# Black brane evaporation through D-brane bubble nucleation

Oscar Henriksson<EMAIL_ADDRESS>Department of Physics and
Helsinki Institute of Physics
P.O. Box 64, FI-00014 University of Helsinki, Finland

###### Abstract

We study the process of black brane evaporation through the emission of
D-branes. Black branes in asymptotically anti–de Sitter spacetimes,
holographically dual to field theory states at finite temperature and density,
have previously been found to exhibit an instability due to brane nucleation.
Working in the setting of D3-branes on the conifold, we construct static
Euclidean solutions describing this nucleation to leading order—D3-branes
bubbling off the horizon. Furthermore, we analyze the late-time dynamics of
such a D3-brane bubble as it expands and find a steady-state solution
including the wall profile and its speed.

††preprint: HIP-2021-20/TH

##  I Introduction

Black holes evaporate due to Hawking radiation [1]. Though this fact is on
firm theoretical footing, there are still many open questions regarding the
precise details of this evaporation, some of which may require a complete
theory of quantum gravity to settle.

String theory provides such a theory. Besides strings, this framework also
introduces other extended objects known as branes. These dynamical objects
gravitate, and a large number of them can be described in the supergravity
limit as a black brane. In analogy with Hawking radiation, one can ask if such
black branes can be unstable to the emission of the individual branes from
which they are formed. This has indeed been found to occur [2, 3, 4, 5, 6],
for example by computing the effective potential for a brane probe and finding
a global minimum _outside_ the horizon. In these cases the horizon and the
true minimum are typically separated by a potential barrier; thus, the
emission can be expected to proceed through bubble nucleation [7, 8]. Such
_brane nucleation_ can provide the main mechanism for black branes to decay,
and are thus important for a complete understanding of their quantum
description. Moreover, through holographic duality [9], the brane nucleation
can be interpreted as bubble nucleation in a dual field theory at strong
coupling, a topic of interest in early-universe cosmology [10].

In this Letter, we provide the first detailed study of a brane nucleation
event. We consider a black brane, built from a stack of D3-branes at a
conifold singularity, which emits a single D3-brane. We focus on two different
aspects; the initial nucleation of the brane as a localized bubble stretching
outward from the horizon, and the late-time steady-state expansion of this
brane into the bulk. We do this by first deriving an effective action
describing the embedding of the D3-brane in the black brane geometry. Then, we
use this to find static configurations with $O(3)$ symmetry, whose Euclidean
action gives the leading semi-classical contribution to the nucleation rate.
Next, assuming a steady-state solution, we solve for the late-time expansion
of the brane. In particular this lets us obtain the terminal velocity and
profile of the brane. Throughout, we use the probe approximation, neglecting
the backreaction of the nucleating D3-brane on the black brane background.

The black branes we study live in an asymptotically anti–de Sitter (AdS)
spacetime, and are holographically dual to states in the Klebanov-Witten (KW)
gauge theory [11] at non-zero temperature and density. The D3-brane emission
is then dual to the spontaneous breaking of the gauge group through the
condensation of a scalar field dual to the radial position of the brane. Had
the minimum of the brane’s effective potential been at a finite radius (as is
the case in some similar setups [5, 6]) the nucleation would have resulted in
a phase transition to a (metastable) Higgs phase, analogous to color
superconductivity in QCD [12]. For our D3-branes the minimum is instead
located at the AdS boundary; this seems to indicate a fatal instability of the
finite density KW theory, with the dual black brane slowly disappearing as one
D-brane after the other nucleates and travels towards infinity (at least until
the combined backreaction of the emitted branes can no longer be ignored).

