#### Instabilities of microstate geometries with antibranes

HJE
Instabilities of microstate geometries with antibranes
Iosif Bena 0
Giulio Pasini 0
Gif-sur-Yvette 0
France 0
0 Institut de physique theorique, Universite Paris Saclay, CEA , CNRS
One can obtain very large classes of horizonless microstate geometries corresponding to near-extremal black holes by placing probe supertubes whose action has metastable minima inside certain supersymmetric bubbling solutions [1]. We show that these minima can lower their energy when the bubbles move in certain directions in the moduli space, which implies that these near-extremal microstates are in fact unstable once one considers the dynamics of all their degrees of freedom. The decay of these solutions corresponds to Hawking radiation, and we compare the emission rate and frequency to those of the corresponding black hole. Our analysis supports the expectation that generic non-extremal black holes microstate geometries should be unstable. It also establishes the existence of a new type of instabilities for antibranes in highly-warped regions with charge dissolved in uxes.
Black Holes in String Theory; AdS-CFT Correspondence
1 Introduction 2 3 4
5
6
Near-extremal black hole microstate geometries
2.1
2.2
Microstate geometries for BPS black holes with large horizon area
Adding metastable supertubes
The instability of near-extremal microstates
The emission rates of non-extremal microstates
4.1
The decay time and the emission rate
Typicality of the decay channel
5.1
5.2
Scaling properties of the -emission rate
How typical is the
decay channel?
Conclusions and outlook
A Construction of the three-charge smooth background B Some physical properties of bubbled solutions 1 5
away the information from the black hole [2{6].
The most advanced e ort to build this kind of horizonless structure in string
theory, oftentimes known as the \microstate geometry" programme, consists of constructing
smooth supergravity solutions that have the same charge, mass and angular momenta of
the black hole but have no event horizon.1 In a microstate geometry the would-be black
hole throat is capped and instead of a horizon one has a number of nontrivial cycles
(bubbles) threaded by uxes [9{15]. Thus, the singularity of the black hole would be resolved at
the scale of the horizon by a geometric transition analogous to the Klebanov-Strassler [16],
Polchinski-Strassler [17] and Lin-Lunin-Maldacena [18] ones.
An elaborate technology has been developed to construct smooth microstate solution
for supersymmetric three- and four-charge black holes in the framework of 11D or type
II supergravities compacti ed to ve or six-dimensions, and by now very large families of
1For other recent attempts to construct stringy structure at the black hole horizon see [7, 8].
{ 1 {
such solutions are known.2 Furthermore, the entropy of these solutions has been argued to
reproduce the growth with charges of the Beckenstein-Hawking entropy of the
corresponding black hole [20], so there is little doubt left that the horizons of supersymmetric black
holes will be replaced by horizonless structure.
However, despite this success, the world of non-BPS non-extremal microstates remains
quite unexplored. The few known exact solutions in the JMaRT class [21{25] and the
Running-Bolt class [26, 27], though horizonless, do not have the right charges to correspond
to a black hole with a classically-large horizon area. Furthermore, their construction is
quite artisanal at this point, although the recent discovery of the Bossard-Katmadas (or
the \Floating JMaRT") system [25] indicates that more systematic methods may be on
HJEP04(216)8
the horizon [28].
In parallel to these e orts, it was argued in [1] that one can systematically obtain
very large classes of microstate geometries for non-extremal black holes by placing a
supertube [29] at metastable minimum inside the known BPS microstate geometries. The energy
M of this supertube then gives the mass above extremality of the solution. Although no
exact solutions in this class are known, one can argue that the supertubes should backreact
into a smooth solution,3 and hence one expects that there should exist a very large number
of bubbled microstate geometries corresponding to near-extremal black holes. Since
supertubes have charges opposite to those of the black hole, these bubbled geometries would
have cycles that are wrapped both by positive and by negative uxes,4 and this is exactly
the structure that Gibbons and Warner have proven to be necessary if one is to replace
non-extremal black holes by stationary horizonless black hole solitons [32{34].
Furthermore, since these microstate geometries have neither inner nor outer horizon,
they should be thought of as resolving the singularity of the non-extremal black hole all
the way to the outer horizon, which is backwards in time from the location of the
singularity. This pattern of singularity resolution is rather extraordinary and, if con rmed by
the construction of fully backreacted non-extremal microstate geometries, it would have
important implications not only for black hole singularities but also for cosmological ones.
There is another important di erence between the non-extremal microstate geometries
in the JMaRT and Running Bolt classes, and the near-extremal microstates constructed
using anti-supertubes. The JMaRT geometry is unstable [35] (and so is the Runnning
Bolt one [
36
]) and this instability, which comes from the existence of an ergoregion, can be
matched precisely to the fact that the D1-D5-P CFT state dual to the JMaRT geometry is
also unstable [37{39]. Furthermore, it has been argued that a similar ergoregion instability
should be present in all non-extremal microstate geometries [
40
].
On the other hand, the near-extremal microstate geometries of [1] are obtained by
placing probe anti-supertubes inside extremal bubbling geometries at metastable minima
2The largest family, parameterized by arbitrary continuous functions of two variables, was recently
constructed in [19].
solution [30].
3For supersymmetric solutions it was shown that the Born-Infeld equations governing the supertube
are equivalent to those ensuring smoothness and absence of closed timelike curves in the fully-backreacted
4Similar to the proposed metastable ux compacti cations of [31].
{ 2 {
of their Hamiltonian, and hence these con gurations could in principle be much longer
lived than one expects for a typical near-extremal microstate by studying the D1-D5 CFT.
Indeed, the near-extremal CFT states consist of a very large number of (supersymmetric)
left-mover momentum modes and a much smaller number of supersymmetry-breaking right
movers, and it seems very di cult to prepare states where the annihilation of these modes
is suppressed such that the decay takes place over very long time scales.
In this paper we resolve this tension by showing that the near-BPS microstate
geometries that one obtains by placing metastable anti-supertubes inside long scaling solutions [1]
can in fact lower their energy when the bubbles of the scaling solution move relative to each
other. Hence, what appears to be a metastable con guration from the point of view of the
action of a probe brane is in fact an unstable one if one takes into account the degrees of
freedom corresponding to the motion of the bubbles.
We study a scaling microstate geometry that is constructed using seven collinear
Gibbons-Hawking centers [14] as well as an anti-supertube probe. If one keeps the GH
centers collinear, the supertube Hamiltonian will have both supersymmetric and metastable
minima [41], corresponding respectively to microstate geometries for supersymmetric and
non-extremal black holes. For the latter, the mass above extremality of the black hole,
M , is simply equal to the value of the Hamiltonian in the metastable minimum.
However, it is well known that supersymmetric solutions that are constructed using
N GH centers have a 2N
2-dimensional moduli space, that is parameterized by the
solutions of the N
1 bubble (or integrability) equations that govern the inter-center
distances [
11, 12, 42
]. Hence, the mass above extremality,
M , is in fact a function of
the position in the moduli space. Thus, if one is to prove that the supertube minimum
corresponds to a metastable black hole microstate, one must also show that
M has a
minimum when the GH points are collinear, and does not decrease as one moves around
the moduli space.5 We show that it does.
Indeed, if one examines
M as a function of certain moduli space directions one nds
that the collinear GH con guration corresponds to a saddle point, and
M can in fact
decrease as some of the centers move o the axis. Our result implies that the instability
is triggered by the motion of the bubbles threaded by
ux. As the con guration moves
away from the saddle point, the kinetic energy of the bubbles increases, and we expect this
to result in gravitational radiation that will relax the system towards a supersymmetric
minimum.
One can also estimate the time-scale characteristic to this instability, as well as the
energy emission rate, as a function of the charges and mass of the solution as well as of the
other parameters of the non-extremal microstate geometry. To compute the time-scale we
estimate the energy of the con guration as a function of the tachyonic rotation angle in the
GH base space of the solution, as well as the kinetic term corresponding to the motion in
5Things are even a bit more complicated: as the GH points move in the 2N
2-dimensional space of
solutions to the bubble equations the SU(2) angular momentum of the BPS solution, JL, changes and hence
the solutions do not correspond to the same black hole. For xed values of the charges and angular momenta
the moduli space of microstates of the corresponding black hole is a constant-JL slice of the 2N
2 space
of solutions to the bubble equations, and has therefore dimension 2N
{ 3 {
the moduli space. The latter calculation is performed by formally interpreting our solution
as a multi-center four-dimensional solution whose constant in the Taub-NUT (D6 brane)
harmonic function has been set to zero. We then argue that the kinetic term corresponding
to the angular motion we consider is the same as the one obtained by replacing some of
the centers of the bubbling solution with the corresponding black hole and black ring. This
allows us to compute this term and to
nd its scaling with the length of the black hole
microstate throat.
We also check whether the microstate decay channel we study is similar to the decay
channel one expects for a typical microstate of a D1-D5-P near-extremal black hole, which
was computed in [43] and nicely follows from Stephen's law in
ve dimensions. There are
three quantities that one can compare: the emission rate, the frequency peak energy and
the typical radius of the microstate [15]. By changing the parameters of the microstate
solutions we scan over, we can get any two of these parameters to agree, but not the
third. This indicates that the particular non-extremal microstate geometries that we are
considering are not typical.
This is not at all unexpected: of all the microstates of the BPS black hole we have
chosen to uplift to a non-BPS one only a particular one, corresponding to seven collinear
GH centers with certain
uxes on them. Furthermore, the starting con guration is clearly
not a typical representative of the microstate geometries of the BPS black hole | one can
nd even more complicated solutions with a GH base where the centers are not aligned,
and we expect from [20] that the typical states that contribute to the entropy of this
black hole come from superstrata excitations of solutions with a GH base, that depend on
arbitrary functions of two variables. Hence, it is hard to expect generically that the uplift
of the non-typical seven-center BPS microstate geometry will give us a typical non-BPS
microstate geometry whose decay rate will match that of the black hole.
Our calculations does open the way to investigate whether in the phase space of
bubbling solutions we can
nd non-BPS microstate geometries that decay as the typical
microstates do. There are two possibilities: either the typical microstates have many more
bubbles than the solution we consider (and then our calculations should reveal that the
decay rate approaches that of the black hole as one increases the number of bubbles), or
the typical microstates will have a smaller number of bubbles, and their entropy will come
from the oscillations of these bubbles that leave the topology intact. The recent
superstratum counting of [20] appear to favor the second possibility, although there are quite a few
subtleties that may incline the balance in favor of the rst one.
Arguments about typicality aside, we believe that our investigation has two important
conclusions: the
rst is that microstate geometries of non-extremal black holes will be
unstable, and therefore the dynamics of these black holes will correspond to a chaotic
motion of mutually-non-supersymmetric centers at a bottom of a black-hole like throat. It
would be interesting to investigate whether one can obtain generic features of this chaotic
behavior, and perhaps compare it to other situations where chaotic behaviors appear in
black hole physics [44].
The second implication of our calculations is the discovery of a new decay channel
for antibranes in solutions with charge dissolved in
uxes. Normally one studies these
{ 4 {
antibranes by considering them as probes in a solution and examining their action while
assuming that the solution remains unchanged. Our investigation shows that this approach
can give misleading results, and in order to determine whether an antibrane is metastable or
unstable one should examine its full interactions with the moduli of the underlying solution.
This paper is organized as follows. In section 2 we brie y review the construction of
the near-BPS microstates found in [1]. In section 3 we explicitly show that these
nearBPS microstates are unstable along a direction in the moduli space. Then in section 4
we compute the energy emission rate for a decay along the channel corresponding to this
direction. In section 5 we study how the emission rate and other quantities scale with
the length of the throat and the charges of the solution. We also compare the microstate
emission rate, emission frequency and radius to those corresponding to the typical states in
the black hole thermodynamic ensemble. We conclude in section 6 and present some future
directions. The detailed construction of a three-charge smooth microstate can be found in
appendix A, and some of the intuition used to determine the kinetic term corresponding to
the motion in the moduli space of BPS solutions is explained in more detail in appendix B.
2
Near-extremal black hole microstate geometries
In this section we brie y review the construction of near-extremal black hole microstate
con gurations by placing a metastable supertube inside a supersymmetric bubbling
solution [1].
We rst present the main features of a smooth scaling microstate solution
that has the charges of a
ve-dimensional BPS black hole with a classically large horizon
area [14]. We then make the microstate near-extremal by placing an anti-supertube probe
at a metastable minimum.
to appendix A.
The full features of the BPS three-charge black hole microstate solution are relegated
2.1
Microstate geometries for BPS black holes with large horizon area
A smooth horizonless microstate geometry that has the same charges and supersymmetries
of a
ve-dimensional three-charge BPS black hole and has a Gibbons-Hawking (GH) base
space has the metric:
ds121 =
(Z1Z2Z3) 32 (dt + k)2 + (Z1Z2Z3) 31 ds42 + (Z1Z2Z3) 31 X3 dx23+2I + dx24+2I
I=1
ZI
where the ZI are the warp factors corresponding to the three charges, k is the
angularmomentum 1-form and ds24 is the metric of the GH base:
ds42 = V 1(d
+ A)2 + V (dy12 + dy22 + dy32)
All the quantities appearing in (2.1) and (2.2) are de ned following the standard
procedure [12, 13] detailed in appendix A. The whole background is determined once one xes
the number N of GH centers, as well as the residues of the four harmonic functions V and
(2.1)
(2.2)
{ 5 {
between the satellites and the central blob are not on scale.
KI at these centers:
V =
N
X vi
where ~y = (y1; y2; y3) and ~gi is the position of the i-th pole in the same subspace. The
particular BPS microstate geometry we will use in this paper has N = 7 GH centers, whose
(vi; kiI ) parameters are symmetric with respect to the GH center in the middle:6
where the meaning of k^ will become clear in a moment.
As explained in appendix A, the distances rij between the GH centers are subject
to N
1 bubble equations that need to be satis ed to prevent the existence of closed
timelike curves:
N
X
same axis. Given that the parameters in (2.4) are invariant under i ! 8
i for i = 1; : : : 4,
the solution of (2.5) will give rise to a collinear con guration, shown in gure 1, with
rij = r(8 i)(8 j)
Hence, the collinear solution is determined completely by r12; r23; r34.
The family of collinear microstate geometries we consider is parameterized by k^, which
controls the depth of our microstate. There is a critical value k^0 at which the solution is
singular,7 and changing k^ around this value gives rise to scaling solutions that have very
small rij and hence a very long throat:
k^ = k^0 + ;
rij = rij + O( 2) ;
6The value of k13 given in [1] di ers from the one we give here and in [15] because of a typo.
7For the curious, k^0
3:17975.
(2.4)
(2.5)
(2.6)
(2.7)
{ 6 {
where the rij are determined by the
uxes on the two-cycles between the centers. As
expected, in the scaling limit the ratios between the distances are
xed [42], and the
physical distances between the GH centers become independent of the distance between
these points in the GH base [14]. However, as
approaches zero the length of the throat
of the microstate geometry diverges as
1
.
We choose to work with a throat that is long-enough to describe the typical sector
of the D1-D5-P black hole but not in nite, and we will x the length of the throat for
now by setting k^ = 3:1667. In section 5 we will relax this condition and examine how
the physics we nd changes as k^ moves towards the critical value. We then use (2.4) to
solve (2.5) and nd
and the ratios
{ 7 {
We discuss in appendix B the physical interpretation of these very large ratios.
The three electric charges of the solution and its SU(2) angular momenta are (A.16):
Q1 = 1:48 105;
Q2 = 1:20 105;
As one can see from equation (A.18), JL vanishes because of the Z2 symmetry of our
con guration (2.6). This solution represents a supersymmetric horizonless microstate of a
BMPV black hole [45] with a classically large horizon area (and hence nonzero entropy) [14].
