İSTANBUL TECHNICAL UNIVERSITY  INSTITUTE OF SCIENCE AND TECHNOLOGY
M.Sc. Thesis  by
Eng. Ahmet Emir GÜMRÜKÇÜOĞLU
Department : Physics Engineering
Programme: Physics Engineering
MAY 2004
GRAVITY ON BRANE WORLDS
İSTANBUL TECHNICAL UNIVERSITY  INSTITUTE OF SCIENCE AND TECHNOLOGY
M.Sc. Thesis  by
Eng. Ahmet Emir GÜMRÜKÇÜOĞLU
(509011051)
Date of submission : 26 April 2004
Date of defence examination: 21 May 2004
Supervisors Prof. Dr. Alikram ALIEV (F.G.E.)
Assoc. Prof. Dr. Neşe ÖZDEMİR
Members of the Examining Committee Prof. Dr. Ömer Faruk DAYI (İ.T.Ü.)
Prof. Dr. Ayşe Hümeyra BİLGE (İ.T.Ü.)
Prof. Dr. Cihan SAÇLIOĞLU (S.Ü.)
MAY 2004
GRAVITY ON BRANE WORLDS
ii
ACKNOWLEDGMENTS
At first, I will give my thanks to Assoc. Prof. Neşe Özdemir who is partly
responsible for my decision in going on in physics and Prof. Alikram Aliyev who
showed me that every abstract mathematical formula can be expressed by simple
words, how ever complex it may be. I would like to express my endless gratitude to
my family who are too many to cite here one by one and were behind me, always
caring in every step I took. I also would like to thank Duygu BALCAN and Tolga
BİRKANDAN for their moral support during the writing of this work.
At last, but not least, I would like to dedicate this work to my partner in life, Suzan
KÖSEOĞLU.
May 2004          Ahmet Emir GÜMRÜKÇÜOĞLU
iii
TABLE OF CONTENTS
ABBREVIATIONS v
LIST OF TABLES vi
LIST OF FIGURES vii
NOTATION AND CONVENTIONS viii
ÖZET ix
SUMMARY x
1. INTRODUCTION 1
1.1. A Journey to the Extra Dimensions 1
1.2. The Kaluza-Klein Picture 2
1.3. Brane World Scenarios 4
1.3.1. Arkani-Hamed, Dimopoulos and Dvali’s (ADD) Brane World Model 5
1.3.2. Randall-Sundrum (RS) Brane World Models 7
1.3.2.1. Randall-Sundrum’s First Brane World Model (RS1)   7
1.3.2.2. Randall-Sundrum’s Second Brane World Model (RS2) 12
1.3.3. Some Other Brane World Models 13
2. A BRIEF REVIEW ON BRANE WORLD GRAVITY 17
3. FOILATION OF THE FIVE DIMENSIONAL MANIFOLD INTO FOUR
DIMENSIONAL TIME-LIKE HYPERSURFACES 20
3.1. Coordinates on the Hypersurface 20
3.2. Unit Normal Vector to the Hypersurface 21
3.3. Induced Metric on the Hypersurface 22
3.4. The Metric in Coordinate System ( yi, z ) 23
3.5. Extrinsic Curvature 24
3.6. The Metric Connections on the Hypersurface 25
3.7. Riemann Tensor on the Hypersurface 26
3.8. Ricci Tensor on the Hypersurface 27
3.9. Scalar Curvature on the Hypersurface 27
3.10. Einstein Tensor on the hypersurface 28
4. BOUNDARY CONDITIONS 29
4.1. The Set-Up 29
4.2. The First Junction Condition 31
4.3. The Second Junction Condition 32
4.4. Junction Conditions in Brane World Scenarios 35
iv
5. GRAVITATIONAL FIELD EQUATIONS ON A 3-BRANE 36
5.1. Effective Field Equations 36
5.2. Evolution Equations 43
6. GRAVITATIONAL FIELD EQUATIONS ON A 2-BRANE 45
6.1. Effective Field Equations 45
6.2. Evolution Equations 46
7. ROTATING BLACK HOLE ON A RS2 BRANE 47
6.1. Black Hole Without Matter on Brane 47
6.2. Black Hole With Gauge Field on Brane 48
8. CONCLUSION 50
REFERENCES 51
CURRICULUM VITAE 56
vABBREVIATIONS
KK : Kaluza-Klein
RS : Randall-Sundrum
GR : General Relativity
ADD : Arkani-Hamed, Dimoploulos, Dvali 
SM : Standard Model
EW : Electroweak
Pl : Planck
vi
LIST OF TABLES 
       Page
Table 1.1. A summary of the non-critical and critical brane-world model
solutions …………………………………………………………
  
16
vii
LIST OF FIGURES 
     Page
Figure 1.1
Figure 1.2
Figure 3.1
: The shape of RS1 model...............................................................
: The shape of RS2 model...................................……....................
: A two dimensional example of a hypersurface family Σz on
M5....................................................................…………………...
 8
 12
 20
viii
NOTATION AND CONVENTIONS 
We use the sign conventions of [40], with a metric with signature (-1,1,1,1).
Unless noted, all the quantities are assumed five dimensional.
Here is the list of frequently occuring symbols.
M5 : Five dimensional manifold
Σz : Four dimensional hypersurface
xA : Coordinates on M5
yi : Coordinates intrinsic to Σz
z : Extra dimension
gAB : Bulk metric
hij : Induced metric
L : Compactification radius
Mpl : Planck scale
MEW : Electroweak scale
dS : de Sitter
AdS : Anti-de Sitter
Λ : Cosmological constant
Kij : Extrinsic curvature
Ni : Shift vector
N : Lapse function
nA : Normal vector
ΓABC : Connections of gAB
λijk : Connections of hij
∇A : Covariant derivative compatible with gAB
Di : Covariant derivative compatible with hij
GNL : Lie Derivative with respect to the shift vector
ix
ZAR DÜNYA ÜZERİNDE GRAVİTASYON
ÖZET
Pek çok modern teoride boyut sayısı alışılagelen 3+1 boyuttan fazladır. Yüksek
boyutlardan efektif 3+1 boyutlu teoriyi elde etme mekanizmaların biri olan “zar
dünya” senaryoları dünyamızı yüksek boyutlu bir “üst-dünya”nın içindeki bir
hiperyüzey olarak betimler. Standart Model etkileşimlerinin zar üzerinde lokalize
olması, ancak gravitasyonun yüksek boyutlara da çıkabilmesi sayesinde
gravitasyonel etkileşimlerin diğer etkileşimlere göre neden daha zayıf olduğuna dair
getirdiği açıklama, 30 TeV mertebesinde enerjilerde yüksek boyut etkilerinin
gözlenebilmesi umutlarını doğurmuştur. Bu etkilerin gözlenmesi, zar dünyaları
öngören sicim teorilerine de deneysel bir destek sağlayacaktır.
Bu çalışmada, ektra boyutlara evrime izin veren genel bir koordinat sistemi için 4+1
boyutlu bir manifoldu, 3+1 boyutlu zamansal hiperyüzeylere dilimleyerek yüksek
boyutlu eğrilik terimlerinin yüzey terimleri cinsinden genel ifadelerini elde ettik.
Daha sonra Israel’in sınır koşullarından faydalanılarak 3+1 ve 2+1 boyutlarda
gravitasyonel alan denklemlerine ulaştık. Bu denklemler, ektra boyutta ivmenin
kaldırılmasıyla Gaussian normal koordinatlarda daha önce bulunmuş sonuçları
vermektedir.
Ayrıca kısıt denklemlerini Randall Sundrum zarı için çözerek, zar üzerinde bir dönen
kara delik çözümü elde ettik.
xGRAVITY ON BRANE WORLDS
SUMMARY
In most of the modern theories, the number of dimensions is larger than the usual
3+1 dimensions. “Brane world” scenarios, which are one of the mechanisms of
dimensional reduction to an effective 3+1 theory, describe our world as a
hypersurface in a higher dimensional “upper-world”. With the help of the
localization of Standard Model interactions on the brane and the free gravitational
fields accessing the bulk, an explanation is found to the weakness of the gravitational
fields compared to others.  This raised hope on observing higher dimensional effects
at order of 30 TeV. The realization of these observations will serve as an
experimental support to the string teories which predict the brane worlds.
In this work, for a general coordinate system which allows acceleration in the extra
dimensions, we sliced 4+1 manifold into 3+1 hypersurfaces and obtained higher
dimensional curvature quantities in terms of the surface quantities. Then applying
Israel’s junction conditions, we reached gravitational field equations in 3+1 and 2+1
dimensions. These equations reduce to known ones when acceleration in extra
dimensions vanishes.
We then solved the constraint equations for a Randall-Sundrum brane, obtaining a
rotating black hole solution on the brane.
11. INTRODUCTION
1.1 A Journey to the Extra Dimensions
Throughout history, the main problem of science was to explain, understand and
explore the nature with as little words as possible or, one may say, “to draw the
boundaries to cover as little area as possible”. The harder the human mind tried to
simplify and restrain nature’s behavior, it always found a way out and confused the
biggest minds. In other words, with every abnormal behavior, the mathematical tools
became more complex and at each step some unexplored behaviors of nature
continued to reveal. Although some behaviors can be summarized by simple
mathematical formulae, the physical insights leading to that formula are mostly
unexpressable by words. The study of extra dimensions takes a step further by being
unimaginable. Even though one could for example, see an eleven dimensional object,
one would only observe the three dimensional projection of it. That is why the spatial
dimensions other than three are referred to as extra. But by the beginning of the
twentieth century, Physics decided to use the calculus of manifolds as a tool, and
everything became more and more complex, yet at the same time, more simple. 
In 1909, inspired by Maxwell’s Electrodynamics and Einstein’s Special Relativity,
Minkowski suggested that time and three space were of the same nature and could be
united as a four dimensional manifold where physics could be expressed in a simple
form [4]. The idea was taken further by Einstein in 1915 [5] with General Relativity,
where he generalized Minkowski’s idea to all four dimensional manifolds.
After the breakthrough created by the revolutionary ideas of Minkowski and
Einstein, other physicists started playing with the number of dimensions of our
universe, hoping to solve long standing problems or to make old theories
mathematically prettier. The mechanism by which one finds an effective four
dimensional theory was more or less the same. These were generally called Kaluza-
2Klein theories, honoring the first physicists who tried this mechanism. The biggest
flaw of this mechanism was the near-impossibility to observe any effect resulting
from the high dimensionality of space. Then in the nineties, a new approach became
popular strengthened by its compatibility with string theories: the brane world
scenarios. Now there is hope, a hope to explore the existence of other dimensions, a
hope to be sure if the route Physics took was the right one. Human mind still moves
on to a sea of complexity without a compass, yet nature gets revealed day by day.
1.2 The Kaluza-Klein Picture1
The idea of an extra spatial dimension was fist introduced by Nordström in 1914 [6]
and independently by Kaluza in 1921 [7], in an attempt to unify general relativity
with electrodynamics in a theory of five dimensions. To be able to recover the four
dimensional physics effectively, Kaluza conceived the size of the fifth dimension as
really small. As for the reason why there is no observational evidence pointing
towards a fifth dimension, both Nordström and Kaluza avoided this question and
simply demanded that the metric is independent of the fifth coordinate. This
assumption is called the cylinder condition which is one of the many cases where
physics progressed without knowing why. In 1926, Klein contributed Kaluza’s work
by giving a physical basis for this condition [8,9]. He showed that the cylinder
condition would arise naturally if the fifth coordinate had a circular topology, thus
the fields would depend on it periodically and could be Fourier-expanded. Also, he
showed that the scale should be small enough, so that the energies above the ground
state would be so high that they would be unobservable.
Kaluza’s technique and Klein’s contribution lead to a new way of exploring the
nature. This technique is called Kaluza-Klein (KK) compactification and can be
applied to any theory with any number of extra dimensions. The manifold formed by
the extra dimensions is taken compact, essentially homogeneous and very small. The
compactness ensures that the spacetime is effectively four dimensional.
A simple case for Kaluza-Klein Compactification is a (4+1) dimensional manifold
where the usual four dimensions of spacetime and the extra spatial dimension are
                                                          