##  II Gravity solutions and field theory dual

We study asymptotically AdS${}_{5}\times T^{1,1}$ black brane solutions of
type IIB supergravity found by Herzog et al. [4]. Defining forms with legs on
$T^{1,1}$,

$\begin{split}\omega_{2}&\equiv\frac{1}{2}\left(\sin\theta_{1}d\theta_{1}\wedge
d\phi_{1}-\sin\theta_{2}d\theta_{2}\wedge d\phi_{2}\right)\\\ g_{5}&\equiv
d\psi+\cos\theta_{1}d\phi_{1}+\cos\theta_{2}d\phi_{2}\ ,\end{split}$ (1)

the ansatz for the 10D metric is

$\begin{split}ds_{10}^{2}=&L^{2}e^{-\frac{5}{3}\chi}ds_{5}^{2}+L^{2}e^{\chi}\bigg{[}\frac{e^{\eta}}{6}(d\theta_{1}^{2}+\sin^{2}\theta_{1}d\phi_{1}^{2})\\\
&+\frac{e^{\eta}}{6}(d\theta_{2}^{2}+\sin^{2}\theta_{2}d\phi_{2}^{2})+\frac{e^{-4\eta}}{9}g_{5}^{2}\bigg{]}\
,\end{split}$ (2)

with $ds_{5}^{2}$ an asymptotically AdS5 line element, and $L$ the
corresponding AdS radius. Meanwhile, the self-dual 5-form field strength takes
the form $F_{5}=L^{4}\left({\cal F}+*{\cal F}\right)$, with

${\cal F}=-\frac{2}{27}\omega_{2}\wedge\omega_{2}\wedge
g_{5}-\frac{1}{9\sqrt{2}}dA\wedge g_{5}\wedge\omega_{2}\ .$ (3)

Here $A$ is a $U(1)$ gauge field.

This ansatz provides a consistent truncation of type IIB supergravity to a 5D
theory containing the metric, the gauge field $A$ and scalar fields $\chi$ and
$\eta$. The resulting equations of motion have asymptotically AdS5 charged
black brane solutions; their 5D metric can be written as

$ds_{5}^{2}=-g(r)e^{-2w(r)}dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}d\vec{x}_{3}^{2}\
,$ (4)

with $g(r_{H})=0$ at the horizon radius $r_{H}$. These solutions were
constructed numerically and studied in [4, 6], and we refer the interested
reader there for more details.

String theory on asymptotically AdS${}_{5}\times T^{1,1}$ is holographically
dual to the KW theory [11], and the supergravity limit we study describes it
in the large-$N$, strong coupling limit. KW theory is an $\mathcal{N}=1$
superconformal theory with gauge group $SU(N)\times SU(N)$ and matter fields
$A_{\alpha}$ and $B_{\tilde{\alpha}}$ ($\alpha,\tilde{\alpha}=1,2$) in the
$(N,\bar{N})$ and $(\bar{N},N)$ representation, respectively. It provides the
low-energy description of $N$ D3-branes placed on the tip of the cone with
base $T^{1,1}$ (the conifold). Among the global symmetries a certain $U(1)$
factor is often referred to as “baryonic” since the only gauge invariant
operators charged under it are heavy, having conformal dimensions of order
$N$. The corresponding conserved current is dual to the $U(1)$ gauge field in
the gravity truncation above. Thus, the black branes we study are dual to
states of KW theory at finite temperature and baryonic chemical potential,
computed by standard methods as $T=e^{-w(r_{H})}g^{\prime}(r_{H})/4\pi$ and
$\mu=\Phi(r\to\infty)$. Since the theory is conformal, the physics depends
only on the dimensionless parameter $T/\mu$.

The lightest baryonic operators in KW theory involve determinants of the
bifundamental matter fields, which are dual to D3-branes wrapped on a 3-cycle
on $T^{1,1}$ [13]. As the effective potential for these wrapped D3’s does not
show an instability [4], many of the usual ways a charged black brane can
become unstable near extremality are ruled out. The only known instability is
instead due to the emission of D3-branes parallel to the original stack, which
is what we study in the following.

##  III D3-brane effective action

We imagine that one of the D3-branes that build up the black brane localizes
somewhere in the bulk. It then acts as a domain wall, which in the dual KW
theory effectively changes the rank of the two $SU(N)$ gauge groups by one
[13]. Thus, if the energy can be lowered by separating a D3-brane from the
black brane, the gauge symmetry in KW theory is spontaneously broken.