2.2
Adding metastable supertubes
To build a near-BPS microstate we add a supertube probe [29] to the BPS solution of
section 2.1. If the supertube is at a supersymmetric minimum, the resulting microstate is
still BPS. However, supertubes can also have metastable minima [41], and these give rise
to microstate geometries of near-BPS black holes [1].
In the duality frame where the black hole has three M2 brane charges (2.1), a supertube
has two types of M2 brane charges, q1, q2, as well as a dipole charge d3 corresponding to an
M5 brane that wraps the ber of the Gibbons-Hawking space [
11
]. The potential energy
of a supertube is [41]:
H =
where
p
Z1Z2Z3V 1 s
respectively, and V and KI are the harmonic functions de ned in (2.3). When the
supertube is at a metastable point of the potential, H > q1 + q2, and supersymmetry is broken.
(2.12)
(2.13)
The total charges of the system are then given by the sum of the charges of the background
and of the probe, while the total mass is
Mtot =
M + X Qbackground + qprobe
where
M is the value of H
(q1 + q2) at the metastable point. Even if the backreacted
solution corresponding to this supertube has not been constructed explicitly, it is possible
to argue that the resulting background is globally smooth in the duality frame where the
charges of the black hole correspond to D1 branes, D5 branes and momentum.
As in [1], we consider a supertube whose charges are much smaller than those of the
background, and whose physics can therefore be captured by the probe approximation:
The fact that q2 and q1 have opposite signs does not automatically imply that
supersymmetry is broken [46]. A supertube with a given set of charges can have both BPS and
metastable minima, and the parameters whose positivity ensures that supersymmetry is
not broken are the q~i.
Let y1 be the coordinate parameterizing the axis of gure 1 centered in g4. Given
y1 !
the symmetric arrangement of the GH points in section 2.1, H in (2.11) is invariant under
y1. The probe potential has several Mexican-hat-type metastable minima, and we
focus on the one in the proximity of g6. The mass above extremality of this supertube is
M = H(23:812; 0; 0)
(q1 + q2)
0:04742
It is important to stress that the existence of metastability does not depend on the
parameter
controlling the depth of the scaling BPS microstate geometry, because when
the depth scales with
as in (2.7) the probe potential (2.11) transforms as
H(ri0j ) !
H(rij ) + O( 2)
3
The instability of near-extremal microstates
In this section we show that the nearly-BPS microstates built in section 2.2 by using probe
anti-supertubes are classically unstable.
We consider a microstate geometry with seven GH centers that have the vi and kiI
parameters as in (2.4).
The location of the GH centers is constrained by the bubble
equations (2.5) and by the requirement that JL is zero, and hence the moduli space of
the solutions is six-dimensional. For simplicity we examine a subset of this moduli space
constrained by the symmetry (2.6), which automatically ensures that JL = 0 and reduces
the number of independent bubble equations from six to three. Furthermore, we will
focus our discussion only on con gurations where all the centers lie on the same plane.8
The multicenter solutions that satisfy all these requirements can be parameterized by two
coordinates ( ; ), that are the angles shown in gure 2.
8We have also analyzed the con gurations where the centers are not on the same plane, but all the
relevant physics is captured by the planar ones.
{ 8 {
g1
g2
g3
g4
g5
g6
g7
HJEP04(216)8
As one changes these angles ( ; ) 2 [0; 2 ] [0; ], the bubble equations (2.5)
determining the inter-center distances are modi ed, and have to be solved again to determine
the new values of the distances. In principle this way of parameterizing the solutions of the
bubble equations can lead to singularities, as the bubbles can collapse for certain critical
values of the angles, but this does not happen for the particular solution we consider.
It is not hard to see rst that as one changes
and
the action of a probe supertube
with charges (2.15) will continue having a metastable minimum in the vicinity of g6 (or g2).
However, the exact value of the energy of the probe, which gives the mass above extremality,
becomes a nontrivial function of the angles:
M ( ; ).
We analyzed
M ( ; ) numerically starting from the collinear con guration ( ; ) = 0.
We found that keeping
xed
M is monotonically increasing with
behavior does not depend on the choice of k^ for the scaling (2.7), as shown in gure 3.
2 [0; ] and this
On the contrary, if one keeps
xed and varies
, the mass of the anti-supertube
M ( ; ) decreases as
starts increasing. This behavior is again independent on the
particular choice for k^ in (2.4), as shown in gure 4.
This proves that the collinear near-BPS microstate con guration obtained in section 2
is classically unstable. Indeed, the initial microstate corresponds to ( ; ) = 0 for the
parametrization of gure 2 and gure 4 shows that
M decreases when
= 0 and
starts
increasing. The maximum relative di erence in
M ( ; 0) is
M (0; 0)
M ( =2; 0)
M (0; 0)
and in the scaling regime (2.17) this does not depend on the choice of k^.
Of course it is interesting to ask whether this con guration will settle into another
metastable minimum somewhere in the moduli space, or simply will keep decaying until
it reaches a BPS minimum. In the slice that we explored there appears to be a minimum
when
=
=2 and
= 0, but this does not imply that this minimum will be metastable.
There could be other instability directions corresponding to other motions of the points in
the moduli space.
{ 9 {
ΔM (β)
regimes. The blue curve is obtained for k^ = 3:1667, while the purple one is obtained for k^ = 3:175,
which corresponds to
= 0:36 in (2.7). For the sake of clarity, the values of the purple curves have
been multiplied by 0:98 times the ratio of the two
M for the collinear con gurations.
2 [0; ] for
= 0 in two di erent scaling
regimes. The blue curve is obtained for k^ = 3:1667, while the purple one is obtained for k^ = 3:175,
which corresponds to
= 0:36 in (2.7). The values of the purple curves have been multiplied by
0:98 times the ratio of the two
M for the collinear con gurations in the two regimes.
= 0 in two di erent scaling
To prove that there will never be any metastable minimum one should investigate
the full 12-dimensional moduli space,9 which seems computationally tricky to perform,
especially because not all solutions to the bubble equations are free of closed time-like
9The moduli space of N centers subject to the bubble equations and subtracting the center of mass
motion is 2N
2 dimensional.
curves. Furthermore, if we eliminate the constraint that the total JL is zero, we can
explicitly nd a direction on the moduli space that leads to a scaling behavior, and as the
con guration moves in that direction the energy of the anti-supertube approaches zero.
Hence, we believe there is good reason to assume that metastable points in the moduli
space are rare, if not altogether inexistent.
4
The emission rates of non-extremal microstates
uli space of solutions to the bubble equations (2.5) with the symmetry (2.6). Classically,
this instability would trigger a motion of the GH centers down the microstate throat and,
in particular, gure 4 represents the potential energy that governs this motion. Quantum
mechanically, this instability triggers a decay process towards extremality that causes the
emission of radiation. From the thermodynamical point of view this is expected: a
nearBPS black hole has a nonzero Hawking temperature and hence emits radiation according to
(the ve dimensional version of) Stephen's law. Our initial collinear near-BPS microstate
would then decay into a series of di erent near-BPS microstates closer to extremality. We
can interpret the symmetrical rotation in
of gure 2 as a possible initial decay channel
of the collinear microstate and using
M ( ) shown in
gure 4 we want to estimate the
energy emitted per unit time
into this particular decay channel at the beginning of the
decay cascade.
We consider the initial decay process from the collinear microstate to the state given
by the con guration
=
=2;
= 0 of gure 2, which represents the minimum energy
state in the slice considered in gure 4. The emitted energy for this process is given by
m
M ( = 0)
M ( = =2)
(4.1)
and is only a small fraction of the initial
M (0) | see (3.1). It is hence correct to say
that
= =2 is only an intermediate state in the decay cascade that brings the microstate
towards extremality.
To de ne an emission rate we need to know how much time the system needs to emit the
energy m. We suppose that this average time is of the same order of the characteristic time
scale
that, classically, governs the motion for small
as seen from an observer at in nity.
Alternatively,
can be seen as the necessary time to have a measurable displacement from
the collinear con guration seen from in nity.