1 The review part is based on [1] and [2]
3represented respectively by coordinates xi (i = 0,1,2,3) and z . To be able to obtain a
four dimensional effective theory at low energies, one must limit the coordinate z
depending on a parameter called compactification radius as z ∈ [0, 2πL]. Now, the
spacetime dimensions are infinite as usual, but the extra dimension is S1. For a free
massless particle, one can write down the Klein-Gordon equation
(5) 0φ =, (1.1)
where (5),  represents five dimensional d’Alembertian. Fourier-expanding φ in
periodic coordinates z shows that the solution of (1.1) is the superposition of these
functions
,
k
k
inzip x L
p n e eφ =G (1.2)
Using this in (1.1) gives
2
2 0
i
i
np p
L
− = (1.3)
From the four dimensional point of view, the second term of this equation is the mass
squared. So the mass of a KK mode is of order L-1, and by setting the
compactification radius small enough, one can truncate to the massless mode in low
energy limit.
Kaluza’s aim was to explore five dimensional gravity without any matter fields.
Taking the Einstein-Hilbert action in five dimensions
2
4 (5)
5 0
1
16
L
S d x dz g R
G
π
π= − −∫ ∫ (1.4)
where ABg  and 
(5) R  are the five dimensional metric and Ricci tensor, respectively. In
order to reach the effective four dimensional action, one must first choose the form
of the five dimensional metric. Generally, the (i,j) part of ABg  is identified with
4ijh (four dimensional metric), the (i,4) part with the four dimensional electromagnetic
potential tensor iA  and the (4,4) part with a scalar field φ . A convenient metric
ansatz would be [1]
2 2 2
2 2
ij i j i
AB
j
h A A A
g
A
κ φ κφ
κφ φ
 + =    
(1.5)
where κ  is a scaling constant used for later convenience. When the cylinder
condition is applied, the integral with respect to the fifth coordinate in (1.4) becomes
trivial. Then expressing the five dimensional Ricci scalar in terms of the four
dimensional one and setting 44 Gκ π= one finds [1]
,
,4 2
2
5 4
1 1
8 4 24
m
mmn
mn
RLS d x h F F
G G
φ φφ φ π φ
 = − − + +   ∫ (1.6)
Comparing (1.6) with (1.4), one sees that 
5
4 2
GG
Lπ= (1.7)
Although we started off to combine gravity with electromagnetism, we ended up
with an additional coupling to the scalar field φ. Kaluza and Klein set φ = 1 although
they were not comfortable about it. Today, the existence of a scalar field is not a
suspicious idea anymore. 
1.3 Brane World Scenarios
The brane world scenarios started as a more realistic alternative to the Kaluza-Klein
picture. In the Kaluza-Klein picture, for an effective four dimensional spacetime, the
size of the extra dimensions must be microscopical. Common sense tells that this
should be about the Planck length (lpl ~ 10-33 cm) although there are works where the
5compactification was made at the length of the electroweak scale [10]. But there is
another mechanism in which, the ordinary matter (SM fields) is trapped to a four dim
ensional submanifold (or  a membrane) of the fundamental space [14, 15]. The size
of the extra dimensions need not be small, in fact they may even be infinite, thus
arising the possibility to observe the extra dimensions. That’s one of the many
reasons why there is so much interest towards the brane world scenarios: a potential
detectability… But the most important feature is that the general idea of the lower
dimensional manifolds (or p-branes in that context) is a natural consequence of the
M-Theory. For example, gauge fields can be localized on D-branes [11]. Although
some realistic brane world scenarios based on M-Theory had been proposed [12,13],
most of the phenomenological models have nothing to do with M-theory’s p-branes,
but there is hope that there will be some counterparts in the fundamental theory.
Hence, the term brane used in this work stands for any four dimensional
hypersurface on which ordinary matter is localized, regardless of the mechanism by
which it is trapped.
So somehow the matter fields (ie. The SM fields) are localized on a (3+1)
dimensional membrane or domain wall, embedded in a (3+1+d) dimensional
manifold. In the brane world picture, the extra dimensions may be large, even
infinite. Although the idea of brane worlds goes back to the sixties [16], a detailed
investigation was made only recently2. 
1.3.1 Arkani-Hamed, Dimopoulos and Dvali’s (ADD) Brane World Model
The model introduced by Arkani-Hamed, Dimopoulos and Dvali (ADD) [18,19] was
the first brane world scenario and intended to solve the hierarchy problem or to put it
in another way, to explain why gravity is weaker than other forces. They started by
neglecting the brane tension and by compactifying the extra dimensions. In a way,
they reintroduced the KK picture, though the size of the extra dimensions L  could be
large. The SM fields may stop four dimensional behavior (according to dynamics on
the brane) much below L , but gravitation becomes multi dimensional just below L .
In the light of recent experiments which verified Newton’s inverse square law at
                                                          
2 see note in [17] for a brief history of brane world scenarios
6distances about 0.2 mm  [22], one can say that the size of the extra dimensions must
be as large as 0.1 mm 3.
This is an opportunity to attack the hierarchy problem. In a theory with d extra
dimensions, the gravitational action can be written as
( ) ( )4 44
4
1   
16
d dd
d
S d xd z g R
Gπ
+ +
+
= − −∫ (1.8)
where G4+d = 1/Md+2 is the (4+d) dimensional Newton’s constant. Note that the multi-
dimensional gravitational scale M is the fundamental mass parameter here, instead of
the four dimensional Planck scale. In ADD model, the graviton zero mode mediates
the four dimensional gravity, hence the wave function is homogeneous over the extra
dimensions. This lets us take the metric independent of the extra coordinates. Doing
the trivial z integration in (1.8), one gets
2
4 (4)
16
d
d
MS V d x h Rπ
+
= − −∫ (1.9)
where Vd is the volume of the extra dimensions. Now taking the four dimensional
gravitational scale as the Planck scale, we can deduce from (1.9) that
2 2d
pl dM M V
+= (1.10)
Now to understand what (1.10) means, one may take the size of all extra dimensions
as L . Then from  (1.10)  one can express the size L in terms of the mass scales as
2
1 2
d
pl
d
M
L
M +
= (1.11)
If L is large compared to the fundamental length 1M − , the Planck mass should be
much larger than the fundamental scale M. This is the breakthrough of the ADD
model. Speculating that there should be only one fundamental scale, they took M to
be the electroweak scale ( )1 TeVEWM ∼  so that the source of the hierarchy between
                                                          
3 Actually until the publishing of [18,19], the inverse square law was established at several  milimeter
7the two scale will be the largeness of the extra dimensions. Using this assumption in
(1.11) one gets
( )
( )
219
17 32
1 2
10  GeV
10  cm
1 TeV
d
d
dL
− +
+ =∼ (1.12)
From (1.12), one sees that if there is only one extra dimension, L has an impossible
value of L ~ 1015 cm, but for 2d = , L has a more acceptable value of about a
milimeter though this distance is still in the range of Newtonian gravity. That is why
astrophysicists and cosmologists exclude the scale M ~ 1 TeV for d = 2, but a more
decent value of  M ~ 30 TeV suggests extra dimensions with the size of one to ten
micrometers. This is another motivation for the experimentalists to explore
deviations from the inverse square law in micrometer range [23-27].
It should be noted that in a more realistic d = 6 choice, the size goes down to 10-12
cm which is still larger than the electroweak scale which is roughly 10-17 cm.
However, one should keep in mind that those figures are obtained by assuming that
all the extra dimensions are of the same size. If some dimensions are smaller than
others then one may observe deviations from the inverse square law for d > 2.
1.3.2 Randall-Sundrum (RS) Brane World Models
1.3.2.1 Randall-Sundrum’s First Brane World Model (RS1)
ADD’s result is remarkable because it opened a door for low energy tests to check
the existence of the extra dimensions. But by solving the hierarchy between gravity
and other forces, they generated another hierarchy between the weak scale and the
compactification scale [28].
1/
1
EWd ML
 (1.13)
                                                                                                                                                                    
range [20,21]. The idea of large extra dimensions inspired the research in [22].
8Randall and Sundrum’s first model (RS1) [28] was another attempt to solve the
hierarchy problem. They started off with two domain walls with opposite brane
tensions and between them they put an AdS5 bulk (see figure 1.1).
The action for this model is
( )
( ) ( )
4 3
4 4
2
vis vis vis hid hid hid
S d x d g M R
d x h V d x h V
π
π
φ
−
= − −Λ +
+ − − + − −
∫ ∫
∫ ∫L L
(1.14)
Where g is the five dimensional metric, hvis and hhid are the four dimensional metrics
on the branes and Vvis and Vhid are constant “vacuum energy” which act as a
gravitational source. The first term of (1.14) is the five dimensional Einstein-Hilbert
action while the others correspond to visible and hidden branes.
The four dimensional metrics on the branes are defined as
( ) ( )
( ) ( )
, 0
,
hid i i
ij AB
vis i i
ij AB
h x g x
h x g x
φ
φ π
≡ =
≡ = (1.15)
By minimizing the action of (1.14), one can find the five dimensional field equations
( )
( )
3
1 1 {
2 4
}
vis i j
AB AB AB vis vis ij A B
hid i j
hid hid ij A B
g R g R gG V h h
M
V h h
δ δ δ φ π
δ δ δ φ
 − − = − Λ − + − −  
+ −
(1.16)
Figure 1.1 The shape of RS1 model. At φ = 0 (hidden brane) the warp factor is
maximum, at φ = π (visible brane) it is minimum.
9The form of the five dimensional metric can be found by requiring four dimensional
Poincaré invariance on the brane. The metric ansatz authors of  [28] chose is
2 2 2 2i j
ij cds e dx dx r d
σφη φ−= + (1.17)
where xj are the usual four dimensional coordinates and φ is the fifth coordinate
which is a finite interval determined by rc. Using this ansatz in the field equations
(1.16) one gets
2
2 3
6
4cr M
σ ′ Λ= − (1.18)
( ) ( )2 3 33 4 4hid visc c c
V V
r M r M r
σ δ φ δ φ π′′ = + − (1.19)
Solving (1.18) and requiring orbifold symmetry φ → -φ  yields
324c
r
M
σ φ Λ= − (1.20)
For (1.20) to have a physical meaning, the cosmological constant must be negative, a
not surprising outcome of  AdS5 bulk.
By considering the metric a periodic function in φ and derivating (1.20) twice, one
can deduce
3
3
24
24
hid visV V M k
M k
= − =
Λ = −
(1.21)
where k is a scale of the order the Planck scale. The final form of the RS1 metric is
then
22 2 2ckr i j
ij cds e dx dx r d
φη φ−= + (1.22)
10
The model introduced, one can attempt to attack the hierarchy problem. Note that in
(1.22) the metrics on the branes are both conformally flat. Let us consider metric
perturbations γij to the flat metric
ij ij ijh η γ= + (1.23)
Using (1.23) in the gravitational action and integrating over the fifth coordinate one
can obtain the four dimensional effective action:
( )
23 4 2
3
2 4
2
2 1
c
c
kr
eff c
kr
S M dx hr R d e
M e dx hR
k
π
φ
π
π
φ −
−
−
= − +
= − − +
∫ ∫
∫
…
…
(1.24)
where R  is the Ricci scalar defined by the metric hij. As the four dimensional
effective scale for gravity is of the order the Planck scale, from equation (1.24), one
can write down:
( )3 22 1 ckrpl MM ek π−= − (1.25)
(1.25) shows that even for large kr, the Planck scale depends only weakly to the five
dimensional gravitational scale. But for the physical masses of SM, things are
different. 
Consider a Higgs field localized on the visible brane, the action will be:
( )224 † 20ijvis vis vis i jS d x h h D H D H H mλ = − − − +  ∫ … (1.26)
where 0m  is the five dimensional mass parameter.
As the visible brane is located at φ = π, the metric on the brane will be
( ) 2, ckriij ijg x h e πφ π −= = (1.27)
11
Combining equation (1.26) with (1.27) and renormalizing the wave function as
ckrH e Hπ→ , one gets
( )22 24 † 20ckrijvis i jS d x h h D H D H H e mπλ − = − − − +  ∫ … (1.28)
The result is not limited to a Higgs field. Generally the effective mass on the brane is
related to the five dimensional mass as:
0
ckrm e mπ−= (1.29)
So, supposing that the bare Higgs mass is of order the Planck scale (~ 1019 GeV), to
be able to observe the mass on the brane at the EW scale (~ 1 TeV), we only need to
set krc ~ 12. This way, there is no very large hierarchy between the fundamental
parameters.
In the argument above, we have set the fundamental scale to the Planck scale, but
this is not mandatory. Consider a coordinate transformation ckri ix e xπ→ . Using this
in the metric of (1.17) then in (1.26),  we see that no rescaling occurs: the Higgs
mass observed on the brane is equal to its physical mass. But the gravitational scale
is different now. Applying the transformation to the effective gravitational action of
(1.24), one sees that the Planck scale is
( )3 22 1ckrpl MM ek π= − (1.30)
This time, the fundamental scale is set to be the EW scale. Again, there is no
additional hierarchy, the scale of krc is of the same order as before. This way,
Randall and Sundrum not only solved the hierarchy problem, their result tells us that
any scale can be fundamental. Also, the non-existence of any extra hierarchy
between  the fundamental parameters raises the importance of their work.
12
1.3.2.2 Randall-Sundrum’s Second Brane World Model (RS2)
The brane world scenario of Section 1.3.2.1 leads to some unexpected results. If  the
gravitational scale is of order the electroweak scale,  we should be able to observe a
five dimensional gravity, though experimentally we know that gravity looks four
dimensional up to two tenths of  a milimeter [22]. Although one can localize gravity
by compactifying the extra dimensions, the RS2 model tackles this problem with an
infinite extra dimension and the solution reproduces Newton’s inverse square law on
the brane [29]. 
The set-up is a little different than the preceding one. In RS1 model, the visible brane
(our world) had negative tension. This ensured an exponential term in scale equations
(1.25) and (1.29). But as the visible brane was at the warp factor’s minimum, gravity
was localized on the hidden one. Forgetting about the hierarchy problem, one may
concentrate on localizing gravity on the brane. So this time, the positive tension
brane is the visible one and the other is brought to infinity. Taking the metric (1.17)
with cr →∞  one gets
22 2k z i j
ijds e dx dx dzη−= + (1.31)
By minimizing the gravitational action, and considering low energy approximations,
the authors of [29] found the KK spectrum of the effective four dimensional theory
Figure 1.2 The shape of RS2 model. Note that there is only one
brane with a positive tension, located at z = 0.
13
and finally deduced a three dimensional gravitational potential due to a point mass
m  on the brane as
( ) 2 211N mr G r r kφ
 = +   (1.32)
(1.32) is the Newtonian potential of four dimensional gravity with Yukawa type
corrections at distances r < k -1 (short distances).
The RS2 model destroyed the prejudice (or lore as the authors stated [29] ) regarding
the compactness and size of the extra dimension. It opened a whole new perspective
on multi-dimensionality and on our place in it.
1.3.3 Some Other Brane World Models
In Sections 1.4.1 - 1.4.3, we reviewed the most important brane world models though
there are two more we should briefly introduce.
The first of these models is an attempt to unify the approach to the hierarchy problem
in RS1 and the localization of gravity on brane in RS2 [30].
In this case, there are two positive non-equal tension branes. The one with larger
tension is called The Planck Brane, and the other, The TeV Brane. Although ordinary
matter is localized on the TeV brane,  gravity is effectively four dimensional in both.
The hierarchy problem is solved just as in [28] when the visible brane is taken as the
TeV brane.
The second model that will be discussed here is actually a collection of models. In
the RS models, the brane cosmological constants are set to zero by (1.21) resulting a
flat metric on the brane. Because of that, those models are called critical brane
worlds. But the recent studies of supernovae claim that the cosmological constant
must be small but positive [32,33]. The unrealistic criticality condition (1.21) is a
consequence of the metric ansatz (1.17), which can be generalized to allow de Sitter
and anti-de Sitter branes [31]
( )2 2 2i jijds a z h dx dx dz= + (1.33)
14
The action for this model is given by
( )4 4(5)14S d xdz g R d xdz h zλ δ
 = − − −Λ − −  ∫ ∫ (1.34)
where λ  is the four dimensional cosmological constant on the brane. The bulk
cosmological constant is taken as Λ = −6k2 [28]. So the solution will satisfy the
boundary conditions on a positive tension brane. The five dimensional Einstein
equations read [34-37]
2
2
2 3 4
a a k
a a a
λ  − − = −  
 
(1.35)
24 4a k
a
− = − (1.36)
where an overdot denotes differentiation with respect to the fifth coordinate.
According to the sign of λ, the solutions of (1.35) and (1.36) will be
( ) ( )10 ,                                     sinh
3
a z kz c
k
λλ > = ± + (1.37)
( )0 ,                                                 kz ca z eλ ± += = (1.38)
( ) ( )10 ,                                     cosh
3
a z kz c
k
λλ < = − ± + (1.39)
where c is a constant of integration.
Imposing Israel junction conditions [38] (see also Chapter 4)
58
3ij ij
GK hπ σ∆ = − (1.40)
15
where Kij is the extrinsic curvature and will be defined in Chapter 3.
Using the general ansatz of (1.33) and imposing 2]  symmetry across the brane one
finds
5
0
4
3z
Ga
a
π σ
+→
= − (1.41)
As we assumed a positive tension, equations (1.37), (1.38) and (1.39) will become
( ) ( )10 ,                                     sinh
3
a z kz c
k
λλ > = − − (1.42)
( )0 ,                                                 kz ca z eλ − += = (1.43)
( ) ( )10 ,                                     cosh
3
a z kz c
k
λλ < = − − (1.44)
The brane tensions for each case will be
0 ,              coth  k c kλ σ> = > (1.45)
                                     0 ,              kλ σ= = (1.46)
0 ,              tanh  k c kλ σ< = < (1.47)
where 54 3Gσ π σ= . If one demands that the factor a to be equal to one on the
brane, one gets
                                       0 ,              sinh
3
k cλλ > = (1.48)
                                       0 ,              0cλ = = (1.49)
16
   0 ,              cosh
3
k cλλ < = − (1.50)
Comparing (1.48) and (1.50) with (1.45) and (1.47), one can deduce the brane
cosmological constants for the non-critical cases as
( )2 23 kλ σ= − (1.51)
The final solutions are summarized in the table below.
Table 1.1 A summary of the non-critical and critical brane-world model solutions. Here, cσ denotes
the tension of the critical brane.
Model Tension
Induced
Metric
The a(z) factor k Cosmological
Constant
RS2 cσ σ=
Minkowski
( )ijη k ze− - 0λ =
Sub-
critical c
σ σ< 4AdS ( )1 cosh
3
c k z
k
λ− − cosh
3
cλ− 0λ <
Super-
critical c
σ σ> 4dS ( )1 sinh
3
c k z
k
λ − sinh
3
c
λ
0λ >
The study of the non-critical branes, and especially of super-critical branes may
provide solutions with effective four dimensional cosmological constant with correct
sign. Although the induced cosmological constant is negative, AdS4 brane in a AdS5
bulk is also interesting and has been investigated [31] for implications for
holography.
17
2. A BRIEF REVIEW ON BRANE WORLD GRAVITY
The result of [28], where gravity is localized on the brane, made a real impact on
theoretical physicists and a search began for answers on what gravitational fields,
due to sources on the brane, look like, on and off the brane.. Here, we will briefly
skim over some important studies on black holes and on four dimensional effective
gravity. 
At first sight, to acquire a black hole on the brane in RS2 model may seem
straightforward. Moreover, replacing the four dimensional flat metric in (1.31) with
any Ricci flat metric, the five dimensional field equations are still satisfied4. The first
thing that comes in mind then is to replace ηij with the Schwarzchild solution, thus
on the brane, one observes a Schwarzschild black hole, which is in five dimensions a
black string in AdS. However, those solutions have Gregory-Laflamme instability
[43] near the AdS horizon. The first RS2-based black hole solution was a
generalization of these black holes and was called the black cigar solution [44].
When this cigar extended all the way down to the AdS horizon, the metric for the
black string was recovered. But the authors walked around the instability by
conceiving that their solution, far from the AdS horizon, looks like a black string, but
has its horizon closed off before reaching the AdS horizon. Their conclusions were
supported by the works of Emparan, Horovitz and Myers, where they calculated
exact black hole solutions, stationary [45] and rotating [46], on a 2-brane by
considering an AdS-C bulk.
There is a number of work on linearized gravity in brane backgrounds, where the
solution of RS2 model (1.32) is inspected more thoroughly and the additions to the
Newton’s Law in order of  r -3 is verified in different approaches [47-50]. Moreover,
there has been works on linearized gravity on alternative backgrounds [51] and
Karch-Randall backgrounds [52] among others, enlarging our knowledge on the
aspects of the gravity on the brane at long distances. But the most fruitful work was
18
due to Shiromizu, Maeda and Sasaki  [42] who wrote down the effective field
equations for a 3-brane (see Chapter 5) and recovered Einstein equations for long
distances, using Gaussian normal coordinates where an acceleration-free condition
is satisfied
0A BAn n∇ = (2.1)
where nA is the unit normal vector to the brane. Using their equations, a solution for a
static spherically symmetrical black hole on the brane was established. Interestingly,
in the absence of a Maxwell field, the solution resembled the Reissner-Nordström
solution [53]. A more detailed study, again using the equations of [42] showed that,
in the presence of a gauge field on the brane, the induced metric is Reissner-
Nordström with two types of charges [54]. Assuming a static, spherically symmetric
metric in form
( ) ( )
2
2 2 2 2
2
drds U r dt r d
U r
= − + + Ω (2.2)
and using it in the 3-brane field equations, they obtained
( ) 2 2 44 2 621 20
G M Q l QU r
r r r
β += − + + (2.3)
where Q is the Maxwell charge and β is the tidal charge resulting from the nonlocal
bulk effects. 
Although the metric form of a static black hole on the brane is now known, there’s
still not much knowledge on what the bulk metric for this case looks like. Linearized
gravity investigations, suggest that a brane-world black hole should be “pancake”
shaped, smoothly going to the bulk then back on the brane. The solution of [54] has
curvature singularities at a finite distances in the extra dimensions.
In the following chapters, we will be constructing effective field equations in a
general coordinate system. This way, we show that some extra terms that may
                                                                                                                                                                    
4 This is discussed for general p-brane solutions in [55] 
19
describe some shaded effects of the bulk (such as acceleration in the fifth dimension)
in [42]  will reveal. We will then find a general solution to the Hamiltonian
constraint equation describing a rotating black hole on the brane. 
20
3. FOILATION OF THE FIVE DIMENSIONAL MANIFOLD INTO FOUR
DIMENSIONAL TIME-LIKE HYPERSURFACES
As a model of a brane world embedded in five dimensional bulk, we chose the
spacelike slicing method used in Hamiltonian formulation of GR (see [39-41] ) in a
coordinate setting in the fashion of Arnowitt, Deser and Misner [56].
We begin with a five dimensional manifold M5 . On M5 we introduce coordinate
system xA with A = 0, 1, 2, 3, 4. Let the line element on M5 be
2 A B
ABds g dx dx= (3.1)
Next, we consider a foilation of the five dimensional spacetime into a family of non-
intersecting time-like hypersurfaces defined by
( ) ConstantAz x = (3.2)
3.1 Coordinates on the Hypersurface
Now, we need to define a coordinate system on  Σz . A priori, the coordinates on Σz
need not coincide with those of  Σz’ . It is, however, convenient to introduce a
relationship as follows.
Consider a congruence of curves γ  intersecting the hypersurfaces  Σz (see Fig. 2.1.1).
Let ( P, Q, R ) , (P’, Q’, R’) and (P’’, Q’’, R’’)  be events on hypersurfaces Σz , Σz’ ,
Σz’’  respectively. Then we may define a coordinate system ( yi , i = 0, 1, 2, 3) on Σz
such that
( ) ( ) ( )i i iy P y P y P′ ′′= = (3.3)
21
This construction defines another coordinate system ( z, yi )on M5. There exists a
transformation between the two coordinates
( ),A A ix x y z= (3.4)
The vector tangent to the curves ( )iyγ
i
A
A
y
xZ
z
 ∂=  ∂ 
(3.5)
is called the evolution vector. The vectors tangent to hypersurfaces zΣ  are
A
A
i i
z
xe
y
 ∂=  ∂ 
(3.6)
3.2 Unit Normal Vector to the Hypersurface
The tangent vectors established, we must define a vector normal to Σz ie. nAnA = 1.
As the value of z changes only on the direction orthogonal to Σz , covector ∂A z is
indeed normal to that hypersurface. The unit normal to Σz is then
Figure 3.1 A two dimensional example of a hypersurface family Σz on M5 . The curves γ
connect the hypersurfaces. Note that the curves do not have to intersect the surfaces
orhogonally.
22
( )( )
A
A A
A
A
zn N z
z z
∂= = ∂
∂ ∂
(3.7)
where the scalar N defined by ( )( ) 1/ 2AAN z z −= ∂ ∂ is called the lapse function. It is
straightforward to see that
0Ai Ae n = (3.8)
3.3 Induced Metric on the Hypersurface
Since the curves γ do not generally intersect Σz orthogonally, ZA is not necessarily
parallel to nA . Therefore, we can decompose ZA in the basis of normal and tangent
vectors. For the normal part, we make use of (3.5) and (3.7)
A
A
A A
x zZ n N N
z x
 ∂ ∂ = =  ∂ ∂  
(3.9)
 and the tangent part is defined as
A i i
AZ e N= (3.10)
where the four-vector N i is called the shift vector.
A A i A
iZ Nn N e= + (3.11)
The displacement on hypersurface Σz can be calculated using (3.1) for constant z
( )2 A BAB z
A B
i j
AB i j
i j
ij
ds g dx dx
x xg dy dy
y y
h dy dy
Σ =
  ∂ ∂=   ∂ ∂  
=
(3.12)
23
where A Bij AB i jh g e e=  is the induced metric (or “first fundamental form”) on Σz .
3.4 The Metric in Coordinate System ( yi, z )
From (3.12), we infer that
i j
AB A B ij A Bg n n h e e= + (3.13)
Using (3.7), we can write down
( )0,0,0,0,An N= (3.14)
To be able to express the five dimensional metric in coordinates ( yi , z ), we first
express dxA in that coordinate as
( )
( ) ( )
A A
A i
i
A A i
i
A A i A i
i i
A i i A
i
x xdx dz dy
z y
Z dz e dy
Nn e N dz e dy
Ndz n dy N dz e
∂ ∂= +∂ ∂
= +
= + +
= + +
(3.15)
Now we use the expression (3.15) in (3.1) to express the line element in coordinates
( yi , z )
( ) ( )( ) ( ) ( )( )
( )( )
( )
2
2 2
2 22
A i i A B j j B
AB i j
i i j j
ij
i j i i
ij i i
ds g Ndz n dy N dz e Ndz n dy N dz e
h dy N dz dy N dz N dz
h dy dy N dy dz N N N dz
= + + + +
= + + +
= + + +
(3.16)
We can express (3.16) in matrix form as
2
ij i
AB
i
j i
h N
g
N N N N
  =   + 
(3.17)
24
The inverse of (3.17) is then
2 2
2 2
1
i j i
ij
AB
j
N N Nh
N Ng
N
N N
 + −  =  −  
(3.18)
Now raising (3.14) with (3.18), we find the components of the unit normal in
contravariant form as
1,
i
A Nn
N N
 = −  
(3.19)
3.5 Extrinsic Curvature
We introduce the 4-tensor
( )
( )
( )
1
2
1
2
A B
ij A B i j
A B B A
A B
n AB i j
K n e e
n n
g e e
= ∇
= ∇ +∇
= L
(3.20)
called the extrinsic curvature or the second fundamental form of the hypersurface. It
describes the extrinsic aspect of the spacetime: the embedding of the hypersurface in
the enveloping spacetime manifold. Using (3.11), we find
( )1
2
A B
ij i j A B B A B A A BK e e Z Z N NN
= ∇ +∇ −∇ −∇ (3.21)
where we defined iA i AN N e= . Projecting the derivatives of the shift vector on the
hypersurface and defining a covariant derivative operator Di compatible with the
induced metric hij , we get
25
( )( )12 A Bij Z AB i j i j j iK g e e D N D NN= − −L (3.22)
In a particular coordinate system where
( )
5
, 0,1, 2,3i ix y i
x z
= =
=
(3.23)
the expression (3.22) reduces to
1
2
ij
ij i j j i
h
K D N D N
N z
∂ = − − ∂ 
(3.24)
From now on, we will use the coordinate system of (3.23).
3.6 The Metric Connections on the Hypersurface
We introduce the metric connections on the hypersurface as
( )1
2
i im
jk j mk k jm m jkh h h hλ ≡ ∂ + ∂ −∂ (3.25)
To be able to calculate the ABg connections in terms of the hypersurface quantities,
let’s first define those
( )1
2
A AM
BC B MC C BM M BCg g g gΓ ≡ ∂ + ∂ −∂ (3.26)
Using the metric expression (3.16) in definition (3.25), we calculated the five
dimensional connections as
i
i i
jk jk jk
N K
N
λΓ = + (3.27)
26
5 1
ij ijKN
Γ = − (3.28)
5 5
5
m
i mi iN D fΓ = Γ + (3.29)
5 5
55 5 5
m
mN fΓ = Γ + ∂ (3.30)
5
5 5
i i i i
j j j jN NK D NΓ = − Γ + + (3.31)
5 2
55 55 5 2
i i i i m i m i
m mN N NK N N D f N D NΓ = − Γ + ∂ + − + (3.32)
where f  ≡ log ( N ). Note that we have used the index 5 to denote the fifth dimension.
3.7 Riemann Tensor on the Hypersurface
We introduce the Riemann curvature four-tensor intrinsic to the hypersurface in
terms of the hij connections
(4) i i i m i m i
jkl k jl l jk jl km jk mlR λ λ λ λ λ λ≡ ∂ − ∂ + − (3.33)
whereas the five dimensional Riemann five-tensor is defined as
A A A M A M A
BCD C BD D BC BD CM BC MDR ≡ ∂ Γ −∂ Γ +Γ Γ −Γ Γ (3.34)
Using the expressions (3.27)-(3.32), we calculated the five dimensional Riemann
tensors in terms of the four dimensional quantities
(4) ijkl ijkl il jk ik jlR R K K K K= + − (3.35)
( )5 mijk mijk k ij j ikR N R N D K D K= + − (3.36)
27
( )
( )
5 5 5 5
2
m m m m
i j imj i jm ij im j mj i
m
i mj j i j i
R N R N N D K K K D N K D N
N K K D D f D fD f
= + −∂ + +
+ − − (3.37)
3.8 Ricci Tensor on the Hypersurface
The Ricci tensor intrinsic to the hypersurface Σz is defined as
(4) (4) mnij imjnR h R≡ (3.38)
Contracting the Riemann tensors of (3.35)-(3.37) with the metric (3.18), we express
the five dimensional Ricci four-tensor as
(4)
5
1 1 2 mij ij i j i j ij ij i mj ijNR R D D f D fD f K K K K KKN N
= − − + − ∂ + −GL (3.39)
( )5 m mi im m i iR N R N D K D K= + − (3.40)
( ) ( )2 255 5 5m m m mn m nm m m mn n mR N R N D D f D fD f N K K N N D K K= − + − + −∂ (3.41)
Where NGL is the Lie derivative with respect to the shift vector and K is defined
as mmK K≡ .
3.9 Scalar Curvature on the Hypersurface
The intrinsic scalar curvature is defined as
(4) (4) mn mnR h R≡ (3.42)
Tracing the expressions (3.39)-(3.41), we calculated the five dimensional scalar
curvatures in terms of the hypersurface quantities as
28
( ) ( )(4) 2 52 2mn m mmn m m NR R K K K D D f D fD f KN= − − − + + −∂GL (3.43)
3.10 Einstein Tensor on the hypersurface
To be able to write down the field equations, we will introduce the four dimensional
Einstein four-tensors
(4) (4) (4)1
2ij ij ij
G R h R≡ − (3.44)
The five dimensional Einstein tensor components are calculated as
( ) ( ) ( )
( )
(4)
5
2
1 1 3
12 2
2
ij ij ij ij ij ij ij i j i jN
m mn m m
i jm ij mn m m
G G K h K K h K KK D D f D fD f
N N
K K h K K K D D f D fD f
= + − − ∂ − − − +
 + + + + + 
GL
(3.45)
( )5 m mi im m i iG N G N D K D K= + − (3.46)
( ) ( )2 2 (4)55 5 1  2m m n mnm n m mnNG N G N K N D K N K R K K= − − + − −GL (3.47)
29
4. BOUNDARY CONDITIONS
In Chapter 3, we took the fifth coordinate z continous, the hypersurfaces Σz are
infinitesimally thin in this approach. In order to write down the effective field
equations on the brane, we must first make sure that physics is continuous there, or
interpret any jump that occurs. The boundary conditions for thin hypersurfaces is due
to Israel [38] and are called Israel junction conditions. The formalism adopted here is
due to Poisson [41].
4.1 The Set-Up
Let Σz partition five dimensional spacetime into two regions M5+ and M5- . Let’s call
the metric on M5±  as gAB± and coordinates as xA±. The problem is to patch the two
parts smoothly on Σz so that the union of metrics gAB±  forms a valid solutions of
gravitational field equations.
Let the coordinates yi on the two sides of Σz be same, be the coordinates on both sides
of the hypersurface. Let’s also choose the unit normal to point from M5- to M5+.
Supposing that in the overlapping region there is a coordinate system xA defined in
the neighbourhood of Σz which is let pierced orthogonally by a congruence of
Figure 4.1 The hypersurface Σz as the boundary between two manifolds M5+ and M5-. Note
that the normal vector’s direction is chosen to point towards M5+.
30
geodesics that intersect it orthogonally. Let also the proper distance along the
geodesic l vanish at the point of intersection. The normal vector is then5
A
A xn
l
∂= ∂ (4.1)
Let the jump across Σz of any tensorial quantity A defined in both M5±  be
[ ] ( ) ( )5 5A A M A M+ −Σ Σ= − (4.2)
As l and xA are continous across Σz , the jump of nA is according to (4.1) is
0An  =  (4.3)
Also, as yi is the same on both sides of Σz , the jumps of the tangent vectors on the
hypersurface according to (3.6) are
0Aie  =  (4.4)
We will use the language of distribution in the fashion of [41]. We introduce the
Heaviside distribution Θ( l ) defined as
( )
1 ,  for 0
0 ,  for 0
indeterminate , for 0
l
l l
l
+ >Θ = < =
(4.5)
and have the following properties
( ) ( ) 0l lΘ Θ − = (4.6)
( ) ( )2 l lΘ = Θ (4.7)
                                                          
5 This is actually the acceleration-free condition (2.1). This condition holds at least in the
neighbourhood of the hypersurface [42].
31
( ) ( )d l l
dl
δΘ = (4.8)
where δ ( l ) is the Dirac distribution. Also note that the product Θ( l )δ ( l ) is not
defined as a distribution.
4.2 The First Junction Condition
The definitions done, we begin by expressing the metric ABg  in the overlapping
coordinates as a distribution valued tensor
( ) ( )AB AB ABg l g l g+ −= Θ +Θ − (4.9)
From (4.9) we will go as far as Einstein tensor in the coordinates xA . The next step
on this track is expressing the connections in the overlapping region, for which we
first calculate
( ) ( ) ( ) ( )( )
( ) ( ) ( ) [ ]
C AB C AB C AB AB AB C
C AB C AB C AB
g l g l g l g l g n
l g l g l n g
δ δ
δ
+ − + −
+ −
∂ = Θ ∂ +Θ − ∂ + + −
= Θ ∂ +Θ − ∂ + (4.10)
in which, we used (4.1) and (4.8). Note that the last term in (4.10) is singular and the
Christoffels will be problematic for they will have terms proportional to Θ(l)δ (l). To
be able to eliminate it, we must impose [ ] 0ABg = , defined only in the overlapping
coordinates. To generalize this, we will make use of (4.4)
[ ] 0
[ ] 0
A B
AB i j
ij
g e e
h
=
= (4.11)
The coordinate independent expression (4.11) is the first junction condition and is a
consequence of a well-defined geometry of the hypersurface.
32
4.3 The Second Junction Condition
In the preceding section, we set the metric continuous in the overlapping coordinates.
We will see what conditions must one satisfy in order to get a continuous curvature
across Σz . Now, using (4.10), one can express the connections in the xA coordinates
as
( ) ( )A ABC BC Cl l+ Α−ΒΓ = Θ Γ +Θ − Γ (4.12)
To express the Riemann tensors, we first have to derivate (4.12)
( ) ( ) ( ) ( )
( ) ( ) ( )
A A A A A
D BC BC D D BC D BC D BC
A A A
D BC D BC D BC
l n l l n l
l l l n
δ δ
δ
+ + − −
+ −
∂ Γ = Γ +Θ ∂ Γ − Γ +Θ − ∂ Γ
 = Θ ∂ Γ +Θ − ∂ Γ + Γ 
(4.13)
Using (4.13), the Riemann tensor is found as
( ) ( ) ( )( )A A A A ABCD BCD BCD BD C BC DR l R l R l n nδ+ −    = Θ +Θ − + Γ − Γ    (4.14)
The last term in (4.14) implies a discontinuous curvature across the hypersurface. We
will investigate further, and if possible, eliminate it. We first try to express the jump
of the connection. If the metric in the overlapping coordinates is continuous across
Σz, its tangential derivatives are also continuous. So, the jump of the derivative of the
metric should be normal to the hypersurface
,AB C AB Cg nκ  =  (4.15)
where ABκ  is the jump of the metric’s derivative in the normal direction. Now, the
jump of the connection expressed in terms of ABκ  is
( )12A A A ABC B C C B BCn n nκ κ κ Γ = + −  (4.16)
Using this in (4.14) yields
33
( ) ( )
( )( )12
A A A
BCD BCD BCD
A A A A
D B C C B D BD C BC D
R l R l R
l n n n n n n n nδ κ κ κ κ
+ −= Θ +Θ −
+ − − + (4.17)
Contracting indices A and C in (4.17) gives the Ricci tensor
( ) ( ) ( )( )M NAB AB AB MA B MB A A B ABR l R l R l n n n n n nδ κ κ κ κ+ −= Θ +Θ − + + − − (4.18)
where AAκ κ≡ .
Contracting (4.18) with the overlapping region metric (4.9) yields the scalar
curvature
( ) ( ) ( )( )MMNR l R l R l nδ κ κ+ −= Θ +Θ − + − (4.19)
Finally, we find the Einstein tensor as
( )
( )( )
( )
1
2
AB AB AB
M M M N
MA B MB A A B AB AB MN AB
G l G l G
l n n n n n n g n n gδ κ κ κ κ κ κ
+ −= Θ +Θ −
+ + − − − + (4.20)
Using the five dimensional Einstein field equations we obtain an expression for the
stress-energy tensor
( ) ( ) ( )AB AB AB ABT l T l T lδ τ+ −= Θ +Θ − + (4.21)
where τAB is defined as
( )25 12 M M M NAB MA B MB A A B AB AB MN ABn n n n n n g n n gκ τ κ κ κ κ κ κ≡ + − − − + (4.22)
where 25κ is the constant of proportionality in five dimensional field equations. Note
that the last term in (4.21) is the stress-energy tensor resulting from a thin layer, in
34
other words, the stress-energy tensor of the hypersurface is τAB. Noting that
0BABnτ = , we conclude that τAB  is tangent to the hypersurface
i j
AB ij A Be eτ τ= (4.23)
Using this in (4.22) yields
2
5
1 1 1
2 2 2
1 1
2 2
A B M N MN
ij AB i j ij MN ij MN
A B kl M N
AB i j ij MN k l
e e h n n h g
e e h h e e
κ τ κ κ κ
κ κ
= − − +
= − +
(4.24)
where we have used (3.13).
On the other hand, we have
[ ]
( )( )12
C
B A AB C
C C
AB C A B B A
n n
n n nκ κ κ
 ∇ = − Γ 
= − +
(4.25)
Projecting (4.25) on the hypersurface, we get the jump of the extrinsic curvature
across Σz 
[ ] 1
2
B A B A
ij B A i j AB i jK n e e e eκ  = ∇ =  (4.26)
Comparing (4.26) with (4.24), we express the stress-energy tensor of the
hypersurface in terms of the jump of the extrinsic curvature
[ ]( )2
5
1
ij ij ijK h Kτ κ  = − −  (4.27)
To have a continuos curvature across the boundary then means to have a non-
jumping extrinsic curvature on the hypersurface. Setting
35
[ ] 0ijK = (4.28)
makes sure that there is no discontinuity at the hypersurface. The equation (4.28) is
called the second junction condition.
4.4 Junction Conditions in Brane World Scenarios
For an infinitesimally thin brane, the first condition (4.11) can be applied, as the bulk
should be a valid solution of the field equations. But in the context of brane world
scenarios, one has a non-zero brane tension and localized matter fields, and the
second condition (4.28) is not applicable. Nevertheless, the result of (4.27) will serve
us well in the task of expressing the effective four dimensional field equations. As
we will see, it is possible to uniquely determine the extrinsic curvature in terms of
the energy momentum tensor on the hypersurface, by imposing 2]  symmetry [42].
But first, let’s reverse the equation (4.27) and determine the jump in the extrinsic
curvature in terms of the stress-energy tensor by tracing it, then placing [K] in (4.27)
again. We find
2
5
1
3ij ij ij
K hκ τ τ   = − −     (4.29)
If we impose 2]  symmetry with the hypersurface as the fixed point, the normal
vector will change direction from the hypersurface point of view, because it always
points towards M5+. In M5- , it points into the hypersurface, whereas in M5+ it points
out of the hypersurface. Using the definition of the extrinsic curvature, we deduce
2
5
1 1
2 3ij ij ij ij
K K hτ τκ
+ −  = − = − −   (4.30)
In the next chapter, we will express (3.45) in terms of the stress-energy tensor, and
determine what shape the field equations take.
36
5. GRAVITATIONAL FIELD EQUATIONS ON A 3-BRANE
5.1 Effective Field Equations
Now having every tool that we need, we will obtain the more general ( than [42] )
field equations. Note that the choice (2.1) is identical to setting
2 1 0iN N− = = (5.1)
Using the equations we calculated in Chapter 3 for dimensional reduction from five
to four dimensions, we find the effective field equations on a 3-brane. To do this, we
first express the intrinsic Einstein tensor in terms of the bulk stress-energy tensor.
Inspecting (3.45), one can see that it contains derivatives with respect to the normal
coordinate which, along with the Lie derivative with respect to the shift vector, can
be written as a normal vector Lie Derivative. Those terms contain the bulk’s effects
on gravity on the brane and we express them in a simpler and cleaner form.
Among equations (3.35)-(3.37), only (3.37) involves terms describing the evolution
into the fifth dimension. We have to replace it with four dimensional quantities on
the brane. We note that
5 5
A C B D
j l ABCD j lR R Z Z e e= (5.2)
which, by means of  (3.11) can be written as
( )25 5 5 5k i kj l jl jkl lkj ijklR N E N R R N N R= + + − (5.3)
where A C B Djl ABCD j lE R n n e e= .
One can also use (3.36) to present (5.3) in the form
37
( )25 5 5 k kj l jl jkl k jl j klR N E R N NN D K D K= + − − (5.4)
Comparing (5.4) with (3.37), we obtain
( )51 1mjl jl jl j ml l jNK K E K K D D NN N− ∂ − = − +G L (5.5)
We also note that
5
A B C D
jkl ABCD j k l
i
ijkl jkl
R R Z e e e
R N NB
=
= +  (5.6)
where A B C Djkl ABCD j k lB R n e e e= .
Comparing (5.6) with (3.36) we see that
[ ]2jkl l k jB D K= (5.7)
It is also worth to note that
5
A k A
i iA ik iAG G Z G N NG n= = + (5.8)
comparing (5.8) with (3.46) we see that
A m
iA m i iG n D K D K= − (5.9)
and finally note that
2
55 2
A B i A k i
AB iA ikG N G n n NG N n G N N= + + (5.10)
From equations (5.10) and (3.47) we conclude that
( )(4) 212A B imAB imG n n R K K K= − − + (5.11)
38
From equation (3.43) for scalar curvature, we have
( ) ( )(4) 251 1 12 im mim mN K R R K K K D D NN N∂ − = − − − −GL (5.12)
Taking into account that
( ) ( )5 51 1 2ik ik ikN NKh h K K KKN N∂ − = ∂ − +G GL L (5.13)
and combining (5.5) with (5.12), we obtain
( )( ) ( )
( )
(4) 2
5
1 1
2
12
ms m
ik ik ik ik ms i mkN
m
ik i k ik m
K Kh E h R R K K K K K
N
KK D D N h D D N
N
− ∂ − − = + − − − −
+ + −
G L
(5.14)
Next, substituting (5.14) into expression (3.45), we transform it into the form
(4) (4)1 1
2 2
m
ik ik ik ik ik i mk ikG G E h R h R K K KK
 = − − − − +  
 (5.15)
This equation is dimension independent.
Using the decomposition of the Riemann tensor in d dimensions
( ) ( )( )[ ] [ ] [ ]2 2  2 1 2ABCD ABCD A C D B B C D A A C D BR C g R g R R g gd n n= + − −− − − (5.16)
for d = 5, we pass from ijE  to ijE  . As a result, we have
( )(4) 21 1 13 4 2 mnij ij ij ij ij mnE E R h R h R K K K = + + − − +   (5.17)
where A C B Dij ABCD i jE C n n e e= .
39
Let us write down the 5D Einstein field equations
2
5 5
1
2AB AB AB AB AB
G R g R T gκ= − = −Λ (5.18)
where 5Λ  is the bulk cosmological constant and
( )(5) AB AB ABT T zτ δ= + (5.19)
where (5) ABT  is any energy-momentum tensor of the bulk. Thus, in general, instead of
(5.18), we have
( )2 (5)5  AB AB AB ABhG g T zgκ τ δ5
 = −Λ + +   
(5.20)
where in some cases, ABτ  can be presented as
0
AB AB AB
A
AB
A B
ij i j AB ij ij
h S
n
e e h S
τ λ
τ
τ τ λ
= − +
=
= = − +
(5.21)
Next, we return to the equations (3.45)-(3.47). We have
( )( )
( )
5
2
2 (5)
5 5
1 2 3
1 2     
2
     