The action of a D3-brane is the sum of a Dirac-Born-Infeld (DBI) and a Wess-
Zumino (WZ) term [14],

$S_{D3}=-T_{3}\int d^{4}\xi\sqrt{-\det P[G]}+T_{3}\int P[C_{4}]\ ,$ (5)

where $P[\dots]$ denotes the pullback of spacetime fields to the brane
worldvolume. The constant dilaton has been absorbed into the tension of the
brane, given by [4]

$T_{3}=\frac{1}{(2\pi)^{3}g_{s}l_{s}^{4}}=\frac{27}{32\pi^{2}}\frac{N}{L^{4}}\
.$ (6)

Letting the brane extend in the directions parallel to the horizon, we
parameterize its worldvolume by the spacetime coordinates $\\{t,\vec{x}\\}$.
The brane will be located on a constant and arbitrary point on $T^{1,1}$,
while the embedding in the radial direction is taken to be a general function
$R=R(t,\vec{x})$. In the following, we denote derivatives with respect to $t$
by a dot and with respect to $x_{i}$ by $\partial_{i}$, $i=1,2,3$. The
components of the 10D metric will be denoted by $G_{\mu\nu}$.

For such an embedding, the induced metric on the brane can be written as

$ds^{2}_{4}=\gamma_{tt}dt^{2}+2\gamma_{ti}dt\,dx^{i}+\gamma_{ij}dx^{i}dx^{j}\
,$ (7)

where the components of the induced metric are given by

$\begin{split}\gamma_{tt}&=G_{tt}+G_{rr}\dot{R}^{2}\\\
\gamma_{ti}&=G_{rr}\dot{R}\,(\partial_{i}R)\\\
\gamma_{ij}&=G_{ij}+G_{rr}(\partial_{i}R)(\partial_{j}R)\ .\end{split}$ (8)

The determinant in the DBI term is then

$\det
P[G]=G_{tt}G_{xx}^{3}\left(1+\frac{G_{rr}}{G_{tt}}\dot{R}^{2}+\frac{G_{rr}}{G_{xx}}(\partial_{i}R)^{2}\right)\
.$ (9)

For the WZ term we need the pullback of the 4-form potential defined by
$F_{5}=dC_{4}$, the relevant component of which is $L^{4}a_{4}(r)\,dt\wedge
dx_{1}\wedge dx_{2}\wedge dx_{3}$, where $a_{4}(r)$ is found by integrating
$a_{4}^{\prime}(r)=4r^{3}e^{-w-\frac{20}{3}\chi}$ with the condition that it
goes to zero on the horizon. Plugging in the metric components and using (6),
the full action becomes

$\begin{split}S_{D3}&=-\frac{27N}{32\pi^{2}}\int
d^{4}x\,\Bigg{\\{}R^{3}e^{-w(R)-\frac{10}{3}\chi(R)}\\\
&\sqrt{g(R)-\frac{e^{2w(R)}}{g(R)}\dot{R}^{2}+\frac{(\partial_{i}R)^{2}}{R^{2}}}-a_{4}(R)\Bigg{\\}}\
.\end{split}$ (10)

In the dual KW theory, the radial position of the brane maps to a combination
of eigenvalues of the matrices $A_{\alpha}$ and $B_{\tilde{\alpha}}$. If the
energy can be lowered by a brane separating from the stack, the dual field
will condense, causing the gauge symmetry breaking $SU(N)\times SU(N)\to
SU(N-1)\times SU(N-1)\times U(1)$. We can then interpret (10) as a quantum
effective action describing the dynamics of the scalar field in the resulting
$U(1)$ sector.

The effective potential is obtained by setting all derivatives of $R$ in (10)
to zero. Evaluating it on the backgrounds discussed above, one finds an
instability — a global minimum at $r=\infty$ — for $T/\mu\lesssim 0.2$ [4, 6].
We now proceed to study the dynamics of this instability.

