{2}^{2},$
(1.2)

with

$f(r)\coloneqq(1+k_{-}^{2}r^{2})+\Theta(r-a(t))(k_{+}^{2}r^{2}-k_{-}^{2}r^{2}),$
(1.3)

On parametrizing the radius of the bubble in terms of the proper time ($\tau$)
for an observer on the shell i.e. , $r=a(\tau)$, the induced metric takes the
FLRW form

$\mathrm{d}s^{2}_{\rm{ind}}=-\mathrm{d}\tau^{2}+a(\tau)^{2}\mathrm{d}\Omega_{3}^{2}.$
(1.4)

Einstein’s equations require the presence of a stress tensor on the shell
given by

$S_{ab}=-\frac{1}{\kappa_{5}^{2}}\left(\left[K_{ab}\right]^{+}_{-}-\left[K\right]^{+}_{-}\gamma_{ab}\right),$
(1.5)

where $\kappa_{5}^{2}\equiv 8\pi G_{5}$, 333We follow the standard conventions
where the coefficient of the five dimensional Ricci scalar in the Einstein
Hilbert action is $M_{5}^{3}=1/(2\kappa_{5}^{2})=1/(16\pi G_{5})$, where
$M_{5}$ is the five dimensional Planck mass. and $\gamma_{ab}$ is the metric
induced on the shell from the bulk, where the indices
$(a,b)\in\left\\{0,1,2,3\right\\}$ run over the shell. $K_{ab}$ is the
extrinsic curvature of the shell as seen from the AdS bulk, and
$\left[\cdot\right]^{+}_{-}\coloneqq(\cdot)_{+}-(\cdot)_{-}$ is the difference
of the corresponding quantity between the outside and the inside of the
bubble. This governs the evolution of the radius of the bubble, and for a
shell of constant tension $\sigma$ it is given by

$\sigma=\frac{3}{\kappa_{5}^{2}}\left(\sqrt{k_{-}^{2}+\frac{1+\dot{a}^{2}}{a^{2}}}-\sqrt{k_{+}^{2}+\frac{1+\dot{a}^{2}}{a^{2}}}\right),$
(1.6)

where $a(\tau)\equiv a$ and
$\dot{a}\coloneqq\mathrm{d}a(\tau)/\mathrm{d}\tau$. For a critical value of
the tension $\sigma_{\rm{crit}}$ given by

$\sigma_{\rm{crit}}\coloneqq\frac{3}{\kappa_{5}^{2}}\left(k_{-}-k_{+}\right),$
(1.7)

the cosmological constant on the shell vanishes, giving a four dimensional
Minkowski brane. For a brane with a slightly sub-critical tension
$\sigma=\sigma_{\rm{crit}}(1-\epsilon)$, equation (1.6), up to linear order in
$\epsilon$, gives the Friedmann equation

$H^{2}\equiv\frac{\dot{a}^{2}}{a^{2}}=-\frac{1}{a^{2}}+\frac{\kappa_{4}^{2}}{3}\Lambda_{4}+\mathcal{O}\left(\epsilon^{2}\right),$
(1.8)

where

$\displaystyle\kappa_{4}^{2}\equiv 8\pi G_{4}$
$\displaystyle\equiv\frac{2k_{+}k_{-}}{k_{-}-k_{+}}\kappa_{5}^{2},$
$\displaystyle\Lambda_{4}$
$\displaystyle\equiv\sigma_{\rm{crit}}-\sigma=\epsilon\sigma_{\rm{crit}}.$
(1.9)

Therefore, an observer living on such an expanding shell experiences a de
Sitter universe with a positive spatial curvature.

In the presence of matter in the five dimensional AdS, the metric is given by
equation (1.1) with
$f_{\pm}(r)=1+k_{\pm}^{2}-\kappa_{5}^{2}M_{\pm}/(3\pi^{2}r^{2})$. This gives
an additional contribution to the Friedmann equation which goes as $1/a^{4}$
and can be identified as a radiation density in four dimensions i.e. ,

$H^{2}=-\frac{1}{a^{2}}+\frac{\kappa_{4}^{2}}{3}\left[\Lambda_{4}+\frac{1}{2\pi^{2}a^{4}}\left(\frac{M_{+}}{k_{+}}-\frac{M_{-}}{k_{-}}\right)\right].$
(1.10)

Four dimensional matter can be obtained from a cloud of strings in the bulk,
which has
$f_{\pm}(r)=1+k_{\pm}^{2}-\kappa_{5}^{2}M_{\pm}/(3\pi^{2}r^{2})-\kappa_{5}^{2}\alpha/(4\pi
r)$, and gives the following contribution to the Friedmann equation

$H^{2}=-\frac{1}{a^{2}}+\frac{\kappa_{4}^{2}}{3}\left[\Lambda_{4}+\frac{1}{2\pi^{2}a^{4}}\left(\frac{M_{+}}{k_{+}}-\frac{M_{-}}{k_{-}}\right)+\frac{3}{8\pi
a^{3}}\left(\frac{\alpha_{+}}{k_{+}}-\frac{\alpha_{-}}{k_{-}}\right)\right].$
(1.11)

In summary, a positively spatially curved four dimensional de Sitter universe
can be modeled as the effective spacetime seen by a four dimensional observer
living on the surface of an expanding shell in an asymptotically five
dimensional AdS. The matter and radiation densities observed in the four
dimensional universe are induced by the presence of matter and a cloud of
strings in the five dimensional bulk.

This model also provides a way of understanding the dS horizon in four
dimensions [6]. A four dimensional observer living on the shell accelerates in
the fifth dimension with the expanding shell and measures an Unruh
temperature. This turns out to be equal to the temperature associated with the
dS horizon in four dimensions. The dS temperature thus has a five dimensional
interpretation in terms of the acceleration of the expanding shell. This means
that the cosmological horizon is a section of the Rindler horizon in five
dimensions.

Gravitational dynamics on the shell can be studied using the Gauss (-Codazzi)
equations, which relate the Einstein equations in the bulk to those on the
shell in terms of its embedding and extrinsic curvature,

$R^{(5)}_{\alpha\beta\gamma\delta}\,e^{\alpha}_{c}e^{\beta}_{a}e^{\gamma}_{d}e^{\delta}_{b}=R^{(4)}_{cadb}+(K_{ad}K_{cb}-K_{cd}K_{ab}).$
(1.12)

Combined with the thin shell junction conditions, it was shown in [6] that the
four dimensional Einstein equations are given by

$\begin{split}\left(G^{(4)}\right)^{a}_{b}=&-\underbrace{2k_{+}k_{-}\left(3-\frac{\kappa_{5}^{2}}{k_{-}-k_{+}}\sigma\right)}_{\equiv\,\kappa^{2}_{4}\left(\sigma_{\textrm{crit}}-\sigma\right)\,\equiv\,\kappa^{2}_{4}\Lambda_{4}}\delta^{a}_{b}-\frac{\kappa^{2}_{5}}{\pi^{2}a(\tau)^{4}}\left(\frac{M_{+}k_{-}-M_{-}k_{+}}{k_{-}-k_{+}}\right)\left(\delta^{a}_{0}\delta^{0}_{b}-\frac{1}{3}\sum_{i=1}^{3}\delta^{a}_{i}\delta^{i}_{b}\right)\\\
&-\frac{3\kappa^{2}_{5}}{4\pi
a(\tau)^{3}}\left(\frac{\alpha_{+}k_{-}-\alpha_{-}k_{+}}{k_{-}-k_{+}}\right)\delta^{a}_{0}\delta^{0}_{b},\end{split}$
(1.13)

The left hand side of this equation is the Einstein tensor, and the right hand
side gives the sources corresponding to the four dimensional cosmological
constant, radiation and matter respectively. Each contribution is easily
identified through its equation of state given by $p=-\rho,p=\rho/3$ and
$p=0$, respectively. Taking the $(\tau,\tau)$ component of the Einstein tensor
in four dimensions, namely
$G^{(4)}_{\tau\tau}=3\left(1/a^{2}+\dot{a}^{2}/a^{2}\right)$ and using the
above equation, gives the Friedmann equation (1.11) as expected.

In the current paper, we will re-derive (1.9) through a computation of the
graviton propagator in momentum space. In this computation, we will explicitly
use intuitions coming from holography [14]. Holography, in particular the
AdS/CFT correspondence [15], relates a gravitational theory in $(d+1)$
dimensional anti de Sitter spacetime to a non-gravitational $d$-dimensional
conformal field theory living on the boundary of the $(d+1)$ dimensional bulk
spacetime. In its full generality, both the bulk and boundary theories are
quantum theories. However, in certain limits 444namely, in the limit of large
number of degrees of freedom, $N\rightarrow\infty$ in the boundary field
theory and large t’Hooft coupling $\lambda=g_{\textrm{\sc qft}}^{2}N\gg 1$,
$g_{\textrm{\sc qft}}$ being the coupling in the boundary field theory. the
gravitational theory becomes classical as well as weakly coupled. In this
limit, there is a well-defined prescription [16, 17] that relates bulk
quantities to boundary data. In particular, the bulk modes that decay with
radial distance near the boundary (termed as the normalizable modes) are
interpreted as the expectation value of the corresponding operator on the
quantum field theory on the boundary, while the other independent set of
relatively slowly growing or constant modes (namely, the non-normalizable
modes) have the interpretation of sources. As an example, for the graviton
propagator, normalizable modes correspond to the stress tensor while the non-
normalizable modes correspond to the metric on the boundary. We will not get
into much of the technical details of holographic prescription in this paper
555Interested readers are referred to the elaborate review [18] and references
therein., but only use this intuition while fixing the constants of
integration when computing the bulk propagator.

Our main focus in this paper will be the following. In the construction of [5,
6], the shell is radially symmetric and is located at a constant radius in the
five dimensional bulk at each instant of time. The presence of four
dimensional Einstein gravity on the brane was shown in the presence of
homogeneous and isotropic matter. However, in the presence of localized matter
on the brane, breaking homogeneity and isotropy, it is expected to bend [19,
20]. This backreaction, which has not been considered so far, is one of the
main aspects that we will focus on in this paper. In addition, we will also
compute the metric on the bent brane as well as the back-reacted five
dimensional metric.

$\xi=0$
---
$x^{a}$
---
$\xi$
---
(a)
---
(b)
---
(c)
---
$k_{-}$
---
$k_{+}$
---
Figure 1: A schematic representation of the brane, showing the effect of
various forms of stress tensor. The brane is shown in black and the sources
are in red. (a) A point mass causes the brane to “sag” i.e. , $\xi<0$. A gauge
transformation to bring the brane back up to $\xi=0$ is physically equivalent
to adding a negative four dimensional mass to the brane to “float” it back up.
(b) A string pulling on the brane from the inside has the same effect, while
(c) a string pulling from outside has the opposite effect and contributes with
a positive four dimensional energy in a gauge where the brane is flat.

In figure 1, we summarize the contribution of matter sources to the bending of
the brane. We will discuss this in detail in section 2 and here we present a
geometrical understanding of the main result. In the coordinates that we are
using, the Poincaré horizon is located at $\xi\to-\infty$ and the boundary of
AdS is at $\xi\to\infty$, so gravity pulls down towards the center of the AdS
throat, implying that matter confined to the brane will make it sag (case (a)
in figure 1). The brane will bend in a similar way with a string pulling
downwards (case (b) in figure 1). As we will argue in subsequent sections, it
is the case with strings pulling the brane upwards (case (c) in figure 1) that
will be the most interesting. The presence of such strings will affect the
metric in the bulk and on the brane in two different ways. There is the change
in the metric sourced directly by the string that is present even without the
brane, as well as a contribution from the bending through the junction
conditions. As we will see explicitly in section 3.2., the direct effect is
subleading at distances much larger than the AdS-scale, where the effect from
the bending dominates.

In order to distinguish between the two effects, let us invoke the covariant
conservation of the stress tensor i.e. , $\nabla_{\alpha}T^{\alpha\beta}=0$.
666The Bianchi identities require the stress tensor to be covariantly
conserved in order for Einstein’s equations to be consistent. This is a
constraint on the components of the stress tensor that we will use here to
find the stress tensor of the end point of the string in terms of the tension
of the string. This is similar to the treatment in [20]. For $\beta=\xi$, the
conservation condition gives (assuming no cross terms in $\xi$ i.e. ,
$T^{a\xi}=0$)

$\partial_{\xi}T^{\xi\xi}+4kT^{\xi\xi}+ka(\xi)^{2}\eta_{ab}T^{ab}=0.$ (1.14)

Recalling that the metric is
$\mathrm{d}s^{2}=\mathrm{d}\xi^{2}+a(\xi)^{2}\eta_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}$,
a string of constant tension $\tau$ stretching only in the $\xi$ direction
(and located at $x^{a}=x^{a}_{0}$) has an energy density given by

$T_{0}^{0}=T^{\xi}_{\xi}=\tau\frac{1}{a(\xi)^{3}}\delta\left(x^{a}-x^{a}_{0}\right).$
(1.15)

This corresponds to an equation of state $p=\rho$ along the fifth direction
and $p=0$ in the transverse direction, which is exactly as expected for a
string stretching out infinitely only in the $\xi$ direction. If, instead, the
string ends on the brane (located at $\xi=\xi_{0}$, say), then $T_{\xi}^{\xi}$
becomes

$T_{\xi}^{\xi}=\tau\frac{1}{a(\xi)^{3}}\delta\left(x^{a}-x^{a}_{0}\right)\Theta\left(\xi-\xi_{0}\right).$
(1.16)

Equation (1.14) then gives

$T_{0}^{0}=\underbrace{\tau\frac{1}{a(\xi)^{3}}\delta\left(x^{a}-x^{a}_{0}\right)\Theta\left(\xi-\xi_{0}\right)}_{\textrm{string
stretching in the
bulk}}+\underbrace{\frac{\tau}{k}\frac{1}{a(\xi)^{3}}\delta\left(x^{a}-x^{a}_{0}\right)\delta\left(\xi-\xi_{0}\right)}_{\textrm{end
point on the brane}}.$ (1.17)

The first term is the expected energy density of the string stretching in the
bulk, but now there is a second contribution on the brane corresponding to the
point where the string ends on the brane. This is the energy density of a
point particle with mass $\tau/k$ exactly as was argued from the Friedmann
equations in [6]. It is useful to think of this as a point mass hanging by a
string in the gravitational well of the AdS-space. As the universe expands,
and the brane moves upwards, the string pulls the point mass so that it can
move along with the brane.

This is the setup that we will be revisiting throughout this paper. The
organization is as follows. In section 2, we will introduce the idea of brane
bending in the presence of matter sources. We will start with a leading order
computation demonstrating the effect of brane bending and then perform a more
detailed analysis to show which way the brane bends in response to being
pulled by strings in the bulk. In section 3, we will compute the graviton
propagator first in momentum space and then in position space. Subsequently,
we will discuss how these computations are related by a gauge choice. Finally,
in section 4, we will highlight some future directions. In particular, we will
discuss the possibility of fully localizing gravity in the shellworld model
using two approaching bubbles, the possibility of realizing black holes on the
shellworld and a realization of holography on a cut-off brane.

##  2 Brane bending: a primer

In this section, we will study the effect of brane bending in response to
matter (both in the bulk and on the brane) that we briefly introduced in
section 1. It will be convenient to work in coordinates where the AdS5 metric
has conformally flat slices along a chosen fifth direction $\xi$ i.e. , AdS5
$\equiv\mathbb{R}\times_{a(\xi)^{2}}\rm{MkW}_{4}$.

$\mathrm{d}s^{2}=\mathrm{d}\xi^{2}+a(\xi)^{2}\eta_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b},$
(2.1)

where $a(\xi)\coloneqq\exp(k\xi)$, $1/k=L$ is the AdS radius, indices
$(a,b)\in\left\\{0,1,2,3\right\\}$ as before, $\eta_{ab}$ is the Minkowski
metric in four dimensions and Greek letter indices are five dimensional i.e. ,
$x^{\alpha}\in(\xi,x^{a})$.

Let us choose two AdS5 spaces with radii $L_{\pm}=1/k_{\pm}$ in the regions
$\xi\gtrless 0$ respectively. In order for this composite spacetime for all
$\xi\in\mathbb{R}$ to be a solution of Einstein’s equations, a source needs to
be present at the discontinuity at $\xi=0$. In this case, the source is a
brane situated at $\xi=0$ which has a stress tensor proportional to the
difference of its extrinsic curvatures as seen from either side. This, along
with the uniqueness of the induced metric on the brane goes by the name of
Israel-Lanczos-Sen’s thin shell junction conditions [21, 22, 23],

$\begin{split}\left[\gamma_{ab}\right]^{+}_{-}&=0,\\\
S_{ab}&=-\frac{1}{\kappa_{5}^{2}}\left(\left[K_{ab}\right]^{+}_{-}-\left[K\right]^{+}_{-}\gamma_{ab}\right),\end{split}$
(2.2)

where the extrinsic curvature $K_{ab}$ is defined (in terms of the normal to
the shell $n_{\alpha}$ and tangents
$e^{\alpha}_{a}\coloneqq\mathrm{d}x^{\alpha}/\mathrm{d}y^{a}$) as
$K_{ab}=n_{\alpha;\beta}e^{\alpha}_{a}e^{\beta}_{b}$. In these coordinates,
the extrinsic curvature becomes $K_{ab}=(1/2)\partial_{\xi}\gamma_{ab}$. Note
that the second condition is the same as the one we used in (1.5) while the
first condition was automatic in section 1 when choosing proper time on the
shell.

In the absence of matter on the brane and in the limit of large proper radius
($ar\to\infty$), the brane can be taken to be flat and lie at $\xi=0$.
However, in the presence of matter, the brane is not expected to be flat any
more. The deformation of the brane in the presence of matter was discussed in
[19, 20] and is usually referred to as brane bending. Before analyzing the
effect of bending in detail for our setup, let us first look at a matter
perturbation and the corresponding bending at leading order.

Let us consider the perturbation of the bulk metric in the presence of a small
amount of matter on the brane (resulting from the presence of strings in the
bulk). Off the brane and away from the strings, there are no sources and the
metric can be chosen to be transverse and traceless and is given by the
junction conditions. There are two perturbation parameters in this case –
first, the matter density ($\textstyle\chi$) and second, the inverse of the
proper radius ($ar$). The metric can be thought of as a double expansion in
these parameters
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}/r$
and $1/(kar)$. Up to leading order in large proper radius, a transverse
traceless perturbation in response to matter density $\textstyle\chi$ is given
by 777We will derive the metric to linear order in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}/r$
and all orders in $(kar)^{-1}$ in section 3.2 and this metric is just the
large proper radius expansion of the metric in equation (3.33) i.e. , expanded
to $(kar)^{0}$.

$\mathrm{d}s^{2}=\mathrm{d}\xi^{2}+a(\xi)^{2}\left[-\left(1-\frac{4\chi}{r}\right)\mathrm{d}t^{2}+\left(1+\frac{2\chi}{r}\right)\mathrm{d}r^{2}+\left(1+\frac{\chi}{r}\right)r^{2}\mathrm{d}\Omega_{2}^{2}\right].$
(2.3)

The position of the brane taking into account the bending is given by
$\xi=\tilde{\xi}+f(r)$. Shifting the radial coordinate by a constant
$r=\tilde{r}-{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}$
and dropping quadratic terms in $f$ or its derivatives gives

$\mathrm{d}s^{2}=\mathrm{d}\tilde{\xi}^{2}+a(\tilde{\xi})^{2}\left[-\left(1-\frac{4\chi}{r}+2kf(r)\right)\mathrm{d}t^{2}+\left(1+\frac{2\chi}{r}+2kf(r)\right)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{2}^{2}\right],$
(2.4)

For this to be a vacuum solution of Einstein’s equations at lowest order in
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}/r$
and $1/(kar)$,

$kf(r)=\frac{\chi}{2r}=\frac{M}{3r},$ (2.5)

where $\chi=2M/3$, and $M$ is the effective mass from the four dimensional
point of view.

Let us now do a more careful analysis, taking into account the presence of
sources. Starting from equation (2.2) and splitting the stress tensor on the
brane into the brane tension $\sigma$ (equation of state $\rho=-p$) and other
matter $T_{ab}$ i.e. , $S_{ab}=-\sigma\gamma_{ab}+T_{ab}$, the second junction
condition becomes

$\frac{1}{2\kappa_{5}^{2}}\left[\partial_{\xi}\gamma_{ab}\right]^{+}_{-}+\frac{\sigma}{3}\gamma_{ab}=-\left(T_{ab}-\frac{1}{3}T\gamma_{ab}\right).$
(2.6)

In the absence of matter on the brane i.e. , $T_{ab}=0$, the brane is
intrinsically flat and has a critical tension given by

$\sigma=\frac{3}{\kappa_{5}^{2}}\left(k_{-}-k_{+}\right).$ (2.7)

Now consider a perturbation of the AdS5 metric due to the presence of some
matter both on the brane and in the bulk. These perturbations can be chosen to
be Gauss normal i.e. , $g_{\alpha\beta}\mapsto g_{\alpha\beta}+\delta
g_{\alpha\beta}$ such that

$\delta g_{\mu\xi}=\delta g_{\xi\xi}=0.$ (2.8)

This causes a change in the induced metric
$\gamma_{ab}\mapsto\gamma_{ab}+\hat{h}_{ab}$, which in our coordinates becomes
$a(\xi)^{2}\eta_{ab}\mapsto a(\xi)^{2}\eta_{ab}+\hat{h}_{ab}$. The second
junction condition then becomes (recall that this is evaluated on the shell)

$\frac{1}{2\kappa_{5}^{2}}\left[\partial_{\xi}\hat{h}_{ab}\right]^{+}_{-}+\frac{\sigma}{3}\hat{h}_{ab}=-\left(T_{ab}-\frac{1}{3}T\eta_{ab}\right),$
(2.9)

where $\sigma$ is the critical tension in equation (2.7).

In the presence of matter, the no longer expected to be at $\xi=0$. This makes
it difficult to apply the junction conditions since the position of the brane
would be given by two different functions $\xi=f_{+}(x)$ and $\xi=f_{-}(x)$ as
seen from either side of the brane. To get around this, we can make a gauge
transformation on the outside and another on the inside of the brane,
following [19, 20]. Demanding that the new coordinates are also Gauss normal
and that the brane is located at $\xi=0$ in the new coordinates, we get,
following [24],

$\xi\mapsto\xi-f(x),\qquad x^{a}\mapsto
x^{a}+\frac{1}{2k}\left(1-a^{-2}\right)\partial^{a}f(x)+q^{a}(x),$ (2.10)

where $q^{a}(x)$ is an arbitrary function of the transverse coordinates. As
mentioned above, we need to apply different coordinate transformations on
either side of the brane to flatten it and so $q^{a}(x),f(x)$ and $k$ are
different on either side of the brane. In these new coordinates, the the
metric perturbation becomes

$\hat{h}_{ab}=h_{ab}+\frac{1}{k}\left(1-a^{2}\right)\partial_{a}\partial_{b}f+2kf\gamma_{ab}-a^{2}\left(\partial_{a}q_{b}+\partial_{b}q_{a}\right),$
(2.11)

which when evaluated on the brane ($\xi=0\Rightarrow a=1$) becomes

$\delta\gamma_{ab}=h_{ab}+2kf\eta_{ab}-2q_{[a,b]},$ (2.12)

where
$q_{[a,b]}\coloneqq(1/2)\left(\partial_{a}q_{b}+\partial_{b}q_{a}\right)$ is
the anti-commutator. Continuity of the induced metric ($\delta h_{ab}=0$)
requires

$\left[h_{ab}\right]^{+}_{-}=\left[kf\right]^{+}_{-}=\left[q_{[a,b]}\right]^{+}_{-}=0.$
(2.13)

It is practical to introduce the function $F\coloneqq kf$ since it is
continuous across the shell. Using this, equation (2.9) becomes

$\frac{1}{2\kappa_{5}^{2}}\left[\partial_{\xi}h_{ab}\right]^{+}_{-}+\frac{\sigma}{3}h_{ab}=-\Sigma_{ab},$
(2.14)

where $\Sigma_{ab}$ is defined as

$\Sigma_{ab}=\left(T_{ab}-\frac{1}{3}T\eta_{ab}\right)-\frac{1}{\kappa_{5}^{2}}\left(\frac{1}{k_{+}}-\frac{1}{k_{-}}\right)\partial_{a}\partial_{b}F,$
(2.15)

which gives the contribution of brane bending on the stress tensor. Therefore,
choosing a bent gauge (or equivalently, considering a bent brane) modifies the
stress tensor on the brane with a gauge dependent factor $F=kf$. This can be
chosen so that $\Sigma_{ab}$ is traceless. This determines $F$ in terms of the
trace of the stress tensor on the brane $T$ and explicitly shows how the brane
bends. This is analogous to the result in [24].

$T=-\frac{3}{\kappa_{5}^{2}}\left(\frac{1}{k_{+}}-\frac{1}{k_{-}}\right)\Box
F.$ (2.16)

Note that the factor $3$ nicely cancels the $1/3$ in $kf(r)$ that we found in
equation (2.5) by matching against Schwarzschild. A point mass on the brane
i.e. , $T_{00}\sim\delta(r)>0\Rightarrow T=-T_{00}\sim-\delta(r)$, gives $\Box
F\sim\delta(r)\Rightarrow F\sim-1/r<0\Rightarrow f<0$. In equation (2.10), we
have defined $f(r)$ as the amount that the brane actually bends and so the
coordinate $\xi$ has to be pulled down by the same amount to get a flat brane.
Placing a point mass on the brane ($T_{00}\sim\delta(r)>0$) implies $f(r)<0$
which means that the brane sags in response to it (case (a) of figure 1). The
same is true for a string that pulls the brane towards the inside (case (b) of
figure 1). With a string pulling on the brane from outside, the opposite is
true i.e. , $f(r)>0$ and the brane is pulled up as shown in case (c) of figure
1.

##  3 Linearized gravity

In this section, we will first present key features of graviton propagators on
our shell-world, starting from a bulk computation in momentum space. We will
then repeat this computation in the position space, but now allowing for the
brane to bend. This will in turn allow the possibility of having massive
structures on the shell as argued before. We will finally discuss how these
two apparently different computations are related via a gauge choice.

###  3.1 Linearized gravity in momentum space

Let us consider perturbations of the AdS metric in response to sources, both
in the bulk and on the brane of the form

$\mathrm{d}s^{2}=\mathrm{d}\xi^{2}+a(\xi)^{2}\left(\eta_{ab}+h_{ab}(\xi,x^{a})\right)\mathrm{d}x^{a}\mathrm{d}x^{b},$
(3.1)

where the perturbations are taken to be Gauss normal. Here
$a(\xi)\coloneqq\exp(k\xi)$, $1/k$ being the AdS length scale.

Taking the transverse coordinates, $x^{a}$, as standard Cartesian coordinates,
the linearized Einstein equations in the world volume directions of the brane
are given by [20]

$\Box\bar{h}_{ab}=a^{-2}\left(-\eta_{ab}\partial^{c}\partial^{d}\bar{h}_{cd}+\partial^{c}\partial_{a}\bar{h}_{bc}+\partial^{c}\partial_{b}\bar{h}_{ac}\right)-\frac{1}{2}\eta_{ab}a^{-4}\partial_{\xi}\left(a^{4}\partial_{\xi}\bar{h}\right)-2\kappa_{5}a^{-2}T_{ab},$
(3.2)

where $a$ is a shorthand for $a(\xi)$ , $\bar{h}_{ab}$ is the trace reversed
perturbation defined as $\bar{h}_{ab}\coloneqq h_{ab}-\frac{1}{2}\eta_{ab}h,$
and $h\coloneqq h_{ab}\eta^{ab}$ is the trace with respect to the unperturbed
metric. Outside the source, a transverse gauge can be chosen [20] i.e. ,
$\partial^{a}\bar{h}_{ab}=0$, which implies Lorenz gauge for the metric
perturbations $h_{ab}$ i.e. , $\partial^{a}h_{ab}=\frac{1}{2}\partial_{b}h$.

It is easier to work with a perturbation which includes the metric factor
$a(\xi)^{2}$ (i.e. , $\gamma_{ab}\coloneqq a(\xi)^{2}h_{ab}$) and its trace-
reversed form $\bar{\gamma}_{ab}=\gamma_{ab}-\frac{1}{2}\eta_{ab}\gamma$.
Rewriting (3.2) in terms of $\bar{\gamma}_{ab}$ and taking a trace gives

$a^{-2}\partial^{2}_{c}\bar{\gamma}+3\left(\partial^{2}_{\xi}-4k^{2}\right)\bar{\gamma}=-2\kappa_{5}^{2}T,$
(3.3)

where traces are taken with respect to the background metric i.e. ,
$\bar{\gamma}\coloneqq\bar{\gamma}_{ab}\eta^{ab}$ and $T\coloneqq
T_{ab}\eta^{ab}$. Subtracting this from (3.2) gives an equation for the
traceless piece

$a^{-2}\partial^{2}_{c}{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}+\left(\partial^{2}_{\xi}-4k^{2}\right){\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}=-2\kappa_{5}^{2}\Sigma_{ab},$
(3.4)

where
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}$
and $\Sigma_{ab}$ are the traceless pieces of $\bar{\gamma}_{ab}$ and $T_{ab}$
respectively i.e. ,

$\begin{split}\bar{\gamma}_{ab}&={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}+\frac{1}{4}\eta_{ab}\bar{\gamma},\\\
T_{ab}&=\Sigma_{ab}+\frac{1}{4}\eta_{ab}T.\end{split}$ (3.5)

We will now try to solve for the traceless perturbation
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}$.
This is easier to do in momentum space. After a Fourier transformation,
equation (3.4) becomes

$\left(-\frac{p^{2}}{a^{2}}+\partial_{\xi}^{2}-4k^{2}\right)\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{ab}(p,\xi)=-2\kappa_{5}^{2}\tilde{\Sigma}_{ab},$
(3.6)

where the tilde represents the Fourier transform in the transverse directions.
$p^{2}=-|p_{0}|^{2}+|\vec{p}|^{2}$ is negative for pure AdS spacetime. $p^{2}$
can be positive for spacelike modes with $|p_{0}|^{2}<|\vec{p}|^{2}$. Such
modes can appear, for example, when there is a black hole in the bulk
spacetime. 888Holographically, these modes are commonly interpreted as thermal
modes in the boundary quantum field theory. Note, in zero temperature field
theory, we can never have such modes because of spectrum condition
($\omega>k$). However, when we connect the system to a heat reservoir,
particles with arbitrarily large momenta can come out of the reservoir or go
in keeping the total energy of the system lie within a narrow band. In our
case, with the source on the shell, we no longer have pure AdS spacetime. Let
us use this fact to postulate that such non-propagating modes do exist in our
case, and let us study the gravitational propagator between two bulk points
across the shell for these modes. We will justify our claim through a position
space computation in the next section.

Equation (3.6) suggests that the structure of propagator for the modes,
$\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{ab}$
should be very similar to that of a minimally coupled massless scalar
propagator in this spacetime. In order to see this similarity explicitly, let
us consider a minimally coupled massless scalar field in AdS5 and the Green
function for the scalar Laplacian in the bulk i.e. ,

${\Box}\Delta_{5}\left(X^{\alpha},\tilde{X}^{\alpha}\right)=\frac{\delta^{5}(X^{\alpha}-\tilde{X}^{\alpha})}{\sqrt{-G}},$
(3.7)

where, as before, $X^{\alpha}\equiv\left\\{\xi,x^{c}\right\\}$. $\alpha$ runs
from $0$ to $4$, $c$ runs from $0$ to $3$ and $\xi$ is the fifth direction. In
the background of pure AdS spacetime,

$\mathrm{d}s^{2}=\mathrm{d}\xi^{2}+a^{2}(\xi)\eta_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b},$
(3.8)

the five dimensional scalar Laplacian takes the following form

$\begin{split}{\square}\Delta_{5}\left(X^{\alpha},\tilde{X}^{\alpha}\right)&=\frac{1}{\sqrt{-G}}\partial_{\alpha}\left(\sqrt{-G}g^{\alpha\beta}\partial_{\beta}\Delta_{5}\left(X^{\alpha},\tilde{X}^{\alpha}\right)\right)\\\
&=\left(\partial^{2}_{\xi}+\frac{4a^{\prime}}{a}\partial_{\xi}+\frac{\partial^{2}_{c}}{a^{2}}\right)\Delta_{5}\left(X^{\alpha},\tilde{X}^{\alpha}\right).\end{split}$
(3.9)

Let us now perform the Fourier transform of the Green function $\Delta_{5}$ in
the transverse directions ($x^{c}$),

$\Delta_{5}\left(X^{\alpha},\tilde{X}^{\alpha}\right)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{ip(x-\tilde{x})}\Delta_{p}(\xi,\tilde{\xi}).$
(3.10)

The Fourier components then satisfy

$\left(\partial^{2}_{\xi}+\frac{4a^{\prime}}{a}\partial_{\xi}-\frac{p^{2}}{a^{2}}\right)\Delta_{p}(\xi,\tilde{\xi})=\frac{\delta(\xi-\tilde{\xi})}{a^{4}}.$
(3.11)

With a change of variables $\hat{\Delta}_{p}\coloneqq a^{2}\Delta_{p}$, this
becomes

$\left(\partial^{2}_{\xi}-\frac{p^{2}}{a^{2}}-\underbrace{2\left(\frac{a^{\prime
2}}{a^{2}}+\frac{a^{\prime\prime}}{a}\right)}_{=4k^{2}}\right)\hat{\Delta}_{p}(\xi,\tilde{\xi})=\frac{1}{a^{2}}\delta(\xi-\tilde{\xi}),$
(3.12)

The left hand side of (3.12) is exactly the same as that of (3.6). Identifying
$\tilde{\Sigma}_{ab}$ appearing in the right hand side of (3.6) as the source
localized on the shell, one can, analogously, write down the scalar propagator
for the traceless modes,
$\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{ab}(p,\xi)$.

$\left(-\frac{p^{2}}{a^{2}}+\partial_{\xi}^{2}-4k^{2}\right)\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}(p;a_{+},a_{-})=\delta(a_{+}-a_{-}).$
(3.13)

This scalar propagator carries most of the relevant features of the bulk
graviton propagator in presence of a localized source on the shell. From now
on, we can therefore safely treat the modes,
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}$
as scalar modes
$\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}$.
The index structure of the actual graviton propagator can be reinstated using
the exact relation between graviton and scalar propagators for localized
sources [20, 25]. The propagator for this mode satisfies equation (3.13) where
the location of the shell is at $a_{+}=a_{-}\equiv a_{s}$. Outside the source,
this is solved piece-wise inside and outside the shell by

$\begin{split}\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{+}(p;a_{+},a_{-})&=A(p,a_{-})\,K_{2}\left(\frac{p}{a_{+}k_{+}}\right)+B(p,a_{-})\,I_{2}\left(\frac{p}{a_{+}k_{+}}\right),\\\
\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{-}(p;a_{+},a_{-})&=C(p,a_{+})\,K_{2}\left(\frac{p}{a_{-}k_{-}}\right),\end{split}$
(3.14)

where $K_{2}$ and $I_{2}$ are modified Bessel functions. $I_{2}(p/ak)$
diverges at small $a$, i.e. , at the Poincaré horizon ($a\to 0$), so
regularity at the horizon excludes it in
$\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{-}$.
Continuity of
$\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}$
across the shell gives the first junction condition and the stress tensor on
the shell determines the jump in the derivative. These are the thin shell
junction conditions

$\begin{split}\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{-}(p;a_{+},a_{s})&=\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{+}(p;a_{s},a_{-})\\\
\frac{1}{2\kappa_{5}^{2}}\left[\frac{\partial}{\partial\xi_{i}}\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{i}(p;a_{+},a_{-})\big{|}_{a_{i}\to
a_{s}}\right]^{i=+}_{i=-}+\frac{\sigma}{3}\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{+}(p;a_{s},a_{-})&=\frac{1}{2\kappa_{5}^{2}}.\end{split}$
(3.15)

Eliminating $A(p,a_{-})$ and $C(p,a_{+})$ from equation (3.13) by imposing
these junction conditions gives

$\begin{split}\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{+}(p;a_{+},a_{-})&=A(p,a_{-})\,K_{2}\left(\frac{p}{a_{+}k_{+}}\right)+B(p,a_{-})\,I_{2}\left(\frac{p}{a_{+}k_{+}}\right)\\\
&=-\left[\frac{1}{g_{K}(p,a_{s})}+B(p,a_{-})\frac{g_{I}(p,a_{s})}{g_{K}(p,a_{s})}\right]K_{2}\left(\frac{p}{a_{+}k_{+}}\right)+B(p,a_{-})I_{2}\left(\frac{p}{a_{+}k_{+}}\right),\end{split}$
(3.16)

where the functions, $g_{K}(p,a_{s})$ and $g_{I}(p,a_{s})$ are defined as

$\begin{split}g_{K}(p,a_{s})&\coloneqq\frac{p}{a_{s}K_{2}\left(\frac{p}{k_{-}a_{s}}\right)}\left[K_{1}\left(\frac{p}{k_{-}a_{s}}\right)K_{2}\left(\frac{p}{k_{+}a_{s}}\right)-K_{1}\left(\frac{p}{k_{+}a_{s}}\right)K_{2}\left(\frac{p}{k_{-}a_{s}}\right)\right],\\\
g_{I}(p,a_{s})&\coloneqq\frac{p}{a_{s}K_{2}\left(\frac{p}{k_{-}a_{s}}\right)}\left[K_{1}\left(\frac{p}{k_{-}a_{s}}\right)I_{2}\left(\frac{p}{k_{+}a_{s}}\right)+I_{1}\left(\frac{p}{k_{+}a_{s}}\right)K_{2}\left(\frac{p}{k_{-}a_{s}}\right)\right].\end{split}$
(3.17)

Let us examine the behavior of the modes $\tilde{\chi}$ and their propagator
near the boundary i.e. , $a\to\infty$. In this limit one can expand the Bessel
functions appearing in the bulk propagator, (3.16)

$\begin{split}\lim\limits_{a\to\infty}K_{2}\left(\frac{p}{ak}\right)&=\frac{2a^{2}k^{2}}{p^{2}}-\frac{1}{2}+\frac{4p^{2}\log(a)-4p^{2}\log\left(\frac{p}{2k}\right)-4\gamma
p^{2}+3p^{2}}{32a^{2}k^{2}}+\ldots,\\\
\lim\limits_{a\to\infty}I_{2}\left(\frac{p}{ak}\right)&=\frac{1}{8}\frac{p^{2}}{a^{2}k^{2}}+\ldots,\end{split}$
(3.18)

where $\gamma$ is the Euler-Mascheroni constant. Terms which vanish at the
boundary are normalizable modes in the bulk and yield energy in the
holographically dual theory on the boundary. On the other hand, divergent
terms at the boundary are non-normalizable modes in the bulk and can be
interpreted holographically as a change in the boundary theory. Since we do
not expect a change of energy in the boundary theory in a nucleation event
such as the creation of the brane, we need the normalizable modes to vanish.
For massless scalar modes (also for the traceless modes for graviton
fluctuations) in AdS5, this corresponds to vanishing of terms that go as
$a^{-2}$. 999The log term in (3.18) is a consequence of working with odd-
dimensional AdS spacetime. From the holographic renormalization, such terms
can be shown to be directly related to the Weyl anomaly of the dual boundary
field theory [26]. This log term term can be ignored so long as the boundary
metric is flat as is in our case. This fixes the constant, $B(p,a_{-})$ as

$B(p,a_{-})=-\frac{\eta}{\eta g_{I}(p,a_{s})+g_{K}(p,a_{s})},\qquad\textrm{
where }\quad\eta\coloneqq 3-4\gamma+4\ln 2.$ (3.19)

With this value of $B(p,a_{-})$, the bulk propagator
$\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{+}(p;a_{+},a_{-})$
in equation (3.16), can be evaluated on the shell in the small momentum limit
to give

$\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{+}(p;a_{+},a_{-})=\,\frac{a_{+}^{2}}{p^{2}}\left(\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\right)+\mathcal{O}\left(p^{0}\right)\Rightarrow\Delta_{\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}}^{\rm{shell}}(p;a_{s},a_{s})=\frac{a_{s}^{2}}{p^{2}}\left(\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\right)+\mathcal{O}\left(p^{0}\right).$
(3.20)

The leading order term is the four dimensional scalar propagator on the shell.
The five dimensional bulk-bulk propagator in the Randall-Sundrum braneworld
[7, 8], as computed in [20], splits into a four dimensional propagator and a
piece corresponding to the Kaluza-Klein modes. In the above result, we have
expanded in small momentum and have recovered the zero mode of the five
dimensional graviton corresponding to the four dimensional graviton localized
to the shell. The other modes would appear as higher order corrections in $p$.

The expression for the graviton mode on the shell corresponding to equation
(3.20), can be obtained by convoluting the propagator with the source,
$-2\kappa_{5}^{2}\tilde{\Sigma}$, given in (3.6). In momentum space this is a
simple product yielding

${\tilde{\chi}}_{s}(p;a_{s})=-\,2\kappa_{5}^{2}{\tilde{\Sigma}}\frac{a_{s}^{2}}{p^{2}}\left(\frac{2k_{-}k_{+}}{k_{-}-k_{+}}\right).$
(3.21)

This is the result that was derived in [5] demonstrating the appearance of
four dimensional gravity with an effective Newton’s constant of
$G_{4}\coloneqq 2G_{5}k_{+}k_{-}/(k_{-}-k_{+})$. It is worth mentioning at
this point that in this derivation we have worked in a gauge where the brane
is placed at $\xi=0$. However, as we discussed in section 2, the presence of
matter causes the brane to bend and we must change coordinates accordingly to
straighten out the brane. We saw that this implies that matter placed directly
on the brane, or caused by a string pulling from inside, contributes with a
negative stress tensor. This implies that $\tilde{\Sigma}_{ab}<0$ and with the
minus sign in equation (3.21), four dimensional gravity is correctly
reproduced. In fact, we can take the analysis further and verify that this,
which comes from the five dimensional junction conditions, correctly
reproduces the four dimensional junction conditions. Fourier transforming
equations (2.16) and (2.12) we get

$\tilde{T}=\frac{3}{\kappa_{5}^{2}}\left(\frac{1}{k_{+}}-\frac{1}{k_{-}}\right)p^{2}\tilde{F},\quad\textrm{
and
}\quad\tilde{h}_{ab}=\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{ab}+2\tilde{F}\eta_{ab}+2ip_{[a}\tilde{q}_{b]}.$
(3.22)

Using equation (3.21) we get

$\begin{split}\tilde{h}_{ab}&=\tilde{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{ab}+2\tilde{F}\eta_{ab}+2ip_{[a}\tilde{q}_{b]}\\\
&=-\frac{\tilde{\Sigma}_{ab}}{\alpha
p^{2}}+2\tilde{F}\eta_{ab}+2ip_{[a}\tilde{q}_{b]}\\\ &=-\frac{1}{\alpha
p^{2}}\left(\left(\tilde{T}_{ab}-\frac{1}{3}\tilde{T}\eta_{ab}\right)+\cancel{\frac{1}{\kappa_{5}^{2}}\left(\frac{1}{k_{+}}-\frac{1}{k_{-}}\right)p_{a}p_{b}\tilde{F}}\right)+2\tilde{F}\eta_{ab}+\cancel{2ip_{[a}\tilde{q}_{b]}},\end{split}$
(3.23)

where

$\alpha\coloneqq\frac{1}{4\kappa_{5}^{2}}\left(\frac{1}{k_{+}}-\frac{1}{k_{-}}\right),$
(3.24)

and the arbitrary function $\tilde{q}_{a}$ is chosen so that is cancels the
term proportional to $p_{a}p_{b}\tilde{F}$ in $\tilde{\Sigma}_{ab}(p)$. Using
(3.22) this gives

$\begin{split}\tilde{h}_{ab}&=-\frac{1}{\alpha
p^{2}}\left(\tilde{T}_{ab}-\frac{\tilde{T}}{3}\eta_{ab}\right)+\frac{\tilde{T}}{6\alpha
p^{2}}\eta_{ab}\\\ &=-\frac{1}{\alpha
p^{2}}\left(\tilde{T}_{ab}-\frac{\tilde{T}}{2}\eta_{ab}\right)=-\frac{2\kappa^{2}_{4}}{p^{2}}\left(\tilde{T}_{ab}-\frac{\tilde{T}}{2}\eta_{ab}\right),\end{split}$
(3.25)

where $\kappa^{2}_{4}\equiv 8\pi G_{4}$. This is precisely the four
dimensional junction condition with the right four dimensional Planck
constant. The minus sign is because of the sign of the stress tensor as
discussed before.

Let us conclude with few more comments on the main result of this section,
equation (3.21). For small momentum, ${\tilde{\chi}}_{s}$ gives the four
dimensional graviton propagator on the shell thereby producing four
dimensional gravity. For large momentum, ${\tilde{\chi}}_{s}$ decays
exponentially fast towards zero. It can be verified from equations (3.16) and
(3.17) that the crossover in the behavior of ${\tilde{\chi}}_{s}$ occurs when
$p\sim a_{s}k_{+}$. This is compatible with a holographic picture where the
dual CFT on a shell at large and fixed $\xi$ is modified only at large
distances when the bulk geodesics corresponding to correlation functions in
the CFT dip down far enough into the bulk to hit the bubble wall. Short
distance correlators in the boundary CFT do not see the shell and remain
unmodified.

###  3.2 Linearized gravity in position space

So far, we have solved for the traceless perturbation in momentum space. In
order to find all components of the metric perturbations in a usable form, we
would like to find the corresponding perturbation in position space. To do
this, let us study perturbations of the AdS5 metric in response to matter on
the brane which, as discussed in the previous section, can be the endpoint of
a string in the bulk. These perturbations
(${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}$)
can be chosen to be Gauss normal and in the Lorenz gauge. Considering
spherically symmetric perturbations on the brane 101010contrary to the
perturbation considered in (3.1)., the metric (changing coordinates from $\xi$
to $z\coloneqq e^{\xi}$ and setting $k=1$) can be written as

$\begin{split}\mathrm{d}s^{2}&=\frac{\mathrm{d}z^{2}}{z^{2}}+z^{2}\eta_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}+\overbrace{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}}^{\rm{perturbation}}\\\
&=\frac{\mathrm{d}z^{2}}{z^{2}}+z^{2}\left[-\left(1+h_{t}(r,z)\right)\mathrm{d}t^{2}+\left(1+h_{r}(r,z)\right)\mathrm{d}r^{2}+\left(1+h_{a}(r,z)\right)r^{2}\mathrm{d}\Omega_{2}^{2}\right].\end{split}$
(3.26)

Further, we make an ansatz $h_{i}(r,z)=z^{n-2}f_{i}(rz)$, where $rz$ is the
proper radius. 111111Choosing $h_{i}(r,z)=z^{n-2}f_{i}(rz)$ makes the
perturbation to the AdS metric
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{ab}\sim
z^{n}f_{i}(rz)$ after including the $z^{2}$ factor in front of the four
dimensional part of the metric. Making a gauge choice that the perturbation is
traceless ($h_{ab}\eta^{ab}=0$) gives the following linearized Einstein’s
equation for $f_{t}(rz)$

$\left(1+k^{2}z^{2}r^{2}\right)f_{t}^{\prime\prime}(zr)+\left(\frac{2}{zr}+\left(2n+1\right)k^{2}zr\right)f_{t}^{\prime}(zr)+(n^{2}-4)k^{2}f_{t}(zr)=0,$
(3.27)

where as usual, primes denote derivatives with respect to the argument. In
terms of $f_{t}$, the radial part $f_{r}$ is given by a remarkably simple
first order equation

$zrf_{r}^{\prime}(zr)+3f_{r}(zr)+f_{t}(zr)=0,$ (3.28)

and the angular part follows from vanishing of the trace

$f_{t}(zr)+f_{r}(zr)+2f_{a}(zr)=0.$ (3.29)

For $n=3$, equation (3.27) has two solutions

$\displaystyle f_{t}^{(1)}(zr)$ $\displaystyle=\frac{1}{zr},$ $\displaystyle
f_{t}^{(2)}(zr)$
$\displaystyle=\frac{3+2z^{2}r^{2}}{\left(1+z^{2}r^{2}\right)^{3/2}}.$ (3.30)

The first solution $f_{t}^{(1)}(zr)$ gives a perturbation
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{tt}=z^{2}h_{t}=z^{3}f_{t}=z^{2}/r$,
while the second solution gives a perturbation which has the following
asymptotics

$\displaystyle\lim\limits_{r\ll
1}{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{tt}^{(2)}$
$\displaystyle\sim z^{3},$ $\displaystyle\lim\limits_{r\gg
1}{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{tt}^{(2)}$
$\displaystyle\sim\frac{z^{2}}{r}.$ (3.31)

The perturbation
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{tt}^{(1)}$
behaves, in fact, like the CHR black string [27] solution at short distances
and is singular at $r=0$. So it needs the presence of string with a delta
function source $\delta(r)$. This is a string with non-constant tension [28]
and is not related to the constant tension string we study. We discard this
solution and focus on
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{tt}^{(2)}$.
This gives the following radial and angular perturbations

$\displaystyle f_{r}$ $\displaystyle=-\frac{1}{\sqrt{1+r^{2}z^{2}}},$
$\displaystyle f_{a}$
$\displaystyle=-\frac{2+r^{2}z^{2}}{2\left(1+r^{2}z^{2}\right)^{3/2}}.$ (3.32)

So, the complete perturbed metric to first order is

$\begin{split}\mathrm{d}s_{\rm{brane}}^{2}=&-z^{2}\left(1+{c_{1}}z\frac{3+2z^{2}r^{2}}{\left(1+z^{2}r^{2}\right)^{3/2}}\right)\mathrm{d}t^{2}+z^{2}\left(1-{c_{1}}z\frac{1}{\sqrt{1+r^{2}z^{2}}}\right)\mathrm{d}r^{2}\\\
&+z^{2}r^{2}\left(1-{c_{1}}z\frac{2+r^{2}z^{2}}{2\left(1+r^{2}z^{2}\right)^{3/2}}\right)\mathrm{d}\Omega_{2}^{2}+\frac{\mathrm{d}z^{2}}{z^{2}}.\end{split}$
(3.33)

This behaves like the CHR solution at large $r$ but does not need a source.
The leading contribution at large distance is what we used in (2.3) to obtain
the Schwarzschild metric on the brane, (2.4). Taking the brane bending into
account, we find that there is an effective source,
$\Sigma_{ab}=\delta(r)/2\kappa_{5}^{2}$, induced on the brane as discussed at
the end of section 2. We conclude that the $n=3$ solution corresponds to
perturbations of the metric due to a point source in the brane world (possibly
induced by a pulling string) at $r=0$. These grow towards the boundary of AdS
and are therefore non-normalizable perturbations.

Having understood the metric sourced by a mass on the brane, let us now study
the metric induced by the stretched string. In the presence of a source in the
bulk, the perturbations can be taken to be cylindrically symmetric around the
string but do not need to be in the Lorenz gauge. They are of the general form
of equation (3.26) with a non-zero perturbation in the $z$ direction. Making
the ansatz $h_{i}(r,z)=z^{n-2}f_{i}(rz)$ as before, but now with $n=0$ for a
string with constant tension, Einstein’s equations can be solved to linear
order to get the metric outside a constant tension string stretching in the
bulk.

$\begin{split}\mathrm{d}s_{\rm{brane}}^{2}=&-z^{2}\left[1-\frac{c_{2}}{z^{2}}\left(3+2r^{2}z^{2}-\frac{2}{rz}\left(1+r^{2}z^{2}\right)^{3/2}\right)\right]\mathrm{d}t^{2}\\\
&+z^{2}\left[1-\frac{c_{2}}{z^{2}}\left(1+2r^{2}z^{2}-2rz\sqrt{1+r^{2}z^{2}}\right)\right]\mathrm{d}r^{2}\\\
&+z^{2}r^{2}\left[1-\frac{c_{2}}{3z^{2}}\left(3+2r^{2}z^{2}-\frac{2}{rz}\left(1+r^{2}z^{2}\right)^{3/2}\right)\right]\mathrm{d}\Omega_{2}^{2}\\\
&+\left[1-{c_{2}}\left(2+\frac{2}{rz}\sqrt{1+r^{2}z^{2}}\right)\right]\frac{\mathrm{d}z^{2}}{z^{2}}.\end{split}$
(3.34)

Away from the string (at large $r$), the perturbations go as $1/z^{2}r^{2}$,
while close to the source (at small $r$), $h_{t}\sim h_{z}\sim 1/rz$.

###  3.3 Final comments on the role of gauge choice

As promised at the beginning of this section, we will conclude with a few
comments on how the computations in the last two sections are connected
through a choice of gauge. We showed that depending on whether we choose the
brane to be straight or bent, the physical interpretation would be quite
different. In particular, we noted that the bent gauge automatically allows
for an effective delta function source on the brane. This is what connects our
computation in momentum space to that in the position space. While in the
first case, we kept the position of the brane fixed, in the later we allowed
brane bending. Noting that the Fourier transform of the solution
$f_{t}^{(2)}(zr)$ in (3.30) is nothing but the modified Bessel function
$K_{2}$ (see appendix A), we intuitively see that if we had chosen to work out
the fluctuations in the momentum space, but taking into account the bending of
the brane, we would have needed to work with the pure $K_{2}$ solution on both
sides of the brane. This would amount to setting $B(p,a_{-})=0$ in (3.14). By
self consistency, then, the brane is bent so that the presence of the $K_{2}$
does not induce any matter on the brane, such that we find a pure
Schwarzschild metric. Alternatively, we can perform a coordinate
transformation, shifting $z$ as a function of the coordinates on the brane, to
obtain a straight brane. This will generate terms of the form $I_{2}$ through
mixing. Again, we should make sure that there is no matter induced on the
brane, which corresponds to there being no normalizable modes. This implies a
specific relative coefficient of the $K_{2}$ and $I_{2}$ so that the
subleading $1/z^{2}$ mode is canceled. This would precisely correspond to the
momentum space computation that we have performed.

##  4 Discussion and outlook

In this work, we have provided a clear understanding of backreaction on our
shellworld due to localized and extended sources in the bulk. This complete
picture, in turn, suggests several interesting directions for further studies
that we are currently pursuing.

###  4.1 One more view from the bulk: possible model with localized gravity

We have argued that there is a consistent holographic picture of the dark
brane, with massive particles realized as strings hanging from the boundary of
AdS. Even though gravity is mediated by the bending of the brane,
gravitational effects extend all the way to the boundary through the non-
normalizable modes. Interestingly, there is an alternative possibility
involving two dark branes where gravity is fully localized in the fifth
direction. Physically, it would correspond to two branes about to collide. As
shown in figure 2, one can imagine strings stretching between the two branes,
making each of the branes bend in response. Our universe, together with the
matter it contains, is supported on this combined structure. Thus, it is
perhaps better to think of the two branes as one thick brane with an internal
structure. As the universe expands, the sandwich gets thinner, implying a
dramatic end point when the bubbles collide. As explained in [6], the distance
between the two branes can be microscopic and still allow for plenty of proper
time on the branes before collision.

In this picture, any gravitational perturbation is localized to the sandwich
universe and decays as we move away from it. The holographic picture
corresponds to slicing the sandwich in two, assuming the two sides to be
mirrors of each other, and inserting a holographic screen. It also requires
that the two branes are sufficiently far away from each other. We will
investigate this setup further in a forthcoming paper.

$x^{a}$
---
$\xi$
---
$k_{-}$
---
$k_{+}$
---
$k_{+}$
---
$k_{-}$
---
$k_{+}$
---
Figure 2: A cartoon representing strings (in red) stretching between two
copies of the world brane (in black) that can be close together. The energy
density of the strings and their endpoints on the branes causes them to bend.
As the four dimensional universe within the brane expands, the branes move
closer to each other (shown with blue arrows). The gravitational perturbation
from the strings is localized to the branes and the false vacuum in between.
The perturbations decays away as we move out from the sandwich universe into
the true vacuum.

###  4.2 Black objects on the brane

An important challenge is to construct solutions corresponding to black holes.
Given that massive particles corresponds to stretched strings, it seems
natural to consider such objects in the context of the sandwich universe
discussed above. Black holes should form if, for instance, a shell of
stretched strings such as in figure 2, were to collide and approach their
Schwarzschild radius. One possible outcome would be a five dimensional black
hole acting as a connection between the two braneworlds. Inspired by [29, 30],
we believe there are also other possibilities that should be investigated. In
particular, the collision of the strings could initiate the nucleation of a
bubble of true vacuum that could prevent the formation of a black hole. The
idea would be that such a structure could remain stable due to Unruh radiation
in parallel to what was proposed in [29, 30]. We will provide details of such
constructions in an upcoming publication.

###  4.3 Holography on a cut off brane

The derivation of the bulk propagator in momentum space presented in section
3.1 shows that the effective four-dimensional gravity on our shellworld is
perfectly consistent with the expectations from holography. By holography we
here mean the description of gravitational physics in the bulk AdS in terms of
the field theory living on its boundary at $a=\infty$. Here, it is worth
mentioning several previous attempts towards investigating the gravitational
duals of collapsing shells in asymptotically AdS spacetime [31, 32, 33, 34].
However, with a view to understand the gravitational dual of the dynamics of
thermalization in the boundary quantum field theory, all of these studies were
centered on collapsing shells. It would nevertheless be straightforward to do
a similar computation of boundary propagators in our case. Although our model
of expanding bubble is physically very different from the scenario of
collapsing shells in AdS, the holographic techniques we need to employ in
order to compute the boundary correlators would be quite similar. In
particular, taking the backreaction into account, it would be worth
investigating the signatures of a possible formation on black objects on the
shell from the perspective of the boundary field theory [35]. However, it
would perhaps be even more interesting and illuminating to understand the
whole picture in terms of the the holography on the shell. Considering the
shell to be a finite radial cut off from the perspective of both the AdS
spacetimes, one possible way to understand this holographically is to consider
a CFT deformed by an irrelevant operator, typically termed as $T\bar{T}$
deformation in literature [36, 37, 38]. This is a very active field of
research in the holography community and would be instrumental in connecting
the five-dimensional bulk physics in our set up to a deformed CFT on the
shellworld. We hope to report on this soon.

## Acknowledgments

We would like thank Giuseppe Dibitetto, Johanna Erdmenger, Miguel Montero,
Lisa Randall, Marjorie Schillo, Cumrun Vafa and Irene Valenzuela for several
stimulating discussions. We would also like to thank the referees for their
valuable comments. The work of S.B. is supported by the Alexander von Humboldt
postdoctoral fellowship.

##  Appendix A Fourier transform of $K_{2}$

The 3 dimensional Fourier transform of the modified Bessel function $K_{2}$ in
spherical coordinates is given by

$\begin{split}\mathcal{F}(K_{2}(p))=\int\mathrm{d}^{3}\vec{p}e^{i\vec{p}\cdot\vec{r}}K_{2}(p)&=2\pi\int\limits_{0}^{\pi}\mathrm{d}\theta\sin\theta\int\mathrm{d}p\,e^{ipr\cos\theta}p^{2}K_{2}(p)\\\
&=\frac{4\pi}{r}\int\mathrm{d}p\,p\sin prK_{2}(p)\\\
&=-\frac{4\pi}{r}\frac{\partial}{\partial r}\int\mathrm{d}p\cos
prK_{2}(p)\end{split}$ (A.1)

$K_{2}(p)$ has an integral representation given by [39]

$K_{2}(p)=-\int\limits_{-\infty}^{\infty}\mathrm{d}r\cos
pr\frac{2r^{2}+1}{2\sqrt{1+r^{2}}},$ (A.2)

which can be inverted to give 121212The Fourier cosine transform and its
inverse are defined with normalization
$F(\omega)=(1/2)\int\limits_{-\infty}^{\infty}f(t)\cos\omega t\,\mathrm{d}t$
and $f(t)=(1/\pi)\int\limits_{-\infty}^{\infty}F(\omega)\cos\omega
t\,\mathrm{d}\omega$ respectively.

$\int\limits_{-\infty}^{\infty}\mathrm{d}p\cos
prK_{2}(p)=-\pi\frac{2r^{2}+1}{\sqrt{1+r^{2}}},$ (A.3)

Using this in (A.1) gives

$\mathcal{F}(K_{2}(p))=\frac{4\pi}{r}\frac{\partial}{\partial
r}\left(\frac{2r^{2}+1}{2\pi\sqrt{1+r^{2}}}\right)=4\pi^{2}\frac{2r^{2}+3}{\left(1+r^{2}\right)^{3/2}}.$
(A.4)

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11institutetext: Centre for Medical Image Computing & Department of Computer
Science, University College London, London, UK 22institutetext: AINOSTICS
ltd., Manchester, UK 33institutetext: UR2NF-Neuropsychology and Functional
Neuroimaging Research Unit affiliated at CRCN – Centre for Research in
Cognition and Neurosciences and UNI - ULB Neuroscience Institute, Université
Libre de Bruxelles (ULB), Brussels, Belgium 44institutetext: GIGA - Cyclotron
Research Centre - In Vivo Imaging, University of Liège (ULiège), Liège,
Belgium

# Eddeep: Fast eddy-current distortion correction for diffusion MRI with deep
learning

Antoine Legouhy 1122 Ross Callaghan 22 Whitney Stee 3344 Philippe Peigneux
3344 Hojjat Azadbakht 22 Hui Zhang 11

###### Abstract

Modern diffusion MRI sequences commonly acquire a large number of volumes with
diffusion sensitization gradients of differing strengths or directions. Such
sequences rely on echo-planar imaging (EPI) to achieve reasonable scan
duration. However, EPI is vulnerable to off-resonance effects, leading to
tissue susceptibility and eddy-current induced distortions. The latter is
particularly problematic because it causes misalignment between volumes,
disrupting downstream modelling and analysis. The essential correction of eddy
distortions is typically done post-acquisition, with image registration.
However, this is non-trivial because correspondence between volumes can be
severely disrupted due to volume-specific signal attenuations induced by
varying directions and strengths of the applied gradients. This challenge has
been successfully addressed by the popular FSL Eddy tool but at considerable
computational cost. We propose an alternative approach, leveraging recent
advances in image processing enabled by deep learning (DL). It consists of two
convolutional neural networks: 1) An image translator to restore
correspondence between images; 2) A registration model to align the translated
images. Results demonstrate comparable distortion estimates to FSL Eddy, while
requiring only modest training sample sizes. This work, to the best of our
knowledge, is the first to tackle this problem with deep learning. Together
with recently developed DL-based susceptibility correction techniques, they
pave the way for real-time preprocessing of diffusion MRI, facilitating its
wider uptake in the clinic.

###### Keywords:

Diffusion MRI Distortion correction Eddy-currents.

##  1 Introduction

In magnetic resonance imaging (MRI), diffusion-weighted imaging (DWI) is
designed to indirectly uncover the intricate microstructure of biological
tissues through the study of the movement of their water molecules. It has
various clinical applications including stroke diagnostic [1] or tumor
characterisation [2]. DWI requires the acquisition of a large number of
volumes with different diffusion sensitization gradients, more than a hundred
for advanced diffusion models. To achieve acceptable scan duration, spin-echo
single-shot echo-planar imaging (EPI) sequences are used [21, 3]. However,
this technique is vulnerable to off-resonance effects leading to tissue
susceptibility and eddy-current induced distortions. Eddy currents are a
result of rapid switching of magnetic field gradients. They induce additional
unwanted spatially-varying magnetic fields that will interfere with the
spatial localization, leading to geometrical distortions [4, 5]. As the nature
and extent of the distortions depend on the strength and direction of the
applied diffusion gradient, this causes misalignment between the volumes of a
single DW acquisition, thus disrupting downstream modelling and analysis.

The essential correction of eddy distortions is typically done post-
acquisition, with image registration. This task is non-trivial because
correspondences between volumes are severely disrupted by the volume-specific
signal attenuations induced by varying directions and strengths of the applied
gradients. In particular, the outer cerebro-spinal fluid (CSF) is a vanishing
boundary beyond very low b-values, very different attenuation patterns appear
in areas of anisotropic diffusivity like white matter, and the signal-to-noise
ratio (SNR) decreases as the b-value increases. As a result, even intensity
similarity metrics designed to handle large differences in contrast, like
mutual information, do not behave well in general. Therefore, although free of
eddy distortions, a $b=0$ image cannot directly be used as target target for
registration.

Strategies have been developed to address this challenge. In FSL Eddy [6],
arguably the most popular tool for correcting eddy-current distortion, instead
of targeting a $b=0$, each volume is iteratively registered to a predicted,
non-distorted, version of itself through a Gaussian process [17]. In Tortoise
[16], the correction strategy is adapted to the b-value of each volume. But
except for the low b-value case, where volumes are corrected by registering
with mutual information as the similarity metric, the essential idea is also
to predict a non-distorted version of each volume, albeit with a different,
b-value-dependent, approach compared to FSL Eddy. These tools achieve good
correction but they are computationally intensive as they rely on traditional
registration which involves restarting a new optimisation from scratch each
time.

Recently, major advances in image processing have been enabled by deep-
learning (DL). DL-based registration methods [15, 13] demonstrated state-of-
the-art alignment accuracy, doing so in a split second at inference. They have
already been used for the correction of susceptibility distortions [22, 23],
but not yet for eddy-current ones because the issue of disrupted
correspondences holds. But DL has also revolutionized the domain of image
translation where conditional GAN (cGAN) [11] architectures like pix2pix [9]
have been used in medical imaging for T1-weighted to $b=0$ image translation
in [12].

In this paper, we propose a novel approach for the correction of eddy-current
distortions that leverages those deep-learning innovations. It is composed of
two models: 1) A 3D pix2pix image translator, a paired and supervised model,
to restore correspondences between volumes. Regardless of its b-value and
gradient direction, each volume is translated to a version that corresponds to
a target b-value and is orientation-averaged. 2) An unsupervised registration
model to align the translated images. Together, they allow us to estimate and
correct for eddy distortion and head motion.

We trained and evaluated five models with various training sizes and
augmentation configurations. Results demonstrate comparable distortion
correction to FSL Eddy, while requiring only modest training sample sizes.
Correction at inference is orders of magnitude faster than traditional
techniques.

##  2 Background

###  2.1 Eddy distortion model

The theory behind distortions induced by unwanted, additional fields to the
$B_{0}$ is detailed in [4]. The particular case of eddy distortions for DWI
have been studied in [5]. It emerges that eddy distortions can reach an
amplitude of several millimeters along the phase-encoding direction (PED) but
are negligible along the remaining directions; only DW images are affected
(not $b=0$) and the distortions varies with respect to the diffusion gradient
strength; and that the induced shift, to the first order approximation, may be
expressed as a linear combination of the original coordinates. However, it has
empirically been shown in [6, 7] that this linear approximation is not always
adequate and that additional higher-order terms may be required. A quadratic
model has been proposed as a good trade-off between increased flexibility and
parsimony.

Let $x=(x_{1},x_{2},x_{3})^{T}$ be the original coordinates of a voxel and
$y=(y_{1},y_{2},y_{3})^{T}$ be the distorted counterparts. Assuming the PED
along the 2nd axis ($y_{1}=x_{1}$ and $y_{3}=x_{3}$) and, the quadratic
distortion model, $y_{2}$ can be expressed as:

$y_{2}=x_{2}+x^{T}Qx+Lx+t\ ,\quad\quad\textrm{ where }Q\in\mathbb{R}^{3\times
3}_{\mathrm{sym}},\ L\in\mathbb{R}^{1\times 3}\textrm{ and
}t\in\mathbb{R}.\vspace{0.1cm}$ (1)

This uni-directional (constrained along the PED) global transformation has 10
degrees of freedom (dof) in 3D.

###  2.2 Accounting for between-volumes head motion

Due to the speed of EPI, within-volume head motion may be considered
negligible. Between-volumes head motion can be estimated using a rigid
transformation (6 dof in 3D). The order matters when composing the estimated
motion matrix $R$ with the estimated eddy distortions $E$. The subject’s head
can move freely in space but distortions are image reconstruction artifacts
that are anchored to the reference frame defined by the imaging gradients
(slice-select, phase-encoding and frequency-encoding). The coordinates
accounting for motion should be written: $y=E\circ Rx$. Eddy and rigid
transformations are disentangled for better interpretability but their
parameters are not all independent.

##  3 Methods

An overview of our approach is shown in Fig. 1. The rest of this section
describes the image translator and the image registration model used in turn
for eddy distortion correction, before discussing the augmentation strategies
for training.

Figure 1: Overview of the generator part of the translator and the
registration model during training.

###  3.1 Image translation

The choice of the target for translation is guided by the following
observations:

  * •

For a given b-value, assuming a uniform sampling on the sphere of the gradient
directions, averaging the distortion free volumes results in an image that has
no direction dependency.

  * •

It is easier to remove extra material around an object (outer CSF signal at
low b-value) than to generate anything that is absent.

  * •

At moderately high b-value, typically between 700 and 3000, the CSF signal is
fully attenuated to noise level, but the signal-to-noise ratio in tissue is
still sufficiently high to preserve edges that drive alignment.

Therefore, we propose an image translator that takes a DW volume of any
gradient direction and b-value (including $b=0$), and produces an homologous
aligned volume corresponding to a moderate b-value $b_{\mathrm{ref}}$,
averaged over all directions. As it is essential that the translation process
does not introduce any shift, the more constrained paired cGAN architecture of
pix2pix [9] is preferred to the cycleGAN [10] alternative.

Given the need of pix2pix for paired training data, in the current
implementation, these are produced using FSL Eddy for simplicity. A moderately
high target b-value $b_{\mathrm{ref}}$ is chosen among the available one.
Then, for each subject, a target image for supervision is built by averaging
the corrected volumes associated with $b_{\mathrm{ref}}$. During training,
corrected data is used as input and the model learns to translate it into the
target image (see Fig. 1). During inference, however, the translator receives
distorted data as input, generating corresponding distorted outputs. Those
will be the inputs of the subsequent registration model. Although eddy
distortion magnitudes are significant enough to affect downstream modeling and
analysis, they remain sufficiently minor for a translator trained on corrected
data to effectively handle distorted one during inference. This is especially
true if spatial augmentation is done on the training data.

###  3.2 Image registration

Once translated, correspondences between volumes are restored, enabling
accurate intensity-based registration. In fact, contrasts are now so similar
that a simple mean squared error (MSE) can be used as similarity criterion to
guide a registration. Recently, the problem of intensity-based image
registration has been tackled using convolutional neural networks (CNN). The
case of deformable registration have been addressed in [15, 13, 14] using a
sequence of U-Net to estimate a deformation field and a spatial transformer to
warp the moving image. In [15], a second architecture was also presented for
affine transformation estimation where the U-Net is replaced by an encoder
followed by dense layers.

We propose a similar approach to estimate the eddy distortion and motion
parameters following the unidirectional quadratic transformation depicted in
Section 2.1. The registration model, illustrated in Fig. 1, is a sequence of
the following blocks:

  1. 1.

Input: A distorted, translated DW volume (moving); and a translated $b=0$
image (reference); concatenated.

  2. 2.

Geometry encoder: Input goes through a CNN encoder learning to extract
features of varying spatial resolutions for localization.

  3. 3.

Dense layers: The flattened encoder output goes through a series of fully
connected layers, where the nodes of the last layer are the parameters of the
rigid and eddy transformations.

  4. 4.

Resampler: A spatial transformer network [19] transform the moving image with
the estimated transformation parameters. During training, the translated
moving DW volume is resampled to be compared to the translated $b=0$ one. At
inference, the original moving DW volume is directly resampled instead for
correction.

###### Remark 1

We freeze the weights of the translator when producing the inputs for the
registration model instead of attempting some joint optimization. Indeed,
without supervision, the translator could for example learn to map everything
to 0 leading to a subsequent perfect 0 MSE loss for the registration model.

###  3.3 Augmentation strategies

Translator: During training, corrected data is used, so an easy contrast
augmentation is to simply create a new input from a weighted average of two
volumes of the same subject. Spatial augmentation can be done by applying the
same affine and deformable transformations to the input and target images.
Registration model: No contrast augmentation is needed as the inputs are
translated images. For spatial augmentation, one can apply the same
transformation to the inputs if it preserves parallelism with the PED.
Assuming the PED being along the second axis, one can use an affine matrix $A$
such that $A_{12}=0$ and $A_{32}=0$, and a deformation field constrained along
the PED. In addition, a rigid transformation can be composed on the right, for
the moving volume only.

##  4 Evaluations

###  4.1 Dataset

As part of an ongoing study investigating memory learning and consolidation,
we acquired paired AP and PA diffusion-weighted EPI data for 90 subjects. Each
are composed of 13 $b=0$ and DW volumes for $b=650$, $1000$ and $2000$; with
respectively 15, 30 and 60 gradient directions; voxel size of 2 mm isotropic.

We split our subjects into mutually exclusive sets for training, validation,
and testing. We used two sizes for training: small (S) with 8 subjects for
training and 4 for validation; medium (M) with 32 and 8 respectively. The
testing set comprises 30 subjects. While it is preferable to use distinct
training/validation sets for the two models, we chose to reuse half of the
training/validation sets of the translator for the registration model to
minimise the combined sizes (if the translator and the registration model both
use size M training/validation sizes, only 48/12 subjects overall are
necessary).

###  4.2 Models

We trained 4 translators with various configurations, all targeting
$b_{\mathrm{ref}}=2000$: $T_{S}$, $T_{S}^{+}$, $T_{M}$ and $T_{M}^{+}$; where
+ indicates that the augmentation in Section 3.3 have been used on the input
with a probability 0.5, $S$ and $M$ refer to the training sample sizes in
Section 4.1. Following the same nomenclature for the training of registration
model $R$, we produced the following 5 full models: $R_{S}\circ T_{S}$,
$R_{S}^{+}\circ T_{S}^{+}$, $R_{S}^{+}\circ T_{M}^{+}$, $R_{M}\circ T_{M}$ and
$R_{M}^{+}\circ T_{M}^{+}$.

###  4.3 Implementation details

Translator: We used a 3D adaption of pix2pix [9]. The generator is a U-Net
[18] with: (32, 64, 128, 256) encoder features, (256, 128, 64, 32, 1) decoder
features; the discriminator is a patchGAN [9] with: (16, 32, 64, 128, 1)
encoder features; both using Leaky ReLU activation and kernel size of 3.
Training parameters: learning rate of $10^{-4}$, batch size of 4, MSE loss for
both the generator and patch discriminator with weights 100 and 1
respectively.
Registration model: The geometry encoder has (16, 32, 40, 48, 56) features,
Leaky ReLU activation and kernel size of 3. The series of fully connected
layers starts with the flattened encoder input, then split in 5 sets of 64
nodes, each fully connected to resp. 6, 3, 1, 3, 3 nodes for resp.: eddy
quadratic, linear and translation; rigid rotation and translation parameters;
using Leaky ReLU activation except for the last layer. The resampler is
similar to the spatial transformer in [13], adapted to handle the quadratic
transformation depicted in Section 2.1. Training parameters: learning rate of
$10^{-5}$, batch size of 4, MSE loss.

The code is available at: github.com/CIG-UCL/eddeep. It uses tools from
VoxelMorph111github.com/voxelmorph/voxelmorph and SimpleITK222simpleitk.org.

###  4.4 Results

Animations for qualitative assessment of the translation and distortion
correction are provided as supplementary material.
Translator: The quality of the translators was evaluated for each b-values
using MSE between the translated image and the target $b_{\mathrm{ref}}$
average image. Results are shown in Fig. 2. As expected, for a given amount of
augmentation, increasing the number of training samples leads to better
results. Interestingly, with only 8 subjects and data augmentation, one can
reach similar performances as with 32 subjects without augmentation.

Figure 2: Translation results by b-value for the testing dataset. Boxplot:
mean absolute error (MAE) between translated and target for various
translators. target, original (intensity normalized) and translated images
with $T_{M}^{+}$.

Registration model: Those models were compared against FSL Eddy, the state-of-
the-art tool for this task. We used, for each b-value, the voxel-wise
intensity standard deviation across volumes near the interface between
cortical gray matter and CSF as an indicator of the amount of misalignment.
This is motivated by the observation that mismatch of this interface across
volumes will generate higher values of standard deviation compared to when
this interface is well aligned across volumes. To quantify this, we extracted
a mask of this interface (subtraction of a dilated (2 iterations) and eroded
(iterations) brain mask from FSL BET [20]). Results are shown in Fig. 3.
Correction is better for full models where the training of the registration
was done with a sample of size M. Among those, interestingly, $R_{M}^{+}\circ
T_{S}$ does not seem to be penalized by the training sample size S of its
translator. Correction with the best performing models lead to output very
similar to FSL Eddy.

Figure 3: Correction results by b-value for the testing dataset using the
standard deviation (std) across volumes. Images show the map of std for raw
data (top), corrected with FSL Eddy (middle) and through $R_{M}^{+}\circ
T_{M}^{+}$ model. The outer brain mask is displayed with yellow lines. The
boxplots are showing the std for uncorrected data and corrected with FSL Eddy
or our various full models.

##  5 Discussion and conclusion

We proposed a novel approach for the correction of eddy-current distortions,
the first using with deep-learning for this task to the best of our knowledge.
We use a sequence of an translator to restore correspondences between images
and a registration model for distortion estimation and correction. We achieved
correction on par with traditional state-of-the-art techniques. Only modest
training sample sizes are necessary, especially for the translator, thus
making the upfront training cost relatively low. Inference is then very rapid.
A limitations of this work is the necessity for the translator to be trained
using data corrected with an external tool. However, even a translator trained
with as few as 8 subjects (plus 4 for validation) leads to good correction
with the subsequent registration model. It seems that, given the low amount of
degrees of freedom of the distortion model, a relatively approximate
translated images are sufficient. Another limitation is the absence of ground
truth distortions which limits evaluation. We plan to address both those
shortcomings in a future work using synthetic diffusion data training input
and for evaluation, in the same vein as in [24] but including higher order
distortion models. Together with recently developed DL-based susceptibility
correction techniques, this work paves the way for real-time preprocessing of
diffusion MRI, facilitating its wider uptake in the clinic.

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