m
ik ik ik i k i mk ikN
ms m
ik ms m
ik ik ik
G K h K D D N K K KK
N
h K K K D D N
N
hh T z
g
κ τ δ
 − ∂ − − + + − 
 + + +  
 = −Λ + +   
GL
(5.22)
2 (5)
5  
m B
m i i iBD K D K T nκ− = (5.23)
( )(4) 2 2 (5)51  2 mn A Bmn ABR K K K T n nκ5− + = Λ − (5.24)
40
Now, going on the brane and substituting (5.14) into (5.22) and (5.24), we have
( )(4) 2
2 (5) (5) (5)
5 5
1
2
1 1         
3 3
ms m
ik ik ms i mk ik ik
A B
ik ik ik AB
G h K K K K K KK E
h T h T n n Tκ
+ − + − +
  = Λ + + −    

(5.25)
It may be useful to introduce the traceless tensor ikW  as
1
4ik ik ik
W E h E= −  (5.26)
where
( )(4) 21  2ik msik msE h E R R K K K= = − − +  (5.27)
Then (5.25) can be written in the following form
( )2 (5) (5) (5)5 5
1 1
2 2
1 1     3 
2 4
m ms
ik i mk ik ms ik ik ik
A B
ik ik ik AB
G K K h K K K K h K W
h T h T T n nκ
   + − − − +      
 = − Λ + − −  
(5.28)
where
( )51 1 14 2
1 1
4
m
ik ik ik i k ik i mkN
ms m
ik ms m
W K Kh D D N KK K K
N
h K K D D N
N
  = − ∂ − − + − +    
 + +  
GL
(5.29)
To relate the tensor Wik to Eik , one can use (5.17) which can be put in the form 
ik ik ikE W U= + (5.30)
where 
41
( )(4) 2
2
(5)  (5)5
1 1  
3 24
1
3 4
ms
ik ik ik ms
ms
ik ik ms
U R h R R K K K
T h h Tκ
= − + + − +
 = − −  
(5.31)
We see that Uik = 0 if the bulk energy-momentum tensor vanishes and Wik coincides
with Eik . 
Next, we shall discuss the conservation equations. We begin with
2 (5) 2
5 5 
m B
m i i iB iD K D K T n Jκ κ− = = (5.32)
Substituting into this equation (4.30), we obtain
2mm i iD Jτ = − (5.33)
 It follows that there is exchange of energy flux between the brane and the bulk.
When Ji = 0, we arrive at the conservation equation for matter on brane. On the other
hand, calculating the divergence of (5.28), we obtain the equation
( )2 25 51 22i mi i i iik kim k i k ik ikD W K B K J KJ h D D Uκ κ ρ= − − + − (5.34)
where (5) A BABT n nρ = , or substituting into this equation, the relation (4.30), we
obtain its alternative form
( ) ( )45
2
5
1 1 12
4 3 3
1 3
2
i mi i i i i
ik k mi i mk k k i i k k
i
k ik
D W D D D J
D D U
κ τ τ τ τ δ τ τ τ δ τ
κ ρ
  = − + − + −    
+ −
(5.35)
We see that when the bulk energy momentum tensor vanishes, the divergence of Wik
is completely determined by the matter distribution on the brane. We have
42
( )(4) 2 45 5 51 32ik ik ik ik ikG h W Uκ ρ π κ= − Λ − − − + (5.36)
where
21 1 1 1
4 3 2 3
m ms
ik i mk ik ik mshπ τ τ ττ τ τ τ  = − − − −     (5.37)
Supposing that 
ik ik ikh Sτ λ= − + (5.38)
where λ  is the brane tension, we have, instead of (5.36), the following equation
(4) 2 4
4 5 3ik ik ik ik ik ikG h S W Uκ κ π= −Λ + + − − (5.39)
where,
4 2 2
4 5 5 5
1 1
2 6
κ λ κ ρ Λ = Λ + −   (5.40)
2 2
4 5
1
6
κ λκ= (5.41)
21 1 1 1
4 3 2 3
m ml
ik i mk ik ik mlS S SS h S S Sπ     = − − − −         (5.42)
In the absence of the bulk energy momentum, equations (5.39)-(5.42) coincide with
the result of  [42].
On the other hand, if the pressure on the brane ρ  is constant that 4Λ  can be thought
of as 4D cosmological constant with contribution from bulk energy-matter. However,
if the brane matter tensor Sik = 0, and 4 0Λ = , we have equations
43
( )(4) 3ik ik ikR W U= − + (5.43)
and
( )3 0i ik ikD W U+ = (5.44)
If we denote the 4D general relativity energy-momentum tensor
( )2 (4)4  3ik ik ikT W Uκ ↔ − +  we see that a stationary general relativity solution with
traceless energy momentum tensor gives rise to non-vacuum brane-world solution in
five dimensional gravity with traceless brane-on components of the bulk energy
momentum tensor.
5.2 Evolution Equations
The tensor Wij of (5.36) is not freely specifiable, but its divergence is constrained by
matter terms. This means that our effective field equations are not closed. To
overcome this we will find the evolution equations describing the evolution of Wik in
the bulk.
The five dimensional Bianchi identities [ ] 0A BC DER∇ =  reduce to four independent
equations
[ ] 0i jk lmR∇ = (5.45)
[5 ] 0jk lmR∇ = (5.46)
[ ] 5 0i jk lR∇ = (5.47)
[ 5] 5 0i j lR∇ = (5.48)
44
These equations give
(4)
[ ] 0i jk lmD R = (5.49)
( ) (4) (4)5 [ ] [ [ ]]
[ ] [ ] [ ]
 2  2 2
             2 2 2 0
m
ijkl ijm k l i ij l kkl jN
kl j i j l k i i k l j
R N R K ND B B D N
B D N K D D N K D D N
∂ − + + +
+ + + =
G  

L
(5.50)
(4)
[ ] [ ] 0
m
i kj l i kj mlD B K R+ = (5.51)
( )5 [ ] [ ]
(4)
[ ]
2 2
                  2  0
m m
jik j i k k jim km j iN
m
k i j ijmk
B ND E NK B NB K
E D N R D N
∂ − + − +
+ + =
G    

L
(5.52)
Using the 5D field equations (5.18) in (5.52) we get
( )
( )
(5)
5 [ ] [ ]
2 2 2
[ ] 5 [ ] 5 [ ] 5 [ 5 ]
2 2
5 [ ] ] [ ] [ ] 5 [ ]
2
5 [
2  2
2 2 2   2
3 3 3
1 1   2 2
2 3 6
5   
6
m m m
jik j i k k jim imjk km j iN
m
k i j m i j k i j k k j iN
m
k i j j k i j k i j m k j i
k i
B ND W NK B C D N NB K
W D N N J K h N J K h J
N h D D T U D N h U D N Th D N
h D
κ κ κ
κ ρ κ
κ ρ
∂ − + − + +
+ + + + ∂ −
 + − + + +  
+
G
G
L
L
] 0j N =
(5.53)
Again, using (5.18) in (5.50), we get
( ) ( ) ( )
( )
( )
2 (4)
5 5 5 5
( )
[  | | ]  [ ]  [ ] ( )
2 2
5 5
1 13  
2 3
3 2
2 2 2
1
4 6 2
mk
ij ij ij mikjN N N
m m m km
j im i j m j im ik jm ij mk
k k k k
k j i i j k j i k i j k ij
ij ij ij ij
W U h T NK R
NK U NK W NK W NK K K K K
ND B B D N B D N K D D N NKW
N Nh K K T Kh K
κ ρ
ρκ κ
 ∂ − = − ∂ − − ∂ − + +  
+ + + + −
+ + + − −
− + + +
G G G
  
L L L
( )56 ij ijN K Kh + Λ −
(5.54)
Those two evolution equations, along with (5.36) form a closed system of equations.
45
6. GRAVITATIONAL FIELD EQUATIONS ON A 2-BRANE
6.1 Effective Field Equations
The equations calculated in Chapter 3 are actually independent of the number of
dimensions. Using those, we repeated the method of Chapter 5 for a 2-brane
embedded in a four dimensional manifold, and found the effective three dimensional
field equations. Note that only in this chapter, all quantities are assumed four
dimensional except where indicated.
The gravitational field equations on a 2-brane are
(3) 2 4
3 3 4 2ik ik ik ik ik ikG h S W Uκ κ π= −Λ + + − − (6.1)
where
2 2 2
3 4 4 4
1 3
3 16
κ ρ κ λ Λ = Λ − +   (6.2)
2 4
3 4
1
8
κ κ λ= (6.3)
21 1 1 1
4 2 2 2
m mn
ik i mk ik ik mnS S SS h S S Sπ   = − − − −     (6.4)
2 (4) (4)
4
1 1
2 3
mn
ik ik ik mnU T h h Tκ  = − −   (6.5)
46
( )4
1
3
1 1 2
3 3
1 1
3
ik ik ik
m
ik ik i k ik i mkN
mn m
ik mn m
W E h E
K h K D D N KK K K
N
h K K D D N
N
= −
  = − ∂ − − + − +    
 + +  
G
 
L (6.6)
6.2 Evolution Equations
To make the three dimensional field equations closed, we repeat the calculations of
Section 5.2 for a 2-brane thus finding
( )
( )
4 [ ] [ ] [ ]
2 2 2
[ ] 4 [ ] 4 [ ] 4 [ 4 ]
2
4 [ ] ] [ ]
2 2 2
              2
2 1              2
3 2
              2
m m m
jik j i k k jim km j i km j iN
m
k i j m i j k i j k k j iN
m
k i j j ijmk k i j
k
B ND W NK B NB K NB K
W D N N J K h N J K h J
N h D D T C D N U D N
h
κ κ κ
κ ρ
∂ − + − + +
+ + + + ∂ −
 + − + +  
+
G
G
L
L
( ) 2[ ] 4 [ ]1 43mi j m k i jU D N T h D Nρ κ+ −
(6.7)
and
( ) ( )
( )
4 [    | | ] 4
   [ ]    [ ] ( )
2
4 4
2 2
4 4
4
2 2 2
2 2 2
2
3
4 1 1
3 4 3 2
1
3
k mn m
ij k i j ij imjn i jmN N
m k k m m
i jm j i k i j k m i j i mj
mn
mn ij ij N
ij ij ij ij
W ND B U NK R NK U
NK U B D N B D N NW K NKK K
NNK K K NKW
N NK Kh T K Kh
NK h
κ ρ
κ ρ κ
∂ − = − ∂ − + +
+ + + + −
− − − ∂ −
   − + + +      
− Λ
G G
G

 
L L
L
ij
(6.8)
47
7. ROTATING BLACK HOLE ON A RS2 BRANE
We will, using the approach of [54], solve the constraint equations to get a rotating
black hole solution on the brane.
7.1 Black Hole Without Matter on Brane
The Hamiltonian constraint equation of (5.11) is
( )2 (4) 25 12 mnAB mnT R K K Kκ = − − + (7.1)
For a Randall Sundrum brane we have
5 2
2 2
5 4
2
5
6
2
6
l
l
l
κ κ
λ κ
Λ = −
=
=
(7.2)
If  the energy momentum tensor of the bulk and the brane vanishes everywhere,
equation (7.1) reduces to 
(4) 0R = (7.3)
To solve (7.3) for a rotating black hole, we propose a metric ansatz in Kerr Schild
form [56] as
( ) ( )
( )
22 2 2 2 2 2 2 2sin 2 sin
, i ji j
ds du dr dr d r a d a drd
h r k k dx dx
θ θ φ θ φ
θ
= − + + + Σ + + +
+
 
(7.4)
48
where a is the rotation parameter, ki = (1, 0, 0, - a sin2θ ) is a null vector and
2 2 2cosr a θΣ = + (7.5)
( )2 2
2 2 2
r a
du dt dr
r a Mr
= − + − (7.6)
2 2 2
ad d dr
r a Mr
φ φ= − + − (7.7)
Using (7.4) and solving (7.3) for ( ),h r θ  gives
( ) 2, Mrh r βθ = − +Σ Σ (7.8)
where, β and M are integration constants. (7.8) is a Kerr type solution with a charge
term. This tidal charge is due to non-local bulk effects resulting from tensor Wij.
7.2 Black Hole with Gauge Field on Brane
Trapping a Maxwell field on the brane with potential one form
( )2sinerA du a dθ φ= − −Σ  (7.9)
will give a four dimensional energy-momentum tensor as
2
4
1
4
k km
ij ik j ij kmT F F h F Fκ = − (7.10)
In this case equation (7.1) reduces to
4
(4)
4
leR = − Σ (7.11)
49
Using the ansatz (7.4), we find a general solution to (7.11) as
( )
( ) ( )
2
4 4 2 2 2 4 2
4
3 5 5
2,
cos 2 cos 5 cos 3 3 arctan cos
8 cos
e Mrh r
ra a a r r r ale
a
βθ
θ θ θ θ
θ
+= − +Σ Σ
 + + + Σ −  Σ  
(7.12)
which, up to linear terms in a2 cos2θ  gives
( ) 2 4 2 2 2 42 6 2 2 62 cos 2 17, 20 140
e Mr le a e M leh r
r r r r r r r
β θ βθ  + +≈ − + − + − +   (7.13)
Setting a = 0 in (7.13), one recovers the non-rotating solution of [54]. 
50
8. CONCLUSION
We have derived covariant gravitational field equations on 2-brane and 3-brane
worlds in the framework of non-geodesic slicing of bulk spacetime.
We have shown that the form of these effective equations on the branes remains the
same as in the case of geodesic slicing of bulk spacetime [42] and coincides exactly
with them when acceleration vanishes.
Since our effective gravitational equations, as in the case of  geodesic slicing, are not
closed, we have derived the evolutionary equations into the bulk identites which
make our system of equations closed.
Thus the gravitational field equations obtained in this work, generalize the equations
of Shiromizu, Maeda and Sasaki [42],  removing the special presuggestion of
Gaussian normal coordinates.
Further, we have studied black hole solutions on a 3-brane, solving the constraint
equations  for a rotating black hole. We have found that the original metric of
rotating black hole acquires a tidal charge on the 3-brane due to the bulk effect of
Weyl curvature tensor.
51
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56
CURRICULUM VITAE
Ahmet Emir GÜMRÜKÇÜOĞLU was born in Istanbul, in 1976. He graduated from
Lycée Saint-Joseph in 1995, and from Physics Department of Istanbul Technical
University in 2001. He is an M.Sc. student in Physics Program of Istanbul Technical
University since 2001, and has been working as a research assistant in the Physics
Department of the same University since November 2001.