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International Scholarly Research Network
ISRN Astronomy and Astrophysics
Volume 2011, Article ID 131473, 10 pages
doi:10.5402/2011/131473
Review Article
Understanding Gravity: Some Extra-Dimensional Perspectives
V. H. Satheeshkumar1,2 and P. K. Suresh1
1School of Physics, University of Hyderabad, Hyderabad 500 046, India
2Department of Physics, Baylor University, Waco, TX 76798-7316, USA
Correspondence should be addressed to P. K. Suresh, pkssp@uohyd.ernet.in
Received 4 November 2011; Accepted 13 December 2011
Academic Editor: T. Boller
Copyright © 2011 V. H. Satheeshkumar and P. K. Suresh. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Gravity is one of the most inexplicable forces of nature, controlling everything, from the expansion of the Universe to the ebb and
flow of ocean tides. The search for the laws of motion and gravitation began more than two thousand years ago but still we do not
have the complete picture of it. In this paper, we have outlined how our understanding of gravity is changing drastically with time
and how the previous explanations have shaped the most recent developments in the field like superstrings and braneworlds.
1. Introduction
Gravity is an immediate fact of everyday experience, but its
fundamental understanding presents some of the deepest
theoretical and experimental challenges in physics today.
Gravitational physics is concerned with some of the most
exotic large-scale phenomena in the Universe. But it is
also concerned with the microscopic quantum structure
of spacetime and the unification of all fundamental forces
of nature. Gravity is thus important on both the largest
and smallest scales considered in contemporary physics and
remains one of the greatest challenges of twenty-first century
science.
Gravity dominates the large-scale structure of the Uni-
verse only by default. Matter arranges itself to cancel
electromagnetism, and the strong and weak forces are
intrinsically short range. At a more fundamental level,
gravity is extravagantly feeble. Where does this outlandish
disparity come from? Perhaps the most unusual thing about
gravity we know about is that, unlike the other forces of
nature, gravity is intimately related to space and time. Why
is it so different from the other interactions? Why has it
not been able to unify it with the rest? Some attempts to
understand quantum gravity have required that we live in
more than four dimensions! If so, why do we not see the
other dimensions? How are these extra dimensions hidden
from our world? Is there a way to detect them? It is the aim of
this paper as to provide a short summary of the present status
of these extra-dimensional theories of gravitation. But before
graduating to extra-dimensional theories, we will have a look
at the well-established theories of gravity due to Newton and
Einstein.
1.1. Newton’s Gravity. The very earliest ideas regarding
gravity must have been based on every day experience. For
example, objects fall unless they are supported and climbing
a hill is harder than walking on a level. Aristotle was the first
to give some reasoning for these observed facts. In his view,
the whole Universe is made up of four concentric spheres, the
innermost being the Earth, then comes Water, Air, and Fire.
Since stone is more of the “Earth” type, it falls down on the
Earth when thrown up! Ptolemy extended this view to the
heavens and ended up with the geocentric theory. Aristotle’s
notion of the motion impeded understanding of gravitation
for a long time. Copernicus’s view of the solar system was
important as it allowed sensible consideration of gravitation.
Kepler’s laws of planetary motion were based on the volumes
of observational data collected by Tycho Brahe. Galileo’s
understanding of the motion and falling bodies was through
his inclined plane experiments. But till then nobody knew
what is the “cause” for such a motion. This sets the scene for
Isaac Newton’s theory of gravity which was presented in his
treatise The Principia [1] in 1686.
Newton started from Galileo’s law of falling objects and
applied it to an unlikely object, the Moon, which seems to
flout the law of gravity. Newton realized that the Moon is
2ISRN Astronomy and Astrophysics
not immune to gravity and is continuously falling towards
the Earth, but it keeps missing it! Newton thus realized that
gravity was not something special to the Earth, but it also
acts in space. This was a profound and revolutionary idea.
According to Aristotle, the laws governing the heavens were
considered to be completely different from the laws of physics
here on Earth. Now, however, if the moon was affected by
gravity, then it made sense that the rest of the solar system
should also be subjected to gravity. Newton found that he
could explain the entire motion of the solar system from
the planets to the moons to the comets with a single law
of gravity. Newton’s Universal Law of Gravity states that
“all bodies attract all other bodies, and the strength of the
attraction is proportional to the masses of the two bodies and
inversely proportional to the square of the distance between
the bodies.” This is called universal because it applies to
all bodies in the Universe regardless of their nature. (We
know that it is not completely “universal” because zero-mass
objects do not feel gravity in the Newtonian picture and
in this sense, apart from many other, Einstein’s theory is
more universal than Newton’s. Of course during the time
of Newton, zero-mass object would have made no sense.) A
modern mathematical way of saying this is
F=GMm
R2,(1)
where Gis Newton’s gravitational constant, Mand mare
the masses of the objects, and Ris the distance between the
objects. This law can be expressed in differential form as
Poisson’s equation
∇2φ=4πGρ,(2)
where φis gravitational potential and ρis the mass density of
the object.
Despite its power in explaining the planetary orbits in
the solar system, Newton was unhappy with the lack of a
mechanism by which gravity worked. Until then, all forces
were believed to be contact forces—except the gravity. The
Newtonian concept of “action-at-a-distance” was profoundly
disturbing to his opponents who attacked his theory as an
“occult”.
From the period immediately following Newton’s dis-
covery of his Universal Law of Gravitation, to about the
turn of the nineteenth century, the theory of gravitation
stayed essentially unchanged. More sophisticated mathemat-
ical tools for understanding the interplay of the planets
were developed, but the underlying theory remained stable.
The excitement during this period mainly came from the
systematic application of the theory of gravity to the heavens,
for example, Halley’s prediction of the return of the comet
that now bears his name; discovery of the Neptune by John
Adams and Urbain Leverrier; William Hershel’s observations
of binary stars and the calculation of the mass of stars; James
Maxwell’s (the same Maxwell known in electrodynamics
and thermodynamics) explanation of rings of the Saturn.
Of course, other advances were made; among the most
important were the experiments of Cavendish who directly
demonstrated the gravitational force between two objects in
the laboratory.
1.2. Einstein’s Gravity. The twentieth century was a time of
tremendous progress in physical science. For the understand-
ing of gravity, the century began with two puzzles. The first
of these puzzles concerned the orbit of the planet Mercury.
The second puzzle was related to a series of experiments
performed by the Hungarian physicist Roland E¨
otv¨
os at the
end of the nineteenth century. E¨
otv¨
os was intrigued by the
curious link between Newton’s laws of gravity and motion.
His experiments showed that the gravitational mass was the
same as the inertial mass to at least a few parts in a hundred
million.
Einstein’s theory of General Relativity [2], published in
1915, is our most detailed mathematical theory for how
gravity works. The foundation stone for the general relativity
is the equivalence principle, which assumes equivalence
between the inertial mass with the gravitational mass. This
implies “the weak equivalence principle,” that is,the effects of
gravitation can be transformed away locally by using suitably
accelerated frames of reference. This can be generalized
to “the strong equivalence principle,” which allows us to
study gravitational interaction by studying only the geometry
of the spacetime. The modern approach to gravity as the
geometry of curved spacetime is based on this theory.
To understand the geometry of spacetime, consider the
distance between two spacetime points in any inertial frame,
ds2=c2dt2−dx2−dy2−dz2.(3)
But if these two points are not connected by a straight line,
the distance can be given by a more general form
ds2=gμνdxμdxν,(4)
where sum over repeated indexes is implied. The indexes
μ,ν=0, 1, 2, 3 run over four spacetime coordinates. The
coefficient gμνis a function of the spacetime coordinate xμ.
This is called the metric and it specifies the geometry of the
spacetime. To study the geometry of any spacetime for an
understanding of the theory of gravity, it is enough to study
the metric gμν.
It follows then from the Principle of Equivalence that
the equations which govern gravitational fields of arbitrary
strength must take the form
Gμν=8πG
c4Tμν,(5)
where Gμνis called the Einstein tensor which has the
geometrical information about the spacetime, Gis the
Newton’s gravitational constant, and Tμνis the energy-
momentum tensor of the matter present. Einstein tensor is
given by
Gμν=Rμν−1
2gμνR,(6)
where Rμνis the Ricci curvature tensor and Ris Ricci scalar.
Ingeneralrelativity,oneperformscalculationstocom-
pute the evolution and structure of an entire Universe at a
time. A typical way of working is to propose some particular
collection of energy and matter in the Universe, to provide
ISRN Astronomy and Astrophysics 3
Tμν. Given a particular Tμν, the Einstein equation turns into
a system of second-order nonlinear differential equations
whose solutions give us the metric of spacetime gμν,which
holds all the information about the structure and evolution
of a Universe with that given Tμν.
General Relativity is perhaps the most beautiful physical
theory and one of the crowning glories of modern physics. It
is powerful, pleasing to the aesthetic sense, and well tested.
General Relativity has survived many different tests, and it
has made many predictions which have been confirmed.
The recently concluded experimental investigation using the
satellite-based mission Gravity Probe B confirms the two
fundamental predictions of general relativity, the geodetic,
and frame-dragging effects [3]. The detection of gravita-
tional waves is one of the most fundamental predictions of
general relativity which has not been confirmed as of today.
Currently many state-of-the-art gravitational wave detectors
are in operation. However none of them have the sensitivity
to directly detect the gravitational waves yet [4]. Other tests
focus on the laboratory-scale measurements to look for signs
of extra-dimensions, such as a deviation from inverse square
[5,6] and missing energy signals in CMS [7]andATLAS[8]
experiments at the Large Hadron Collider of CERN. Data
from any of these experimental studies will greatly improve
our understanding of gravity and will show us how to go
beyond the mathematics of General Relativity to create an
even better theory.
The unification of quantum theory and general relativity
has been a major problem in physics ever since these theories
were proposed. The problem is that since all fields carrying
energy are affected by gravity, gravity contributes to its
own source. When trying to do calculations on the energy
scale where gravity is usually thought to be similar in
strength as the other forces, the graviton self-coupling causes
fluctuations which introduces infinities in the calculations.
This has led many theories to accommodate the idea of
extra dimensions to get the quantum gravity. A recent
nontechnical review of extra-dimensional theories can be
foundin[9]. One of the early possibilities for such a
unification of the then known interactions, that is, gravity
and electromagnetism, was suggested by Kaluza [10]and
Klein [11,12]. But historically, it was Nordstr¨
om [13]who
brought the idea of extra spacial dimension into physics.
2. Kaluza-Klein Theory
An early proposal to unite general relativity and classical
electrodynamics was given by [10] in 1921. He showed that
the gravitational and electromagnetic fields stem from a
single universal tensor and such an intimate combination
of the two interactions is possible in principle, with the
introduction of an additional spacial dimension. Although
our rich physical experience obtained so far provides little
suggestion of such a new spacial dimension, we are certainly
free to consider our world to be four-dimensional spacetime
of the bigger five-dimensional spacetime. In this scenario,
one has to take into account the fact that we are only aware
of the spacetime variation of state quantities, by making their
derivatives with respect to the new parameter vanish or by
considering them to be small as they are of higher order. This
assumption is known as the cylindrical condition.
The five-dimensional line element is given by
ds2=gμνxμ,ydxμdxν(7)
with yas the additional spatial coordinate. The five-
dimensional metric can be expressed as
gμν=⎛
⎝gμνgμ5
g5νg55⎞
⎠,(8)
where all unhatted quantities are four-dimensional and all
hatted quantities are five-dimensional.
Once we have a spacetime metric, like in standard general
relativity we can construct the Christoffel symbols Γμ
νρ, the
Riemann-Christoffel curvature tensor Rμ
νρσ , the Ricci tensor
Rμν, the curvature invariant R, and then the field equations.
This approach gave a striking result, the fifteen higher-
dimensional field equations naturally broke into a set of ten
formulae governing a tensor field representing gravity, four
describing a vector field representing electromagnetism, and
one wave equation for a scalar field. Furthermore, if the
scalar field was constant, the vector field equations were just
Maxwell’s equations in vacuo, and the tensor field equations
were the 4-dimensional Einstein field equations sourced by
an electromagnetic field.
In one fell swoop, Kaluza had written down a single
covariant field theory in five dimensions that yielded the
four-dimensional theories of general relativity and electro-
magnetism! But many problems plagued Kaluza’s theory. Not
the least of which was the nature of the fifth dimension.
There was no explanation given for Kaluza’s ad hoc assump-
tion, the cylindrical condition.
In 1926, Klein [11,12]providedanexplanationfor
Kaluza’s fifth dimension by proposing it to have a circular
topology so that the coordinate yis periodic, that is, 0 ≤y≤
2πR,whereRis the radius of the circle S1. Thus the global
space has topology R4×S1. So Klein suggested that there is
a little circle at each point in four-dimensional spacetime.
This is the basic idea of Kaluza-Klein compactification.
Although there are four space dimensions, one of the space
dimensions is compact with a small radius. As a result, in
all experiments we could see effects of only three spacial
dimensions. Thus Klein made the Kaluza’s fifth dimension
less artificial by suggesting plausible physical basis for it
in compactification of the fifth dimension. The theory of
gravity on a compact spacetime is called Kaluza-Klein theory.
A detailed pedagogical account of this is given in [14–16].
We introduce the following notations,
g55 =φ,
g5μ=κφAμ,
gμν=gμν+κ2φAμAν.
(9)
4ISRN Astronomy and Astrophysics
Hereby the quantities gμνare reduced to known quantities.
Now, the new metric can be written as
gμν=φ−1/3⎛
⎝gμν+κ2φAμAνκφAμ
κφAνφ⎞
⎠, (10)
where the field φappears as a scaling parameter in the
fifth dimension and is called the dilaton field. The fields
gμν(x,y), Aμ(x,y), and φ(x,y)transform,respectively,asa
tensor, a vector, and a scalar under four-dimensional general
coordinate transformations.
The Einstein-Hilbert action for five-dimensional gravity
can be written as
S=1
2
k2d5x−g
R, (11)
where
kis the five-dimensional coupling constant and
Ris
the five-dimensional curvature invariant. We can get the field
equations of gravity and electromagnetism from the above
action by variational principle.
As Klein suggested, the extra dimension has become
compact and satisfies the boundary condition
y=y+2πR. (12)
All the fields are periodic in yand may be expanded in a
Fourier series
gμνx,y=
+∞
n=−∞
gμνn(x)ein·y/R,
Aμx,y=
+∞
n=−∞
Aμn(x)ein·y/R,
φx,y=
+∞
n=−∞
φn(x)ein·y/R.
(13)
The equations of motion corresponding to the above
action are
∂μ∂μ−∂y∂ygμνx,y=∂μ∂μ+n2
R2gμνn(x)=0,
∂μ∂μ−∂y∂yAμx,y=∂μ∂μ+n2
R2Aμn(x)=0,
∂μ∂μ−∂y∂yφx,y=∂μ∂μ+n2
R2φn(x)=0.
(14)
Comparing these with the standard Klein-Gordon equation,
we can say that only the zero modes (n=0) will be massless
and observable at our present energy and all the excited
states, called as Kaluza-Klein states, will have masses
mn∼|n|
R(15)
as well as charge
qn=√2κn
R(16)
as shown by Salam and Strathdee [17], where nis the mode
of excitation. So, in four dimensions we shall see all these
excited states with mass or momentum ∼O(n/R). Since we
want to unify the electromagnetic interactions with gravity,
the natural radius of compactification will be the Planck
length
R=1
Mp
, (17)
where the Planck mass Mp∼1019 GeV.
Since the Kaluza-Klein metric is a 5×5 symmetric tensor,
it has 15 independent components. However, because of
various gauge fixings we will have only 5 independent degrees
of freedom. Whereas in four dimensions we have only 2
degrees of freedom for a massless graviton. This implies that
from four-dimensional point of view a higher-dimensional
graviton will contain particles other than just ordinary four
dimensional graviton. The zero-mode of five-dimensional
graviton contains a four-dimensional massless graviton with
2 physical degrees of freedom; a four-dimensional massless
gauge boson with 2 physical degrees of freedom and a real
scalar with 1 physical degree of freedom. Whereas the non-
zero mode of five-dimensional graviton is massive and has 5
physical degrees of freedom.
Kaluza and Klein’s five-dimensional version general
relativity, although flawed, is an example of such an attempt
to unite the forces of nature under one theory. It led to
glaring contradictions with experimental data. But some
physicists felt that it was on the right track, that it in fact
did not incorporate enough extra dimensions! This led to
modified versions of Kaluza-Klein theories incorporating
numerous and extremely small extra dimensions. The three
main different approaches to higher dimensional unification
areasfollows.
(1) Compactified Approach. In this scenario extra dimen-
sions are forbidden for us to experience as they
are compactified and are unobservable on presently
accessible energy scales. This approach has been suc-
cessful in many ways and is the dominant paradigm
in the higher-dimensional unification. This has led
to new theories like 11-dimensional supergravity,
10-dimensional superstring theory, the latest 11-
dimensional M-theory and Braneworld theory.
(2) Projective Approach. Projective theories were designed
to emulate the successes of Kaluza-Klein theory with-
out epistemological burden of a real fifth dimension.
In this way of unification, the extra dimensions
are treated as mathematical artifacts of a more
complicated theory. The fifth dimension is absorbed
into ordinary four-dimensional spacetime by replac-
ing the classical tensors of general relativity with
projective ones, which in turn alters the geometrical
foundation of general relativity itself.
(3) Noncompactified Approach. This approach prefers to
stay with idea that the new coordinates are physical.
Following Minkowski’s example, one can imagine
coordinates of other kinds, scaled by appropriate
ISRN Astronomy and Astrophysics 5
dimension transporting parameters to give them
units of length. In this approach the extra dimensions
may not necessarily be spacelike. This takes the
observable quantity such as rest mass as the extra
dimension.
Here we discuss only the compactified approaches and the
interested readers can refer to [18] for the detailed review and
the comparative study of these three approaches.
3. String Theory
After the discovery of nuclear interactions, physicists found
that it no longer seemed that the Kaluza-Klein theory with
one extra dimension was a viable candidate to include all the
gauge interactions. The electromagnetic interaction could
be accommodated with only one extra dimension. But the
strong, weak and electromagnetic interactions, that is, the
SU(3) ×SU(2) ×U(1) gauge theory requires more degrees
of freedom than a 5-metric could offer. However, the way
in which to address the additional requirements of modern
physics is not hard to imagine; one merely has to further
increase the dimensionality of theory until all of the desired
gauge bosons are accounted for. Then how many dimensions
do we need to unify modern particle physics with gravity
via the Kaluza-Klein mechanism? The answer comes from
N=8 supersymmetry which contains spin-2 particle. When
N=8 supersymmetry is coupled with general relativity,
one has 11-dimensional supergravity theory [19].Butitwas
realized that it is not possible to get all the gauge interactions
and the required fermion contents of the standard model
from this theory [19]. Then there were attempts to consider
11-dimensional theories with gauge groups. Of course, the
main motivation of obtaining all gauge interactions and
gravity from one Einstein-Hilbert action at 11 dimensions
would be lost, but still this became an important study for
sometime. In this construction the main problem was due to
new inconsistency, the anomaly.
This problem was tackled with String theory [20]. Briefly,
the origin of string theory was the discovery by Vendramin
[21] and Virasoro [22]ofsimpleformulasasamodelfor
describing the scattering of hadrons. These formulae revealed
a rather novel mathematical structure which was soon
interpreted by the physical picture based on the relativistic
dynamics of strings by Nambu, Nielsen, and Susskind [23–
25]. This string interpretation of “dual resonance model”
of hadronic physics was not influential in the development
of the subject until the appearance of the 1973 paper by
Goddard et al. [26]. It explained in detail how the string
action could be quantized in light-cone gauge. Interestingly,
among the massless string states, there is one that has spin
two. In 1974, it was shown by Scherk and Schwarz [27]and
independently by Yoneya [28] that this particle interacts like
graviton, so the theory actually includes general relativity.
This leads them to propose that string theory should be
used for unification rather than for hadrons. This implied, in
particular that the string length scale should be comparable
to the Planck length, rather than the size of hadrons, that is,
10−15 m, as it was previously assumed. All this made string
theory a potential candidate to be a theory of quantum
gravity.
String theory replaces all elementary point particles that
form matter and its interactions with a single extended
object of vanishing length. Thus every known elementary
particle, such as the electron, quark, photon, or neutrino
corresponds to a particular vibration mode of the string. The
diversity of these particles is due to the different properties
of the corresponding string vibrations. In fact the laws of
quantum mechanics tell us that a single elementary string
has infinite number of vibrational states. Since each such
vibrational state behaves as a particular type of elementary
particle, string theory seems to contain infinite types of
elementary particles. This would be in contradiction with
what we observe in nature were it not for the fact that most
of these elementary particles in string theory turn out to be
very heavy, and not observable in present experiments. Thus
there is no immediate conflict between what string theory
predicts and what we observe in actual experiments. On the
other hand, these additional heavy elementary particles are
absolutely essential for getting finite answers in string theory.
The possible advantage of string theory is that the
anomalies faced by Supergravity are fixed naturally by the
extended nature of strings. The analog of a Feynman diagram
in string theory is a two-dimensional smooth surface, and
the loop integrals over such a smooth surface lack the zero-
distance, infinite momentum problems of the integrals over
particle loops. In string theory infinite momentum does not
even mean zero distance, because for strings, the relationship
between distance and momentum is roughly like
ΔL∼
p+αp
.(18)
The parameter αis related to the string tension, the
fundamental parameter of string theory, by the relation
Tstring =1
2πα.(19)
The above relation implies that a minimum observable
length for a quantum string theory is
Lmin ∼2√α.(20)
Thus zero-distance behavior which is so problematic in
quantum field theory becomes irrelevant in string theories,
and this makes string theory very attractive as a theory of
quantum gravity.
If string theory is a theory of quantum gravity, then this
minimum length scale should be at least the size of the Planck
length, which is the length scale made by the combination of
Newton’s constant, the speed of light, and Planck’s constant
Lp=GN
c3=1.6×10−35 m.(21)
All was well, but this was only consistent if the dimension
of spacetime is 26 and had only gauge bosons in it. More-
over these bosonic string theories are all unstable because
6ISRN Astronomy and Astrophysics
the lowest excitation mode, or the ground state, is a tachyon.
Adding fermions to string theory introduces a new set of
negative norm states or ghosts. String theorists learned that
all of these bad ghost states decouple from the spectrum
when two conditions are satisfied: the number of spacetime
dimensions is 10, and theory is supersymmetric, so that there
are equal numbers of bosons and fermions in the spectrum.
The resulting consistent string theories are called Superstring
theories and they do not suffer from the tachyon problem
that plagues bosonic string theories.
A very nice feature of such superstring theories is that,
in 10 dimensions the gauge and gravitational anomalies
cancel for E8×E8group and the SO(32) group. It was
then found that when the extra six-dimensional space is
compactified, the four-dimensional world contains all the
required fermions and the standard model gauge groups.
Supersymmetry could remain unbroken till the electroweak
scale to take care of the gauge hierarchy problem. This the
appears to be the unified theory of all know interactions.
At that time (1984-85), string theorists believed there were
five distinct superstring theories. They differ by very general
properties of the strings [29].
(i) In the first case (Typ e I ) the strings are unoriented
and insulating and can have boundaries in which case
they carry electric charges on their boundaries.
(ii) In two theories (the Type IIA and Type IIB) the strings
are closed and oriented and are electrical insulators.
(iii) In two theories (the heterotic superstrings with gauge
group SO(32) and E8×E8) the strings are closed,
oriented, and superconducting.
But now it is known that this naive picture was wrong,
and that the five superstring theories are connected to one
another as if they are each a special case of some more
fundamental theory. In the mid-nineties it was learned that
various string theories floating around were actually related
by duality transformations known as T-duality and S-duality.
T-duality is a symmetry of string theory, relating type IIA and
type IIB string theory, and the two heterotic string theories.
S-duality relates Type I string theory to the heterotic SO(32)
theory. Using various known dualities between different
compactification of different string theories one can now
argue that all five string theories are different ways of
describing a single theory. These ideas have collectively
become known as M-theory,whereMis for membrane,
matrix, or mystery, depending on your point of view!
In string theory, we assume that the particles we see
around us are actually like strings. Since the entire string
propagates with time, we have to apply boundary conditions
to the end points for consistency. This lead us to either open
or closed strings, which have different boundary conditions.
When this theory was extended to a membrane (or brane for
short), one has to apply boundary conditions to its boundary
surfaces. This can then be extended to higher n-dimensional
branes. In general, branes are static classical solutions in
string theories. A p-brane denotes a static configuration
which extends along p-spatial directions and is localized in
all other directions. A p-brane is described by a (p+1)-
dimensional gauge field theory. Strings are equivalent to 1-
branes, membranes are 2-branes, and particles are 0-branes.
A special class of p-branes in string theory are called D-
branes. Roughly speaking, a D-brane is a p-brane where the
ends of open strings are localized on the brane. D-branes
were discovered by investigating T-duality for open strings.
Open strings do not have winding modes around compact
dimensions, so one might think that open strings behave like
particles in the presence of circular dimensions.
Although these theories now appear to be far from any
experiments, it is now established that these theories have
the prospect of becoming theory of everything. The scale at
which this theory is operational is close to the Planck scale.
This makes it experimentally nonviable for a very long time,
or probably at any time!
4. Braneworld Models
Considering the large separation between the weak scale
(103GeV) and the traditional scale of quantum gravity, the
Planck scale (1019 GeV) is one of the most puzzling aspects
of nature. This is known as the hierarchy problem.One
theoretical means of solving this problem is to introduce
supersymmetry. Alternatively one may hope to address the
hierarchy by exploiting the geometry of spacetime. An
extremely popular theory which cures the hierarchy problem
by changing the geometry of spacetime with extra space
dimensions is the so-called braneworld scenario.
This phenomenological model has been motivated by the
work of Horava and Witten [30,31], who found a certain
11-dimensional string theory scenario where the fields of the
standard model are confined to a 10-dimensional hypersur-
face, or brane. In this picture, the nongravitational degrees
of freedom are represented by strings whose endpoints reside
on the brane and on the other hand, gravitational degrees of
freedom in string theory are carried by closed strings, which
cannot be tied down to any lower-dimensional object. Hence,
the main feature of this model is that the standard model
particles are localized on a three-dimensional space called
the brane, while gravity can propagate in 4 + ndimensions
called the bulk. It is usually assumed that all ndimensions are
transverse to the brane and have a common size R.However,
the brane can also have smaller extra dimensions associated
with it, of size rRleading to effects similar to a small finite
thickness.
The three main features of braneworld models are as
follows.
(1) Localization of standard model particles on the
brane: a first-particle physics application of this idea
was put forwarded by Rubakov and Shaposhnikov
[32] and independently by Akama [33].
(2) Localization of gauge fields on the brane: a mech-
anism for gauge field localization within the field
theory context was proposed by Dvali and Shiffman
[34]. Localization of gauge fields is a rather natural
property of D-branes in closed string theories [35].
ISRN Astronomy and Astrophysics 7
(3) Obtaining four-dimensional gravity on the brane: all
the existing braneworld models obtain the laws of
(3 + 1) dimensional gravity on the brane as their low
energy approximation.
The size and geometry of the bulk, as well as the types of
particles which are allowed to propagate in the bulk and on
the brane, vary between different models. Some important
braneworld models are discussed briefly here in the order
of their appearance in the literature. Somewhat detailed
discussion is given in [36].
4.1. Braneworlds with Compact Extra Dimensions. Here, to
obtain (3 + 1) dimensional gravity on the brane the idea of
KK compactification is combined with braneworld idea. This
was proposed in 1998 by Arkani-Hamed et al. [37]along
with Antoniadis et al. [38]. The additional dimensions are
compact, may be as large as as micrometer! As one of its
attractive features, the model can explain the weakness of
gravity relative to the other fundamental forces of nature.
In the brane picture, the other three SM interactions are
localized on the brane, but gravity has no such constraint
and “leaks” into the bulk. As a consequence, the force
of gravity should appear significantly stronger on small
say, submillimeter scales, where less gravitational force has
“leaked”. This opens up new possibilities to solve the Higgs
mass hierarchy problem and gives rise to new predictions
that can be tested in accelerator [7,8], astrophysical [39–44],
and table-top experiments [5,6].
The action for gravity in (4 + n) dimensions is given by
S4+n=M2+n
∗
2d4x2πR
0dnyGR4+n
+d4xg(T+LSM),
(22)
where M∗∼(1–10) TeV, g(x)=G(x,y=0) and T+
LSM=0. The low effective four-dimensional action for a
zero mode takes the form
S=M2+n
∗2πRn
2d4xgzmRzm
+d4xg(T+LSM).
(23)
Comparing it with standard four-dimensional pure gravity
action we get
M2
pl =M2+n
∗(2πR)n.(24)
Postulating that new quantum gravity scale is at a few TeV,
we find the size of the extra dimensions to be
R=1030/n−17 cm.(25)
For one extra dimension, n=1, the size of extra dimension
would be R∼1013 cm. This is excluded since it would have
modified gravity in solar system scale. For n=2wegetR∼
10−2cm, which is interesting since it predicts modification of
four-dimensional laws of gravity at submillimeter scale.
Two static sources on the brane interact with the
following nonrelativistic gravitational potential
V(r)=−GNm1m2
n=+∞
n=−∞
Ψny=0
2e−mnr
r, (26)
where Ψn(y=0) denotes the wave function of nth KK mode
at a position of the brane and mn=|n|/R.IfrLfrom the
above expression we get
V(r)=−GNm1m2
r.(27)
This recovers the conventional four-dimensional law of
Newtonian dynamics. In the limit rLwe get
V(r)=−m1m2
M2+n
∗r1+n.(28)
This is the law of (4 + n)-dimensional gravitational interac-
tions. Therefore, the laws of gravity are modified at distances
of order R.
4.2. