Download Black Spot.

Master’s thesis
University of Utrecht
august 2005
Black Spot.
Is the D1-D5 fuzzball
a black hole?
G.P.J.C. Wijnker
Supervisor : dr. S.J.G. Vandoren
Department of physics
University of Utrecht
1
Abstract
In this thesis, we examine Samir Mathur’s fuzzball proposal. He conjectures
that in this proposal the entropy paradox of black holes is solved by replacing
the old black hole picture by the fuzzball picture.
In chapter 2, we first look at the black holes in general relativity and the
entropy paradox. Then, in chapter 3, we see how certain D-brane configurations
can become black holes in supergravity.
In chapter 4, we focus on the D1-D5 fuzzball, the main subject of this thesis.
It is again a solution of the field equations of supergravity. We see how the
distinctive geometries of the fuzzball arise from the dual F-P system and discuss
briefly some concepts of the proposal. In chapter 5 we look at other research
published on the fuzzball proposal, and in chapter 6 we conclude that at this
stage, a final verdict over the status of the fuzzball picture cannot be made;
but some other interesting concepts can be extracted from this proposal and
investigated independently of this status.
Contents
1 Introduction
5
2 Black holes in general relativity
2.1 General relativity . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Einstein equations . . . . . . . . . . . . . . . .
2.1.2 Geodesics: the path of particles . . . . . . . . . . .
2.1.3 Kaluza-Klein compactifications . . . . . . . . . . .
2.2 Classical black holes . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Schwarzschild solution . . . . . . . . . . . . .
2.2.2 Schwarzschild in other coordinates . . . . . . . . .
2.2.3 The Reissner-Nordström solution . . . . . . . . . .
2.2.4 The extremal Reissner-Nordström solution . . . . .
2.2.5 The hairless Kerr-Newman solution . . . . . . . . .
2.2.6 Carter-Penrose diagrams . . . . . . . . . . . . . . .
2.3 Black hole thermodynamics . . . . . . . . . . . . . . . . .
2.3.1 Classical laws . . . . . . . . . . . . . . . . . . . . .
2.3.2 Hawking radiation: a first step towards quantum
holes . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The information paradox and the entropy paradox
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3 Black branes in supergravity
3.1 String theory . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 The bosonic string . . . . . . . . . . . . . . . . . . .
3.1.2 Quantization . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Worldsheet fermions + bosonic string = superstring
3.1.4 String diagrams and CFT . . . . . . . . . . . . . . .
3.2 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 From worldsheet to space-time . . . . . . . . . . . .
3.2.2 Type IIB supergravity . . . . . . . . . . . . . . . . .
3.3 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 S-duality . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 T-duality . . . . . . . . . . . . . . . . . . . . . . . .
3.4 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Solitons in supergravity . . . . . . . . . . . . . . . .
3.4.2 Compactification . . . . . . . . . . . . . . . . . . . .
3.4.3 Complementarity of the descriptions . . . . . . . . .
3.5 BPS black branes . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 BPS state . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.2 Black brane metrics . . . . . . . . . . . . . . . . . . . . .
3.5.3 D-brane superpositions . . . . . . . . . . . . . . . . . . .
The three charge black hole . . . . . . . . . . . . . . . . . . . . .
4 From black spot to fuzzball
4.1 Introduction . . . . . . . . . . . . . . . . . . .
4.2 The ’naive’ solution . . . . . . . . . . . . . .
4.2.1 D1-D5 system . . . . . . . . . . . . . .
4.2.2 Black spot . . . . . . . . . . . . . . . .
4.2.3 Duality transformations . . . . . . . .
4.2.4 The F-P system . . . . . . . . . . . . .
4.3 The F-P fuzzball solution . . . . . . . . . . .
4.3.1 Microstates . . . . . . . . . . . . . . .
4.3.2 Fields . . . . . . . . . . . . . . . . . .
4.4 The D1-D5 fuzzball solution . . . . . . . . . .
4.4.1 Example of the displacement function
4.4.2 Size end entropy of the fuzzball . . . .
4.4.3 The singularities . . . . . . . . . . . .
4.5 Other concepts in the fuzzball proposal . . .
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5 Recent research on the fuzzball picture
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6 Conclusion
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A Differential forms
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A.1 Hodge duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B Charges and couplings of gauge fields
B.1 Maxwell fields . . . . . . . . . . . . . .
B.1.1 The charged particle . . . . . .
B.2 Higher rank form fields . . . . . . . . .
B.3 Geometrized charge . . . . . . . . . .
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C S-dualities and T-dualities for the fuzzball solution
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C.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C.2 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2
Conventions and notations
conventions
The metric used here has (−++...+) signature, and is written as gµν . The
determinant det gµν is written as g. ηµν is the metric tensor of flat space-time.
In the first chapter, all metric is written in Einstein frame. In other chapters,
the string frame metric is used, unless noted differently.
The units used here are natural units, so c = kB = ~ = 1 (respectively the speed
of light, Boltzmann’s constant and Planck’s constant devided by 2π).
Most of the time, Newton’s gravity constant is absorbed into the mass or the
charge of the black hole; the mass/charge is geometrized (this term is used in
[6]). See appendix B for more on this.
Einstein summation convention is used. This means: Vµ V µ ≡
P
µ
Vµ V µ
Other short hand notations for summations are:
P
i
Vi V i
=
i Vi V , summation over all spatial indices ;
(∂Φ)2
=
∂µ Φ∂ µ Φ = g µν ∂µ Φ∂ν Φ ;
∇2
=
∇µ (g µν ∇ν ) ;
(F (n) )2
=
Fµ1 ...µn F µ1 ...µn ;
Fµ1 ...µn
=
∂µ1 Aµ2 ...µn + cyclic permutations ;
dx·dx
=
summation over the extended spatial directions:
dz·dz
=
summation over compact spatial directions:
²
=
total antisymmetric Levi-Civita tensor, normalized as
²0123... = 1 in Minkowski space-time
or ²123... = 1 in Euclidean space ;
²̃ ijkl
=
Levi-Cevita tensor of flat space .
3
P
P
a (dz
i 2
i (dx )
a 2
) ;
;
notations
The notations used in the books and articles often differ from each other. Here
the notations used in this thesis for abbreviations, the names of solitons in
supergravity, the harmonic functions in the fields of the D1-D5-P black hole,
and the indices of coordinates are given.
abbreviations
GR
SUSY
SUGRA
SUGRA IIA/B
d
=
=
=
=
=
general relativity ;
supersymmetry ;
supergravity ;
type 2A/B supergravity ;
number of dimensions ;
solitons
F(-string)
NS5(-brane)
Dp(-brane)
superstring carrying NS charge,
(sometimes called NS1 or NS1-brane) ;
brane with 5 spatial dimensions carrying NS charge,
(sometimes called S5 brane) ;
brane with p spatial dimensions carrying R-R charge .
harmonic functions
harmonic function for D1-brane (sometimes called H −1 [39], f1 [27]) ;
harmonic function for D5-brane (sometimes called (K − 1)[39], f5 [27]) ;
harmonic function for the momentum in a compact direction
(sometimes called W [22], HK [34], fk [27]) .
H1
H5
HP
indices
For space-times with both compact and extended dimensions, these coordinates
are used:
µ, ν, . . .
t
µ̃, ν̃, . . .
i, j, . . .
a, b, . . .
all spatial and the timelike coordinates ,
the timelike coordinate ,
all extended —spatial and timelike— coordinates ,
the extended spatial coordinates ,
the compact coordinates .
For the D1-D5 and F-P system these indices are used:
xi , xj ; i,j = 1,2,3,4
za , zb ; a,b = 1,2,3,4
y
u(=t+y), v
(=t-y)
extended spatial coordinates ,
compact spatial coordinates
transverse to the F-string and D-string ,
the compact coordinate
parallel to the F-string and the D1-string ,
the left and right light-cone coordinates
of the F-string and D1-string .
4
Chapter 1
Introduction
Black holes are very peculiar objects. Their physics gives rise to a lot of questions
concerning the quantum theory of gravity. Mathur and others claim that in
[38, 39], they found a solution to several black hole paradoxes. In this proposal
the black hole picture changes radically.
The conventional black hole picture coming from general relativity is a singularity surrounded by a horizon and empty space between (see figure 1.1a).
All mass resides in the singularity, where the curvature of space-time becomes
infinite; the horizon is the point of no return: everything falling through the
horizon is bound to fall on the singularity.
In Mathur’s proposal this picture is replaced by the fuzzball picture : a black
hole is a fuzzball of D-branes with no curvature singularity. The new horizon is
the boundary of the fuzzball (see figure 1.1b).
1(a)
1(b)
Figure 1.1: (a) The conventional picture of a black hole
‘fuzzball’ picture.
(b) the proposed
What do the black hole solutions in general relativity (and supergravity) look
like? What are the paradoxes of black hole physics? What are D-branes? What
does the fuzzball solution look like and how is it obtained? How does it solve
some black hole paradoxes? Is this really a black hole? These are the main
questions that will be examined in this thesis after a short introduction in the
next pages.
5
In GR several black hole solutions were found. Wheeler’s phrase ’black holes
have no hair ’ summarizes a lot of the research on the uniqueness of the black
holes. It means that a four dimensional space-time containing one stationary
black hole is uniquely determined by the conserved quantities of the black hole:
mass, charge, angular momentum.
This can be generalized to higher dimensional space-times with charged,
nonrotating black holes.
The dynamics of black holes can be summarized in four laws [7] that have a
remarkable resemblance to the laws of thermodynamics. This analog became
even more powerful by the proposal of Bekenstein [9, 10] and Hawking [11]:
black holes have entropy. This Bekenstein entropy SB is proportional to the
horizon area AH :
SB =
AH
,
4GN
where GN is Newton’s gravity constant.
Bekenstein argued that if black holes wouldn’t have entropy, this would
lead to a violation of the second law of thermodynamics: Throwing systems
with entropy into a black hole would lead to the decrease of the entropy of the
universe.
Hawking looked at quantum fields near the horizon, and found that a black
hole is physically black: it radiates like a black body at the Hawking temperature
and has Bekenstein entropy due to the first law of black hole thermodynamics;
the mass of the black hole is the source for the energy of this radiation: the
black hole becomes lighter.
The entropy conjecture lead to two important paradoxes: the entropy paradox
and the information paradox.
The entropy paradox comes from the fact that entropy obtained from thermodynamics is a measure for the number of microstates described by statistical
mechanics. Therefore a black hole should have a large number of microstates,
contradicting the no-hair theorem. We will focus on this paradox in this thesis.
Secondly, in a unitary quantum theory, the information of the initial states
cannot be destroyed totally. When a black hole has evaporated totally, some
information of the matter that fall in, must have returned into space-time.
Probably the radiation from the black hole contains some information and isn’t
as random as the thermal radiation from Hawking’s calculations.
The microstates of ordinary matter were found after the quantization of energy,
phase space, spin, etc. It is assumed that quantum gravity will solve the black
hole paradoxes (or lead to even complexer paradoxes).
String theory is a quantum theory of gravity that should provide answers
to these paradoxes. It has a lot of new fundamental objects: F-strings (with
one spatial dimension), NS5-branes (with five spatial dimensions), Dp-branes
(with p spatial dimensions). This offers many ways to construct a black hole:
F-strings, NS5-branes, D-branes, Kaluza-Klein compactification, etc. Sen [26]
investigated black holes made out of F-strings.
6
Strominger and Vafa [32] constructed a black hole in supergravity with the right
number of microstates according to the Bekenstein entropy. SUGRA is the low
energy limit of string theory; both quantum effects and higher order curvature
are small in this limit.
This classical black hole is a configuration of D-branes, which are the central
objects in string theory since the second string revolution of 1995. They carry
Ramond-Ramond charge.
Other solutions followed. The best known example is the three charge black
hole in 5 dimensions: a system of compactified D1-branes and D5-branes, and
momentum charge P in a compactified direction.
The Bekenstein entropy and the number of microstates could be compared because this configuration is a BPS-state. The degeneracy of a BPS-state is the
same in two complementary descriptions, because it is a supersymmertric ground
state.
The microstates can be counted by taking the string coupling times the
number of charge quanta gs N small, such that string pertubation theory is valid
and the degeneracy of the D-brane configuration can be counted with Cardy’s
formula.
By increasing gs N , the SUGRA description becomes valid, and black hole
formation is expected. Now the Bekenstein entropy can be calculated.
BPS black holes in SUGRA are identified as extremal black holes. These
don’t radiate since their Hawking temperature is zero. From the extremal black
hole, the near-extremal black hole can be obtained by pertubation. This is used
to study the Hawking radiation.
Now the main problem is that we cannot follow how the black hole forms out of
the branes when gs N increases. The relation between the microstates in both
descriptions is unclear.
Next to that, the Hawking radiation and the information paradox cannot be
studied properly. The near-extremal case doesn’t tell how the information gets
from the singularity to the horizon in the SUGRA description.
A lot of D-brane configurations that form a BPS-state, have a large degeneracy according to calculations in string pertubation theory. Still most configurations, e.g. the D1-D5 system, become null-singularities in SUGRA, ’black
holes’ with zero horizon areas and therefore zero Bekenstein entropy. Probably
the SUGRA solution breaks down near the branes due to higher order curvature
corrections.
The fuzzball proposal for the D1-D5 system [38, 39, 45] is a solution of the
SUGRA field equations. It might solve a few of the paradoxes of black hole
physics. This solution can be obtained from a dual CFT theory, by the AdSCFT correspondence. But in this thesis it will be constructed from the F-P
system by several S-dualities and T-dualities.
The microstates are obtained by looking at the vibration profile of the Fstring. The transverse modes make the string bend away from r = 0. In this
new solution, there is no special place where the curvature singularity can be
located.
By taking a classical limit of the vibration profile, classical smooth geometries are found in the D1-D5 system. But the generic microstate must be ob7
tained from a quantized string profile, which is expected to give large fluctuations of the metric. Therefore the name ’fuzzball’. We will only focus on the
classical limit.
These solutions have no horizon. Mathur states firmly, that microstates
cannot have a horizon; if they would have a horizon, they would also have
entropy and microstates with horizons, etc. ; the total entropy would become
an infinite sum. The horizon of the fuzzball is placed at the boundary of the
generic fuzzball, the most probable microstate.
Several other concepts are briefly introduced here, like fractionation, density
of degrees of freedom, etc. These concepts make the fuzzball concept a rich but
complicated proposal, with a lot of entries for new research.
The fuzzball solution is said to solve parts of the paradoxes of black holes. Only
the method is rather radical. The main question stays wether this solution does
really describe a black hole, as Mathur conjectures in his proposal. If a large
part of the proposed solution survives in the future, it would be a great victory
for string theory.
8
Chapter 2
Black holes in general
relativity
Although even in the 18th century, the English geologist John Michell (1784)
and Pierre Simon de Laplace (1796) independently came up with the idea of an
object which such a high density, that even light couldn’t escape from it, it is
rather conventional to start with general relativity, formulated in 1915 by Einstein, before talking about black holes. These black holes in GR are sometimes
called classical black holes, since these black holes are states without quantum
effects. Still GR is a theory with features one could hardly call classical.
9
2.1
General relativity
Black holes automatically pop up as solutions from GR. A short introduction is
therefore inevitable. One can find more about GR are [1, 2, 3, 4].
2.1.1
The Einstein equations
The most popular way to impress people who are not familiar with GR, is to
talk about the concept of space-time curvature and then jump to black holes
and the expanding universe.
curvature
The curvature of a mathematical space is described by the Riemann theory of
curved spaces with the Riemann curvature tensor Rµνκλ as central object. Einstein connected pseudo-Riemannian space-times with gravity through the Einstein equations eq.(2.1); these postulate that space-time is curved (l.h.s. Gµν )
by energy (r.h.s with Tµν ). These equations hold in any number of dimensions
larger than two (d > 2).
Gµν + gµν Λ =
Gµν
≡
κ2 Tµν ,
1
Rµν − gµν R ,
2
(2.1)
here gµν is the metric tensor; Tµν the energy-momentum tensor; Gµν the
Einstein tensor; Rµν = Rαµαν the Ricci tensor, a contraction of the Riemann
α
tensor; R = Rα
is the Ricci scalar, a contraction of the Ricci tensor; Λ is the
cosmological constant; κ2 is the coupling of gravity to curvature.
When the non-relativistic, weak-field limit is set equal to Newtonian gravity,
κ2 must be identified with 8πGN , GN Newton’s gravity constant. This is true
(d)
in d=4. In d dimensions, we use GN (see 2.1.3).
The energy-momentum tensor for a perfect fluid with pressure p and energy
density ρ is given by:
Tµν = (p + ρ)ẋµ ẋν + pgµν ,
xµ , ẋµ ≡ dxµ /dλ are the coordinates and the coordinate velocity; λ is the
affine parameter (see 2.1.2). When Λ is positive, it acts as a negative intrinsic
universal pressure. It is important for de Sitter and anti-de Sitter spaces (see
2.2.4).
Flat space-time is only a solution of eq.(2.1) when Λ is zero. Because black
holes are defined in asymptotic flat space-times, Λ is choosen to be zero from
the next subsection.
When considering the metric of the vacuum Tµν = 0, the equations reduce to:
2
Λ=0.
d−2
For d = 2 the Einstein equations eq.(2.1) aren’t dynamical anymore, they only
depend on the topology of the space-time.
Rµν − gµν
10
Einstein-Hilbert action
The Einstein equations of vacuum can also be derived from the Einstein-Hilbert
action eq.(2.2) by varying the metric gµν :
Z
√
1
SEH =
dd x −g (R − 2Λ) .
(2.2)
(d)
16πGN
√
It contains an invariant measure ( −gdd x , with g = det gµν ) and a curvature
scalar, the Ricci scalar R.
The energy-momentum tensor is obtained from the action of the matter fields
by varying the metric:
2 δSmatter
.
Tµν = √
−g δgµν
Other gravity theories can be constructed from combinations of curvature scalars,
like Rµν Rµν , Rµνκλ Rµνκλ , etc. Such extensions would make the theory even
more complicated then it already is, so we stick to eq.(2.2).
general coordinate transformations
In GR, there are no global inertial frames, so the relativity postulate of special
relativity must be generalized. This leads to another leading concept in GR:
general coordinate invariance.
In GR one can switch between coordinate systems, while the physics should
stay the same. In other words, under general coordinate transformations: xµ →
x̃µ (x) the equations must transform covariantly (like scalars, vectors, tensors).
Therefore the coordinates do not immediately have a physical meaning. The
r and t in Schwarzschild metric do not for all observers measure the radial
distance and the time. The time measured by a (massive) observer is defined by
the proper time (see 2.1.2). In space-times with asymptotic flatness (see 2.2.1)
the coordinates can have direct physical meaning, like the radius and time as
measured by an observer at infinity.
This freedom to choose the coordinate frame reflects the unphysical degrees of
freedom and corresponds to some symmetries of the Riemann tensor. Like in
gauge theories, this can be summarized in a Bianchi identity for the Einstein
tensor:
∇µ Gµν = 0 ,
(2.3)
∇µ is the covariant derivative (see 2.1.2), replacing the ordinary derivative.
This identity holds not only for GR, but for all gravity theories with an action
invariant under general coordinate transformations.
Using eq.(2.3), eq.(2.1) and the affine connection (see 2.1.2), the energy-momentum
conservation in GR can be formulated as:
∇µ T µν = 0 .
It is not possible to define an invariant notion of energy in a general space-time.
11
2.1.2
Geodesics: the path of particles
From the Einstein equations eq.(2.1) one can derive the space-time metric, which
fixes the invariant length s of a test particle travelling through this space-time:
ds2 = gµν dxµ dxν .
In general, s can be chosen equal or proportional to an affine parameter λ; for
massive particles, s can be chosen equal to the proper time τ . The invariant
geodesic action for a massive test particle is taken proportional to s:
Z
Z
p
S = −m ds = −m dλ −g µν ẋµ ẋν ,
(2.4)
and give the EOM of test particles eq.(2.6) by varying xµ . These describe
geodesics: the shortest path between two point. But only for massive particles.
By introducing an auxiliary field e(x) called the ’einbein’ and representing the
metric on geodesic, eq.(2.4) can be modified in such a way that it will also hold
for massless particles:
Z
¡
¢
S̃ = dλ e−1 g µν ẋµ ẋν − em2 .
(2.5)
The EOM of the field e(x) is a constraint. When putting this constraint into
eq.(2.5) one gets eq.(2.4); S and S̃ describe the same massive particles.
In GR, affine connection (see below) transforms the EOM of xµ into the
geodesic equation.
ẋν ∇ν ẋµ = ∇ẋµ = 0 ,
(2.6)
µ
ẋµ = dx
dλ is the coordinate velocity. Again we see the covariant derivative
(see below). This equation replaces Newton’s second law, with the covariant
derivative already containing the gravitational force. Other nonzero forces (f µ )
will replace the zero in eq.(2.6).
covariant derivative
A normal derivative ∂µ doesn’t transform covariantly due to the curvature, so
this is replaced by the covariant derivative ∇µ , which does transform as a tensor.
By demanding this covariant derivative to be linear and to obey Leibniz rule,
the covariant derivative working on a vector can be written as:
dVµ
− Γκµν Vκ ẋν ,
∇µ Vν = ∂µ Vν − Γκµν Vκ ,
dλ
where the Christoffel symbol Γκµν is defined through affine connection. The
effect of ∇ on a tensor can be derived, by writing the tensor as a product of
vectors and applying Leibniz rule.
∇Vµ ≡
12
affine connection
In GR, the freedom left to choose λ or e(x), is fixed by affine connection (e =
constant; s equal to τ or λ). This is equivalent to taking the metric parallel
transported (∆ρ . . . = 0) over the geodesics:
∇ρ gµν = 0 .
It is this relation that makes it possible to add to the Einstein tensor a cosmological constant in eq.(2.1) without breaking the Bianchi identity eq.(2.3). The
Cristoffel symbol is fixed by affine connection:
Γκµν =
1 κλ
g (∂µ gλν + ∂ν gλµ − ∂λ gµν ) .
2
Usefull equations are:
√
√
∂µ ( −g V µν1 ...νn ) = −g ∇µ (V µν1 ...νn ) ,
when V µν1 ...νn is antisymmetric in its indices.
symmetries: Killing vectors
When the geodesic action eq.(2.5) is stationary under a translation
(xµ → xµ − αkµ (x), for small α), the following equation holds:
∇µ kν (x) + ∇ν kµ (x) = 0 .
(2.7)
This is the Killing equation, and kµ (x) is a Killing vector field. It is associated
with a symmetry of the space-time and gives a conserved quantity of the test
particle:
Q = k µ (x)ẋµ ,
which is invariant on the geodesic due to the geodesic equation eq.(2.6) and
the Killing equation eq.(2.7). A Killing vector means the metric can be written
independent of a coordinate, and the Killing vector can be written in terms of
the differential operator of this coordinate:
k = k µ (x)
∂
.
∂xµ
A general metric will have no Killing vectors, flat space has the maximum of 10
Killing vectors (the Poincaré-invariance).
An example of Killing vectors is the timelike vector (∂t ), when the metric
is stationary; the conserved quantity is the energy of the test particle on the
geodesic. Also one can find spacelike Killing vector(s) in a stationary metric;
this is connected to the conservation of angular momentum.
13
2.1.3
Kaluza-Klein compactifications
Since the Einstein equations hold in almost any number of dimensions, Kaluza
came up with the idea of getting a d-dimensional space-time from a higher
dimensional space-time. Klein refined this idea later. It defines a reduction
program: the unwanted dimensions are taken to be compact, often a torus, so
the space-time is R4 × S (d−4) , with periodic coordinates z a = z a + 2πR(a) ; R(a)
is the radius of the compact dimension za .
One can decompose the mixed d=5 metric of R4 × S 1 as follows:
¡
¢2
(4)
(5)
ds2 = gµν
dxµ dxν = gµ̃ν̃ dxµ̃ dxν̃ + e2σ dz a + Aµ̃ dxµ̃ ,
(2.8)
−1
gµ̃z .
where e2σ = gzz ; Aµ̃ = gzz
The full theory must still be invariant under a transformation like:
xµ → xµ + ²µ (x) ,
gµν → gµν − ∂ν̃ ²µ − ∂µ̃ ²ν ,
where the tilde stands for all the extended dimensions.
Rotations around the compact dimension z only depending on the noncompact
space-time: (²µ̃ = 0, ²z (xµ̃ )), the terms gzz and gµ̃ν̃ are invariant, but:
gµ̃z → gµ̃z − ∂µ̃ ²z (xµ̃ ) .
From the d=4 point of view and in the decompostion of (2.8), σ is a scalar, called
the Kaluza-Klein scalar, gµ̃ν̃ is proportional to the d=4 metric, and Aµ̃ a vector
invariant under U (1) gauge transformations (∂µ̃ ²z (xµ̃ )), called the Kaluza-Klein
vector.
The five dimensional Ricci tensor can be decomposed, and the metric redefined
such that the d=5 Einstein-Hilbert action is transformed into d=4 EinsteinHilbert action with extra gauge and scalar fields. For lateron we need the
relations:
(5)
GN =
(4)
GN
,
2πRz
q
p
g (d=5) = eσ
(4)
gµ̃ν̃ .
(2.9)
In a quantum theory, the wavefunction must bean one-valued-function, and
therefore becomes periodic in the compact directions:
ψ(x4 ) = ψ(x4 + 2πR4 ),
The momentum will get quantized: p4 = n/R4 , n ∈ N . This quantized
momentum can be identified as charge.
Compactification is of great importance to string theory, where only ten or
eleven-dimensional space-times are possible.
14
2.2
Classical black holes
The first exact solution of GR, the vacuum around a static, spherically symmetric object in an asymptotic flat space-time, was found by Schwarzschild in
1916. Generalizations followed: charged object (Reissner (1916) and Nordström
(1918)); higher dimensional Schwarzschild (Tangherlini 1925); rotating objects
(Kerr, 1963) and finally the charged, rotating object (Newman e.o. 1965).
When all mass resides at one point, the metric is valid almost everywhere. This
option was only investigated as a serious physical solution from the 1960s. A
lot of research went to the examination of black holes after gravitional collapse,
and how general the Schwarzschild and other solutions are. In 1967 Wheeler
finally came with the name ’black hole’.
2.2.1
The Schwarzschild solution
This solution eq.(2.10) describes the vacuum around a static, spherically symmetric massive point.
2M 2
2M −1 2
)dt + (1 −
) dr + r2 dΩ2(2) ,
(2.10)
r
r
M is an integration constant, here identified as the geometrized mass: the
(4)
mass in newtonian gravity times the Newton’s gravity constant: GN MN ; dΩ2(2) =
dθ2 + sin2 θdφ2 , is the line element of a two sphere.
The Schwarzschild metric has four Killing vectors: a timelike vector (∂t )
connected to conservation of energy, and three spatial vectors connected to the
conservation of angular momentum of the test particle, so the testparticle moves
in one plain (e.g. θ=π/2) and has still one Killing vector left (∂φ ).
ds2 = −(1 −
singularities
At both r = 0 and r = 2M an element of the metric blows up or goes to zero.
For an extended object like an star, both singularities lie within this object
and this vacuum solution won’t hold; another metric must be obtained, so no
problem there.
For a Schwarzschild black hole, this solution is valid almost everywhere, so
one has to have closer look at the singularities. All the mass of the black hole
resides at r=0, this is a curvature singularity. The Riemann tensor blows up,
likewise the volume of space-time containing r=0. In [1, 4] precise definitions of
singularities can be found. For now, a black hole solution contains a curvature
singularity settled behind another singularity, the horizon.
horizon
The horizon at the Schwarzschild radius rSW = 2M is a coordinate singularity.
It can disappear after a coordinate transformation (see 2.2.2). Nothing special
happens when something goes through this sphere; only the gtt and grr change
sign, and ’time’ t and ’space’ r exchange their role. A particle passing the
horizon is then bound to fall into the singularity. The horizon is the point of
no return. It is therefore called the event horizon and has a special role in GR,
especially in thermal properties of black holes (see 2.3).
15
asymptotic flatness
The black holes described here are defined in space-times with aymptotic flatness. This means that at spatial infinity (r→∞) the metric is Minkowskian
and entities like mass, angular momentum can be attributed to the black hole
system, when the connection is made to the physics of flat space-times. In
other space-times, like in an anti-de Sitter space time (AdS), the cosmological
constant is nonzero and the so-called ADM-mass is not defined.
Birkhoff ’s theorem
Birkhoff’s theorem says that any spherically symmetric metric in d=4 is automatically static (so we can choose gti = 0), which effectively implies that it
must be Schwarzschild.
Schwarzschild black holes can also be found in higher dimensions and Tangherlini did prove that this is again an unique solution, so Birkhoff’s theorem holds
in dimensions d≥4. Only then:
(d)
−1
gtt = grr
=1−
.
16πGN
MN
,
(d − 2)ωd−2 rd−3
where MN is the mass as measured flat space, here used to show the de(d)
pendence on GN ; ω(n) = 2π (n+1)/1 Γ(n + 1/2), the volume of a n-sphere embedded in Rn+1 (S n ⊂ Rn+1 ); Γ is the Euler-gamma
function, for which
√
Γ(n) = (n − 1)Γ(n − 1) and Γ(1) = 1, Γ(1/2) = π.
2.2.2
Schwarzschild in other coordinates
The Schwarzschild solution can be written in other coordinates frames. In some
frames the coordinate singularity disappears.
isotropic coordinates (t,ρ,θ,φ) :
Ã
2
ds = −
2M
ρ )
2M
+ ρ
(1 −
(1
!2
µ
¶4
2M
dt + 1 +
(dρ2 + r2 dΩ2(2) ) ,
ρ
2
2
r → ρ where r = ρ (1 + M/2ρ) . Notice that this solution does only describe
the space-time outside the horizon.
Eddington-Finkelstein coordinates (v,r,θ,φ):
µ
¶
2M
ds2 = − 1 −
dv 2 + 2M dvdr + r2 dΩ2(2) .
r
(2.11)
First define the Regge-Wheeler radial coordinate: dr∗2 ≡(grr /gtt )dr2 ; for the
Schwarzschild: r∗ = r + 2M ln |r/(2M ) − 1|. The ingoing E-F coordinates are
defined by: v = t + r∗ with −∞ < v < ∞. These initially only describe the
space-time outside the horizon, but can be analytically continued for all r > 0.
The outgoing E-F coordinates are obtained by u = t − r∗; the metric then will
be almost the same as eq.(2.11), with v replaced by u and +dv by −du. These
16
outgoing E-F solution describes a so-called white hole, the time reverse of a
black hole: matter can only pass the horizon when it comes out the white hole,
so nonexisting in our universe.
Kruskal-Szekeres coordinates (U, V, θ, φ):
32M 3 − r
e 2M dU dV + r2 dΩ2(2) ,
(2.12)
r
(u, v) → (U, V ) where: U = −exp(−u/4M ), V = exp(v/4M ). This describes again only the space-time outside the horizon. Notice that U ≤ 0 and
V ≥ 0. Now r(U, V ) is given implicitly by :
ds2 = −
r − 2M r/2M
e
.
2M
When we have the coordinates U and V running from −∞ to +∞, while the
relation for r(U, V ) still holds, four different regions are obtained, describing
both a black hole and white hole. The so-called Kruskal-diagram plots the
(U, V ) part in a diagram (figure 2.2).
The lines of constant U and V (outgoing or ingoing radial null geodesics)
0
are plotted at 45 , so the spacetime diagram now looks like:
U V = −er∗/2M = −
singularity
r=0
U
..............
.........
.....
.....
....
.
.
...
.....
........
........ ..... ....... . . . . .. .. .. .. .. .. .. .. .. .. ...... .. ....... ... ........
.
.
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.. .........
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....... ...
..... ......... .
........... . .. .. .. . . . . .. .. ...........
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............ . . . .. .. .. .. .
.....
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...........
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.........................
V
r < 2M
II
IV
r = 2M
I
r > 2M
U <0
V >0
III
singularity
r=0
Figure 2.1: Kruskal diagram. There are four regions in a Kruskal space-time,
depending on the signs of U and V . Region I is the space-time outside the
horizon of a Schwarzschild black hole; region II is the space-time between the
horizon and the singularity at r=0 (UV =1). The regions III and IV are the
space-time inside and outside the horizon of a white hole. Light rays are drawn
0
as lines at an angle of 45 .
17
2.2.3
The Reissner-Nordström solution
Reissner and Nordström generalized the Schwarzschild solution by adding charge
to this spherically symmetric object. Although this black hole wouldn’t survive
in our universe, it has interesting properties.
Einstein-Maxwell theory
Our starting point is the sourceless Einstein-Maxwell action:
Z
¢
√ ¡
1
d4 x −g R − F 2 ,
S=
(2.13)
16πGN
with Fµν = ∂µ Aν − ∂ν Aµ and Aµ is the vector potential. Note that the
coupling of the electromagnetic part is 1/16πGN .
Einstein equations with the electromagnetic energy-momentum tensor become:
1
Gµν = 2Fµρ F ρν − gµν F 2 .
2
The curved Maxwell equations are the field equations and the Bianchi identity
for this vector potential:
∇µ F µν = 0 ,
∇µ ∗ F µν = 0 ,
(2.14)
where the hodge dual * is used to show the nice symmetry between the field
equations and Bianchi identity. One could easily interchange these equations.
This duality continues to hold in the case of electric sources, when magnetic
charges are introduced. This duality will be used in supergravity (3.3.1).
charged black hole
The Reissner-Nordström solution is due to a generalization of Birkhoff’s theorem, the unique solution for the vacuum around a charged spherically symmetric
black hole. This solution is easily expanded with magnetic charges to get a dyonic object:
ds2
=
At
=
∆
=
∆ 2 r2 2
dt + dr + r2 dΩ2(2) ,
r2
∆
Q
, Aφ = −P cos θ ,
r
r2 − 2M r + Q2 + P 2 ,
−
(2.15)
Q and P are the electric and magnetic charge of the black hole. These can
be defined without asymptotic flatness in the language of forms (see appendix
B):
Z
Z
1
1
Q=
∗F , P =
F
(2.16)
4π
4π
Source terms can easily be added to eq.(2.13).
The original RN solution can be obtained by setting P =0. For the black
hole solution, all the (charged) mass of the black hole is at r = 0 and again
gtt (r = 0) blows up; this is again the curvature singularity.
18
two horizons, three solutions
There can be two horizons in this solution. One can rewrite gtt as
∆ = (r − r+ )(r − r− ),
r± = M 2 ±
p
M 2 − Q2 − P 2 .
One of features that makes this solution interesting is the fact that one can
separate three Reissner-Nordström solutions.
non-extremal black hole (M 2 > Q2 + p2 ): this solution has two real
horizons where grr blows up. The r+ is called the event horizon; here t and r
are interchanging roles again, so matter passing the event horizon, must pass
the horizon at r− .
extremal black hole (M 2 = Q2 + p2 ): this is the type examined a lot in
string theory, since it is an equilibrium state (see 2.2.4 and 3.5.1).
naked singularity (M 2 < Q2 + p2 ): this solution is regarded as unphysical
and is forbidden by the Cosmic Censorship Conjecture.
Cosmic Censorship Conjecture
For M 2 < Q2 + p2 the r± get complex and there is no horizon. This is forbidden
by the Cosmic Censorship Conjecture. This proposal by Penrose forbids naked
singularities to exist in this universe, except for the Big Bang. This also comes
back in the third law of black hole thermodynamics (see 2.3.1). So it excludes
solutions of GR. Unfortunately there is no proof in GR, but supersymmetry
may save the day.
2.2.4
The extremal Reissner-Nordström solution
The extremal black hole has a lot of interesting features. For now, we set P
equal to zero, but of course the dynonic case can be treated equally.
The two horizons at r+ and r− coincide at M (=|Q|) and the metric looks a bit
simpler:
ds2 = −(1 −
|Q| 2 2
|Q| −2 2
) dt + (1 −
) dr + r2 dΩ2(2) .
r
r
Here one can define the so-called Schwarzschild radius rSW = |Q|.
In isotropic coordinates (t, ρ, θ, φ) the fields become:
ds2
At
H
³
´
= −H −2 dt2 + H 2 dρ2 + ρ2 dΩ2(2) ,
= −(H −1 − 1) ,
|Q|
= (1 +
),
ρ
with r → ρ = r − Q, so the horizon is located at ρ = 0.
19
(2.17)
harmonic function
Here we see the harmonic function H, which is normally a solution of the Green
equation ∂µ ∂ µ H(x) ∝ δ(x − x0 ). In our case it is the solution of the field
equations H −1 ∂ 2 H(x) = 0.
From the harmonic function, the Schwarzschild radius rSW can be obtained;
this is used in supergravity.
multi-center extremal black holes
Now in these coordinates, the solution eq.(2.17) can easily be generalized to a
solution with more black holes at arbitrary ’fixed’ positions:
H =1+
X
i
|Qi |
,
| ρ − ρi |
where ρi are positions of the horizons. These are Majumdar-Papapetrou solutions. They are not spherically symmetric, except for the Reissner-Nordström
case. The charges must be of the same sign; then the gravitational attraction
is equal to the electromagnetic repulsion. The extremal black hole is clearly an
equilibrium state.
extremal RN in d=5
A mulitple-charged extremal black hole in isotropic coordinates is the first black
hole described by D-branes, with the right properties. Only this was derived in
d=5; extremal Reissner-Nordström in d=5 becomes:
ds2
A
H
³
´
= −H −2 dt2 + H dρ2 + ρ2 dΩ2(3) ,
= −(H −1 − 1) ,
Q(d=5)
= (1 +
),
ρ2
(2.18)
here, (r2 → ρ2 − Q(d=5) ).
anti-de Sitter space-time
When one has a closer look at the geometry near the horizon (ρ → 0) of an
extremal black hole, one discovers the metric approximates:
ds2 = −
ρ2 2 Q2 2
dt + 2 dρ + Q2 (dθ2 + sin2 θdφ2 ) .
Q2
ρ
This is the Bertotti-Robinson metric. It is the 2 dimensional anti-de Sitter
space-time times the two sphere (AdS 2 × S 2 ).
Both the de Sitter and the anti-de Sitter are highly symmetrical spaces for
non-vanishing cosmological constant; for de Sitter positive cosmological constant
and curvature and for anti-de Sitter negative. This identification is important
for the AdS/CFT-correspondence (subsection 3.5.3) in string theory.
20
The vacuum equations for nonzero Λ become:
2(d − 2)
.
Λ
For a (+++... ) signature and positive λ, this would give the metric of a sphere.
The spherically symmetric solution for a (−+++..) metric is:
Rµν = 4l−2 gµν ,
l2 =
¶
¶−1
µ
µ
±r2
±r2
2
dr2 + r2 dΩ2(d−2) ,
ds = − 1 − 2
dt + 1 − 2
l
l
2
(2.19)
The + sign gives the de Sitter space-time; the − sign anti-de Sitter.
The Λ sets a length scale l. For small Λ, the l is large; and for small distances
(r << l), the metric of eq.(2.19) looks like flat space.
The AdS d can be rewritten by choosing r = l sinh θ:
ds2 = − cosh2 θdt2 + l2 dθ2 + l2 sinh2 θdΩ2(d−2) .
Another well known metric for AdS d is obtained by looking at the (d-2)sphere in
the metric as if it is embedded in Rd−1 ; then an extra variable can be introduced:
(xd−1 )2 = l2 −
d−2
X
x2i .
i=1
By taking xi ∼ 0 and xd−1 ∼ l the angular part of the metric can be rewritten:
l
2
dΩ2d−2
=
d−2
X
dx2i
−2
+ (xd−1 )
i=1
d−2
X
i=1
xi dxi ≈
d−2
X
dx2i ,
i=1
For r >> l, this gives AdS d in so-called local coordinates:
ds2 =
l2 2
r2
2
2
2
+
·
·
·
+
dz
)
+
(−dt
+
dx
dr .
d−2
1
l2
r2
(2.20)
The two metrics of AdS d can also be obtained from the metric of a d-dimensional
hyperboloid slice, both with another range for the coordinates then the hyperboloid.
Near the extremal black hole, the cosmological constant for this AdS space-time
generated by the electric flux, as can be seen by comparing eq.(2.20) with the
Bertotti-Robinson metric.
21
2.2.5
The hairless Kerr-Newman solution
The Kerr-Newman solution is the most general black hole solution for axisymmetric, asymptotic space-time in d=4. Several uniqueness theorems (Carter,
Israel, ... these are reviewed in [6]), state that these are stationary rotating,
charged black holes. The Kerr-Newman solution in Boyer-Lindquist coordinates
is:
µ
ds2
=
+
At
=
a =
∆
=
−
µ
µ 2
¶
r + a2 − ∆
dt2 − 2a sin2 θ
dtdφ
Σ
¶
(r2 + a2 )2 − ∆a2 sin2 θ
Σ
sin2 θdφ2 + dr2 + Σ2 dθ2 ,
Σ
∆
∆ − a2 sin2 θ
Σ
¶
−Qra sin2 θ + P (r2 + a2 ) cos θ
Qr − P a cos θ
,
Aφ =
,
Σ
Σ
J
,
M
r2 − 2M r + a2 + Q2 + P 2 , Σ = r2 + a2 r cos2 θ ,
(2.21)
J represents the angular momentum in the z-direction (z=r cos θ); M the
mass; Q and P the electric and magnetic charge. When J=0 and therefore a=0
this solution reduces to the Reissner-Nordström solution. Likewise it reduces to
the Kerr solution when Q,P=0 and to the Schwarzschild solution when both Q,
P and J are zero.
p
Again there are possibly two horizons at r± =M 2 ± M 2 − Q2 − P 2 − a2 , two
coordinate singularities, and therefore three different solutions: a non-extremal
black hole, an extremal black hole and a naked singularity.
The metric is independent of t and φ; it has two Killing vectors: k=∂t and
m=∂φ . The surfaces at r± are Killing horizons (this means χµ χµ (r± )=0) for
the Killing field χµ (x):
χ = k + ΩH m ,
ΩH =
dφ
gtt
a
|r =
|r = 2
.
dt + gφφ + r+
+ a2
ring singularity
A Kerr or Kerr-Newman solution hasn’t got a singularity at one point, but at
the ring r=0, θ=π/2. This can be seen when taken the limit Q , J and M to
zero, but a is kept constant. Flat space is then obtained, only in ellipsoidal
coordinates, where r=0 represents a circle in the θ=π/2 surface. So it can be
analytically continued to negative r.
This solution has a new kind of problems: closed timelike curves. These show
when taking a curve, going through the inside of the black ring, negative r.
Closed timelike curves are unphysical since causality is violated by them. For
the naked singularity, this problem becomes even larger.
22
ergosphere
In the metric of the Kerr-Newman metric there is another interesting thing: the
ergosphere. It is located at:
p
r = M + M 2 − Q2 − p2 − a2 cos2 θ .
Inside this volume (called the ergoregion), nothing can stand the rotation of the
metric. This means that when the black hole rotates clockwise, even massless
particles that travelled counterclockwise at spatial infinity, will travel clockwise
inside the ergoregion as measured by an observer at spatial infinity. This is
called frame-dragging.
Penrose proces
Penrose discovered how to extract energy from the black hole by a process inside
this ergosphere. In the ergoregion, the Killing vector k isn’t timelike. Therefore
the conserved energy (E=−pµ k µ ) associated with k doesn’t have to be positive.
Particles going in and out the ergoregion, can increase their energy. It has a
close analog in superradiance: the scattering of radiation by the black hole.
This Penrose process made clear that rotating black holes eventually become
non-rotating. It also lead to lots of research on black holes and the discovery of
the black thermodynamics (section 2.3).
no-hair theorem
Generalizations of Birkhoff’s theorem can be summarized into Wheelers ’nohair theorem’: if the final state of a gravitational collapse in d=4 is a black
hole, then its space-time is uniquely described by its mass, charges and angular
momentum: black holes have no hair. In higher dimensionsional space-times,
this uniqueness theorem only holds for nonrotating black holes.
conserved quantities
The conserved quantities that give the state of the black hole, are not all defined
equally. The charges can be defined by an integral over F or ∗F . This is
invariant because of the integration over a form. The other two parameters,
mass and angular momentum are defined with help of so-called Komar-integrals.
These are integrals over a three-sphere at spatial infinity. Without asymptotic
flatness, the ADM-energy and ADM-angular momentum cannot be defined.
2.2.6
Carter-Penrose diagrams
A graphic representation of black holes are Carter-Penrose diagrams. Penrose’s
definition of a black hole -”region of space-time from which no signal can escape
to infinity”-, becomes more clear when the space-time is conformally compactified, without changing the causal structure:
ds2 → ds̃2 = Λ2 (x)ds2 ,
Λ 6= 0 ,
Λ must be chosen so, that infinity becomes finite, so Λ(r → ∞) → 0 and
Λ(t → ±∞) → 0. These points at infinity do not originally belong to the
original space-time, but can be added to make a nice diagram.
23
For a spherically symmetric space-time it is enough to draw one direction, so
we get a 2d diagram of t and r.
For Minkowski space-time, take the coordinate transformations:
tan U
tan V
= t−r ,
= t+r ,
−π/2 < U < π/2 ,
−π/2 < V < π/2 ,
r ≥ 0 → V ≥ U, .
Then the metric becomes:
−2
ds2 = (2 cos U cos V )
£
¤
−4dU dV + sin2 (V − U )dΩ22 ,
and it is easy to guess that it can be compactified by choosing Λ = 2 cos U cos V .
Then the next table can be made for the points at infinity:
spatial ∞
io
⇒
r→∞
t finite
⇒
U = −π/2
V = π/2
past/future
temporal ∞ i±
⇒
r finite
t → ±∞
⇒
U = ±π/2
V = ±π/2
past null ∞
I−
⇒
(r, t) → (∞, −∞)
r+t finite
⇒
U = −π/2
|V | 6= π/2
future null ∞
I+
⇒
(r, t) → (∞, ∞)
r-t finite
⇒
|U | 6= π/2
V = π/2 .
Now introducing new variables, τ = V + U and χ = V − U :
ds̃2 = −dτ 2 + dχ2 + sin2 χdΩ22 .
Now the Carter-Penrose diagram can be drawn: take τ on the vertical axis, and
χ on the horizontal axis, and one gets the triangle of compactified flat space
without the less interesting angular part (see figure 2.3).
This procedure can be repeated for Kruskal space-time, with a black and white
hole. One looks first at one region
of Kruskal space-time, and have a transformaR
tion of r → r∗, where r∗ ≡ drgrr /gtt , such that radial lightlike geodesics are
0
again straight lines with a 45 angle in the coordinates of (t, r∗). Then (almost)
the same transformation and conformal compactification as for flat space-time
follow.
After this, repeat it for the other regions, identify the borders (like the
horizon) and glue the separate regions together. That way, the diagram 2.4 can
be constructed.
24
timelike geodesic
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i+
V
U
p
ppppppp
pp ppp
ppppp
p
p ppp
pp p p
ppp pp
pppp
I+
i0
ppppp
p ppp
ppppp
p
ppppp
p
I
r=0
t = constant
hypersurface
radial null
geodesic
−
i−
Figure 2.2: Carter-Penrose diagram of Minkowski space-time. Each point rep0
resents a 2-sphere, except points on r = 0 and i0 , i± . Light rays travel at 45
from I − through r = 0 and then out to I + .
r constant < 2M
.
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i+
I+
II
IV
r = 2M
I
i0
III
I−
r constant > 2M
i−
singularity at
r=0
Figure 2.3: Carter-Penrose diagram for Kruskal space-time, with a black hole
(region I and II) and a white hole (III and IV).
25
2.3
Black hole thermodynamics
One can extract energy from a rotating black hole by the Penrose Proces. This
lead to an intensive study on general properties of black holes.
In the early 1970s researchers ( Carter, Hawking, Christodoulou, ...) derived
some features of black holes, that can be summarized in four laws [7]. The name
black hole thermodynamics is given, because of their strong formal analog with
thermodynamics. They establish a relation between properties of the horizon
and the conserved quantities.
The formal analog was turned into a physical correspondence by Bekenstein
[9, 10] and Hawking [11] by attributing entropy and temperature to black holes.
Now the black hole is physically black: it radiates like a black body.
This lead immediately to two paradoxes: one about the information contained by the black hole radiation and one about the microstates of black holes.
In [12] Wald generalized some of these laws to more general gravity theories.
Black hole thermodynamics are quite technical, therefore only the main results
and the ideas behind them are discussed. More on this subject can be found in
[6].
2.3.1
Classical laws
properties of the horizon
First a few properties of the horizon are given.
The surface gravity κs is defined with help of the Killing vector field k µ (x) at
the horizon:
∇µ k 2 |H = −2κs kµ |H
⇔
k ν ∇ν k µ (x) |H = κs k µ (x) |H .
It is the acceleration of a static particle near the horizon, as measured at spatial
infinity. Next we need the co-rotating electric potential (ΦH = χµ Aµ |H ), the
angular velocity of the horizon (ΩH ) and the horizon area (AH ).
For a black
phole with conserved charges (M, Q, J = aM ), horizons located
at r± = M ± M 2 − Q2 − a2 and the event horizon at r+ , these quantities are
given by:
κs
=
ΦH
=
ΩH
=
AH
=
p
r+ − r−
(2 M 2 − Q2 − a2 )
p
,
2 + a2 ) =
2(r+
4M 2 − 2Q2 + 4M M 2 − Q2 − a2
p
Qr+
Q(M + M 2 − Q2 − a2 )
p
,
2 + a2 =
r+
2M 2 − Q2 + 2M M 2 − Q2 − a2
p
ar+
a(M 2 + M 2 − Q2 − a2 )
p
,
2 + a2 =
r+
2M 2 − Q2 + 2M M 2 − Q2 − a2
Z
p
√
2
gθθ gφφ dθdφ = 4π(r+
+ a2 ) = 4π[2M 2 − Q2 + 2M M 2 − Q2 − a2 ] .
r=r+
Note that for the extremal black holes, r+ and r− coincide, so the surface-gravity
disappears for extremal black holes.
26
thermodynamics
The laws of black hole dynamics are summarized in four laws. The proof goes
beyond the scope of this thesis; they are given:
Law
Zeroth
First
thermodynamics
Black Holes
T is constant throughout body
in thermal equilibrium
κs constant over horizon
of stationary black hole
dE = T dS + work terms
dM =
κs
8π dAH
+ ΩH dJ + ΦH dQ
Second
δS ≥ 0 in any process
δA ≥ 0 in any process
Third
Impossible to achieve
T = 0 by a physical
process
Impossible to achieve
κ = 0 by a physical
process
Here, S and T are the entropy and temperature. The exact statements of
these laws are more delicate.
The second law definitely needs a description of what a process can be. The
second law is valid in asymptotic flat space-times when some energy condition
(the weak energy condition) and the cosmic censorship are fulfilled. The δA
means in any process evolving in positive time-direction. A consequence of this
law is that one black hole can’t split in separate black holes.
One sees that the first law confirms a strong relation between quantities,
measured at infinity (M , J), and properties of the horizon (κs , AH , ΩH , ΦH ).
This law will be modified in the case of rotating black holes embedded in a
space-time with an odd number of dimensions.
The last one look a bit odd, because the extremal black hole is just as good
a solution of GR as others. But explicit calculations [8] show that it gets harder
to reach extremality step by step. Planck’s version of the third law (if T ↓ 0
then S ↓ 0) isn’t valid for black hole thermocynamics (when κs ↓ 0 , AH won’t
↓ 0), but this phrasing can be seen as a law that works for some examples but
not all.
These laws become the thermodynamic laws, when κs is proportional to temperature, A to entropy, M to energy. It is natural to identify mass with energy,
but identification of surface gravity with temperature and horizon area with entropy is a whole different category and these analogues were just thought to be
coincidence. They were thought to be unphysical because a black hole doesn’t
radiate and its temperature is therefore zero. Besides that, the no-hair theorem
says there is only one black hole state for certain (M , Q, J), and the entropy
must be zero.
27
2.3.2
Hawking radiation: a first step towards quantum
black holes
The black hole dynamics were found to be not only the mathematical analog
of ordinary thermodynamics, but also physical: black holes have entropy and
black body temperature.
generalized second law
A hairless black hole can violate the second law of thermodynamics. When
throwing a system with some entropy into the black hole, the quantities (M, Q, J)
of the black hole may change, but due to the no-hair theorem, there is still one,
unique state and therefore the entropy is zero. So one of the two laws must be
invalid: the no-hair theorem or the second law.
Bekenstein’s proposal [9, 10] was to attribute entropy to the black holes and
to generalize the second law, such that the total entropy of both the universe
and the black holes are not allowed to decrease in time. He provided arguments
for the entropy to be proportional to the area of the horizon of the black hole.
The second law of black hole thermodynamics forbids this Bekenstein entropy
to decrease. By the principles of statistical mechanics, entropy is connected to
the number of possible microstates (’hair’).
holographic principle
Entropy being proportional to the boundary (horizon area) of a system is remarkable; since other thermodynamic systems (like local field theories), the
number of microstates and therefore the entropy would be related to the volume. This made ’t Hooft come up with the idea of holographic principle [15]:
gravity in d dimensions is somehow equivalent to a local field theory in d-1
dimensions without gravity.
Hawking temperature
Hawking did not agree with Bekenstein’s proposal, but while he was busy investigating radiation near a black hole (see [5] for a more extended history) he found
that black holes do radiates as a perfect black body at Hawking temperature:
κs
.
2π
To derive this temperature, Hawking used a semi-classical approach: quantum
fields in a classical black hole geometry. In the vacuum near the horizon virtual
particle pairs are created. When one particle falls into the hole, recombination
cannot occur and the other particle becomes real and possibly escapes to infinity. Due to gravitational effect there is greybody factor which prevents some
radiation to escape to infinity.
TH =
Another effect combining quantum fields with GR is the Unruh effect. A observer accelerating in a flat space-time vacuum, feels like he is moving in a
thermal bath of particles of certain temperature, the Unruh temperature. This
effect has been measured in electron storage rings. This makes clear that the
identification of quantum field excitations with particles, isn’t always valid in
GR.
28
Bekenstein entropy
For this Hawking temperature the black hole entropy should be:
SB =
AH
(d)
4GN
.
(2.22)
This is true for any number of dimensions. The entropy is a macroscopic entity
which is obtained by coarse-graining over an ensemble of microstates. In this
case, black holes which are equivalent at macroscopic level differ from each other
on the microscopic level. So the Bekenstein entropy describes an ensemble of
black holes.
For Schwarzschild, extremal Reissner-Nordström and d=5 extremal ReissnerNordström black holes the Hawking temperature and Bekenstein entropy become:
TH
SW
exRN
d=5 ex RN
1
8πM
0
0
SB
4πM 2 /GN
πM 2 = πQ2
3
2π 2 Q 2
The Hawking temperature is proportional to the inverse of the mass, so radiation
of existing black holes in the universe is highly infrared.
Notice that for extremal black holes the surface gravity is zero, likewise the
temperature. These black holes don’t radiate. The BPS-state in supergravity
is a supersymmetric ground state and this has been identified with the nonradiating extremal black hole.
black hole entropy as Noether charge
From the proof of the classical black hole thermodymics, it looks like these are
special for black holes in GR. But Wald derived the first law for more general
theories of gravity [12]. In more general theories black hole entropy is obtained
from the langrangian as a Noether charge. This feature is specially attractive,
because now one can calculate the entropy also in higher order curvature gravity.
Since in string theory GR is just a low energy approximation and gets higher
order corrections at higher energy scale, Wald’s proposal is very useful.
29
2.3.3
The information paradox and the entropy paradox
Besides that this is a great victory, coupling GR with quantum field theory, two
paradoxes show up.
information paradox
The energy of the Hawking radiation must come from the black hole itself.
The longer it radiates, the lighter it gets, the more it radiates again. Will this
black hole evaporation stop at some state of the black hole or will the black
hole disappear completely? If it does evaporates completely, where has the
information of the initial states gone that first created the black hole?
In a quantum theory, states evolve in time by a unitary transformation.
Therefore information cannot be lost totally.
But the Hawking radiation is totally random black body radiation, so these
mixed states don’t contain any information about the pure states that made
the black hole. Does the black hole give away information, maybe at the end
of the evaporation or is the semi-classical approach too rough for this radiation
and does it contain subtle information; or is information lost forever and do we
have to reformulate quantum mechanics?
Finally in 2004 even Hawking admitted that the information cannot be lost.
Also the holographic principle and the AdS/CFT duality would predict that
Hawking radiation is a unitary process, since in the corresponding local field
theory without gravity, everything transforms unitary, and therefore this should
happen also in the dual gravity theory.
Some nonlocal information (quantum ’tunneling’) transfer can occur to solve
this problem. Only it has to deal with a large distance like the Schwarzschild
radius.
entropy paradox
The no-hair theorem says that the metric around a stationary black holes with
certain macroscopic properties (M,Q,J) is uniquely defined, and will have only
1 microstate and zero entropy. But we just said that the entropy is proportional
to the horizon area. So what are really the microstates of the black holes; where
is their hair?
The identification between macroscopic and microscopic entropy could in
daily life physics only be made after the invention of quantum theory. So the
entropy paradox will maybe be solved by a quantum theory of gravity. Also the
information paradox shouts for quantum black holes. These problems set a goal
for this theory.
30
Chapter 3
Black branes in
supergravity
During the formation of black hole thermodynamics, new ideas concerning gravity were born on the other side of the spectrum of physics, at very small length
scale. In 1968 the Veneziano formula was introduced to describe certain properties of the strong nuclear force; Nambu and Goto discovered this formula had
to do with objects with one spatial direction: strings.
In the battle over the strong force, string theory lost, QCD won. But this early
born child survived in the incubator and by the input of some new species it
became the foremost candidate for the grand unifying theory; all elementary
particles became strings and gravity could be quantized. Main ideas were the
identification of a two rank particle with the graviton (Scherk, Schwarz and
Yoneya), the length of the string with planck length, worldsheet superssymmetry
by the introduction of fermionic modes on the superstring (Ramond, Neveu,
Schwarz), and space-time supersymmetry (Gliozzi, Scherk, Olive).
GR is here embedded in the low energy limit of string theory: supergravity.
31
3.1
String theory
String theory is a quantum theory containing GR. It should provide some solutions for problems obtained in GR. The entropy and the information paradox
are important issues for this theory. A more complete introduction can be found
in [16, 17].
3.1.1
The bosonic string
In the old string theory, all particles are described by strings. A generalization of
the geodesic action eq.(2.4) to the worldsheet action of one string, gives already
a lot information about the strings.
The action of a d-dimensional bosonic object is proportional to the invariant
worldvolume Σ:
Z
Sd
=
Z
−Td
dΣ = −Td
dd σ
p
−detgαβ ,
Σ
ν
gαβ = gµν ∂α X µ ∂β X ,
gαβ is the metric of the worldvolume, the so-called pull-back metric induced
by the space-time metric gµν ; {σα } the worldvolume coordinates; Td the tension
of the object.
worldsheet action
When considering a freely moving string of length ls in flat space, this will
give us the Nambu-Goto action eq.(3.1), proportional to the two-dimensional
worldsheet:
Z
SN G = −T
Z
d2 σ
dΣ = −T
q
(g αβ ∂α X µ ∂β X ν ηµν ) ,
(3.1)
Σ
1
0
the string tension T = 2πα
0 ; α the Regge-slope; Σ the worldsheet. This
action will give trouble at quantization, because it is difficult to formulate the
squareroot of an operator. Again an auxiliary field hαβ (σ i ), the metric on the
worldsheet can be introduced to get the Polyakov action:
SP = −
T
2
Z
√
d2 σ −hhαβ ∂α X µ ∂β X ν ηµν .
(3.2)
Σ
But because the number of physical degrees of freedom cannot change, this
action has more constraints than eq.(3.1); these are the EOM of this induced
metric:
Tαβ ∝
δSP
=0
hαβ
⇒
hαβ (σ i ) = C(σ i )∂α X µ ∂β Xµ .
Substituting this constraint into eq.(3.2) gives back eq.(3.1), so both actions
are equal upto the EOM. It is said, that this also holds at quantum level if
space-time has the critical dimension (see 3.1.2).
32
symmetries
Action eq.(3.2) looks like the free field theory d independent scalar fields X µ
living on the worldsheet. It has several symmetries:
space-time Poincaré invariance:
X̃ µ = Λµν X ν + aµ ,
Λµν
h̃αβ = hαβ ,
µ
are the Lorentz transformations; a are the translations. This looks like
an internal symmetry from the worldsheet point of view.
worldsheet reparametrization invariance:
X̃ µ (σ̃) = X µ (σ) ,
h̃α0 β 0 (σ̃) = hαβ (σ)
∂σ α ∂σ β
,
∂σ α0 ∂σ β 0
0
under the transformation σ β →σ̃ β (σ). This looks quit natural for a
theory containing GR.
conformal or Weyl invariance:
X̃ µ (σ) = X µ (σ) ,
h̃αβ = 2Λ(σ)hαβ .
Conformal invariance is only a symmetry of eq.(3.2), not of eq.(3.1). By
the conformal invariance, the worldsheet of one open (closed) string can be
deformed into a circle (sphere) in the complex plain.
In the conformal gauge, the parametrization is fixed by choosing flat metric
on the worldsheet with an arbitrary conformal factor, with coordinates(τ, σ):
−∞ < τ < ∞, timelike; and σ0 < σ < σ0 + ls , spatial.
The vanishing of Tαβ leads to the Virasoro constraints:
∂τ X µ ∂σ Xµ = 0 ,
∂τ X µ ∂τ Xµ + ∂σ X µ ∂σ Xµ = 0 ,
Choosing hαβ = η αβ there is still some gauge freedom (see EOM in light-cone
coordinates).
boundary conditions
From varying δX µ , we obtain, besides to the EOM, a surface integral proportional to:
Z ∞
√
∝
dτ −h∂σ X µ δXµ |σ=l
σ=0 .
−∞
To get rid of this integral, one can choose to take the Neumann boundary
conditions eq.(3.3). Open strings have boundary conditions for their endpoints:
∂σ X µ (τ, 0)
= ∂σ X µ (τ, ls ) = 0 ,
Neumann b.c. ;
µ
= ∂τ X µ (τ, ls ) = 0 ,
Dirichlet b.c. .
∂τ X (τ, 0)
(3.3)
The Neumann boundary conditions are the ones needed for freely moving open
string. The Dirichlet boundary conditions are important for D-branes (see 3.4).
33
The other possibility to get a vanishing surface integral, are periodic boundary
conditions:
X µ (τ, 0) =
∂σ X µ (τ, 0) =
hαβ (τ, 0) =
X µ (τ, ls ) ,
∂σ X µ (τ, ls ) ,
hαβ (τ, ls ) .
(3.4)
This gives the closed string solution.
EOM in light-cone coordinates
Other popular worldsheet coordinates are light-cone coordinates in flat space:
σ ± = τ ± σ; in these coordinate one can separate a left and right moving part
in the EOM of X µ :
∇2 X µ = ∂ + ∂ − X µ = 0
µ
⇒ X µ (σ + , σ − ) = XLµ (σ + ) + XR
(σ − )
The EOM of the string is just the wave equation, so the string spectrum can be
seen as an object with independent left and right moving Fourier modes. The
residual symmetry left in the gauge hαβ = η αβ is:
σ + → σ̃ + (σ + ) ,
σ − → σ̃ − (σ − ) ,
and this gives rise to the infinite dimensional algebra of conformal field theories in two dimensions. By solving the Virasoro constraints in these coordinates:
∂+ X µ ∂+ Xµ = ∂− X µ ∂− Xµ = 0
⇒
µ
∂τ X R
∂τ XRµ = ∂τ XLµ ∂τ XLµ = 0 ,
one finds that the string has only transverse degrees of freedom.
open strings
The EOM and the Neumann conditions eq.(3.3) in light-cone coordinates, the
left and right moving modes become connected. The open string solution looks
like:
XLµ
µ
XR
Xµ
X1
√
+
1 µ √ 0 pµ +
x + α
σ + i 2α0
αnµ e−iπnσ /ls ,
2
2
n
n=o
µ
X
√
√
1 µ
p
1 µ −iπnσ− /ls
x + α0 σ − + i 2α0
αn e
,
=
2
2
n
n=o
X1
√
√
= xµ + α0 pµ τ + i 2α0
αnµ e−iπnτ /ls cos(πnσ/ls ) ,
n
n=o
=
(3.5)
xµ /pµ is the centre of mass position/momentum (at τ = 0); αnµ are the
Fourier coefficients and will become like annihilation (n>0) and creation (n<0)
operators in the quantum theory.
The endpoints can carry some non-dynamical, conserved quantum numbers, the
Chan-Paton factors.
The mass is determined by the oscillations of the string, but it will be modified by quantization and by fermionic modes.
34
closed strings
The periodic boundary conditions with the EOM give the closed string solution:
XLµ
=
µ
XR
=
Xµ
=
r
1 µ √ 0 pµ +
α0 X 1 µ −iπnσ+ /ls
x + α
σ +i
α e
,
2
2
2 n=o n n
r
α0 X 1 µ −iπnσ− /ls
1 µ √ 0 pµ −
x + α
σ +i
α̃ e
,
2
2
2 n=o n n
r
´
√
α0 X 1 ³ µ −inπσ+ /ls
xµ + α0 pµ τ + i
αn e
+ α̃nµ e−iπnσ− /ls (. 3.6)
2 n=o n
Again by the Virasoro constraints, only the transverse modes represent physical
degrees of freedom. The coefficients of the left and right moving modes (αnµ /α̃nµ
only depend on each other by:
X
X
i
i
N≡
nα−n
αi,n =
nα̃−n
α̃i,n ≡ Ñ ,
(3.7)
n>0,i
n>0,i
coming from the translation invariance (σ 0 = σ+ constant) on the closed
string.
additions to the Polyakov action
There are two possible additions to eq.(3.2).
First the Gauss-Bonnet term:
λ
λχ =
4π
Z
√
−hR
(2)
Σ
λ
+
2π
Z
dsK ,
(3.8)
∂Σ
R(2) is worldsheet Ricci scalar; Σ, ∂Σ are the worldsheet and its boundary;
K is trace of extrinsic curvature on the boundary; λ is an arbitrary parameter.
And the cosmological term:
Z
Θ=T
√
−h ,
Σ
χ is invariant is under Weyl transformations, but Θ is not, so this term will not
be included in general.
The boundary term of χ is only important for the open string. The first part
of χ will give the Einstein equations eq.(2.1) for two dimensions. But as said in
subsection 2.1.1, there are no dynamics associated to χ. It is said to be equal
to the Euler number, which depends on the topology of the worldsheet only.
Anomalies can occur in the quantized theory.
35
3.1.2
Quantization
Quantization of the string worldsheet can be done with help of commutation
relations or with path integrals. These two methods are briefly summarized. For
a complete introduction on these methods, ghosts and anomalies, see [16, 17].
commutation relations
In the old covariant approach, equal τ Poisson brackets in the classical theory
are replaced by equal τ commutators in the quantum theory:
{X µ (σ, τ ), P ν (σ 0 , τ )}P.B. = δ(σ−σ 0 )η µν
⇒
[X µ (σ, τ ), P ν (σ 0 , τ )] = iδ(σ−σ 0 )η µν ,
where P ν is the conjugate momentum of X ν obtained as the Noether current
of translations (δX µ = aµ ); η is metric tensor of flat space; δ(σ −σ 0 ) is the Dirac
delta function. This leads to the next nonzero commutation relations:
[xµ , pν ] =
iη µν ,
µ
[αm
, αnν ] = mδm+n η µν ,
µ
[α̃m
, α̃nν ] = mδm+n η µν ,
where δm+n is the kronecker delta.
The Virasoro constraints haven’t been solved yet. For this, Goddard, Goldstone,
Rebbi and Thorn used the formalism of the light-cone quantization. In contrast
with the covariant approach, this is not manifestly covariant, but other features
turn out clearly.
√
First define light-cone coordinates in space-time: X ± = (X 0 ± X d−1 )/ 2.
−
Then identify X + = x+ + p+ τ . Using the Virasoro constraints, αm
and α̃n− are
totally determined by the transverse modes αoi and α̃pj . So only the transverse
oscillators are quantized by the next commutation rules:
i
[αm
, αnj ] =
αni |0 > =
mδ ij δm+n,0 ,
α̃ni |0 >= 0 , n > 0 .
The string vacuum |0 > is not the real vacuum, but one string with no
oscillations.
path integral quantization
Here the Polyakov path integral is used.
Z
Z = DXDhe−iS[X,h] .
The gauge-fixing of the symmetries is done with the Fadeev-Popov mechanism. Ghost fields come out of the Fadeev-Popov determinant. One uses BRST
(Becchi-Rouet-Stora-Tyutin) techniques to find a gauge between the anticommuting ghost fields and the original fields, but this goes beyond the scope of
this thesis.
36
critical dimension
Space-time Poincaré invariance is only maintained when the bosonic string lives
in 26 space-time dimensions, the critical dimension. For the superstring, this
will become 10 dimensions. To retain 4 dimensions, one can compactify with the
Kaluza-Klein reduction for example on a n-dimensional torus T n or a CalabiYau manifold.
mass spectrum
Now the mass spectrum of particles generated by oscillations of the string is
quantized:
2
MN
=
2
MN
=
1
(N − a) ,
α0
2
(N + Ñ − 2a) ,
α0
(open strings) ;
(closed strings) ,
MN is not geometrized; a is some constant coming from the Virasoro constraints working on the vacuum. For the bosonic string it is equal to 1. Therefore this string contains a tachyon, an object with negative mass squared. This
unphysical object is removed from string theory when considering superstrings.