In order to
nd
we need to estimate the kinetic and the potential energies
corresponding to the classical motion of the GH centers in the moduli space of solutions to the
small velocity expansion of the kinetic energy of the bubbles, d2E
.
bubble equations (2.5). We continue focusing on Z2 symmetric solutions (2.6), and we
need to nd the small- expansion of the potential energy shown in gure 4, as well as the
For small , the mass above extremality behaves as
d _ 2
M ( ) =
M0 1
c2
2
(4.2)
where c2 is a coe cient that can be found numerically and whose dependence of the charges
of the solution and of the scaling parameters of the solution is discussed in detail in section 5.
The strategy for computing the kinetic term is much more involved. The few obvious
guesses about how to do it, involving for example treating one of the GH centers as a probe
in the background sourced by the others, do not give sensible answers. The essential reason
is that the energy of a bubble does not come only from the GH centers but also from the
uxes wrapping this bubble, and to compute the total energy brought about by the slow
motion of the bubbles one has to compute the full energy of all the
uxes as well, and
integrate the result over the full highly-warped spacetime.
Our strategy is to rather use the fact that some bubbles are much smaller than others,
and therefore the collective motion of a certain small bubble has the same energy as the
motion of the black ring that undergoes a geometric transition to form the small bubble.
One can then \falsely compactify" the solution to a four-dimensional one, by adding a small
constant in the harmonic function describing the Gibbons-Hawking base, and compute the
kinetic energy corresponding to the motion of the black ring. The
nal result is clearly
independent on the small constant we are adding, and hence does not change when taking
this constant to zero and recovering the asymptotically 4+1 dimensional solution.
Hence, the strategy we use can be summarized in the following recipe:
1. We compactify the three-charge solution (2.1) to four dimensions along the ber
by adding a constant to the V harmonic function.
2. We compute a Lagrangian for the motion in the -direction excising GH centers g1
and g2 from the background and replacing them with a singular black ring having
the same mass Mbr and charges at the same distance R
r23 from the center, as
explained in the physical interpretation of our solution in appendix B. The black
ring in 4D is treated as a massive point particle that rotates in
in the background
sourced by the other centers and under the e ect of the potential in (4.2) generated
by the anti-supertube probe closed to g6.
3. We assume that the four-dimensional metric is not a ected by the slow rotation in
, and hence that all the possible corrections to the background
elds caused by
this motion are negligible. In addition, the electromagnetic interactions between the
black ring and the background can be neglected when the rotation is slow.
The rst hypothesis helps to avoid useless computations as it allows to consider just
point particles instead of extended objects. The supergravity solution (2.1) is
asymptotically ve-dimensional. The further compacti cation along
requires some caution. Indeed
one needs to modify the asymptotic behavior of the GH space from R4 to R
3
S1 by
introducing a constant V in the function V in (2.3). The radius of the compacti ed S1,
r
1= V 2, can be thought of as a modulus of the solutions. In addition, one is also allowed
to introduce constants
KI in the de nitions of the harmonic functions KI in (2.3), which
modify the bubble equations (2.5) | see [47] for more details. While all these moduli are
completely arbitrary, they only specify the asymptotics of our solution and do not a ect
the computation we are interested in. The phenomenon that we want to study takes place
deep into the black-hole-like throat and hence is not a ected by the details of the
asymptotic elds of the background. Therefore we can compactify to four dimensions with no
risk of ambiguity.
The second hypothesis is really the key point of our computation. We have veri ed that
taking away centers g1 and g2 from our system does not substantially modify the solution
to the bubble equations (2.5). Using a hat to denote quantities computed without g1 and
g2 we have rb56 ' r56 = r23, rb34 ' r34, up to corrections of order 0:1%. Most importantly,
we veri ed numerically that the probe anti-supertube still has a metastable point close to
g6 and the behavior under rotation is similar to that of the complete system:
Mc( ) =
Mc0(1
b
where c2 was introduced in (4.2).10 This means that we can excise the GH centers g1
and g2 from the whole solution and replace them by a BPS black ring whose charges are
determined by the sum of the residues in the K; L and M harmonic functions of the points
we excised. This is the inverse of the bubbling black ring transition described in [
11
], and it
is not hard to check that the distance of the black ring from the central blob given by g3; g4
and g5 is essentially the same as the original distance between the blob and the excised GH
centers. Since the points g6 and g7 have not been excised, we can put the anti-supertube
probe with charges (2.15) at the metastable location close to g6.
Our strategy is to compute the kinetic term corresponding to the rotation of the black
ring center in
(described in
gure 2), by treating the black ring center as a probe in the
background sourced by the other GH centers.
Finally, the third hypothesis is reliable because
and its derivatives with respect to
time are small and hence modi cations to the four dimensional metric become
higherorder corrections. Therefore we can safely use the four-dimensional metric generated by
the collinear solution to estimate the kinetic term. Note that since the background is kept
xed to rst order in , the electromagnetic interactions between the point-like black ring
and the background are negligible.
4.1
The decay time and the emission rate
We compactify the eleven-dimensional supergravity solution (A.1) on T6 and
to a
fourdimensional solution whose metric is
1
J4 = Z1Z2Z3V
2V 2
1
ds42 = J4 2 (dt + !)2 + J42 [dr2 + r2(d 2 + sin2
d 2)]
(4.4)
and the four-dimensional dilaton is constant [12]. Note that the positivity of J4 is one of
the fundamental requirements for the construction of the solution [48], and comes from the
absence of closed time-like curves in the eleven-dimensional geometry.
10The factor in the relation between bc2 and c2 comes from the fact that the energy reduction caused
by the motion of the black ring alone is half of that caused by the movement of both the black ring and
the bubbling black ring given by the points g6 and g7. Similarly, the kinetic energy corresponding to the
motion of the black ring is half of that corresponding to the motion of both the black ring and the GH
points g6 and g7.
The GH centers g1 and g2 have been excised from the background (4.4) and we denote
with a hat all the quantities computed using only the centers g3; : : : ; g7 with parameters
as in (2.4). The black ring corresponds to a point-like particle at a distance
from g4 substituting g1 and g2 in gure 1. Its mass Mbr corresponds to the mass associated
with g1 and g2 and is nothing but the mass of the black ring microstate that these centers
represent, as explained in appendix B. The latter was found in [14]
Mbr = Qb1r + Qb2r + Qbr
3
QbIr = CIJK dJ f K
where we have introduced the parameters
dI = 2 k1I + k2I
f I = 6k0I + 1 +
k1I + 1
k
I
2
k0I =
k3I + k4I + k5I
If we then let the system rotate along
, the re-inserted point particle interacts with
the potential (4.3). To compute the full Lagrangian we need to
nd the kinetic term
corresponding to the slow motion of the black ring in the moduli space, and for this we
1
v2
1
3
Equation (4.3) allows us to approximate 2c2
c2 and hence the characteristic time scale
of this di erential equation is
can use the classical GR action:
S =
Mbr
Z r
g^
dx dx
dt dt
=
Mbr
Z q
J~ 21
4
1
J~42 R2 _ 2
Z 1
2
3
MbrJ~44 R2 _ 2 + const
where we have used the time t measured by an observer at in nity to parameterize the
worldline. Note that a tilde above J4 means that this quantity is evaluated at the location
of the black ring, namely at a distance R from the center g4.
Thus, the full Lagrangian corresponding to the motion of the GH centers that triggers
the decay of the metastable supertube
and the associated equation of motion to rst order in
is
L =
1
2
MbrJ~443 R2 _ 2
Mc( )
MbrJ~443 R2
2c2
b
M0
= 0
=
b
s
3
MbrJ~44 R2
M0c2
=
m
This equation is the main result of our paper. Since
M0 parameterizes the initial
energy of this solution above the BPS bound we see that the closer our solution is to the
BPS bound the bigger
is.
Using (4.1) and (4.11) following the arguments presented in section 4 we de ne the
emission rate of our microstate in the -channel to be
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
scales under a scaling of the conserved charges QI and
of the distances in the R3 base of the Gibbons-Hawking space underlying the BPS solution.