##  IV Bubble nucleation

To compute the nucleation rate of the transition, we analytically continue to
Euclidean time and take the time direction to have periodicity $1/T$. We
search for static solutions with a spherical $O(3)$ symmetry — such solutions
give an accurate estimate for the nucleation rate at sufficiently high
temperatures, and provide an upper bound at low temperatures. The $t$-integral
can be done right away, giving a factor $1/T$. Due to the anticipated $O(3)$
symmetry we switch to spherical coordinates $\\{\rho,\alpha,\beta\\}$ on the
worldvolume, take $R=R(\rho)$, and integrate over $\alpha$ and $\beta$ as
well. This gives us the Euclidean action

$S_{D3}=\frac{27N}{8\pi T}\int
d\rho\,\rho^{2}\left\\{\frac{R^{2}\sqrt{gR^{2}+(R^{\prime})^{2}}}{e^{w+\frac{10}{3}\chi}}-a_{4}\right\\}\
,$ (11)

where a prime denotes a $\rho$-derivative. Here and below we suppress the
radial dependence of $g$, $w$, $\chi$ and $a_{4}$. Given a solution to the
equation of motion (EoM) resulting from this action, the nucleation rate per
volume is estimated as $\Gamma/V\sim e^{-S_{D3}}$ [7, 8].

The EoM is a second-order ordinary differential equation. Solutions should
satisfy $R^{\prime}(0)=0$ to be smooth at the origin. We can find a series
solution near $\rho=0$ with $R(0)$ a free parameter. We furthermore expect our
solutions to hit the horizon located at $R=r_{H}$ at some finite
$\rho=\rho^{*}$. We can then also find a series solution near this point of
the form

$R(\rho)=r_{H}+R_{1}(\rho^{*}-\rho)+R_{2}(\rho^{*}-\rho)^{2}+\cdots\ ,$ (12)

where everything except the value of $\rho^{*}$ is fixed by the EoM. Using
these two series expansions, we construct the full solutions by a two-sided
numerical shooting method, fixing the parameters $R(0)$ and $\rho^{*}$ by
matching the solutions and their first derivative at some intermediate radius.
The resulting bubble cross sections at three different $T/\mu$ are shown on
the left in Fig. 1.

Figure 1: Left: Bubble cross sections at $T/\mu=0.01$ (solid blue),
$T/\mu=0.09$ (dashed orange), and $T/\mu=0.18$ (dotted green), using the
radial coordinate $z=r_{H}/r$ and with the field theory direction $x$ rescaled
in each case such that the bubble radius is 1. Right: The action evaluated on
the bubble solutions as a function of $T/\mu$.

We then evaluate the action (11) on these bubble solutions. Note that it
diverges in the large-$N$ limit, suppressing the nucleation rate. This is to
be expected as the large-$N$ limit is a classical limit on the gravity side.
On the right in Fig. 1 we show the action divided by $N$ evaluated on our
bubble solutions as a function of $T/\mu$. Approaching the critical value
$T/\mu\approx 0.2$ it diverges (even for finite $N$), as expected on general
grounds. At low temperatures, it approaches an $\mathcal{O}(1)$ number,
implying that nucleation is still greatly suppressed for $N$ large. Recall
however that at low temperatures our static solutions in general only provide
an upper bound on the action, and one should also consider solutions with
dependence on Euclidean time. We leave this for future work.

Note that the static bubble solutions discussed so far can be “completed” in
two distinct ways, due to the fact that a brane laying flat on the horizon is
static in the relevant time coordinate (because of the infinite time
dilation). One way is to add such a horizon brane stretching from the edge of
the bubble out to infinity in the $\vec{x}$ directions. The other way is to
add a horizon _antibrane_ inside the bubble. We interpret the former
completion as the emission of a D-brane from the horizon, and the latter as
brane-antibrane pair creation with the antibrane falling into the horizon and
the brane escaping to infinity. Since an (anti)brane on the horizon has zero
action, neither of these completions affect the nucleation rate and both are
equally likely to occur.

##  V Late-time expansion

As a bubble nucleates, a small perturbation might cause it to start expanding.
At late times, the resulting bubble wall can be approximated as planar, moving
in (say) the $x_{1}$-direction. Then it is useful to parameterize the
worldvolume by its $r$-coordinate instead of its $x_{1}$-coordinate, and work
with the embedding function $X_{1}\equiv X(t,r)$. This switch results in the
action

$\begin{split}S_{D3}=&-\frac{27N}{32\pi^{2}}\int
dt\,dr\,dx_{2}\,dx_{3}\Bigg{\\{}r^{2}e^{-w-\frac{10}{3}\chi}\\\
&\sqrt{1-\frac{e^{2w}r^{2}}{g}\dot{X}^{2}+r^{2}g\,X^{\prime
2}}+X^{\prime}a_{4}\Bigg{\\}}\ ,\end{split}$ (13)

where a prime now denotes an $r$-derivative. Since only derivatives of $X$
appear in the action, there is an associated conserved current describing the
brane’s momentum density, and the EoM is just the conservation of this
current.

At late times we expect the bubble wall to reach a terminal velocity, with
outward pressure being balanced by friction. Thus we search for steady-state
solutions of the form

$X(t,r)=vt+\xi(r)\ .$ (14)

The calculation now proceeds much like the classical drag force calculations
[15, 16]; see also [17]. The ansatz leads to the simplified EoM
$\partial_{r}P^{r}=0$, with the radial component of the momentum current being

$P^{r}=-\frac{r^{4}g\,e^{-w-\frac{10}{3}\chi}}{\sqrt{1-\frac{e^{2w}}{g}r^{2}v^{2}+r^{2}g\,\xi^{\prime}(r)^{2}}}\,\xi^{\prime}(r)-a_{4}\
.$ (15)

Solving for $\xi^{\prime}(r)$ gives

$\xi^{\prime}(r)=\pm\frac{P^{r}+a_{4}}{g}\sqrt{\frac{e^{2w}v^{2}-g/r^{2}}{\left[P^{r}+a_{4}\right]^{2}-r^{6}e^{-2w-\frac{20}{3}\chi}g}}\
.$ (16)

Before attempting to integrate this equation, let us discuss what we expect
our solution to look like. As a brane nucleates and then starts to expand, it
gains energy and momentum from the region near its center, which is at a
larger radius where the potential energy is lower. This accelerates the brane,
while some energy is dissipated into the near-horizon region — this is the
gravity dual of the friction between the bubble and the plasma on the field
theory side. The end result at late times should be a solution extending from
the horizon towards the true minimum at infinity. Thus, we must impose that
$\xi^{\prime}(r)$ diverges as $r\to\infty$. This can only happen if the
denominator in the square root of (16) goes to zero there, fixing $P^{r}$ to
be

$P^{r}=\lim_{r\to+\infty}-a_{4}\pm r^{3}e^{-w-\frac{10}{3}\chi}\sqrt{g}\ ,$
(17)

where we must pick the plus sign to get a finite result. This sets $P^{r}$
equal to the minimum of the effective potential.

Having fixed $P^{r}$ thus, we notice that the denominator inside the square
root of (16), in addition to going to zero at infinity, also crosses zero at
some finite $r$. This forces us to fix $v$ such that the numerator crosses
zero at the same point, ensuring that $\xi^{\prime}(r)$ is everywhere real.
Thus we arrive at a numerical value for the wall speed of the expanding bubble
at asymptotic times, shown as a function of $T/\mu$ on the left in Fig. 2. As
expected, the speed goes to zero at the critical temperature. For a
relativistic theory at zero density the speed would approach the speed of
light at small temperatures; here however, it approaches a smaller value of
about 0.38. Note that the wall speed remains well below the speed of sound of
the background plasma, which is fixed to the conformal value of $1/\sqrt{3}$.
In the usual parlance of first order phase transitions this would thus be
classified as a deflagration (as opposed to a detonation), although in the
probe approximation this is a somewhat artificial distinction since the bubble
doesn’t backreact on the plasma.

Figure 2: Left: The terminal wall speed as a function of $T/\mu$. Right: The
wall profile as a function of the radial coordinate $z=r_{H}/r$ at
$T/\mu=0.01$ (solid blue), $T/\mu=0.09$ (dashed orange), and $T/\mu=0.18$
(dotted green). Note that in all cases the curves diverge to infinity at the
horizon, though this is not visible for the green curve.

With the parameters fixed, (16) can be numerically integrated to find the
profile $\xi(r)$ of the moving wall. Assuming $v>0$, $\xi(r)$ must diverge to
negative infinity at the AdS boundary. This fixes the sign ambiguity in (16),
but only partially, since the derivative $\xi^{\prime}(r)$ always goes to zero
at some point in the bulk. (This can be seen from (16), recalling that $P^{r}$
is negative and $a_{4}(r)$ increases monotonically from zero at the horizon.)
At this point we are free to change the sign used in (16) without creating a
discontinuity in the derivative. Thus, there are two possible late-time wall
profiles; we fix this final ambiguity by requiring that the part of the brane
near the horizon trails behind the front of the wall, much like in [15, 16].
The resulting wall profile is shown on the right of Fig. 2. Note that $\xi(r)$
diverges to negative infinity also at the horizon, as can be seen by expanding
(16) there.

##  VI Discussion

For the charged black branes studied in this Letter, brane nucleation appears
to be the leading channel of decay. One might thus expect the resulting
evaporation process to lead to an information paradox in much the same way as
with the more familiar Hawking radiation [18]. In fact, brane nucleation
offers a way of studying the disappearance of a large black brane (or a black
hole [5]) in AdS without modifications at the boundary [19]. This can lead to
new ideas for solving the information paradox, and new ways of testing such
ideas.

From the point of view of the dual field theory, we have studied barrier
penetration through bubble nucleation at strong coupling. While in this case
the nucleation does not appear to result a new (meta)stable phase in the field
theory, our methods extend straightforwardly to other setups where this would
be the case (since the minimum of the effective potential is at a finite
radius) [5, 6]. There, brane nucleation mediates a genuine first order phase
transition. Much recent work has leveraged holographic duality to study bubble
nucleation [20, 21, 22, 23, 24, 17], motivated by early-universe phase
transitions and the gravitational waves they could produce [10]. This Letter
is to our knowledge the first where both nucleation rates and the bubble wall
speed have been computed from first principles, using simple numerical tools
and without any additional approximations beyond treating the nucleating brane
as a probe.

When a metastable endstate exists, another interesting quantity which can
readily be computed is the surface tension between two coexisting phases.
Furthermore, one could use the effective action (10) to study soliton
solutions and Wilson line configurations along the lines of [25, 26, 27]. We
also hope to study brane solutions with general time-dependence, which would
give a more complete picture of the expansion of the bubble and, in the case
of Euclidean time, a better estimate of the nucleation rate at low
temperatures. Going away from the high-temperature limit might also lead to a
clearer separation between the two possible completions of the static bubble
solutions, which we have interpreted as brane emission and brane-antibrane
pair creation, respectively.

Lastly, it would be very interesting to include backreaction from the bulk
D-branes on the geometry, at least in some approximate way, as this is the
only way to search for the true end state of a brane nucleation instability.

###### Acknowledgements.

Acknowledgments.— It is a pleasure to thank Francesco Bigazzi, Aldo Cotrone
and Carlos Hoyos for helpful comments and suggestions, and Fëanor Reuben Ares,
Mark Hindmarsh, Niko Jokela, Otto Karhu, Huaiyu Li and Niilo Nurminen for
useful discussions and collaboration on related work. The work was supported
by the Academy of Finland (grant number 330346), the Ruth and Nils-Erik
Stenbäck foundation, and the Waldemar von Frenckell foundation.

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