zero modes
The massless states for closed strings will, due to the constraint (N = Ñ ),
contain two indices and will be obtained by working with αµ and α̃µ on the
string vacuum |0 >; These can be separated into three fields: an antisymmetric
form field, a traceless symmetric tensor and the trace:
graviton
dilaton
Bµν
Gµν
Φ
1
2 (αµ α̃ν − αν α̃µ )|0 >
1
(α
α̃
+ αν α̃µ − 2δµν ακ α̃κ )|0
µ
ν
2
αµ α̃µ |0 >
>
Here one of the fields that show up in every superstring theory, is the graviton
field, responsible for gravity. Next to that, the dilaton will always arise, which
will be coupled to the string coupling strength. Bµν arises at IIA/B string
theory, in the low effective energy action SUGRA IIA/B.
From string interactions (see 3.1.4), a so-called vertex-operator is introduced
to describe interaction with the string and therefore the interaction with background fields. From this interactions, the Polyakov action eq.(3.2) can be generalized to the nonlinear sigma model:
T
Sσ = −
2
Z
´
³
√
d2 σ −h hαβ gµν ∂α X µ ∂β X ν + i²αβ Bµν ∂α X µ ∂β X ν + α0 ΦR(2) .
(3.9)
This action descibes a string moving in background of these fields in the conformal gauge. The string carries the charge of the Bµν .
37
3.1.3
Worldsheet fermions + bosonic string = superstring
Fermionic degrees of freedom on the string are introduced by Majorana spinors
(ψ µ ) living on the worldsheet.
µ
¶
Z
T
1 µ γ
µ
2
S=−
d σ ∂α X ∂β Xµ − ψ ρ ∂γ ψµ ,
(3.10)
2
T
ργ represent the worldsheet Dirac matrices; eq.(3.10) is in the conformal
gauge, which means flat worldsheet metric with arbitrary conformal factor.
On this Ramond-Neveu-Schwarz string ψ µ and X µ are connected by what now
is called supersymmetry. For local worldsheet SUSY one finds a supercurrent
J α which must disappear. J α is the superpartner of Tαβ and from here we
obtain the superconformal algebra of the constraints.
Ramond and Neveu-Schwarz sector
Two different boundary conditions are obtained. The fermionic degrees of freedom can be periodic or antiperiodic on the string:
ψ µ (τ, σ + 2ls )
ψ µ (τ, σ + 2ls )
= +ψ µ (τ, σ) ,
= −ψ µ (τ, σ) ,
Ramond conditions ;
Neveu-Schwarz conditions .
The open superstring contains therefore two different sectors (R and NS); and
the closed superstring has four separate area’s, two for left and for right (R-R,
NS-NS, R-NS, NS-R). The NS, R-R, NS-NS sectors give space-time bosons; R,
R-NS , NS-R sectors space-time fermions. The NS-NS gauge fields couple to
fundamental strings; the R-R charge is carried by D-branes (see 3.4).
fermionic modes
µ
µ
With ψ+
and ψ−
as the elements of the worldsheet spinor, the EOM in light-cone
coordinates become:
µ
∂+ ψ−
=0.
µ
=0,
∂− ψ+
So again the left and right moving degrees of freedom can be separated. Together
with the boundary conditions, these give the left and right solution decomposed
in Fourier modes:
1 X µ −irσ+
µ
ψ̃r e
,
ψ+
=√
2 r
1 X µ −irσ−
µ
ψ−
=√
ψr e
,
2 r
r ∈ Z Ramond sector and r + 1/2 ∈ Z for Neveu-Schwarz sector. By the
Virasoro constraints, only the transverse modes represent physical degrees of
freedom.
38
constraints on the modes and critical dimension
Again for the open string ψrµ = ψ̃rµ ; for the closed string:
NL =
NR ,
X
X
i
i
αi,n +
rψ−r
ψi,r ,
nα−n
NL ≡
n>0
NR ≡
X
r>0
i
nα̃−n
α̃i,n
+
n>0
X
i
rψ̃−r
ψ̃i,r .
r>0
To be consistent with the superPoincaré algebra, this quantized superstring
must live in an d=10 space-time. In d=10 spinors contain 32 complex elements.
In d=10 one can lay both Weyl and Majorana conditions on this spinor; so we
are left with 16 real independent fermionic degrees of freedom in space-time.
GSO projection
Gliozzi, Scherk and Olive suggested that an extra condition can be added, known
as the GSO projection. This Dirac-like equation with a space-time Dirac spinor
on the states
Γ|ψ >= +|ψ > ,
reduces the number of independent space-time spinor elements to 8 real
fermionic degrees of freedom, so we can construct space-time SUSY as well.
Now the space-time spinors have became chiral. Also did the GSO-projection
made the tachyon disappear completely and the mass formulas have become:
2
MN
=
2
MN
=
1
(NL + NR ) , (open string) ;
2α0
2
(NL + NR ) , (closed string) ,
α0
MN is the non-geometrized mass. From here five consistent superstring
theories were found, which differ by the number supercharges (1 or 2), the
relative chirality of the gravitino’s, and wether there are fermionic degrees on
only one side of the closed string (heterotic superstring theories), or not (the
others).
Now these problems are solved, one can look at interacting strings.
39
3.1.4
String diagrams and CFT
Here again we will only summarize the concepts used, without details concerning
the calculation. Introductions can be found in [16, 17].
CFT
Since string theory without cosmological constant is conformal invariant, the
study of conformal field theories is very important for the theory as a whole.
Next to that, the features of these CFTs have consequences for the fields generated by these strings. While these fields are obtained by pertubation series (see
3.2), the CFT has consequences for the whole theory. The Virasoro constraints
from the worldsheet energy-momentum tensor, can be written as infinite algebra, the Virasoro algebra. The algebra is characterized by its central charge
c.
For black holes physics, the coupling of CFTs to the AdS space-times near
extremal black holes are interesting (see 3.5.3).
vertex operators
In figure 3.2 a scattering proces of four strings with one loop is drawm. The
worldsheet can be conformally transformed to a torus with four punctures for
the four external strings. When calculating such a diagram with a four point
correlation function, these punctures show up as vertex operators acting on
these points on the worldsheet.
With these vertex-operators one can observe the scattering proces and construct the scattering matrix. From here the space-time field equations are obtained.
x
x
x
x
b)
a)
Figure 3.1: a) worldsheet of closed relevant for the one-loop contribution to
four-point scattering; b) The same diagrams conformal transformed to a torus.
The external strings are transformed into four dots; the vertex operators act on
these points in the correlation function.
sum over topopogies
Since the worldsheet is conformal invariant, the ’Feynmann’ diagrams of open
(closed) strings can be drawn as circles (spheres), annulus (torus), or graphs
40
with more boundaries (handles). The diagrams are only distinctive if they have
different topology. Therefore there are less distinctive string diagrams then
different Feynmann diagrams for e.g. QED.
string coupling
Adding an open (closed) string to the diagram means adding a boundary (handle), lowering the Euler number by 1 (2) (see figure 3.3). When considering the
path-integral of the Gauss-Bonnet term eq.(3.8), amplitudes will be weighted
by a term e−λχ ; χ is the Euler number. So closed strings are coupled to each
other by a factor eλ
After calculations of scattering of string in flat space, other fields set to zero
except for the constant dilaton field Φ = Φ0 in eq.(3.9), the string coupling
constant gs for closed strings is related to this field, the dilaton:
gs = eΦ0 ,
so the λ of the eq.(3.8) is also related to this field. The string coupling will
be defined by this equation.
δχ = −1
δχ = −2
Figure 3.2: worldsheet topology and Euler number change due to emission and
reabsorption of open and closed strings
41
3.2
Supergravity
The string worldsheet is a generalization of the geodesic from GR. Now the
space-time field equations like the Einstein equations eq.(2.1) must be formulated. To get the right field equations, one looks at strings at scattering processes.
The low energy limit of string theory does give d=10 supergravity. The low
effective energy action is obtained in the limit of small α0 (weak curvature) and
small gs (no string loop diagrams).
The subject of supergravity is so large, that even the word ’introduction’ is not
the right word for the next section. See [19] for a real introduction.
Supersymmetry
SUGRA is an extension of GR with supersymmetry. A theory is supersymmetric
when the supercharges Qa leave the action invariant:
µ
B
F
¶
µ
→
B0
F0
¶
µ
= Q̂a
B
F
¶
µ
=
B + δB
F + δF
¶
,
where B/F are the bosonic/fermionic fields. It is called supersymmetric
because this transformation mixes bosonic and fermionic fields: δB ∝ F and
δF ∝ B. SUSY algebra is the Poincaré algebra extended with some extra (anti)commutation relations between the momentum, the angular momentum and
the supercharges. The supercurrent associated with SUSY is the superpartner
of the energy-momentum tensor of GR. Two local SUSY transformations give
a local coordinate transformation.
Now by these transformations one can also define Killing spinors, as supersymmetric partners of Killing vectors, for defining symmetries in the theory.
N gives the number of supercharges. SUGRA can only exist in d≤11. Above
eleven dimensions, the spinors get too large for SUSY.
3.2.1
From worldsheet to space-time
To maintain Weyl invariance for the quantized string in a nontrivial background
eq.(3.9), the full analysis in pertubation series in α0 of this action gives equations
for the background fields. The equations in the leading order in α0 are the field
equations of SUGRA. Higher orders in α0 will give higher order curvature.
The comparison between the coupling of the graviton in string theory and
the coupling of GR, leads to the conclusion, that the mass gap between the
massless modes and the lightest massive string states are of the order of the
planck mass. Therefore calculations with only the massless modes are useful.
The low effective energy action of the NS-NS part of SUGRA II contains the
same fields as the bosonic string:
µ
¶
Z
√
1
1
SS,II = 2
d10 X −gS e−2ΦS RS + 4(∂ΦS )2 − H 2 ,
2κS
12
where ΦS is the dilaton; Hµνρ =∂µ Bνρ +cyclic permutations; RS the Ricci
scalar of the graviton field gSµν ; the S refers to the string frame.
42
string frame and Einstein frame
The action obtained from string scattering, does look like the Einstein-Hilbert
action eq.(2.2), but not completely. There is a difference between the Einstein
frame (GR) and the string frame (string theory). For the transition between
the two, one has to redefine the metric and the dilaton:
ds2E
RE
ΦE
4
4
= gEµν dxµ dxν = e− d−2 ΦE gSµν dxµ dxν = e− d−2 ΦE ds2S ,
µ
¶
4
4(d − 1) 2
4(d − 1)
∇ ΦE −
(∇ ΦE )2 ,
= e d−2 ΦE RS +
d−2
d−2
= ΦS − Φ0 ,
S/E refers to string/Einstein frame.
In ten dimensions this would give:
1
1
2κ2S
Z
ds2E = e− 2 ΦE ds2s
√
d10 X −gS e−2Φs RS
Z
1
d10 X
2κ2E
,
=
√
−gE
[RE + . . .] ,
where the coupling can in string theory be identified as:
(10)
16πGN
for which gs =e
Φ0
= 2κ2E ≡ 2κ2S gs2 = (2π)7 α04 gs2 ,
(3.11)
.
The Bekenstein entropy eq.(2.22) is formulated in Einstein frame like other
properties (energy-momentum tensor, ...) first treated in GR. From now on, all
will be written string frame, unless stated differently.
3.2.2
Type IIB supergravity
For the D1-D5 system (see 4) we need SUGRA IIB, with 2 supercharges and
therefore two gravitini’s, in this case with the same chirality. From the R-R
sector SUGRA IIB gets a scalar C, a two form Cµν and a four form Cκλµν . The
bosonic part of SUGRA IIB in string frame yields three different parts [21]:
SN S−N S
=
SR−R
=
SC−S
=
¸
·
Z
1
1 2
10 √
−2Φ
2
d X −ge
R + 4(∇Φ) − H
2κS
12
¸
·
Z
√
1
1
1
1
10
(1) 2
(3) 2
(5) 2
−
d X −g (G ) + (G ) + +
(G )
2κS
2
12
480
Z
1
1
+
(C (4) + B (2) ∧ C (2) ) ∧ F (3) ∧ H (3) ,
(3.12)
4κS
2
where the gauge fields are written in the language of forms (see appendix
A). The R-R field strengths are: G(1) = F (1) ; G(3) = F (3) + C (0) ∧ H (3) ;
G(5) = F (5) + H (3) ∧ C (2) ; and Fµ1 ...µn = ∂µ1 Cµ2 ...µn + cyclic permutations.
43
The field strengths are formulated such that the action is gauge invariant; the
gauge transformations of these fields aren’t as simple as the ones from electrodynamics. SC−S is the Chern-Simons term, needed for SUSY.
The four form could cause some trouble, because its field strength is selfdual.
Therefore SUGRA IIB was only discovered as the low energy limit of string
theory IIB. The field equations were derived in [20] by using its SL(2, R) symmetry (see 3.3.1). To write down a covariant action, the most used trick to
handle the selfduality, is to treat C (4) as a normal field and take the selfduality
as a constraint at the level of the field equations.
The NS-R and R-NS parts of the action both give fermions: two spin 21 dilatino’s
ψ and two spin 23 gravitino’s ψµ . Because they aren’t used in the fuzzball
proposal, they will be taken to be zero (see 3.5.1).
field equations F-P system and D1-D5 system
The field equations for these fields [20] are more clearly reviewed in [28]. For the
D1-D5 system, we only need the equations with gµν , Φ and C (2) fields turned
on. For the dual F-P system we need the field gµν , Φ, B (2) :
D1-D5 system
Rµν
=
(∂Φ)2
=
∇µ F µνρ
=
−2 ∇µ ∂ν Φ +
1
1 2Φ
e Fµκλ Fνκλ −
gµν e2Φ F 2 ,
4
24
1 2
1
∇ Φ + e2Φ F 2 ,
2
24
0,
F-P system
Rµν
(∂Φ)2
∇µ (e−2Φ H µνκ )
= −2 ∇µ ∂ν Φ +
=
=
1
Hµκλ Hνκλ
4
1 2
1
∇ Φ + H2 ,
2
24
0,
These equations give classical solutions, so no string loop diagrams are taken
into account. For the string tree level the string coupling gs is taken to be very
small.
electric and magnetic sources: p-branes
In d=4 electrodynamics, a particle (0-brane) is the source for the 1-form gauge
field. In SUGRA a p-brane, with p spatial dimensions, carries the charge density
of the (p+1)-form gauge field (see appendix B).
The SUGRA IIB has a 0-form field, two 2-form fields and a 4-form field;
so there are (-1)-branes, 1-branes and 3-branes. A (-1)-brane looks odd. It is
recognized as an instanton.
44
In string theory there is a distinction between the NS-NS and R-R fields.
The NS-NS charge is carried by the fundamental string (a 1-brane). The R-R
charge is carried by D-branes. These are supersymmetric p-branes, where the
mass is completely determined by the charge (see 3.5.1).
For the Maxwell theory without sources eq.(2.14), the Bianchi identity and field
equations of the vector potential Aµ are of the same form, so we can interchange
F and ∗F , or even take a mixture of them, without breaking eq.(2.14).
This duality can be extended to the Maxwell theory with charges, if magnetic
charges are added. The charges become quantized, due to the Dirac charge
quantization (see 3.3.1)
This can easily be generalized to higher rank form fields. The field strength
of the magnetic field is defined by the hodge-dual of the electric field strength:
Fµ1 µ2 ...µp+2 → ∗Fµ1 µ2 ...µp̃+2 =
1
²µ µ ...µ ν ν ...ν F ν1 ν2 ...νp+2 .
(p + 2)! 1 2 p̃+2 1 2 p+2
The potential A(p+1) couples electrically to a brane with p spatial dimensions.
Now ∗F(p̃+2) is the field strength of form field Ã(p̃+1) that couples electrically to
branes with p̃(= d − p − 4) spatial dimensions. Now A(p+1) couples magnetically
to p̃-branes. For d=4 Maxwell theory, both electric and magnetic charges are
then point particles.
For d=10 SUGRA IIB the magnetic charges are carried by 3-branes, 5branes and 7-branes. The d=10 3-branes, like particles in d=4 can be dyonic.
The NS5-brane is the magnetic counterpart of the fundamental string.
compactification
Lower-dimensional SUGRA can be obtained by compactifications. Some important compactifications are compactifications on a n-torus (T n = S 1 × . . . × S 1 ,
n times compactification on a circle) and Calabi-Yau compactifications.
From a (d-n)-dimensional point of view, the d-metric gµν split after compactification on a m-torus into different fields (see eq. (2.8):
(d)
gµν
→
(d−m)
1 metric tensor gµ̃ν̃
,
(a)
m vector gauge fields Aµ̃ ,
(m + 1)m/2 scalars Φ(a,b) (= Φ(b,a) ) .
Also the form fields split in different fields due to the compactifications. One
compactification on a circle would make a d-dimensional n-form field C( µ1 . . . µn )
become an one (d-1)-dimensional n-form field Cµ̃1 ...µ̃n and a (d-1)-dimensional
(n-1)-form field Cµ̃1 ...µ̃n−1 a .
Due to the antisymmetry of the form fields, other possible fields are totally dependent on the (n-1) form field (like Cµ̃1 ...µ̃n−2 a µ̃n ), or disappear (like
Cµ̃1 ...µ̃n−2 a a ).
45
The compactification on a m-torus gives:
in d dimensions
n-form Cµ1 ...µn
→
in d − m dimensions
1 n-form Cµ̃1 ...µ̃n ,
m (n − 1)-form Caµ̃2 ...µ̃n ,
(m-1)m/2 (n − 2)-form Cabµ̃3 ...µ̃n ,
(m-2)(m-1)m/6 (n − 3)-form Cabcµ̃4 ...µ̃n ,
... .
After one compactification a p-brane can become a p-brane or a (p-1)-brane
from a (d-1)-dimensional point of view, depending on wether the direction of
compactification is transverse or parallel to the p-brane.
46
3.3
Dualities
A dual theory should describe the same physics. It was found that the 5 different
d=10 superstring theories together with a d=11 theory (M-theory) are related
to each by several dualities. S-duality and T-duality are used for a comparison
between a D1-D5 system and a F-P system describing the same black hole
solution.
3.3.1
S-duality
Dirac showed that in quantum theories electric charge get quantized when there
are magnetic charges around. He used a Dirac string and found the relation
between electric and magnetic charge:
QP ∝ n ,
n∈Z
with Q/P electric/magnetic charge. So by this, one can formulate a fundamental electric and magnetic charge for n = 1.
This quantization tells us, that strong fields with a large fundamental charge
Q in the electric theory become weak fields with small fundamental charge P in
the magnetic theory. This strong/weak-duality is S-duality.
In string theory S-duality is formulated by inversing the string coupling:
gs
→
gs0 =
1
.
gs
(3.13)
SL(2, Z) symmetry
The IIB stringtheory is selfdual under S-duality. This is reviewed in [22],by
looking at the action of SUGRA IIB in Einstein frame; this action is invariant
under a transformation of the form:
τ = C (0) + ie−Φ →
aτ + b
,
cτ + d
with the condition ad − bc = 1. These transformations form a group isomorphic to SL(2, R), the group of real 2 × 2-matrices with determinant 1. The next
transformation gives:
τ →−
1
τ
⇒
eΦ → e−Φ
⇒
gs = eΦ0 → gs0 =
1
.
gs
So the S-dual of SUGRA IIB is again SUGRA IIB, only there are some field
redefinitions:
Φ → −Φ ,
gEµν → gEµν ,
gsµν → e−Φ gsµν ,
Bµν → Cµν ,
Cµν → Bµν .
Due to quantum effects like charge quantization, this symmetry is broken to
SL(2, Z).
47
3.3.2
T-duality
When one dimension is compactified, the momentum in that direction is quantized: pa = n/Ra , where Ra is the radius of the compact dimension, and n is
an integer.
Next to that, an extended object like a string can wind around this compact
dimension: X a (σ + ls ) = X a (σ) + 2πRa w, where w is number of windings. The
integer w of a closed string is conserved; this is not true for the open string
because its endpoints can move freely.
The mass-spectrum and the relation between left moving and right moving
modes of a closed string around a compactified dimension become:
m2
=
0
=
n2
w 2 R2
2
(N
+
N
−
2)
+
+
,
L
R
α0
R2
α02
nw + NL − NR .
(3.14)
The mass spectrum is invariant under the interchange of the winding number
and the momentum number, when one replaces R by R0 = α0 /R as the radius of
the compact dimension. This is T-duality and is thought to hold for the whole
string theory.
SUGRA II A/B duality: the Buscher’s rules
By T-duality on a torus SUGRA IIA and IIB interchange, so after an even
number of T-dualities one gets again the same theory.
Different values of the fields are identified with each other to give the same
SUGRA actions we are used to; T-duality transforms the old fields into new
ones. These transformations rules are summarized in generalized Buscher’s rules
for the compactification on a circle. The rules for transforming from type IIA
to IIB and visa versa, under T-duality in the y-direction are:
0
gyy
0
Bµν
Φ0
0(n)
=
1
,
gyy
= Bµν −
0
gyµ
=
0
gµν
= gµν −
1
(gµy Byν + Bµy gyν ) ,
gyy
= Φ − log
(n+1)
1
Byµ ,
gyy
√
1
(gµy gyν + Bµy Byν ) ,
gyy
0
Byµ
=
1
gyµ ,
gyy
gyy ,
(n−1)
Cµ...ναβ
= Cµ...ναβy + nC[µ...να gβ]y − n(n − 1)
0(n)
Cµ...ναy
(n−1)
= Cµ...να
− (n − 1)
1 (n−1)
C
g|α]y ,
gyy [µ...ν|y
1 (n−1)
C
B|α|y g|β]y ,
gyy [µ...ν|y
(3.15)
this is written down in string frame; µ, ν, . . . means here all directions except the y-direction; [. . . | . . . | . . .] means antisymmetric and normalized in these
indices, except for indices between lines.
48
3.4
D-branes
The new string theory does not only contain one-dimensional objects (strings),
but also D-branes. These were identified with the p-branes of SUGRA by
Polchinski. The Dp-branes are embedded into superstring theory by T-duality
on open strings. For more on D-branes, see [21].
T-duality on open strings
A freely moving open string has no conserved winding number; therefore Tduality in the a-direction is defined by the transformation:
a
0a
a
(XLa , XR
) → (XL0a , XR
) = (XLa , −XR
),
Using eq.(3.5), the dual solution for the open string becomes:
X1
√
a
X 0a = XLa − XR
= c + α0 pa σ + i 2α0
αna e−iπnτ /ls sin(πnσ/ls ) .
n
n=o
where the term c can be added without disturbing the open string solution
eq.(3.5). In this T-dual theory, the Neumann conditions in the a-direction have
changed into Dirichlet conditions (eq.(3.3)). This means the open string has the
endpoints fixed in this direction and can have conserved winding number, while
they are free in the other directions.
But the Dirichlet conditions means that momentum transport is possible at
the endpoints. The explanation for this, is that the endpoints are attached to
extended objects called D(irichlet)-branes on which the endpoints can freely
move. The other way around, the open strings describe the excitations of the
D-branes. These branes can have any number of dimensions between 0 and 10.
x’ 9
2π R’
0
Figure 3.3: The endpoints of the T-dualised open string are attached to the
D-brane. The shaded D-branes are periodically identified. The thick string has
winding number 1, while the thin string has winding number 0. The other way
around: open strings describe the excitations of the D-brane
49
3.4.1
Solitons in supergravity
By the annexation of the Dp-branes by string theory, the coupling of the R-R
fields to D-branes can be expressed in string constants.
By T-duality, D-branes bring open strings into the pure closed string theories
of SUGRA II. The string theory of open strings has half the number of independent supercharges as string theory II. Therefore the D-branes in SUGRA II
are BPS states with half the SUSY of the original theory (see 3.5.1).
In d=10 SUGRA IIB there are electric charged D(-1)-branes and D1-branes,
magnetic D5-branes and D7-branes and dyonic D3-branes; SUGRA IIA in d=10
has electric D0-branes and D2-branes, magnetic D4-branes and D6-branes.
The coupling of a D-brane to the NS-NS fields, is described by the Dirac-BornInfeld action:
Z
q
Sp = −Tp dp+1 ξe−Φ det(gαβ + Bαβ + 2πα0 Fαβ ) ,
Tp is the brane tension or mass density; {ξ} are worldvolume coordinates of
the brane; gαβ /Bαβ /Fαβ is pull-back of the metric, two form field and the field
strength of Aµ to the worldvolume. The factor e−Φ shows that the coupling
here is of the strength of open strings, not of closed strings e−2Φ like in the
SUGRA IIB action.
The coupling of np Dp-branes to the R-R fields is different, since they carry
R-R charge:
Sp = −
Z
gs2
(10)
16πGN
Z
F(p+2) ∧ ∗F(p+2)
+ np µp
Mp+1
C(p+1) ,
(3.16)
µp is the fundamental coupling (density) of the brane under the (p+1)-form
C(p+1) (see appendix B); M(p+1) is the worldvolume. This is not valid for the C4
field since its field strength is selfdual. The couplings of D-branes are completely
determined by the tensions (see 3.5.1).
metric of D-branes
In isotropic coordinates np Dp-branes (p ≤ 6) at ~xt =0 generate the fields:
ds2s
−1
1
= −Hp 2 (dt2 − dxp · dxp ) + Hp2 dxt · dxt ,
p−3
e−2Φ = Hp 2 ,
C01...p = −(Hp−1 − 1) ,
(3.17)
where C(p+1) is the (p+1)-form field; xt and xp give the (d−p−1) transverse
and p parallel spatial coordinates of the D-branes. Hp is a harmonic function:
Qp
,
(3.18)
r7−p
Qp is the geometrized charge (see appendix B). Hp is the spherically sym2
metric solution of the Laplacian ∂{x
in the transverse directions.
t}
Hp = 1 +
50
tension and charge density in string theory
The fundamental string carries the charge of the B (2) field. The charge density
is given by the coupling of the field to the worldsheet. But in string theory this
coupling is well known eq.(3.9), and can be expressed in string constants:
Z
Z
(2)
µF
B =T
d2 σBµν ²αβ ∂α X µ ∂β X ν ,
WS
WS
thus the fundamental coupling µF is equal to the tension T , 1/2πα0 . From
the Dirac charge quantization for both NS-NS p-branes and Dp-branes:
κ2s µp µ6−p = π ,
κ2s = 64π 7 α04 ,
the coupling charge of the NS5-brane is obtained: µN S = 1/32π 5 α03 . From
this the fields generated by nF F-strings:
ds2s
−HF−1 (dt2 − dx2p ) + dxt · dxt ,
=
e−2Φ = HF ,
Bty = HF−1 ,
and by nN S NS5-branes are obtained:
ds2s
−dt2 + dxp · dxp + HN S dxt · dxt ,
−1
e−2Φ = HN
S ,
1
Hijk = ²̃ijkl ∂l HN S ,
2
=
where the ²̃ijkl is the flat space Levi-Civita antisymmetric tensor in the
transverse spatial directions. Both HF and HN S are harmonic in its transverse
directions, and can be written as:
HF = 1 +
QF
,
r6
HN S = 1 +
QN S
,
r2
(10)
where the QF = nF GN µF ≡ nF cF and equal for QN S ; so the fundamental charge cN S and cF can be obtained from the coupling (appendix B). The
charges and tensions of the D-branes are obtained by S-duality and T-duality
transformations relating them to the F-string. This can be summarized as:
F
N S5
µF 1 = 1/2πα0
µN S5 = 1/32π 5 α03
charge
cF = 32π 2 gs2 α03
cN S = α0
tension
TF = µF
TN S = µN S gs−2
coupling
Dp
µp = (2π)−p α0−
cp = 25−p π
5−p
2
Γ
p+1
2
¡ 7−p ¢
2
gs α0
7−p
2
Tp = µp gs−1
where Γ is the Euler-gamma function. The total dependence of the tension
on the charge makes these D-branes BPS-states.
51
3.4.2
Compactification
Different compactifications of a space-time with D-branes are divided in two
categories: wrapping and transverse compactification.
wrapping
To get finite mass and charge, the branes must be of finite length since Tp
and cp are densities. This can be done by wrapping them around compactified
dimensions. This way the branes become particles at large distance. The next
diagram illustrates a compactified D3-brane:
D3
t
+
x1
-
x2
-
x3
-
x4
-
x5
-
x6
-
z1
+
z2
+
z3
+
where + is parallel direction; - a transverse; {x} are the noncompact dimensions; {z} the compact ones. It now couples to a 1-form field, from a lower
dimension point of view.
transverse compactification
Simple dimensional reduction is compactifying a direction transverse to worldvolume, so the a D-brane in d dimension, stays a D-brane in d-1 dimensions. In
[34, 22] procedures show what happens to the harmonic function in eq.(3.17).
Most important is that the harmonic function becomes again a harmonic
function:
(d)
Hp(d) = 1 +
(d−1)
Qp
| ~xt |d−p−3
Hp(d−1) = 1 +
⇒
Qp
,
| ~xt |d−p−4
at large distances ~xi /Ra , in one dimension less after identification of the
charge:
Q(d−1)
≡ Q(d)
p
p /Ra .
(d)
Qp will be the charge density in ten extended dimensions, Qp the charge (density) in d noncompact dimensions.
This brane is then delocalized, smeared out in the compact dimension. This
is a nice property when discussing D-branes with compactified transverse dimensions, because now the solution is independent of this internal space. The
diagram for the D3-brane above, after another compactification will then become:
D3
t
+
x1
-
x2
-
x3
-
x4
-
x5
-
z1
∼
z2
+
z4
+
z5
+
∼ is a compact direction in which the brane is smeared out. See appendix
C how constants of D-branes transform under S-duality and T-duality.
52
M
g
Figure 3.4: Initially point-like configuration of D-branes in flat space become
black branes with non-trivial geometry as the string coupling gs increases
3.4.3
Complementarity of the descriptions
The two descriptions of D-branes (open strings and SUGRA solitons) are complementary (see figure 3.1.4). Both sides can compared to each other when they
are BPS states.
For SUGRA to be valid, both gs and α0 must be small. But α0 has dimension,
so the weak curvature condition is phrased as:
rSW >>
√
α0 ,
gs np >> 1 ,
where the second equation is obtained by filling in the Schwarzschild radius
7−p
7−p
rSW
= Qp = gs np α0 2 from eq.(3.18).
The SUGRA field equations are valid at string tree level, gs < 1; then the
SUGRA description is valid when np is large.
For open strings, every time a D-brane is added to a string diagram, this gives
an extra boundary and this gives an extra factor gs ; when there are np Dpbranes, the diagram gets a factor ∼ gs np . So open string pertubation theory is
good as long as:
gs np < 1 ,
rSW <<
so the space is almost flat.
53
√
α0 ,
3.5
BPS black branes
Within string theory, black holes can be made out of strings, D-branes and
special topologies. In [22, 27, 28, 29] one can find the entries to the several
areas of string theory, where black holes are studied.
One of most succesfull black hole descriptions in string theory comes from
BPS states. Researchers succeeded to obtain a match between microscopic
entropy and Bekenstein entropy for black holes constructed from D-branes.
3.5.1
BPS state
The Bogomolnyi-Prasad-Sommerfield state is one which is invariant under a
nontrivial subalgebra of the full sumpersymmetry algebra of the theory. A
1 1
1 1
2 / 4 /· · · BPS state preserves 2 / 4 /· · · of the full supersymmetry. To get black
holes like in GR we take the fermionic fields to disappear. This theory can still
contain supersymmetry under certain conditions:
µ
B
0
¶
µ
→
B0
F0
¶
µ
= Q̂a
B
0
¶
µ
=
B + δB
δF
¶
,
where Q̂a are the supercharges constructing this subalgebra; the B and F
are the bosonic and fermionic fields, respectivily. Since the δB ∝ F = 0, this is
rather trivial. But δF ∝ B doesn’t just disappear trivially. It lays conditions on
the Killing spinors and therefore less supersymmetric transformation parameters
are independent than in the full theory.
When compatible with conditions on the Killing spinors, the theory can be
fixed at a gauge with no fermions. This helps a lot getting simpler equations
and comparison with the results in GR.
D-branes carry R-R-charge. The mass is due to the subalgebra completely
determined by the charges. Therefore in the case of a black object, we can
identify it with an extremal black object.
We already know that extremal objects have vanishing temperature and
don’t radiate. This BPS state is therefore a sort of supersymmetric groundstate, and therefore a step in the direction of the proof of the cosmic censorship
conjecture.
non-renormalization theorems
In GR, multi-centered extremal black holes have no interaction, because of the
classical cancellation between gravitational attraction and electrostatic repulsion. The BPS condition comes from the supersymmetry algebra, and is protected against quantum corrections. From here non-renormalization theorems
are derived, stating that specific quantities (like the degeneracy of states) do
not depend on the string coupling strength. This is what we need to compare
microscopic and macroscopic entropy.
54
entropy of BPS black holes
The macroscopic entropy of BPS black branes is calculated in SUGRA. Remember that for SUGRA to be valid, gs np >> 1, so there is extremal black
object. From here one can calculate in Einstein frame the horizon area and the
Bekenstein entropy.
This must be compared with the degeneracy of D-brane states. For this one has
to go to the pertubative open string description at gs np << 1. For simplicity
we can take gs ↓ 0. Now Cardy’s formula [30, 31] is used to get the density
of states in a two-dimensional CFT for large np of open strings ending on the
D-branes.
Now comparing these two with help of the non-renormalization theorems, Strominger and Vafa discovered the first D-brane black hole with right number of
mircostates. [32]; other D-brane solutions followed [37]. However at this moment, one is not able to follow what happens when gs grows; the black hole
formation is invisible and one has to rely blindly on the non-renormalization
theorems.
near-extremal and non-extremal black holes
The blindness in the intermediate region is major disappointment, since some
interesting black holes aren’t extremal. For example, Schwarzschild black holes
cannot become extremal and therefore aren’t in some supersymmetric groundstate. Quantum loops will become important when the black hole forms, and
the entropies cannot be compared easily.
For example, when Susskind tried to relate the high degeneracy of the massive string with a Schwarzschild black hole. The entropy of the string is linear
with the mass M, while the Bekenstein entropy of d-dimensional Schwarzschild
d−2
black hole is M d−3 .
Still Susskind proposed that there is an one-to-one correspondence between the
(uncharged) fundamental string states and the (uncharged) black hole states.
This correspondence principle was generalized by Horowitz and Polchinski to a
wide range of charged black hole states. Also there are ideas that non-extremal
black holes are made of brane/anti-brane pairs. For more on this see [33, 27].
Now the nearest one can do interesting calculations with in string theory is the
near-extremal black hole which is derived from the extremal one by pertubation.
This will be needed when one wants to examen what happens to a particle when
it falls into the black hole, and what happens to the information. During the
transition between the unstable state and the BPS ground state, the object
radiates Hawking radiation.
3.5.2
Black brane metrics
In the SUGRA description black BPS brane solutions were found, which are
translational invariant in the p spatial directions, and have a black hole geometry
in the other directions. A geometry of np Dp-brane at one position (eq.(3.17))
describe such a geometry. Unfortunaly, except for the D3-branes, these are
null-singularities.
55
null-singularity
Most geometries in SUGRA generated by D-branes are null-singularities. This
means that they have zero horizon area; thus the horizon coincides with the
curvature singularity; this is not a naked singularity.
Since they have zero horizon area, they have zero entropy, so it doesn’t look
like the solution to look for black hole mircostates.
Besides that, the SUGRA description will get higher order α0 -corrections
near the branes. Probably this will give a nonzero horizon.
3.5.3
D-brane superpositions
To construct black holes from D-branes with a finite horizon, one not only
looks at multi-center configurations of D3-branes, but one can also look at more
general and more complicated solutions with different D-branes intersecting.
To get again a BPS-state, the conditions for the Killing spinors of the separate
D-branes must be compatible.
For D-branes that are parallel or have rectangular intersections to get a BPSstate, the number n of relative transverse dimensions must be a multiple of 4
[34]:
n = 4k,
k∈Z.
Relative transverse direction means that the open stings ending on one kind of
branes have Neumann boundary conditions in that direction, while the strings
ending on the other kind of branes have Dirichlet boundary conditions.
For k = 0 we get a 12 BPS; for k = 1 we get a 41 BPS state; configurations with
D1-branes embedded in D5-branes or D2-branes embedded in D6-branes are 14 .
For other black brane superpositions, see [22]. Still many of these configurations
give null-singularities in SUGRA.
Ads/CFT correspondence
After considering black holes in string theory, Maldacena came with this remarkeble conjecture [35]: a 10 dimensional string theory in AdS5 × S 5 , has a
correspondence with a conformal gauge field theory in lower dimensions. Nowadays it is the name for a wide range of dualities between gravity theories and
gauge theories without gravity (for a review [36]). It is also called the gravitygauge theory correspondence.
It is related to two ideas in physics. First the holographic principle, where the
gravity theory is related to a gauge theory in 1 dimension less. Secondly, the
idea that large a N gauge theory is equivalent to a string theory.
The case which gave birth to the correspondence is one with D3-branes, which
have a finite horizon area. These D3-branes can be described by closed strings in
SUGRA, which near the horizon have AdS5 geometry; or by open strings living
on the D3-brane, so in a 4 dimensional space-time. The open strings describe a
N=4 superconformal Yang-Mills theory.
56
3.6
The three charge black hole
The D1-D5-P system forms a black hole with the right degeneracy of D-brane
configurations according to the Bekenstein entropy [37].
D-brane configuration
The charges of this black hole come from n1 D1-branes, n5 D5-branes and nP
momentum quanta along the compactified y-direction. The D1-branes and D5branes are compactified on a T4 × S1 with 2πR the length of S1 and (2π)4 V the
volume of T4 , and momentum is added along the direction of D1-brane; only
right moving momentum to maintain the BPS-state:
D1
D5
P
u
+
+
v
+
+
→
x1
-
x2
-
x3
-
x4
-
z1
∼
+
z2
∼
+
z3
∼
+
z4
∼
+
+/-/∼ is parallel/transverse/delocalized direction; → the direction of the
momentum; u=t+y and v=t−y are the light-cone coordinates for the D1-string.
The fields are independent of the compactified directions, so the solution looks
like a multi-charged point again in d=5 SUGRA.
Because this system conserves 81 of the supersymmetry, the nonrenormalization
theorems protect it from quantum corrections, and can be compared at different
coupling strength. The charges are given in [34] by:
(5)
Q1 = n1
gs α03
,
V
(5)
Q5 = n5 gs α0 ,
(5)
QP = np
gs2 α04
.
V R2
It is not clear how these can be obtained with the properties obtained in earlier
subsections.
microstates
First gs ↓ 0, such that the D-brane become infinitely massive for nongeometrized
mass, the excitations of collections of the D-branes can be described by pertubative open strings. Using Cardy’s formula , this configuration gives a degeneracy:
D ∼ e2π
√c
6N
= e2π
√
n1 n5 nP
,
(3.19)
with central charge c = 6n1 n5 and N = nP .
black hole in d=10
In d=10 SUGRA, the D1-brane couples electrically and the D5-brane magnetically to the two form potential C (2) . For large charges, the Φ, gµν and C (2)
fields in string frame become:
57
1
1
= −(H1 H5 )− 2 [dt2 − dy 2 − (1 − HP )(dt − dy)2 ] + (H1 H5 ) 2 dx · dx
ds2S
1
−1
+H12 H5 2 dz · dz ,
e−2Φ = H5 H1−1 ,
Cty = −(H1−1 − 1) ,
Hi = 1 +
Qi
r2
Fijk =
1
²ijkl ∂l H5 ,
2
i = 1, 5, P ,
Now these branes give rise to the following metric in Einstein frame:
ds2E
−1
1
1
=
e− 2 Φ ds2s = H1 4 H54 ds2s
=
−H1 4 H5 4 [dt2 − dy 2 − (1 − HP )(dt − dy)2 ] + H14 H54 [dr2 + r2 dΩ2(3) ]
−3
1
−1
1
3
−1
+H14 H5 4 dz · dz .
From this metric in Einstein frame the horizon area can be calculated at constant
r and t:
A
=
=
=
AT 4 AS 1 AS 3
Z
Z
Z
p
9
3
−3
−1
lim
d4 zH1 H5−1
dΩ(3) r3 H12 H58
dy 2 − HP H1 8 H5 8
r↓0 T 4
1
S3
p S
p
7
2
7
(2π) V R lim(r H1 H5 (HP − 2)) = (2π) V R Q1 Q5 Qp
r↓0
black hole d=5
Since the Bekenstein relation is valid for all dimensions, this black hole can also
be obtained in d=5. First Kaluza-Klein compactification will give extra factors
in the action (see eq.(2.9)), which are absorbed into the the lower dimensional
dilaton:
e−2Φ
(5)
= e−2Φ
(10)
eσz1 eσz2 eσz3 eσz4 eσy ,
where exp[σz1 ] is the Kaluza-Klein scalar of the z1 compactification. The decomposition of the metric eq.(2.8) will be carried out and the metric in Einstein
frame in d=5 will become:
4
(5)
2
1
ds2E = e− 3 Φ ds2s = −(H1 H5 Hp )− 3 dt2 + (H1 H5 Hp ) 3 dx · dx .
Then the area in d=5 becomes:
(5)
AH = 2π 2
q
(5) (5) (5)
Q1 Q5 Qp .
58
Bekenstein entropy
From both d=10 and d=5 point perspective the Bekenstein entropy becomes:
(10)
SB =
(5)
AH
(10)
4GN
(5)
=
AH
(5)
4GN
(10)
√
= 2π n1 n5 np ,
(3.20)
with GN = GN /32π 5 V R = πgs2 α04 /4. From here we see that the logarithm of the number D-brane configurations eq.(3.19) match with the Bekenstein
entropy.
59
Chapter 4
From black spot to fuzzball
Mathur conjectures that the fuzzball solution is a black hole solution. Only one
with different properties and solutions for several paradoxes: a new black hole
picture. A new proposal for a black hole would normally replace a real black
hole. In this case that would be the D1-D5-P black hole.
But at the time of being, the fuzzball proposal for the D1-D5-P system is
still under construction (see chapter 5), therefore we examine the simpler D1-D5
system, which is a null-singularity in the so-called ’naive’ solution.
The fuzzball proposal is a solution of SUGRA IIB without α0 -corrections and
looks like a black hole at large distance. But closer to the center, it has different
microstates in the shape of different geometries. These are regulated by the
displacement function.
Not only do the microstates become visible at classical level, also the curvature singularity and the horizon disappear. To obtain the Bekenstein entropy,
the horizon is chosen to be the boundary of the fuzzball.
Mathur also claims that this provides a good solution for the information paradox. This claim, together with the vanishing singularities and the distinctive
geometries, make this proposal a most interesting model. Only is it a good
alternative for the old black hole?
60
4.1
Introduction
Up till now, the fuzzball solution for the D1-D5 system is the most successful
one. It was first found in [38], but nicely reviewed in [39].
First the naive D1-D5 solution is examined, together with the dual F-P system.
The F-P system is a string winded around a compact dimension with momentum
in this direction. The momentum will make the string oscillate.
After this, the fuzzball solution is obtained. The oscillations will change the
geometry, and the fields will depend on the modes of these oscillations. But the
momentum can be partitioned over the modes in a lot of different ways. The
distinctive microstates of the fuzzball solution are obtained from the different
displacement profiles of the string.
When dualizing back, the D1-D5 fuzzball is obtained; or actually only a part
of the whole solution, in a ’classical’ approximation. A simple example is given
in 4.4.1. It has capped geometry and no horizon. It approximates the naive
solution at large distance.
The Bekenstein entropy should be proportional to the area of the horizon.
First the location of the horizon should be chosen in the fuzzball picture. This
is done in the F-P system.
In the F-P system, the generic microstate is taken; the horizon is taken to
be the boundary of the volume occupied by this oscillating string with the most
probable vibration profile. The horizon area is calculated in the naive solution,
since the fuzzball solution approximates the naive solution rapidly outside the
horizon.
For the D1-D5 system this means that the horizon is the boundary behind
which different microstates start to have distinctive geometries; actually for the
most probable microstates only. So the fuzzball lies behind the horizon.
But the most probable microstate is not achieved in the classical approximation. Therefore it is likely to expect large fluctuations occurring in the metric;
this gives the name fuzzball.
In the last section of this chapter, some ideas occurring in the articles on the
fuzzball, are briefly summarized. These ideas are used to give the fuzzball a
status of a black hole, but might be studied independently of this status.
61
4.2
4.2.1
The ’naive’ solution
D1-D5 system
A system of n1 D1-branes and n5 D5-branes can form a 14 BPS state (see 3.5.3).
It has extremal black hole geometry at large distance. But since it is a nullsingularity, this is probably not the right solution near the branes.
In the naive solution the D1-branes and D5-branes are compactified on T4 × S1 ,
with volume (2π)4 V × 2πR, such that the branes are have finite size, mass and
charge, and the system looks like a point in 5 dimensions:
D1
D5
t
+
x1
-
x2
-
x3
-
x4
-
z1
∼
+
z2
∼
+
z3
∼
+
z4
∼
+
y
+
+/-/∼ is parallel/transverse/delocalized direction.
As with the three charge black hole only the d=10 SUGRA IIB field equations
with the C (2) , gµν , Φ fields turned on, are used:
Rµν
=
(∂Φ)2
=
∇µ F µνρ
=
−2 ∇µ ∂ν Φ +
1 2Φ
1
e Fµκλ Fνκλ −
gµν e2Φ F 2 ,
4
24
1
1 2
∇ Φ + e2Φ F 2 ,
2
24
0,
with the field strength Fµνρ = ∂µ Cνρ + cyclic permutations. The so-called
naive D1-D5 solution is:
ds2
¢
1 ¡
1
= (H1 (x)H5 (x))− 2 dt2 − dy 2 + (H1 (x)H5 (x)) 2 dx · dx
1
1
+H1 (x) 2 H5 (x)− 2 dz · dz ,
e−2Φ
= H5 (x)H1−1 (x) ,
1
²̃ijkl ∂l H5 (x) ,
(4.1)
2
²̃ijkl is the flat space Levi-Civita tensor in the noncompact spatial dimensions
{xi }; (x) means dependence on {xi }; these are isotropic coordinates (compare
with eq.(2.18)), so the horizon sits at r = 0, and the metric describes the horizon
and the space-time outside it.
By the Bianchi identity and the field equations of F (3) , H1 and H5 must be
harmonic in the noncompact spatial directions:
Cty
= −(H1−1 (x) − 1) ,
−1 2
H(i)
∂ H(i) = 0
Fijk =
⇒ H(i) = 1 +
X
j
Q(i)
|r − rj |2
i = 1, 5 ,
P4
r2 = i=1 x2i ; rj are the horizons of black holes; Q(i) the charges in string
constants. From now on, only the single-centered solution is investigated.
62
4.2.2
Black spot
The metric of the D1-D5 system in Einstein metric is:
−3
1
ds2E = e− 2 Φ ds2s
−1
1
3
= −H1 4 H5 4 (dt2 − dy 2 ) + H14 H54 dx · dx
1
−1
+H14 H5 4 dz · dz .
As already told, this D1 D5 system isn’t a real black hole solution but a
null-singularity. This means that the area of the horizon is zero:
AH ∝ V R lim(r3
r↓0
p
H1 H5 ) = V R lim(r
r↓0
p
(r2 + Q1 )(r2 + Q5 )) = 0 .
This means that the horizon is located at the singularity.
This SUGRA solution is normally not that interesting, because higher order
α0 -corrections will show up near the branes, and classical SUGRA IIB is no
longer valid. Also its Bekenstein entropy will be zero, so this doesn’t look like
the right spot to look for black hole microstates. But this changes when looking
at the fuzzball solution.
4.2.3
Duality transformations
The fuzzball solution is best introduced in the F-P system. This system is dual
to the D1-D5 system by some S-dualities and T-dualities:
D5 D1
S
→
Ty
→
Tz1
−→
S
→
Tz1 Tz2 Tz3 Tz4
N S5 F
N S5 P
N S5 P
D5 P
−−−→
D1 P
S
FP ,
→
(4.2)
where the F-P system is a winded F-string around a compact dimension
with momentum P; the winding and the momentum will give this system two
distinctive charges.
To keep the coupling in Einstein frame in 5 dimensions invariant under the
dualities, some of the constants change radically after a duality (α0 is set to 1
for compactness):

  0 


Ry V /Rz1 gs
gs
gs
 Q5 
 Q5 V R2 /gs   Q0w 
√

 



 R →
 ≡  R0  ,

  0 

 √ V
 Rz1 

V /gs   Rz1 
V
V0
V /Rz21 gs
here the primes give the constants in the F-P system; Qw is the charge of
the winded F-string, and is linear with the total charge of the D5-branes, Q5 .
For the transformations of the fields, see appendix C.
63
4.2.4
The F-P system
To have a bound F-P state and not two separate states, the momentum P lies
in the y-direction on the F-string. These are the travelling waves on this string.
These travelling waves can in string theory only be tranverse ones. This is one
of the main points in the fuzzball proposal.
To get a bound BPS state, the waves can only travel in one direction [26].
Waves travelling in both directions would destroy the space-time SUSY. This is
then interesting for near-extremal and non-extremal black holes.
In the F-P system there is still the compactification over T4 × S1 ; the F-string
lies in the y-direction:
F
P
u
+
v
+
→
x1
-
x2
-
x3
-
x4
-
z1
∼
z2
∼
z3
∼
z4
∼
(u, v) = (t + y, t − y) are left and right light-cone coordinates of the string.
This F-P system is a solution of the SUGRA IIB field equations with the fields
gµν , Φ and B (2) turned on:
Rµν
= −2 ∇µ ∂ν Φ +
(∂Φ)2
=
∇µ (e−2Φ H µνκ )
=
1
Hµκλ Hνκλ
4
1 2
1
∇ Φ + H2 ,
2
24
0,
The naive F-P system becomes:
ds2
=
e−2Φ
=
Buv
=
¡
¢
Hw−1 (x) −dudv + (Hp (x) − 1)dv 2 + dx · dx + dz · dz ,
Hw (x) ,
1
− (Hw−1 (x) − 1) .
2
(4.3)
Again Hw (x) and Hp (x) must be harmonic in the extended spatial dimensions:
−1 2
H(i)
∂ H(i) = 0
⇒
Q(i)
r2
(gs0 )2 α03
Qw = w
,
V0
H(i) = 1 +
s = w, p ,
Qp = np
(4.4)
(gs0 )2 α04
.
V 0 (R0 )2
Since this solution is flat in almost all directions, it is easier to handle.
When studying the naive solution of a F-P system (in
√ a heterotic SUGRA),
Sen found [26], that the curvature gets large near r ∼ α0 ; he proposed to put
a ’stretched horizon’ at this location; the Bekenstein entropy of this ’stretched
horizon’ agreed with the microscopic entropy.
64
4.3
The F-P fuzzball solution
The fuzzball solution as proposed by Mathur [38, 39], can be nicely introduced
by starting with the naive F-P solution and then having a closer look at what
must happen with the geometry near this string when it oscillates due to the
momentum; this was already studied in [40]. When dualising this solution back
eq.(4.2), we get the new D1-D5 system.
4.3.1
Microstates
the displacement function
To get a bound F-P system, the momentum Py must be bound to the string.
The F-string can only have modes transverse to y, and must then bend away
from the point r = 0. Questions do rise about the location of the singularity.
For now, let the fundamental string only oscillate in the extended directions.
Therefore the displacement of the string is given by a displacement function
Fi (v) with i = 1, 2, 3, 4.
To get a BPS state, this function is restricted to have only left moving Fi (u) or
right moving modes Fi (v) [26]. Next to that there is the restriction of the total
momentum and the winding number of the string.
The different ways of partitioning the product of the charges over the separate modes described by F~ (v), determine the geometry inside the horizon and
therefore the microstates.
In [39] three different ways to count microstates are given.
oscillations in string theory When setting NL = 0, the constraints
for the string mass and oscillations eq.(3.14) become:
NR = np w
⇒
2
MN
n2p
NR
w2 R02
= 2 0 + 02 +
=
α
R
α02
µ
np
wR0
+
R0
α0
¶2
.
with MN the nongeometrized charge. So the displacement function is
restricted by this condition. The mass of this fuzzball is determined
totally by the charges as we expect from a BPS state:
(5)
M = GN MN = w
(5)
(g 0 )2 α04
(gs0 )2 α03
+ np 0s 0 2 = Qw + Qp ,
0
V
V (R )
(10)
where GN = GN /(2π)5 V 0 R0 . Then Cardy’s formula is used again:
√c
√
D ∼ e2π 6 N = e2π np w ,
with c = 6 (bosonic degrees: 1×4 directions; fermionic: 1/2× 4 directions)
and N = NR .
65
partitions in Fourier modes For large R0 , the small transverse vibrations
can separated in Fourier modes, according to Mathur. Now let the total
momentum be P = np /R, which can be decomposed over several modes
of the string. Each excitation of a Fourier mode k carries energy
and momentum:
ek = pk =
k
2πk
= 0 ,
ls0
Rw
such that P =
X
np
=
mk pk ,
0
R
k
where ls0 = 2πR0 w is the total length of the string; mk is the occupationP
number of mode k. Now the total momentum is equal to both k 2πmk k/ls0
and 2πnp w/ls0 , therefore:
X
mk k = np w .
(4.5)
k
The displacement function can be in the lowest mode(k = 1, mk = np w);
or the highest mode (k = np w, mk = 1). It has a large/small amplitude,
respectively. So this function regulates the different microstates.
p
From the number of partitions in one direction (∼ exp(2π np w/72))
√
the total degeneracy can be calculated (∼ exp(2π np w)).
gas in a box The modes can be investigated as a gas of massless quanta
living in box (string) of length ls0 = 2πR0 w, travelling in one direction.
From here one can formulate the partition function Z for the 4 bosonic
modes and 4 fermionic modes, and calculate several thermodynamic quantities like the entropy:
√
SF −P = 2π np w ,
(4.6)
which is the same entropy as obtained by the other two procedures.
generic state
From the gas of quanta model, one can calculate also other quantities like temperature:
√
T ∼
np w
.
ls0
The average energy of a quantum will be similar to this temperature. This gives
a generic wavenumber:
kg ∼
√
np w .
(4.7)
To localize the horizon, one takes a string with fixed w, np and localize the horizon radius where the string has its maximal displacement for this generic mode.
In the D1-D5 system this is equal at the place where the naive solution becomes
a good/bad approximation for the generic D1-D5 solution outside/inside of this
new horizon.
66
4.3.2
Fields
How does this displacement function change the naive metric eq.(4.3) of the
F-P system? Oscillating F-strings are well known from chiral null geometries
[40] and give this metric:
ds2
¡
¢
Hw−1 (x) −dudv + (Hp (x) − 1)dv 2 + 2Ai (x)dxi dv
+ dx · dx + dz · dz ,
=
Here the same coordinates and compactification manifold as for the naive solution are used. Also other compact manifolds and oscillations in compact
dimension have been investigated in [42]. From the field equations one finds
that Hw and Hp must be harmonic in the extended spatial dimensions:
Hw ∂ 2 Hw (x) = 0 ,
Hp ∂ 2 Hp (x) = 0 ,
∂ i (∂i Aj (x) − ∂j Ai (x)) = 0 .
For one winding, the string would give the next (harmonic) functions:
Hw (x)
Ai (x)
=
1+
= −
Qw
,
| ~x − F~ (v) |2
Hp (x) = 1 +
˙
Qw | F~ (v) |2
,
| ~x − F~ (v) |2
Qw Ḟi (v)
,
| ~x − F~ (v) |2
where Ḟi =dFi /dv.
Since these functions are harmonic, a multi-centered solution can be made:
Hw (x)
=
1+
w
X
s=1
Ai (x)
= −
w
X
s=1
(s)
Qw
,
| ~x − F~(s) (v) |2
(s) (s)
Qw Ḟi (v)
| ~x − F~ (s) (v) |2
Hp (x) = 1 +
w
˙
(s)
X
Qw | F~ (s) (v) |2
,
| ~x − F~ (s) (v) |2
s=1
.
By connecting these different strands, one can make one long string, one bound
BPS-state of the F-P system.
In the large charge limit (w, np >> 1) the SUGRA equations are used to describe
the black branes and for small wavenumber k an approximation can be made:
k
kg
=
√
k
<< 1
np w
⇒ λ=
2πR0 w
>> 2πR0 ,
k
when np and w are of the same order; λ is the wavelength of a mode k.
67
When comparing the amplitude ∆x of one strand of the string with the length
of the strand δx:
˙
∆x ∼| F~ (s) | λ ,
˙
δx =| F~ (s) | 2πR0 ,
δx
<< 1 .
∆x
⇒
For small k neighboring strands give almost the same contribution to the harmonic function, since the wavelength λ is much larger than the length of one
strand. Then the summation can be replaced by an integral:
w
X
s=1
Z
Z
w
→
2πR0 w
ds =
s=0
y=0
ds
dy =
dy
Z
ls0
v=0
dy dv
dv .
ds dy
Fill in:
ds
dy
= (2πR0 )−1 ,
2πR0 w = ls0 ,
(s)
Qw = Qw /w
w
X
⇒
Q(s)
w →
s=1
Qw
ls0
Z
ls
dv .
0
classical solution
In this approximation the fuzzball solution becomes:
ds2
=
e−2Φ(x)
=
Buv
=
Hw (x) =
Ai (x) =
¡
¢
Hw−1 (x) −dudv + (Hp (x) − 1)dv 2 + 2Ai (x)dxi dv
+ dx · dx
+ dz · dz ,
Hw (x) ,
1
− (Hw (x)−1 − 1) , Bvi = Ai (x)Hw−1 (x) ,
2
Z 0
1
Qw ls
dv
,
1+ 0
ls 0
| ~x − F~ (v) |2
Z 0
Qw ls
Ḟi (v)
− 0
dv
.
ls 0
| ~x − F~ (v) |2
Qw
Hp (x) = 1 + 0
ls
(4.8)
Z
0
ls0
dv
˙
(F~ (v))2
,
| ~x − F~ (v) |2
Also calculations on other compact manifolds have been carried out. The naive
F-P system eq.(4.3) is not a microstate in this solution, but at large distance
(| ~x |>>∆x) then | ~x − F~ |−2 ≈| ~x |−2 , the fuzzball solution approximates the
’naive’ solution. The solution approximates the naive solution quite rapidly
outside the region occupied by the vibrating string.
This approximation for small k is called by Mathur classical. The quantized
solution has smeared out to smooth classical geometries. According to ideas in
[38, 41, 45], for these solutions, there will be no information problem since there
is no horizon(see 4.5).
Every F~ (v) function will give an extremal D1-D5 geometry, so every classical
profile of the oscillating string in the F-P system gives a classical D1-D5 geometry. It is expected that every quantum state of the string will give a quantized
D1-D5 metric state.
68
(4.9)
4.4
The D1-D5 fuzzball solution
Here is the D1-D5 solution from [38], although there is a mismatch with the
solution obtained in appendix C. Maybe some field redefinitions hasn’t been
taken into account.
ds2
=
¢
1 ¡
−(H5 (x)H1 (x))− 2 (dt − Ai (x)dxi )2 − (dy + Bi (x)dxi )2
1
1
1
+(H1 (x)H5 (x)) 2 dx · dx + (H1 (x)) 2 (H5 (x))− 2 dz · dz ,
e−2Φ
Cti
Cij
=
=
=
H5 (x)(H1 (x))−1 ,
Bi (x)(H1 (x))−1 , Cty = H1−1 (x) − 1 , Ciy = −Ai (x)(H1 (x))−1 ,
0
(x) + (H1 (x))−1 (Ai (x)Bj (x) − Aj (x)Bi (x)) ,
Cij
(4.10)
with B (1) /C 0(2) as the four-dimensional flat space hodge dual of A(1) /H5 :
dB(x) = − ∗ dA(x) ,
dC 0 (x) = − ∗ dH5 (x) ,
The harmonics H1 (x), H5 (x) and Ai (x) are given by:
H5 (x)
Ai (x)
=
1+
= −
Q5
ls
Z
Q5
ls
Z
ls
dv
0
ls
dv
0
1
,
| ~x − uF~ (v) |2
H1 (x) = 1 +
Q5
ls
Z
ls
dv
0
(uḞ (v))2
,
| ~x − uF~ (v) |2
uḞi (v)
,
| ~x − uF~ (v) |2
where Q5 = u2 Qw , u2 = V 0 R0 /(gs0 )2 Rz0 1 . The fields have become much more
complicated, due to the displacement function. Different F~ generate different
geometries. Mathur and Lunin [39, 44] state that for a generic state the classical approximation cannot be made, and the quantized string generates large
fluctuations in metric: the fuzzball (see figure 4.2 and 4.3).
1(a)
1(b)
Figure 4.1: (a) The classical black hole with a curvature singularity and a horizon; (b) the fuzzball picture : no curvature singularity and the horizon is the
boundary of the generic fuzzball.
69
4.4.1
Example of the displacement function
A good and simple example to illustrate some of the features of the fuzzball is
to take the next displacement function in the F-P system:
F1 = mk cos(kv/wR0 ) ,
F2 = mk sin(kv/wR0 ) ,
F3 = F4 = 0 ,
where by the constraints on F~ (v), mk =np w/k. This is just a string oscillat√
ing in one mode k in two directions. For eq.(4.10) to be valid, k<<kg = np w.
The next functions for the F-P system are obtained:
Qw
m2 kQw
,
Hp (x) = 1 + 2 k02
,
kf
w R
f
Qw m2k
sin θ
, A x3 = 0 ,
sin φ p
kRw
f r2 + m2k
Hw (x) =
1+
Ax1 (x) =
Ax2 (x) =
−
Qw m2k
sin θ
,
cos φ p
kRw
f r2 + m2k
Ax4 = 0 .
(4.11)
where f =r2 + m2k cos2 θ.
Here the coordinates are defined by:
x1
x3
≡
≡
r̃
≡
r̃ sin θ̃ cos φ , x2 ≡ r̃ sin θ̃ sin φ
r̃ cos θ̃ cos ψ , x4 ≡ r̃ cos θ̃ sin ψ
q
r2 + m2k sin2 θ , r̃ cos θ̃ ≡ r cos θ .
For the solution of the D1-D5 system, the solution becomes:
H5 (x) =
Aφ (x) =
Bψ (x) =
Q5
Q1 k
, H1 (x) = 1 +
,
kfu
fu
p
sin2 θ
−mk,u Q1 Q5
, Aψ = Ar = Aθ = 0 ,
fu
p
cos2 θ
−mk,u Q1 Q5
, Bφ = Br = Bθ = 0 ,
fu
1+
where
mk,u
≡
umk ,
Q5
=
u2 Qw ,
fu ≡ r2 + m2k,u cos2 θ ,
Q1 = u2 Qp =
70
u2 m2k,u
Qw .
w2 R02
(4.12)
metric of the example
The metric of the D1-D5 system becomes complicated, totally different from
the naive solution eq.(4.1):
ds2
= −H −1 (dt2 − dy 2 ) + Hfu (dθ2 +
dr2
)
r2 + m2k,u
+ H(r2 + m2k,u fu−2 H −2 Q1 Q5 cos2 θ) cos2 θdψ 2
+ H(r2 + m2k,u − m2k,u fu−2 H −2 Q1 Q5 sin2 θ) sin2 θdφ2
1
− 2mk,u fu−1 H −1 (Q1 Q5 ) 2 (cos2 θdydψ + sin2 θdtdφ)
1
1
+ (kfu + Q1 k 2 ) 2 (kfu + Q5 )− 2 dz · dz
where H ≡
√
1
H1 H5 = [(1 + Q1 k/fu )(1 + Q5 /fu k)] 2 .
For large r >> (Q1 Q5 ):
At large distance from the D1-D5 system, the approximations
q
H∼
(n)
(n)
H1 H5
= (1 +
Q1 k 1
Q5 1
) 2 (1 + 2 ) 2 ,
r2
kr
fu ∼ r 2 ,
r2 + mk,u ∼ r2 ,
~ and B
~
show that the fields approximate the naive fields eq.(4.1) with A
vanishing:
ds2
=
¢
1
(n) (n)
(n) (n) 1 ¡
−(H1 H5 )− 2 (dt2 − dy 2 ) + (H1 H5 ) 2 dr2 + r2 (dθ2 + sin2 θdφ2 + cos2 θdψ 2 )
(n)
1
(n)
1
+(H1 ) 2 (H5 )− 2 dz · dz ,
(n)
(n)
e2Φ
=
H1 (H5 )−1 ,
Cty
=
(H1 )−1 − 1 ,
(n)
Cti = 0 ,
0
Cij = Cij
,
Cyi = 0 ,
(n)
dC 0 = −d ∗ H5
.
Only for k = 1 the naive solution (see 4.2.1) is approximated. For other k, the
solution looks like a naive solution at large distance, but with different charges:
(n)
(n)
Q1 = kQ1 and Q5 = Q5 /k.
71
1
For small r << (Q1 Q5 ) 4
It is stated that the geometries of the D1-D5 fuzzball are all capped. In this
simple, classical example this can be shown explicitly by taking some approximations and rewriting the metric:
2
ds
¸
·
p
(r02 + 1) 2
r02 2
1
02
dt + 02 dy + 02
dr
=
Q1 Q5 −
R02
R
r +1
s
p
¤
£
Q1 k 2
+
Q1 Q5 dθ2 + cos2 θdψ 02 + sin2 θdφ02 +
dz · dz ,
Q5
where
r0 ≡
r
mk,u
mk,u
, ψ0 ≡ ψ − √
y , φ0 ≡ φ − √
t.
mk,u
Q1 Q5
Q1 Q5
This is AdS 3 × S 3 × T 4 . The AdS 3 metric is more clearly obtained by absorbing
R0 into t and y: (t0 , y 0 ) = (t/R0 , y/R0 ). Then the metric is clearly the same as
the one of eq.(2.19) for three coordinates (one periodic) with l2 =1 and with the
− sign, as it should for AdS 2 :
ds2{t0 ,r0 ,y0 } =
p
£
¤
Q1 Q5 −(1 + r02 )dt02 + (1 + r02 )−1 dr02 + r2 dy 02 .
Trouble in the definition of ψ 0 can arise, since both ψ and y are periodic, but
there is a consistent lattice identification of ψ 0 and y, when one fills in the value
for mk,u . The full geometry is therefore flat at spatial infinity, has a throat
approximating the naive geometry for smaller r, and is said to end in a smooth
cap. This particular geometry was derived in [38].
In [42] one can find more general examples like a string also vibrating on the
compact dimensions.
72
4.4.2
Size end entropy of the fuzzball
The naive solution has vanishing horizon. So from this no Bekenstein entropy is
obtained. But the system has a huge number of microstates. Mathur suggests
that a horizon is obtained by coarse-graining over the microstates.
quantized string: the fuzzball
The string is normally quantized. The classical profile is said to be obtained
by partitioning the momentum over a few harmonics with high amplitude and
construct states with a narrow peaked Gaussian wavefunction around a mean
classical string profile.
But from the comparison of the string with the gas of massless quanta (4.3), a
√
generic state has energy in the order of the temperature (ekg ∼ β −1 ∼ np w/ls0 )
and therefore the occupation number becomes:
< mkg >=
1
∼1.
1 + e−βekg
So the generic state cannot be approximated by the classical limit. It is expected
that quantizing the F-P system will give a quantized D1-D5 geometry with large
fluctuations of the metric (δg/g), due to the not sharply peaked wavefunction.
For this generic geometry the term fuzzball is used.
where to place the horizon?
Now the size of the fuzzball depends on the mean value kg , the generic wavenumber. A small k means a large amplitude mk , and therefore a small throat in
D1-D5 system and small travelling time (see 4.5).
To attribute a horizon in the F-P system we look at the generic wavenumber.
It is expected that the distribution of micorstates is sharply peaked around
generic state, like in other statistical system with large charges. The classical
solutions of small wavenumber (k << kg ) are used to estimate the size of the
generic state. The size is said to depend only on the mean value and not on the
fluctuations.
λ=
2πRw
ls
∼√
∼R
k
np w
r
w
np
˙
By filling in λ, Qp ∼ Qw | F~ |2 and eq.(4.4) the amplitude of the string becomes:
˙
˙
∆x ∼| F~ | λ ∼| F~ | R
r
s
w
∼R
np
√
Qp w
∼ α0 ,
Qw np
(4.13)
The solution approximates the naive solution quite rapidly outside the region
occupied by the vibrating string. Here we do place the horizon.
73
Bekenstein entropy
Since there is in general no correlation between the direction of the oscillations,
√
the np w oscillations are randomly distributed over the directions. For the classical SUGRA, both charges are large and leading orders (dipole, quadrupole,..)
cancel.
Therefore
√ we compute the horizon area in the naive metric eq.(4.3), located
at | ~x |= α0 . For this we go to spherical coordinates in the extended space:
dx · dx = dr2 + r2 dΩ23 . The naive Einstein metric of the F-P system:
¤
3
1 £
(n)
ds2E(n) = (Hw(n) )− 4 [−dudv + (HP − 1)dv 2 ] + (Hw(n) ) 4 dr2 + r2 dΩ2 + dz · dz .
Then the horizon area in the limit Hs ≈ Qs r−2 ∼ Qs α0−1 becomes:
3
9
AS 3 ∼ Qw8 α0 8 ,
1
√
p
Qp Qw R0 V 0 α0
√
= (gs0 )2 α04 np w ,
AH ∼
1
AT 4 ∼ Qw2 α0− 2 V 0 ,
p
1
−3
AS 1 ∼ Qp Qw 8 α0− 8 R0 ,
⇒
where again we used the expressions for the charges. Using eq.(3.11), the
Bekenstein entropy will become:
(10)
SB =
A(E)
(10)
4GN
√
gs02 α04 np w √
= np w .
∼
gs02 α04
(4.14)
So when we use these assumptions, we get the microscopic eq. (4.6) and Bekenstein entropy with same dependence on charges and other constants, and of the
same order. Does this make the fuzzball a black hole?
the size of the fuzzball
The area of surface r = constant measured in planck units (lp ) is said to be
invariant under the dualities. Therefore the size of the F-P system and D1-D5
are the same in d=5 Einstein frame:
√
AH ∼ (gs0 )2 α04 np w .
First define the radius and the d=5 planck length:
Ã
R∼
(5)
AH
2π 2
! 13
(5)
1
lp(5) ≡ (GN ) 3 .
,
Now with eq.(4.14) the radius in planck units become:
(5)
1
1
R = (2π −2 GN SB ) 3 ∼ (np w) 6 lp(5) .
This radius is in both the F-P system and the D1-D5 system the same when
expressed in the planck length lp , but according to Mathur not when the radius
is expressed in string length ls . The string would in the F-P system become
large, while in the D1-D5 system it would stay of the order of the planck length.
74
4.4.3
The singularities
curvature singularity
The fuzzball has no geometrical singularities. When the strings sweeps out
in the {x}-directions it cannot be a point singularity at r = 0. The different
metrics are stated to become all capped, so there is no singularity left. Why
would this still be a black hole if it has no horizon or curvature singularity?
Now it still looks like there is singularity at ~x = uF~ (v). For the F-P system,
there is the string, but for the D1-D5 system it is not clear that there should
be this singularity. But it has been proven in [42] that this is only a coordinate
singularity.
Follow the curve that ~x = uF~ (v) describes in the space spanned by {x}.
From a point v0 on the curve, the new coordinate | c | measures the distance
along this curve in the flat metric; the other direction are spanned by (ρ, θ, φ),
˙
sperical polar coordinates. Now | c |≈ u | F~ (v0 ) | (v − v0 ), the functions are
then given by:
H5
≈
H1
≈
Z ∞
Q5
Q5 π
d|c|
,=
,
2
2
˙
˙
u2 ls | F~ (v0 ) | −∞ ρ + | c |
u2 ls | F~ (v0 ) | ρ
˙
Q5 π | F~ (v0 ) |
Q5 π
Q5 π
, A|c| ≈ −
, Bφ = −
(1 − cos θ) .
ls ρ
uls ρ
uls
The separate parts of the metric become:
ds2
→
for (y, ρ, θ, φ)
≈
for (t, | c |)
≈
for ({za })
≈
µ
¶2
ρuls
Q5 π
dy −
(1 − cos θ)dφ
Q5 π
uls
¡
¢
Q5 π
+
dρ2 + ρ2 (dθ2 + sin2 θdφ2 ) ;
ρuls
ρls u 2
−
dt − 2dtd | c | ;
Q5 π
˙
u | F~ (v0 ) | dz a · dz a .
The y-direction gets mixed with the noncompact directions inside the fuzzball.
The (y, ρ, θ, φ) part of the metric, this is said to be the metric near the core of
a Kaluza-Klein monopole; it is smooth when:
2πR = 4π
Q5 π
,
uls
which is satisfied, with our constants. The metric is regular in all parts and
is aymptotic flat. It is said [44] that the geometry is capped before r ↓ 0, so the
branes cannot be located at r = 0
Also the fact that a particle that fall into the throat cannot fall onto the
singularity, but gets reflected [43, 38, 44], supports the idea that there is no
curvature singularity.
75
Flat space
"Throat"
r=0
2(a)
2(b)
Figure 4.2: (a) The naive geometry of extremal D1-D5 (b) the actual geometries; the area of the surface denoted by the dashed line reproduces the microscopic entropy.
stretched horizon
The fuzzball metric has no singularites. So it has no horizon in the sense of
classical black holes. Still we define a horizon to be able to calculate the Bekenstein entropy. The horizon is placed at the boundary of the generic fuzzball;
here the different metrics approximate the naive solution.
The horizon obtained this way looks more like a stretched horizon. But Sen
used this when the curvature becomes string scale and SUGRA isn’t valid anymore. Here the curvature doesn’t become string scale. Wether this generic-state
horizon is still a stretched horizons, is unclear. Lunin [48] and Mathur [39] seem
to not fully agree on this.
why no horizon?
Mathur states that the different microstates cannot have a horizon. Like in
standard thermodynamics, the entropy as a macroscopic property is connected
to number of microstates. Therefore a microstate cannot have entropy itself.
In spite of early attemps to look at small pertubations on the metric of planck
size localized at the singularity or horizon, the right haircut wasn’t found. Next
to that, if every microstate would have a horizon, then it again would have a
Bekenstein entropy. This would mean, they would again have microstates for
themselves. This method will only gain even more entropy. This makes no
sense.
So the horizon is a macroscopic property for an ensemble of different metrics
and can be found after coarse-graining over the microstates.
This idea that the horizon is not just a coordinate singularity, but the boundary
of an object, is supported by other ideas in string theory. In [37] there is the
figure, that when a particle falls into the horizon of a black brane, this is equal
to the description of a string that gets bounded to the brane.
76
4.5
Other concepts in the fuzzball proposal
The details about the concepts introduced in this section, have not been published yet, or go beyond the scope of this thesis. This is just a brief introduction
to interesting concepts introduced mainly by Mathur in combination with the
fuzzball picture.
As already said, in falling matter supports the idea there is no curvature
singularity and that the horizon is a boundary of a finite throat instead of the
point of no return, so it provides ideas about the structure of the fuzzball. Also,
more information about Hawking radiation and a solution for the information
paradox should come out. The in falling matter is low energetic, such that the
back reaction can be ignored, and the BPS state becomes lightly distorted; this
near-extremal black object can be obtained by pertubation from the extremal
case.
traveling time of a particle
Since the metric of different functions F~ (v) have different geometries inside
the hole, a particle falling in will notice this. The different geometries will be
measurable, because uncharged particles falling in, would make the black hole
unstable, and it will start to radiate. This total energy of the radiation should
be of the same size of the in falling particle, unless the charges are changed.
In [43, 38] calculations show that a wavepacket that gets trapped in the
throat, will not fall onto the singularity at ~x = F~ (v), but gets reflected. In
particle description: the particles bounces against the cap
AdS/CFT duality
In the CFT system dual to the D1-D5 system [38], obtained by AdS/CFT
conjecture, something similar happens; a particle falling into the throat of the
AdS 3 ×S 3 ×T 4 geometry of the D1-D5 system has similar features as the energy
of that particle being absorbed by the a string as left and right moving modes;
the SUSY of the CFT is broken, and the near-extremal black hole starts to
radiate: the particle can escape again.
The CFT dual is a 1+1 CFT: a string wrapped N (= n1 n5 ) times around the ydirection in a certain target space (T4N /S N ). Here the string can be partitioned
into component strings. The partition is determined by twist operators σk±± ,
which are associated with component strings with k windings. The CFT state
can be compared with the state of the F-P system:
σk±±
. . . σk±±
|0 >
1
m
(αki11 )† . . . (αkimm )† |0 >
↔
P
with i ki = N . So the wavenumbers of the harmonics of the F-string become winding numbers of components strings. A particle in the D1-D5 system
falling into throat, reflected on the cap, back the beginning of the throat, becomes in the dual CFT energy divided over a left and right moving mode on
a component string, having maximal distance between them on the string, and
meeting each again.
77
For a special classical geometry it is calculated that the time for a particle
to travel up and down the throat is equal to the travelling time for the left and
right moving modes to meet again on the string:
∆tCF T = πRk = ∆tSU GRA ,
k is the winding number of the component string. Remember how larger k,
the deeper the throat, the less the classical limit holds. In the naive SUGRA
solution, the throat is infinite, and therefore the time to escape is much larger.
Also the radiation rates for the two match: the little change of a particle escaping
after a few times bouncing up and down the throat, is equal to the change of
the left and right modes colliding to become a ’free particle’ again.
Now the time that matter is trapped inside the throat of the fuzzball is given
∆t ∼ 1/~, so for classical black holes, this is almost infinite. Then the fuzzball
can be compared with a classical black hole with trapping horizon.
Now Lunin and Mathur conjecture that this AdS/CFT duality holds for all
classical geometries. The breakdown of the duality comes when enough energy
is thrown on the CFT, that every component string is at least excited once in the
lowest possible mode of the component. For large k, there are little components
strings, and the the breakdown is reach soon. This can be compared with D1D5 fuzzball with large k, where the classical limit is not valid anymore. Now
nonlocal information can occur.
density of degrees of freedom
To solve the information paradox in this quantum state, Mathur suggest [41, 45]
that there must be some nonlocal information transfer over large distances.
This means a breakdown of the semiclassical approach of Hawking’s calculation.
Mathur summarizes the conditions for the semiclassical approximation to be
valid and adds a condition:
a spacelike slice, next to the intrinsic and the extrinsic geometry, is specified
by a number of degrees of freedom and this number is conserved.
When the slice stretches (see figure 4.4), for example due to black hole formation,
the density of degrees of freedom drops.
(a)
(b)
(c)
Figure 4.3: (a) An initial hypersurface with a high density of degrees of freedom;
(b) Stretching dilutes these degrees; (c) The degrees of freedom are so sparse
that usual local physics breaks down.
78
When there are more bits of date than degrees of freedom on the slice, the data
has to be transferred to some other place: the nonlocal information transfer.
The maximal length of the throat before the breakdown is:
rmax ∼
V
(d)
GN
,
where V is volume in flat space of a ball with Schwarzschild radius of this
fuzzball; d is the number of extended dimensions (see figure 4.5).
(a)
(b)
Figure 4.4: (a) Flat spacetime in the absence of the branes; (b) The presence
of branes stretches the region marked in (a) to a long throat.
Overstretching means a breakdown of the semiclassical approach and Hawking radiation with no information is invalid here.
Fractionation
Consider the F-string wrapped once around a compact dimension y. An excitation with no netto momentum will have a minimum of energy 1/R+1/R = 2/R.
When the string is wrapped w times, the minimum energy decreases to 2/wRa .
So the larger the charge, the lighter the excitations. For a D1-D5 system with
momentum along the y-direction, the effective size of the bound state of these
fractional degrees of freedom is according to Mathur equal to radius of fuzzball.
Fractionation gives bound states of branes that equal the horizon radius, a fuzz
of brane-antibrane pairs through the whole interior of the black hole. These can
be used for information transfer.
79
Chapter 5
Recent research on the
fuzzball picture
Up till now, the fuzzball for a D1-D5 system is most successful. But this is
normally a null-singularity. So the obtained number of microstates cannot be
compared with the entropy of the old D1-D5 system. So it is a bit arbitrarily
to choose the right place for the horizon such that the Bekenstein entropy and
microscopic entropy match. Therefore this system has the status of a toy model,
waiting to be generalized to other SUGRA solutions.
The D1-D5-P black hole seems to be the first real black hole hole to test the
fuzzball proposal on, since it is closely related to the two charge black hole.
Unfortunaly, there is some difficulty in formulating the such a solution. Again
Mathur and Giusto start with a CFT, related to the black hole via AdS/CFT
correspondence.
Next to that, research should be done on other black hole solutions, and
wether these can also be described in a dual theory through S-dualities and
T-dualities, or through AdS/CFT duality.
Also the effect of higher order curvature corrections are investigated.
Only when a naive extremal black hole solution can be replaced by a fuzzball
solution, this proposal will become a serious candidate for replacing the old
black hole picture. And then one could look for the fuzzball equivalent for
non-extremal black holes.
Therefore this research will probably be of great importance for the future
of this proposal as a whole.
80
F-NS5-P solution
The three charge black hole descibed in 3.6 made of D1-branes, D5-branes and
momentum in the y-direction is dual to the F-NS5-P system:
S Ty Tz1 S Tz1 Tz2 Tz3 Tz4 S
D5 D1 P − − − −
− − −− −→ F P N S5 .
It is expected that some fuzzball picture can be constructed by looking at the
vibrations of the F-string and the NS5-brane.
Like for the D1-D5 system, Mathur and Giusto searched for a CFT dual to
the D1-D5-P system [46, 49] to obtain the three charge fuzzball geometries.
The dual CFT for the D1-D5-P system is a CFT of component strings (dual to
the D1-D5 system), only now added with left moving momentum P along the
components strings.
The geometry dual to the generic CFT state can not yet be made. Only
some subsets are constructed, e.g. with all component strings have one winding.
These are obtained by constructing the geometry for ’inside’ the throat and
’outside’ the fuzzball. These have a domain of overlap and will be made to
match.
These geometries are said to be capped before r ↓ 0 and have no horizon.
Also other concepts like fractionation can be projected on this fuzzball solution.
It turns out that the surface r=constant asymptotes to a constant down the
throat. This would be interesting, since the three naive charge black hole has
already got the right horizon size according to the Bekenstein relation and the
number of microstates. It would be much interesting to see what is behind the
horizon, where the fuzzball solution no longer looks like the naive one.
higher order curvature corrections
In [49] Mathur and Giusto investigate how curvature corrections R4 from both
α0 effects and gs effects do influence the fuzzball. Higher curvature corrections
seems to change some null-singularities into black holes with finite horizon size.
The Bekenstein relation gets modified: A/2GN .
The divergence that would occur for the naive D1-D5 system, seems not to
occur in the fuzzball. In the naive solution this happens because the circle of
the y-direction shrinks to zero when r ↓ 0. But in the case of the fuzzball this
S 1 mixes with the S 3 of the noncompact dimensions of the AdS 3 × S 3 × T 4
geometry of the throat. This saves the day according to Mathur and Giusto.
It would be interesting when in some order f alpha, the naive D1-D5 system
change into a black hole with finite horizon area. Then the D1-D5 fuzzball
system will get another status than the one of a toy model.
81
Chapter 6
Conclusion
The D1-D5 fuzzball is a solution of the SUGRA IIB field equations. Its microstates are determined by displacement functions F~ , that describe the vibration profiles of the winded F-string in the dual F-P system. Different profiles give
different geometries. A lot of technical details make it a rich but complicated
proposal; lengthy calculations are needed to answer some questions properly.
For classical oscillating profiles, the geometries are smooth. The generic state
is not classical; the name ’fuzzball’ comes from the fact that this most probable
microstate is assumed to give large fluctuations in the metric.
The fuzzball has no curvature singularity and no horizon. To obtain the Bekenstein entropy, the horizon is stated to be the boundary of the generic fuzzball.
Outside this boundary, the solution approaches the naive solution. In this rough
approximation, the obtained Bekenstein entropy is of the same order as the microscopic entropy.
The information paradox for a smooth geometry is solved: there is no horizon,
so virtual particle pairs can recombine again and there will be no Hawking
radiation; and the black hole doesn’t evaporate.
For a generic state, nonlocal information transfer can occur according to
Mathur, when a spacelike slice of space-time stretches too much. Also, in falling
matter will give rise to fractional excitations on the branes; these excitations
can have the effective size of the order of the horizon radius; this provides a
vehicle for the information transfer.
For the three charge fuzzball, only subsets of full solution are obtained.
82
The goal Mathur sets himself is to find a solution for the black hole paradoxes.
For this he invented a new picture for black holes. This new picture as it rises
from the work of Mathur, Lunin and Giusto, is a rich proposal, full of interesting
concepts: AdS/CFT correspondence, fractionation, dualities, stretched horizon,
classical/quantized string-geometry duality, density of degree of freedom, the
horizon as a boundary of a string/brane system.
The most striking part of this proposal, is that Mathur claims the fuzzball to
be a black hole without having a curvature singularity or a horizon. Can this
be a black hole?
The naive D1-D5 solution must be replaced, since its number of microstates is
large, but the naive solution has vanishing Bekenstein entropy.
The new solution provides at the level of classical SUGRA an alternative for the
null-singularity. It has a fresh way of dealing with the information paradox and
the entropy paradox. This entropy is obtained by choosing the most probable
microstate, a way of coarse-graining like in other statistical systems.
The fuzzball solution approximates the naive solution at large distance,
where both have black hole geometry. It is said that when particles pass this
new horizon, they are trapped for a time t ∼ 1/~; this looks like a point of no
return from the classical perspective (~ ↓ 0).
For these reasons, the fuzzball solution does not only look like a black hole,
but also provides answers to the entropy paradox and the information paradox.
But a black hole solution at the level of classical SUGRA without a curvature
singularity or a horizon sounds strange and gives rise to a lot of questions. If
this is the correct picture, how does the horizon (and the singularity) get into
black hole solutions of GR and SUGRA? How can the mighty geometry of a
black hole (infinite curvature) become so weak (capped and smooth)?
This solution may have Bekenstein entropy of the right order and with the
right dependence on the charges, but the position of the horizon was determined
by some plausible assumptions within this solution. What is needed, are some
arguments from outside this solution that provide evidence that this position of
the horizon is correct. Now this choice stays a bit arbitrarily and the solution
becomes a black hole because we call it that way.
For the D1-D5 system, there is no good naive black hole description, with a
horizon and Bekenstein entropy.
Before the fuzzball solution can become a serious alternative for a general black
hole, it must provide a solution for a naive black hole with nonzero horizon area,
like the D1-D5-P black hole. Then the horizon areas can be compared. The
obtained results for the D1-D5-P fuzzball are hopeful, but the general solution
hasn’t been found yet.
The fuzzball in the D1-D5 system can be obtained by some simple dualities.
But can other fuzzball black holes be obtained in the same way? Even for the
closely related D1-D5-P black hole it is not clear how to get the general solution.
83
A lot of assumptions are made to get an approximation of the Bekenstein entropy
of the fuzzball. Also other features that make this picture look like a black hole,
are obtained by assumptions. Next to that, a lot of details that are claimed in
the articles of Mathur haven’t been published in detail yet.
At this moment, this doesn’t give a satisfying black hole picture. It might be
that lengthy calculations will make this picture more plausible, but at this moment, it is not a bad idea to focus more on the inspiring concepts, incorporated
by the proposal.
The solution itself is already interesting: a set of complicated geometries that
is governed by rather arbitrarily functions, all with the same charges and mass.
Also, it has a large number of microstates.
The microstates are obtained by AdS/CFT duality. This is an interesting
supplement to the spectacular AdS/CFT correspondence.
Also it is expected that a quantum profile of the F-string, gives rise to large
quantum fluctuations in the metric of the D1-D5 system. So at the level of the
field equations of SUGRA, quantum effects in gravity are expected? Doesn’t
this give rise to new paradoxes? Does this occur more often?
Both fractionation and overstretching of space-time help solving the information paradox. This must be examined more closely, by investigating the
dynamics of black branes.
Mathur states that a microstate of a black hole, cannot have a horizon, otherwise it will have entropy itself. This is a legal question about the status of
the Bekenstein entropy. Do microstates have no horizon; or is the Bekenstein
relation not valid for microstates? And where does the relation come from?
And what is the status of the stretched horizon? Is the fuzzball boundary
also a stretched horizon and does it occur more often when ’averaging’ over
microstates of geometries?
Mathur states firmly that this fuzzball is a black hole. There are some good
reasons for that, but the amalgam of dazzling ideas that the proposal has become, might look like every idea is good enough to save the black hole status of
the fuzzball.
At this moment, the only mature fuzzball solution, the D1-D5 system (and
its dual systems), is most likely a toymodel. The question wether this is the right
black hole picture for a general black hole, cannot be answered at this stage.
The research on the D1-D5-P system will probably be of great importance for
the future of this proposal as a whole.
Therefore I prefer the way Lunin presents this solution [48] above that of
Mathur’s, because Lunin doesn’t write about this proposal as the only right
black hole picture. When the proposal doesn’t survive as a black hole solution,
there are still enough other ideas that look interesting enough to examine in
depth independently of the black hole status.
84
.
85
Appendix A
Differential forms
Gauge fields and potentials like the Maxwell field strength Fµν and the potential
Aµ can be written in the language of forms: F and A. Here the notation from
[21] is used.
A form is an antisymmetric tensor, and therefore independent of the coordinate
frame. The zero form is just an ordinary function; an one-form A is a vector
and can be expanded in the basis of dxµ :
A = Aµ dxµ ,
where Aµ are the components of the vector. For the higher rank forms the
wedge product ∧ is needed. This wedge product is defined by the antisymmetric
tensor product:
dxµ1 ∧ dxµ2 ≡ dxµ1 ⊗ dxµ2 − dxµ2 ⊗ dxµ1 = −dxµ2 ∧ dxµ1 ,
dxµ1 ∧ dxµ2 ∧ dxµ3
≡ dxµ1 ⊗ dxµ2 ⊗ dxµ3 + dxµ3 ⊗ dxµ1 ⊗ dxµ2 + dxµ2 ⊗ dxµ3 ⊗ dxµ1
−dxµ2 ⊗ dxµ1 ⊗ dxµ3 − dxµ3 ⊗ dxµ2 ⊗ dxµ1 − dxµ1 ⊗ dxµ3 ⊗ dxµ2 ,
which is clearly antisymmetric for an odd number of permutations, and
symmetric for an even number of permutations. From this, higher orders can
be derived, by taking all permutations and giving a minus sign for an odd
number and plus sign for an even number of permutations.
Now a p-form is defined by:
1
Fµ ...µ dxµ1 ∧ . . . ∧ dxµp .
p! 1 p
The exterior derivative makes a (p+1)-form out of a p-form:
Fp ≡
dF ≡
1 ∂
Fµ ...µ dxν ∧ dxµ1 ∧ . . . ∧ dxµp .
p! ∂xν 1 p
d2 working on a form, will always give zero, since two partial derivatives are
symmetric in its indices, while the wedge product guarantees antisymmetry for
the indices; therefore the product of these two gives zero.
86
Forms are rather natural to integrate over a manifold M .
Z
Z
M
F(p)
≡
=
1
Fµ1 ...µp dxµ1 ∧ . . . ∧ dxµp
p!
ZM
Z
F1...p dx1 ∧ . . . ∧ dxp =
F1...p dp x .
M
M
This will only give nonzero, when the dimension of the manifold is p. To calculate the volume, F is taken to be the volume form (F1...p = 1). Here the metric
is not required to define an invariant quantity. This is usefull for defining conserved quantities like charges.
A (p+q)-form from a p-form and a q-form is straightforward defined:
(A(p) ∧ B(q) )µ1 ...µp+q ≡
(p + q)!
A[µ1 ...µp Bµp+1 ...µp+q ] ,
p!q!
where the [], mean antisymmetric in these indices. From this we can obtain:
A(p) ∧ B(q) = (−1)pq B(q) ∧ A(p) ,
This relation is needed to proof that a selfdual field strength can give trouble
when formulating an action.
A.1
Hodge duality
The space of independent p-forms on a d-dimensional space-time, is of the same
dimension of that of (d−p)-forms. The map between these form bases is given
by Hodge duality, which is important for electric/magnetic duality:
∗(dxµ1 ∧ · · · ∧ dxµp )
≡
√
−g µ1 ...µp
²̃
µ( p+1)...µd
p!
dxµ( p+1) ∧ · · · ∧ dxµd ,
where the ²̃ is the flat space Cevi-Levita tensor. ² is the curved antisymmetric
tensor, for which the metric tensor is used to raise and lower the indices. The
hodge dual of forms:
∗Gµ1 ...µp
≡
√
−g
µ p+1)...µd
²̃ µ1 ...µp (
Gµ(p+1) ...µd ,
p!
which is used in defining the action of free form fields, by the inner product
of forms over a d-dimensional manifold M :
¢
¡
A(p) , B(p) ≡
Z
M
Z
A(p) ∧ ∗B(p) = p!
M
√
−gAµ1 ...µp B µ1 ...µp dx1 ∧ . . . ∧ dxd .
But remember the convention: Fµ1 ...µn = ∂µ1 Aµ2 ...µn + cyclic permutations;
(F (n) )2 = Fµ1 ...µn F µa ...µn .
87
Appendix B
Charges and couplings of
gauge fields
B.1
Maxwell fields
The best known gauge field is the one from the Maxwell equations. The action
in curved space is:
Z
SEM = − (F ∧ ∗F + A ∧ ∗j) ,
where F = dA, the field strength; ∗j (j = jµ dxµ ) is the source current. Both
are written in language of forms. The Maxwell equations with magnetic current
j become:
dF = j ,
d ∗ F = ∗j .
Through the continuity relation (d ∗ j = 0), the total charge can be written as:
Z
1
Q=
∗j ,
4π Vt
with Vt is the volume at some time t containing the current. Then by the
Maxwell equations, this can be rewritten as:
Z
Z
1
1
Q=
∗F , P =
F ,
4π S 2
4π S 2
where S 2 is a two sphere around the source; P is the magnetic charge.
B.1.1
The charged particle
Now the most interesting current for now is the charged particle:
Z
µ
j = µQ
Ẋ µ
dλ √ δ (4) (xµ − X µ ) ,
−g
γ
where γ is path followed by the particle, parametrized by X µ ; Ẋ µ = dX µ /dλ;
λ is the affine parameter; µQ is the coupling of the particle to the field.
88
The interaction term in the Maxwell action, can be written as an integral over
the worldline of the 1-form:
Z
Z
Z
A ∧ ∗j = µQ
dλAµ Ẋ µ = µQ
A,
(γ)
(γ)
where the first integration is over the whole space-time.
The worldline action of a massive electrically charged particle is the geodesic
action with an extra Wess-Zumino term:
Z
Z
q
µ
ν
S = −m dλ gµν Ẋ Ẋ − µQ dλAµ Ẋ µ ,
which can be used as a source, but also a description of a charged particle
in gravitational/electromagnetic background.
In static spherically symmetric coordinates, the source a r = 0 generates a
electromagnetic field strength and potential:
Ftr =
B.2
Q
,
r2
At =
Q
.
r
Higher rank form fields
In SUGRA higher rank form fields arise. Their sources are p-branes and carry
mass density and charge density. The coupling of the p-brane is given by:
Z
Z
q
Sp = −Tp
dp+1 ζ det(gαβ ) + µp A ∧ ∗j ,
M(p+1)
where Tp is the brane tension; M(p+1) the worldvolume; {ζ} the worldsvolume coordinates; gαβ the pull-back of the metric. Here ∗j of a charged p-brane
will be:
Z
j µ1 ...µp+1 = µp
dX µ1 ∧ . . . ∧ dX µp+1
M(p+1)
δ (d) (xµ − X µ )
p
,
| g(X) |
so the coupling term of np coinciding Dp-branes becomes:
Z
np µp
A(p+1) .
(p+1)
Now the charge density is defined by:
Z
Z
1
∗F(d−p−2) ,
Qp ≡
∗j =
ω(d−p−4) S (d−p−2)
B (d−p−1)
where B (d−p−1) is a capping volume, whose boundary is (topologically)
S
such that the p-brane lies inside this volume and is capped by the
boundary; ω(s) is the surface of a unit s-sphere.
(d−p−2)
89
Then for large distance r in the transverse directions of the p-brane the
potential and its field strength become:
Qp
d−p−3
r
At1...p '
B.3
,
Ftr1...p '
Qp
d−p−2
r
.
Geometrized charge
Now the action of form fields in curved spaces used for SUGRA IIB:
Z
Z
Z
√
1
A(p+1) .
S= 2
dd x −gR − F ∧ ∗F − np µp
2κs
(p+1)
But since the charge is geometrized by the definition above, the coupling isn’t
the same the charge:
Qp ≡ np cp = np κ2s µp .
The fundamental charges cp are summarized in 3.4.1. Also the momentum
charge and winding charge are geometrized:
(10) nP
QP = GN
= 8π 6
R
Qw is obtained by T-duality.
gs2 α04
nP ,
R
90
Qw = QP
R
= 8π 6 gs2 α0 3w ,
α0
Appendix C
S-dualities and T-dualities
for the fuzzball solution
The D1-D5 system is obtained from the F-P system by duality transformations:
S
F P → D1 P
Tz1 Tz2 Tz3 Tz4
−−−→
Tz
S
Ty
S
1
D5 P → NS5 P −→
NS5 P → NS5 F → D5 D1 .
How these transformations change the fields and the constants will be written
down here explicitly.
C.1
Fields
For S-duality, the fields of d=10 SUGRA IIB in string frame transform as:
0
eΦ = e−Φ ,
0
= e−Φ gSµν ,
gSµν
0
= Cµν ,
Bµν
0
= Bµν .
Cµν
For Ty -duality, the fields of d=10 SUGRA IIA/B in string frame transform by
the Buscher’s rules:
0
gyy
=
1
,
gyy
0
Bµν
=
Bµν −
=
gyy e−2Φ ,
Cµ...ναβ
=
Cµ...ναβy + nC[µ...να gβ]y − n(n − 1)
0(n)
Cµ...ναy
=
(n−1)
Cµ...να
− (n − 1)
0
e−2Φ
0(n)
(n+1)
0
gyµ
=
1
Byµ ,
gyy
1
(gµy gyν + Bµy Byν )
gyy
1
=
gyµ ,
gyy
0
gµν
= gµν −
1
(gµy Byν + Bµy gyν ) ,
gyy
(n−1)
0
Byµ
1 (n−1)
C
B|α|y g|β]y ,
gyy [µ...ν|y
1 (n−1)
C
g|α]y ,
gyy [µ...ν|y
where the prime stands for the fields of the obtained dual solution; this is written
down in string frame; µ, ν, . . . means here all directions except the y-direction;
[. . . | . . . | . . .] means antisymmetric and normalized in these indices, except for
indices between lines.
91
The oscillating winded string F P generate the next fields:
ds2
e−2Φ
=
=
Buv
=
Hw−1 [−dudv + (Hp − 1)dv 2 + 2Ai dxi dv] + dx · dx + dz · dz ,
Hw ,
1
− (Hw−1 − 1) , Bvi = Ai Hw−1 ,
2
with u = t + y and v = t − y.
S
→ D1 P
ds2
e−2Φ
=
=
Cuv
=
−1
1
1
Hw 2 [−dudv + (Hp − 1)dv 2 + 2Ai dxi dv] + Hw2 dx · dx + Hw2 dz · dz ,
Hw−1 ,
1
− (Hw−1 − 1) , Cvi = Ai Hw−1 .
2
Tz1 Tz2 Tz3 Tz4
− − − → D5 P
ds2
e−2Φ
=
=
Cuvz1 z2 z3 z4
=
−1
1
−1
Hw 2 [−dudv + (Hp − 1)dv 2 + 2Ai dxi dv] + Hw2 dx · dx + Hw 2 dz · dz ,
Hw ,
1
− (Hw−1 − 1) , Cviz1 z2 z3 z4 = Ai Hw−1 .
2
The six form field C (6) can be transformed into a two form field C (2) by the
Hodge duality of the field strength F (7) ; here the one used by Mathur:
F µ1 ...µd−m =
²µ1 ...µd−m ν1 ...νm (m)
√
Fν1 ...νm .
m! −g
The determinant of the metric and the epsilon tensor are:
√
−g =
1 − 12
Hw ,
2
²tx1 x2 x3 x4 yz1 z2 z3 z4 = 1 → ²uvx1 x2 x3 x4 z1 z2 z3 z4 = 2 .
The three form becomes:
Fijk
=
Fijv
=
3
Hw2 F ijk = Hw2 ²ijkluvz1 z2 z3 z4 Fluvz1 z2 z3 z4 = 2Hw2 ²̃ijkl ∂l Hw−1 = −2²̃ijkl ∂l Hw ,
1 1
− Hw2 F iju = −Hw ²ijukvlz1 z2 z3 z4 Fkvlz1 z2 z3 z4 = 2Hw ²̃ijkl ∂l (Ak Hw−1 ) .
2
According to Mathur’s D1-D5 fuzzball solution, the next fields are obtained by
defining fields with the d=4 flat spatial Hodge duality dC 0 = − ∗(4) dHw and
dB = − ∗(4) dA:
0
Cij = Cij
,
Cti = −Bi ,
Cyi = Bi .
It is unclear where the mismatch with the calculations of this appendix comes
from.
92
S
→NS5 P
ds2
= [−dudv + (Hp − 1)dv 2 + 2Ai dxi dv] + Hw dx · dx + dz · dz ,
= Hw−1 ,
0
= Cij
, Bti = −Bi , Byi = Bi .
e−2Φ
Bij
Tz1
→ NS5 P
ds2
e−2Φ
Bij
= [(Hp − 2)dt2 + Hp dy 2 − 2(Hp − 1)dtdy + 2Ai dxi dt − 2Ai dxi dy]
+ Hw dx · dx + dz · dz ,
= Hw−1 ,
0
, Bti = −Bi , Byi = Bi .
= Cij
The fields don’t change in this step, because gz1 z1 = 1. To prepare the next
duality, a coordinate transformation has been done (u, v) → (t, y).
Ty
→NS5 F
−1
ds2
=
e−2Φ
Bti
=
=
Hp 2 [−dt2 + 2Ai dxi dt + dy 2 + 2Bi dxi dy + 2(Bi Bj − Ai Aj )dxi dxj
−(Bi Bi − Ai Ai )dxi dxi ] + H5 dx · dx + dz · dz ,
Hw−1 Hp ,
−Bi Hp−1 , Bty = −(Hp−1 − 1) ,
Byi
=
−Ai Hp−1 ,
0
Bij = Cij
+ Hp−1 (Ai Bj − Aj Bi ) .
S
→D5 D1
ds2
−1
−1
=
Hw 2 Hp 2 [−(dt − Ai dxi )2 + (dy + Bi dxi )2 ]
e−2Φ
=
+Hp2 Hw2 dx · dx + Hp2 Hw 2 dz · dz ,
Hw Hp−1 ,
Cti
=
−Bi Hp−1 ,
Cty = −(Hp−1 − 1) ,
Cyi
=
−Ai Hp−1 ,
0
Cij = Cij
+ Hp−1 (Ai Bj − Aj Bi ) .
1
1
1
−1
When Hp and Hw are identified with H1 and H5 , respectively, this solution is
almost the one given by Mathur, but not totally. Some parts of the gauge field
obtained here differ by a minus sign with solution published by Mathur: Cti ,
Cty and Cyi .
93
C.2
Constants
For the identification of the harmonic functions some constants will change with
the dualities.
When performing a T-duality, Newton’s constant in the noncompact dimension
b is taken to be invariant:
(10)
(b)
GN =
GN
gs2 α04
=
,
Mb
Mb
where b is number of noncompact dimensions; Mb is the internal, compact
manifold. When this manifold is a torus, and a T-duality is performed in the
(b)
y-direction, gs must also transform to keep the GN invariant:
Ry0 =
1
Ry
⇒
gs0 =
gs
,
Ry
with α0 = 1. For a S-dual theory to have the same spatial infinity, the charge
and radii of compact dimension have to change:
gs0 =
1
gs
⇒
R
R0 = √ ,
gs
Q0 =
Q
.
gs
For the transformation in the fuzzball proposal, this means that the string
strength gs , charge Q, the radius of the y-direction R, the radius of the z1 direction Rz1 and the volume of the four torus V change radically (α0 =1):

gs0


 Qw



 R0


 0
 Rz
1


V0















1/gs0


 Q1 /gs0


p
S 
0
0
→
 R / gs


p
 R0 / g 0
s
 z1

V 0 /(gs0 )2








 Tz1 Tz2 Tz3 Tz4
 − − − −→








 Qw /gs0



p
 R0 / gs0


 p

gs0 /Rz0 1


(gs0 )2 /V 0
Rz0 1 /R0


 Qw V 0 /(gs0 )2

Ty Tz1 
√
− −→ 
 gs0 /(R0 V 0 )

√
 R0 / V 0
z1


(Rz0 1 )2


gs0 /V 0







 Qw V 0 /(gs0 )2




√
 S 
 →  R0 V 0 /gs0





 √


V 0 /Rz0 1




V0
R0 /Rz0 1




 Qw V 0 R0 /((gs0 )2 Rz0 )
1




 S 
p
→
gs0 / R0 Rz0 1 V 0






p 0 0 √ 0


R Rz1 / V



(R0 )2
Mathur says that also the string length ls ∼
from the articles how this happens.
94
√

V 0 /gs0


gs
 
 
  Q5
 
 
 
≡ R
 
 
 
  Rz1
 
 
V
α0 changes, but it is not clear




















 ,






.
95
Acknowledgements
The support I received from friends, family and other people during the writing
of this thesis has been of great importance to me.
First of all, I would like to thank dr. S.J.G. Vandoren of the University of
Utrecht, for being my supervisor. He introduced me to black hole physics and
to Mathur’s proposal. It was encouraging for me to work on such fresh new
ideas.
Besides that, he was so kind to spend so much time working with me on this
thesis, trying to push me in the right direction and pointing out how research
is done and spread within the academic field.
I enjoyed the conversations (about physics and other subjects) with, and the
anti-intellectual statements made by other students: Gijs, Tristan, Gerben, Martijn, Wessel, Alex, Manouk, Michiel, Sietse, Bruce and Jaap.
I would like to thank my boyfriend Jan Carel, my sisters Mara and Gerry, my
brother Charles, my parents Marens and Gerard and the rest of my family, my
friend Ewold, my housemates Hester, Floris, Idsert, Juriaan, Laura, Margonda,
Dieuwertje, for their care, love and support I received from them.
96
.
97
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