These results will be used to check whether this particular kind of emission from the initial
collinear microstate is typical in the thermodynamical ensemble.
5
behaves under two di erent types of scaling of the
background. The results of this section will be used in the next one to compare
with the
thermal emission rate found by the authors of [43] for a D1-D5-P non-extremal black hole
and thus gain information about the typicality of the -decay channel studied in the
preHJEP04(216)8
vious sections.
We are interested in two separate scalings that involve the parameters of our microstate.
The rst one corresponds to the scaling of the depth of the throat of our microstate (2.7)
while keeping the charges and uxes essentially xed:
!
1
2
LMS !
1
LMS
The second one corresponds to scaling all the magnetic uxes in (2.4)
rij !
rij
k !
k
and modi es the charges and the mass as (A.16):
QI !
2QI
M !
2M
Note that
M0 is considered as a free parameter of the system and it does not scale.
This quantity is the energy (2.16) brought about by placing an anti-supertube probe at
metastable point close to g6 and can be kept xed while performing the scalings above by
suitably tuning the anti-supertube charges (2.15). For the particular anti-supertube we are
using, we only need to tune the probe charge q2 and keep q1 and d3 xed as in (2.15). This
is because the charge q2 has opposite sign with respect to the corresponding background
charge and it is responsible for supersymmetry breaking and metastability.
in (5.1) one can use equations (2.17) and (4.2)
to deduce that c2 in (4.11) does not transform, which is also shown in gure 4. Then using
the formulas in appendix A it is easy to determine that J~4 !
3J~4, R2
!
2R2 and hence
As pointed out in [1], the physical importance of is to scale the (metric) length LMS of
the microstate throat as11
11The metric length of the microstate [1] is given by LMS =
suitable cuto .
Z zneck V 12 (Z1Z2Z3) 61 dz; where zneck is a
z7
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
Because of equations (5.4) and (5.5) we see that the decay time corresponding to the
rotation
becomes longer as the length of the throat becomes longer; thus, the closer to
BPS the con guration is the slower it decays.
in (5.2) it is important to observe that after
the scaling the bubble equations (2.5) are exactly solved by rij !
2rij . Then it is easy to
verify that
Mbr !
2Mbr
R
2
!
4R2
~
J4 !
2J~4
(5.6)
Unlike for the -scaling, we could not evaluate analytically the scaling properties of c2
in (4.11). To infer them numerically one can start from a ( = 1) solution with charges
given in (2.10), and in order to keep
M0 in (2.16)
xed while varying
one needs to
change the q2 charge of the anti-supertube probe in (2.15):
By repeating for each value of the evaluation that leads to the potential shown in gure 4,
one nds that
and therefore the overall scaling of
with
under (5.2) is:
1:2
1
2
3
4
q2
50
63:522
117:92
186:355
254:958
c2 !
1
2 c2
!
5
2
m !
0:8 m
!
1
2
!
3:3
In the next section this result will be used to compare the emission rate
in the -channel
with the emission rate of the thermodynamical ensemble to check whether this decay
is typical.
Finding the scaling properties of m is much easier. This quantity does not scale with .
Indeed, the probe hamiltonian (2.11) does scale with , as described in equation (2.17),
but as we tune the probe charge to keep
M0 constant it turns out that
m also remains
constant. This does not happen for the -scaling (5.2) and as there is no analytical formula
for m it is necessary to perform another numerical interpolation. Tuning the probe charge
q2 as before we nd
Finally, using the de nition (4.12) with (5.4), (5.8) and (5.9) one determines the scaling
properties of
under (5.1) and (5.2):
(5.7)
(5.8)
(5.9)
(5.10)
In this section we want to compare the emission process of our microstate with the thermal
emission of a near-BPS
ve-dimensional Reissner-Nordstrom black hole. In particular, we
want to check whether the
channel emission has the features that one expects from
thermodynamics.
We have three fundamental pieces of data about the radiation emission of the
thermodynamical ensemble coming from general relativity and brane technology. The rst is
the computation [43] of the emission rate for a near-BPS three-charge
ve dimensional
black hole in the D1-D5-P frame, where a tiny amount NL of left-moving momentum is
inserted on a string of length N1N5 with NR
NL right-moving momentum, so to break
supersymmetry. The energy emission rate for closed strings was found to be
where TH is the Hawking temperature of a ve-dimensional Reissner-Nordstrom black hole
computed from its surface gravity:
th
TH
pQ1Q2Q3TH5
1 r
Re
M0
M
ms = 2 R
p
Here Re and M are the horizon radius and mass of the black hole, and
M0 is the mass
above extremality, which is assumed to be much smaller than M . Given that the horizon
area for this class of black holes is proportional to A /
simply Stephen's law in
ve dimensions.
The second piece of data for the comparison of energy emission rates is Wien's law,
which follows from the ve-dimensional version of Planck's law and has the same form in
four and ve dimensions:
pQ1Q2Q3 we see that (5.11) is
max;th
TH
where max;th is the peak-frequency of energy emission from the ensemble and TH is the
black hole temperature (5.12).
The third and
nal piece of data is the di erence
L between the depth of the
microstate throat Lms and the throat Lbh of the black hole corresponding to the microstate.12
Since supersymmetric black holes have in nite Lbh, this comparison is meaningful only for
non-supersymmetric black holes. In [1] it was shown that the for class of near-BPS
microstates we discuss one can arrange Lms to be arbitrarily larger or smaller than Lbh.
However, we expect that typical microstate geometries of the black hole will have Lms
comparable to Lbh, and hence
L
The parameter ms in (5.14) is given by
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
The scaling properties of ms under (5.1) and (5.2) are easily found from the results
of section 5.1:
ms !
1
2 ms
ms !
ms
It is straightforward to determine how (5.11), (5.12) and (5.16) scale with
under (5.3):
TH !
2TH ;
th !
7 th;
bh !
1
2 bh
The thermal quantities do not scale with .
Given the three equations describing the thermal emission of the ensemble (5.11) and
the size of the microstate, (5.13) and (5.17), we can argue that a given decay process of a
microstate into a particular channel is typical if its , max and ms match those given by
these equations. The only missing information about our decay channel is
max. We can
argue that the energy emitted during the decay of the nonextremal microstate geometry
is given by the di erence between the highest and the lowest values of the mass above
extremality during the rotation in :
max =
m
bh = 2 ms
(5.17)
where R is the distance of the outermost GH center (4.5), while bh is given by:
which is the horizon radius of the corresponding non-extremal black hole. The parameter
neck in (5.14) corresponds to a certain cuto
needed to measure the throat lengths, but is
irrelevant if when one imposes
L = 0, which implies
Given the scaling properties of our solutions (5.1), (5.2), (5.4), (5.8) and (5.18) we can
start from the initial solution (with charges given in equation (2.10)) and check whether
there is any value of
and
(or alternatively of Q01; Q02; Q03 and 0ms) for which the decay
of our solutions matches the thermal decay:
0th = 0
m0ax;th = m0ax
0bh = 2 0ms
Unfortunately, this is not possible, which implies that the solution we started from
and the families of non-extremal microstate solutions obtained by scaling its depth and
uxes via (5.1) and (5.2) are not typical. This is not surprising | after all, we started
from a very speci c seven-center solution that has a lot of symmetry, and a very large
ratio between certain inter-center distances, and we examined a non-extremal microstate
geometry obtained by adding a certain type of anti-supertube to this solution. It would
have been in fact much more surprising if this decay process had been thermal-like.
It is also possible to parameterizes the departure of the non-extremal microstate we
consider from typicality by introducing a quantity, , that can be thought of as modifying
the estimation of
of section 4.13 If one multiplies
by
in all the equations above,
the system (5.21) can always be solved for some th; th; th. Plugging in the numbers
we nd that
th
1:8 105
(5.22)
This therefore gives an estimate of the departure from typicality of the microstate
we have considered. The nal result th
1 has two possible implications that are not
mutually exclusive. On one hand, it can imply that a typical microstate should have a
that is much smaller than the one of our emission process, or a much larger characteristic
decay time .
This could be for example realized by considering non-extremal microstates that have
more centers, and whose moment of inertia for the rotation in the direction that lowers the
energy of the microstate is much bigger than in our solution. On the other hand (5.22)
suggests also that the emitted energy
m should be much smaller than the one we found.
This is not helped at all by considering microstate geometries that have more centers,
because the potentials of these geometries will be generically less abrupt. Hence, to decrease
the emitted energy one should rather consider creating non-extremal microstates by adding
supertubes to BPS microstates with simple topology (like the superstrata of [20]). It would
be interesting to investigate which of the two options is the best for producing more typical
non-extremal microstate geometries.
6
Conclusions and outlook
We analyzed the near-BPS smooth collinear microstate built in [1] by adding a probe
anti-supertube at a metastable minimum of its potential in the background of the smooth
microstate for a BMPV black hole with large horizon area [14].
We proved that this
microstate is unstable by showing that the potential energy of the supertube
M decreases
if the GH centers of the microstate solutions rotate with respect to the axis of the initial
collinear con guration.
We also estimated in section 5.2 the typical instability time and the emission rate
corresponding to this decay, and compared the results to those expected from
thermodynamics. We used three known thermodynamic quantities: the emitted energy rate, the
peak frequency and the typical radius of microstates, and we showed that they cannot be
all matched by the features of the decay channel we have found. We also estimated the
mismatch to be of order 105, which can be thought of as a measure of the departure from
typicality of the nonextremal black hole microstate solution we started from.
It is clearly important to construct non-extremal microstate geometries that have a
more typical decay. One way to do this is to build near-BPS microstates with more that
13The choice of multiplying is not arbitrary: we fully trust the assumptions that enter in the computation
of
of section 4, but given that we have not investigated the motion of the GH centers in the full
sixdimensional moduli space, we do not know whether other decay directions exist where the decay time is
faster or shorter. For example, if one would rather rotate the segment formed by the central points of
the solutions, g3; g4 and g5, the inertia momentum would be much smaller than the one corresponding to
rotating the black ring centers, and hence the decay time would be much faster.
seven centers, or whose centers are more evenly spaced as the ones in our solutions (which
are at distances whose aspect ratio is of order 1000). Indeed, the seven-centers solution
analyzed in this paper represents somehow the minimal interesting model that one can
build [14]. Furthermore, the moment of inertia corresponding to the motion that
destabilizes our solution is very large as it corresponds to moving an entire bubbled black ring
that has about one third of the total mass of the microstate. Finding a more sophisticated
solution, though more involved, will allow us to look for analogous decay patterns that
have a parametrically smaller moment of inertia or parametrically larger
m, such that
and decay time that is fast-enough to be in the typical range.
From a more general perspective, our result con rms the intuition of [
40
], that most
HJEP04(216)8
of the microstate solutions of non-extremal black holes should be unstable, and hence the
dynamics of these black holes will display a chaotic behavior, corresponding to microstates
being formed and immediately decaying into other microstates, which in their turn decay
very fast. It would be interesting to see if our solutions can be used to shed light on some
of the features of the chaotic behavior of non-extremal black holes discussed in [44].
It is also important to understand the physics of this instability in the decoupling
limit.14 The Callan-Maldacena emission rate goes to zero in this limit, but there are other
instabilities of supergravity solutions that do not [
50, 51
]. Our instability concerns the
dynamics deep down the throat of scaling solutions, and hence it is always present if one
uses the time at the bottom of the throat. On the other hand, emission probabilities are
computed using the time coordinate that is good at the asymptotically- at in nity and in
the scaling limit the relation between these two times becomes degenerate. Hence, in the
scaling limit our emission rate becomes zero. Therefore, from this perspective our
instability appears closer to the Callan-Maldacena one, and di erent from the ones in [
50, 51
],
even though our instability would be set to zero by a qualitatively di erent process than
the Callan-Maldacena one. There is a subtlety concerning the possible decoupling limits of
our system: there are two of them. One is similar to the scaling limit, and corresponds to
throwing away the constants in all the harmonic functions corresponding to the black hole
charges, and obtaining a long throat that has an AdS2
S3
T 6 form in 11D supergravity.
There is another decoupling limit, that one can only take in the duality frame where the
charges of the black hole correspond to D1 branes, D5 branes and momentum, and which
corresponds to throwing away the constant in the D1 and D5 harmonic functions but
keeping the constant in the harmonic function corresponding to the momentum. This yields an
asymptotically-AdS3
S3
T 4 geometry. We have not worked out what happens in this
limit, but we expect that solutions are not a ected by it, as happens for the instabilities
of [
50, 51
]. A deeper analysis of the fate of our instability in the decoupling limit is hence
needed to fully understand its nature.
Besides its implications for black hole physics, our result may also have important
consequences for the program of uplifting AdS vacua obtained from generic ux
compacti cations to obtain de Sitter space in string theory. There is a direct analogy between the
uplift of AdS solutions to de Sitter by adding antibranes [
52, 53
] and the uplifting of BPS
14We thank the referee of this paper for suggesting this discussion.
microstate geometries to microstates of non-BPS black holes by adding anti-supertubes [1].
In both constructions one used the action of the probe antibranes to argue that they have
metastable vacua. However, our investigation reveals that the result of the probe
calculation can be misleading, and that the metastable supertube can be in fact destabilized by
the motion in the moduli space of the underlying geometry. However, unlike microstate
geometries, ux compacti cations usually come with all the moduli stabilized. Nevertheless,
it is possible that even stabilized moduli can be destabilized, especially when their mass is
very low. It would be interesting to understand whether this happens when investigating
antibranes [53{55] in the Klebanov-Strassler warped deformed conifold solution [16].
Acknowledgments
We are grateful to David Turton and Nick Warner for useful discussions and
suggestions. This work is supported in part by the ERC Starting Grant 240210 String-QCD-BH,
by the John Templeton Foundation Grant 48222 and by a grant from the Foundational
Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community
Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was administered by
Theiss Research).
A
Construction of the three-charge smooth background
In this appendix we review the construction of smooth three-charge black hole microstate
solutions. Some aspects of physics of these solutions are discussed in the next appendix,
and more details can be found in [32, 48].
The smooth three charge solution preserves N = 4 supercharges in 11D supergravity
compacti ed on three tori. The metric and the three-form potential are given by:
ds121 = (Z1Z2Z3) 32 (dt + k)2 + (Z1Z2Z3) 31 ds42 + (Z1Z2Z3) 31 X3 dx24+i + dx25+i
i=1
Zi
A(3) = A ^ dx5 ^ dx6 + A ^ dx7 ^ dx8 + A ^ dx9 ^ dx10
1 2 3
where k is the angular momentum vector, Zi are the three warp factors associated to the
electric conserved charges and dx24+i + dx25+i for i = 1; 2; 3 is the standard metric on a
torus. The metric of the base space of this solution, ds24, is chosen to be a multi-center
Gibbons-Hawking/Taub-NUT metric:
where
ds42 = V 1(d + A~ d~y)2 + V (dy12 + dy22 + dy32)
~
r
A~ = r~V
and the Taub-NUT ber
has period 4 . All the functions that appear in the
background (A.1) and in (A.2) depend on (y1; y2; y3), and the full solution is completely
determined by specifying four harmonic functions:
V = XN vj
j=1 rj
KI = XN kj
I
j=1 rj
I = 1; : : : 3
rj = j~y
~gjj
(A.1)
(A.2)
(A.3)
(A.4)
HJEP04(216)8
where N is the number of GH centers located at ~gi and (vj ; kjI ) are parameters to specify.
Notice that one can potentially add some constants V and
KI to the functions in (A.4)
so that the GH space asymptotes R
3
S1 and the KI generate some Wilson lines for the
three form in (A.1). As we want the GH space to asymptote R4 these constants are taken
in (A.2) asymptotes to at R4 if one also requires
When vj 2 Z, the GH centers become benign orbifold singularities. The geometry
The GH metric (A.2) is then ambipolar, meaning that its signature switches from +4 to
4. However, as shown in [
11, 12
], this is not a problem for the full solution (A.1), which
is everywhere smooth and Lorentzian.
The warp factors ZI of the solution are
where the LI are harmonic functions in the GH space. Requiring the ZI to be smooth at
the GH centers and xing their asymptotic value to 1 imply that
(A.5)
HJEP04(216)8
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
N
X vj = 1
ZI = LI +
1
LI = 1
1
XN kjI kjK 1
j=1
vj rj
~k = (d
+ A~) + !~
=
CIJK KI KJ KK
6V 2
KI LI + M
2V
M = m0 +
1
XN kjI kjJ kjK 1
j=1
v
2
j
rj
m0 =
j=1 I
X k
I
j
The last form to de ne in (A.8) is !~, which is given by:
!~ =
1
i;j=1
X vivj iIj iJj iKj !~ij
I
ij
k
I
j
vj
k
I
i
vi
where CIJK
j"IJK j. The BPS solution for the angular momentum vector k is written as
where A is de ned in (A.3) and
is given by
with M another harmonic function. The vanishing of
at the GH centers determines M as
where m0 is a constant whose value is found requiring that k vanishes at in nity:
where, choosing a coordinate system with ~yi = (0; 0; a), ~yj = (0; 0; b) with a > b and
de ning tan
= y2=y1, one has:
To avoid the existence of closed-timelike-curves (CTCs) it is necessary that
!~ij =
y22 + y12 + (y3
a + ri)(y3
b
(a
b)rirj
rj ) d
Z1Z2Z3V
2V 2
0
where
The expression for the di erence of the angular momenta depends also on the positions,
~gi, of the GH centers [12]:
Finally the electric three-form in (A.1) is speci ed by
holds everywhere in the GH space.
Furthermore, to avoid Dirac-Misner strings, the solution must satisfy the bubble
equations that constrain the distances between the GH centers [
11, 12, 42
]:
N
X
summing the l.h.s. of (A.15) over i one gets zero as the
1 out of N bubble equations are independent: indeed
iIj are anti-symmetric in ij. For
smooth microstate solutions constructed using GH centers only (A.7), (A.10) this condition
is equivalent to the vanishing of
in (A.9) at every GH center [14].
Expanding the warp factors ZI in (A.6) and the momentum vector of (A.8) it is possible
to read the three electric charges QI and the two angular momenta J1 and J2 preserved by
this background. In particular once the parameters (vj ; kjI ) have been speci ed the charges
QI , and the sum of the angular momenta are:
QI =
2CIJK X vj 1k~jJ k~jK
J1 + J2 =
3
4 CIJK X v 2~I ~jJ k~jK
j kj k
j
k
I
j
vj
X k
I
s
s
J1
J2 = 8j D~j
D~ =
3
X
N
X k~jI~gj
I=1 j=1
dAI =
I
d
dt + k
ZI
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
where
I are the dipole eld strengths
3
a=1
I =
+ A) ^ dya +
V "abc dyb ^ dyc]
(A.20)
1
2
In the formula above A is the three form computed in (A.3). In the next section we give
a physical interpretation of some bubble solution and explain the role of the dipole eld
strengths.
B
Some physical properties of bubbled solutions
In this section we explain how to use the technique presented in appendix A to build
microstates of BPS black holes and black rings. The solution (A.1) is a bubbling one,
which indicates the fact that the black hole or black ring solutions, that one would get if
the metric ds24 in (A.1) were that of at R4, are replaced by horizonless solutions that we
built using a Gibbons-Hawking metric (A.2) that has the same asymptotics but nontrivial
topology.15 The singularities of black objects are replaced by bubbles threaded by uxes:
Z
ij
I =
I
ij
(B.1)
where the
I are the dipole two-forms de ned in (A.20) and the
uxes
iIj are de ned
in (A.12). These uxes prevent the various two-cycles
ij from collapsing. Now, suppose
that the number of centers N and the GH charges vj subject to (A.5) have been
xed.
Then the uxes
iIj depend on the parameters kjI . Since these uxes determine completely
the distances between the colinear centers by the bubble equations (A.15), if one wants to
increase or decrease the distances between the N centers one can just modify the values of
kjI . This is the key ingredient to build smooth microstates of black holes and black rings
using the procedure of appendix A.
In [13] a smooth microstate was built for a maximally-spinning extremal horizonless
BMPV black hole by taking a blob of GH centers ensuring that the (vi; kiI ) parameters are
roughly of the same order. In the same paper it was shown that if one takes a huge number
of centers and randomly assigns the (vi; kiI ) one inevitably ends up with a microstate of a
maximally-spinning black hole.
On the other hand, in [
11
] it was shown that to build a bubbled solution for a
maximally-spinning (zero-entropy) black ring it su ces to take a blob of GH centers (at
least two) with zero total v-charge and a far-away GH center with v = 1. Such a solution
can be obtained by choosing vj approximatively of the same order, while taking kjI
a for
A crucial di erence between these microstate solutions and the corresponding black
holes and black rings is that the latter have an in nite AdS throat, while the throat have
15This topology is easy to see: the periodic coordinate
of (A.2) is bered over a curve connecting two
GH centers gi and gj and shrinks at both ends of this curve. This gives a nontrivial cycle
ij that is
topologically a two-sphere.
a nite throat that ends in smooth cap. There exists furthermore a limit in which the
length of the throat can become in nite, and this scaling behavior (2.7) can be achieved
for colinear solutions by tuning the vj and the kjI [14], and for non-colinear solutions by
changing the angles between the GH centers [15, 42].
Scaling solutions have also proved to be necessary ingredient for building microstates
for BMPV black holes with large horizon area. This was done in [14] by merging a black
ring blob with a black hole blob at its center. After a suitable choice of the black hole and
black ring charges, this process can results in a microstate for a BMPV black hole with
large horizon area.
We are now able to give a complete physical interpretation of the microstate of section 2.
HJEP04(216)8
The parameters (2.4) are chosen so that centers g1 and g2 together with their counterparts
g6 and g7 via (2.6) are far away from the central blob (2.9). The central blob g3
g4
g5
has total GH charge one and hence represents a bubbled maximally-spinning black hole.
The two satellites have zero GH charge, and represent two symmetric bubbled black rings.
Because of the symmetry (2.6) the full solution has J1 = J2 = J , and furthermore one can
check that Q1Q2Q2 > J 2. Hence the solution represented in gure 1 can be interpreted as
a microstate of a BMPV black hole with a macroscopically large horizon area.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] I. Bena, A. Puhm and B. Vercnocke, Non-extremal Black Hole Microstates: Fuzzballs of Fire
or Fuzzballs of Fuzz?, JHEP 12 (2012) 014 [arXiv:1208.3468] [INSPIRE].
[2] S.D. Mathur, Fuzzballs and the information paradox: A summary and conjectures,
arXiv:0810.4525 [INSPIRE].
(2009) 224001 [arXiv:0909.1038] [INSPIRE].
[3] S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26
[4] S.L. Braunstein, S. Pirandola and K. Z_yczkowski, Better Late than Never: Information
Retrieval from Black Holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190]
[5] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or
Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
[6] A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls,
JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
[7] M. Dodelson and E. Silverstein, String-theoretic breakdown of e ective eld theory near black
hole horizons, arXiv:1504.05536 [INSPIRE].
[8] A. Puhm, F. Rojas and T. Ugajin, in preparation.
[9] S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates,
Nucl. Phys. B 701 (2004) 357 [hep-th/0405017] [INSPIRE].
Phys. B 710 (2005) 425 [hep-th/0406103] [INSPIRE].
(2006) 066001 [hep-th/0505166] [INSPIRE].
black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].
HJEP04(216)8
[16] I.R. Klebanov and M.J. Strassler, Supergravity and a con ning gauge theory: Duality
cascades and chi SB resolution of naked singularities, JHEP 08 (2000) 052
[hep-th/0007191] [INSPIRE].
[17] J. Polchinski and M.J. Strassler, The String dual of a con ning four-dimensional gauge
theory, hep-th/0003136 [INSPIRE].
10 (2004) 025 [hep-th/0409174] [INSPIRE].
[18] H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1=2 BPS geometries, JHEP
[19] I. Bena, S. Giusto, R. Russo, M. Shigemori and N.P. Warner, Habemus Superstratum! A
constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463]
75 (2007) 124026 [hep-th/0604110] [INSPIRE].
(2006) 042 [hep-th/0608217] [INSPIRE].
States, JHEP 10 (2014) 140 [arXiv:1406.4506] [INSPIRE].
[21] V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries
and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [INSPIRE].
[22] S. Giusto, S.F. Ross and A. Saxena, Non-supersymmetric microstates of the D1-D5-KK
system, JHEP 12 (2007) 065 [arXiv:0708.3845] [INSPIRE].
[23] J.H. Al-Alawi and S.F. Ross, Spectral Flow of the Non-Supersymmetric Microstates of the
D1-D5-KK System, JHEP 10 (2009) 082 [arXiv:0908.0417] [INSPIRE].
[24] S. Banerjee, B.D. Chowdhury, B. Vercnocke and A. Virmani, Non-supersymmetric
Microstates of the MSW System, JHEP 05 (2014) 011 [arXiv:1402.4212] [INSPIRE].
[25] G. Bossard and S. Katmadas, Floating JMaRT, JHEP 04 (2015) 067 [arXiv:1412.5217]
[26] I. Bena, S. Giusto, C. Ruef and N.P. Warner, A (Running) Bolt for New Reasons, JHEP 11
(2009) 089 [arXiv:0909.2559] [INSPIRE].
[27] G. Bossard and S. Katmadas, A bubbling bolt, JHEP 07 (2014) 118 [arXiv:1405.4325]
[28] I. Bena, G. Bossard, S. Katmadas and D. Turton, Non-BPS multi-bubble microstate
geometries, JHEP 02 (2016) 073 [arXiv:1511.03669] [INSPIRE].
[hep-th/0103030] [INSPIRE].
Born-Infeld Meets Supergravity, JHEP 07 (2009) 106 [arXiv:0812.2942] [INSPIRE].
Holography, Nucl. Phys. B 789 (2008) 382 [hep-th/0610249] [INSPIRE].
Grav. 31 (2014) 025016 [arXiv:1305.0957] [INSPIRE].
HJEP04(216)8
[INSPIRE].
Grav. 25 (2008) 135005 [arXiv:0711.4817] [INSPIRE].
(2009) 065 [arXiv:0906.2015] [INSPIRE].
[34] P. de Lange, D.R. Mayerson and B. Vercnocke, Structure of Six-Dimensional Microstate
Geometries, JHEP 09 (2015) 075 [arXiv:1504.07987] [INSPIRE].
Quant. Grav. 26 (2009) 035006 [arXiv:0810.2951] [INSPIRE].
[41] I. Bena, A. Puhm and B. Vercnocke, Metastable Supertubes and non-extremal Black Hole
Microstates, JHEP 04 (2012) 100 [arXiv:1109.5180] [INSPIRE].
[42] B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites,
[43] C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl.
JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].
Phys. B 472 (1996) 591 [hep-th/9602043] [INSPIRE].
[arXiv:1306.0622] [INSPIRE].
[44] S.H. Shenker and D. Stanford, Black holes and the butter y e ect, JHEP 03 (2014) 067
[45] J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, D-branes and spinning black holes,
Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [INSPIRE].
138 [arXiv:1407.6007] [INSPIRE].
[46] S. Massai, G. Pasini and A. Puhm, Metastability in Bubbling AdS Space, JHEP 02 (2015)
[47] V. Balasubramanian, E.G. Gimon and T.S. Levi, Four Dimensional Black Hole Microstates:
From D-branes to Spacetime Foam, JHEP 01 (2008) 056 [hep-th/0606118] [INSPIRE].
[48] I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys.
755 (2008) 1 [hep-th/0701216] [INSPIRE].
[49] M. Cvetic and D. Youm, General rotating ve-dimensional black holes of toroidally
compacti ed heterotic string, Nucl. Phys. B 476 (1996) 118 [hep-th/9603100] [INSPIRE].
JHEP 03 (2012) 094 [arXiv:1108.0411] [INSPIRE].
nonsupersymmetric eld theory, JHEP 06 (2002) 021 [hep-th/0112197] [INSPIRE].
HJEP04(216)8
Klebanov-Strassler, JHEP 09 (2010) 087 [arXiv:0912.3519] [INSPIRE].
02 (2015) 146 [arXiv:1410.7776] [INSPIRE].
[10] S. Giusto , S.D. Mathur and A. Saxena , 3 -charge geometries and their CFT duals , Nucl.
[11] I. Bena and N.P. Warner , Bubbling supertubes and foaming black holes , Phys. Rev. D 74 [12] P. Berglund , E.G. Gimon and T.S. Levi , Supergravity microstates for BPS black holes and [13] I. Bena , C.-W. Wang and N.P. Warner , The Foaming three-charge black hole , Phys. Rev . D [14] I. Bena , C.-W. Wang and N.P. Warner , Mergers and typical black hole microstates , JHEP 11 [15] I. Bena , C.-W. Wang and N.P. Warner , Plumbing the Abyss: Black ring microstates , JHEP [20] I. Bena , M. Shigemori and N.P. Warner , Black-Hole Entropy from Supergravity Superstrata [29] D. Mateos and P.K. Townsend , Supertubes, Phys. Rev. Lett . 87 ( 2001 ) 011602 [30] I. Bena , N. Bobev , C. Ruef and N.P. Warner , Supertubes in Bubbling Backgrounds: [31] M. Aganagic , C. Beem , J. Seo and C. Vafa , Geometrically Induced Metastability and [32] G.W. Gibbons and N.P. Warner , Global structure of ve-dimensional fuzzballs, Class . Quant.
[33] P.A. Haas , Smarr's Formula in Eleven-Dimensional Supergravity , arXiv: 1405 .3708 [35] V. Cardoso , O.J.C. Dias , J.L. Hovdebo and R.C. Myers , Instability of non-supersymmetric smooth geometries , Phys. Rev. D 73 ( 2006 ) 064031 [ hep -th/0512277] [INSPIRE].
[36] S.G. Avery , private communication.
[37] B.D. Chowdhury and S.D. Mathur , Radiation from the non-extremal fuzzball, Class . Quant.
[38] S.G. Avery , B.D. Chowdhury and S.D. Mathur , Emission from the D1D5 CFT , JHEP 10 [39] B. Chakrabarty , D. Turton and A. Virmani , Holographic description of non-supersymmetric orbifolded D1-D5-P solutions , JHEP 11 ( 2015 ) 063 [arXiv: 1508 .01231] [INSPIRE].
[40] B.D. Chowdhury and S.D. Mathur , Non-extremal fuzzballs and ergoregion emission , Class.
[50] I. Bena , B.D. Chowdhury , J. de Boer, S. El-Showk and M. Shigemori , Moulting Black Holes, [51] B.D. Chowdhury and B. Vercnocke , New instability of non-extremal black holes: spitting out supertubes , JHEP 02 ( 2012 ) 116 [arXiv: 1110 .5641] [INSPIRE].
Rev . D 68 ( 2003 ) 046005 [ hep -th/0301240] [INSPIRE].
[52] S. Kachru , R. Kallosh , A.D. Linde and S.P. Trivedi , de Sitter vacua in string theory, Phys.
[53] S. Kachru , J. Pearson and H.L. Verlinde , Brane/ ux annihilation and the string dual of a [54] I . Bena , M. Gran~a and N. Halmagyi, On the Existence of Meta-stable Vacua in [55] I. Bena , M. Gran~a, S. Kuperstein and S. Massai, Giant Tachyons in the Landscape , JHEP