Bubbling solutions, entropy enhancement and the fuzzball proposal

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by  Clement Ruef

Bubbling solutions, entropy enhancement and the fuzzball proposal
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aInstitut de Physique Th´eorique,

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CEA Saclay Orme des Merisiers 91191 Gif-sur-Yvette
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In this short note we explain the main idea of the work done in [1,2]. We present a family of black hole
1
2 microstates, the bubbling solutions. We then explain how supertubes placed in such backgrounds have their
entropyenhancedbythepresenceofthebackgrounddipolecharges. Thisindicatesthiscouldaccountforalarge

] amount in theentropyof thethree charge black hole.
h
t
-
p
e
h 1. Introduction very complicated, very fuzzy, but stay neverthe-
[ less completely regular and look like a black hole
Itisawell-knownfactthateveryblackholehas
outside the (would be) horizon. We also don’t
1
anentropy,proportionaltotheareaofitshorizon.
v knowaprioriifthesupergravity(SUGRA)regime
But the microscopic origin of this entropy is still
7 isenoughtodescribeallthismicrostatesorifthey
3 unclear, despite the fact that much progress has are fully stringy. The black hole should be seen
2 been done in this direction over the last decades,
in this picture as a coarse grained description of
3 in particular in the context of string theory. In-
thesegeometries. Thisproposalhasbeensuccess-
.
1 deed, Strominger and Vafa were able to compute fully verified in the case of the two-charge black
0 the entropy of a class of supersymmetric (SUSY)
hole. However,thetwo-chargeblackholedoesnot
9 black holes from a microscopic point of view [3],
have classically a real macroscopic horizon, one
0
by counting bound states of strings and branes
: has to define a “stretchedhorizon”,with entropy
v at small effective coupling constant. SUSY then
Xi protects the counting when this coupling grows, 2hπol√e.2QD1Qiff2e,rtehnetQgir’osubpesinhgatvheecbheaenrgeasboleftthoebcaocnk-
and the entropy matches the macroscopic one at
r struct and count the corresponding microstates,
a large coupling, when the black hole exists. So
in the context of SUGRA, and to reproduce the
this counting, despite its success, doesn’t tell us
correct entropy 2π√2Q1Q2 [5,6]. What remains
anything about the nature of a “black hole mi-
tobedoneistounderstandthethree-chargecase,
crostate” at large coupling.
where the 5D black hole has a real macroscopic
Another very useful tool to count the micro-
horizon, with entropy 2π√Q1Q2Q3.
scopic entropy of black holes is the AdS/CFT
correspondance. If one wants to describe a black
hole in AdS space, or a hole in flat space but 2. Bubbling solutions
with an AdS throat in its near-horizonlimit, the
One veryinteresting class of three-chargesolu-
correspondance allows one to work in the dual
tions is the one of the multicenter solutions, or
CFT, where it is easier to count the number of
bubbling solutions [7,8,9]. This class describes a
states. Once againthis approach,despite its suc-
very large spectrum of different black hole con-
cess, doesn’t explain what is a black hole mi-
figurations: black holes, black rings, concentric
crostate.
black rings or black saturns as well as smooth
Onewaytoanswerthisquestionisthesocalled
horizonless geometries. Let’s concentrate on the
fuzzballproposal,proposedbyMathur(seeforex.
smooth solutions. In this type of configurations,
[4]). It states that every black hole microstate is
the main idea is that the external space isn’t flat
related to a horizonless solution, which can be
anymore but topologically non trivial: there are
1
2
a lot of Taub-NUT like centers, which all carry wecanhavethisentropybecomingverylargeand
electric and magnetic charges. This creates non eventually behaving like pQ3, where Q is a typ-
contractible two-spheres, or bubbles, held by the ical value of the electric charges. In other words,
fluxes between the centers. The electric charges weexpectthatthiscouldaccountforalargefrac-
seen frominfinity arethen createdby these mag- tion of the black hole entropy.
netic fluxes. The smoothness of the solution is
Othersworks[2,11]shedanewlightonthisre-
ensured by the bubble equations - one for each
sult. In [11] we learn that through spectral flow,
center - relating the charges of the centers and
each center of a bubbling configuration can be
the distance between them
turned into a supertube. So the previous anal-
X <Γi,Γj > =<h,Γi >, (1) ysis is in fact more general: each center can be
r dualized into a supertube and can have its en-
j,j6=i ij
tropy enhanced. Finally, in [2], we compare the
where i, j denotes the different centers, rij is the probe Born-Infeld approach and the full backre-
distance between them, Γi encodes the charges actedSUGRAapproach. Welearnthattheprobe
of the center i and h the charges at infinity, and approach captures all the physics of the problem
<, > is a symplectic product between the cen- and confirm that the probe computation of [1]
ters, representing the interactions. Since it will will give rise to smooth SUGRA solutions.
be useful in the next section, let us also men-
tionthatthis bubbling solutionsdo nothavejust REFERENCES
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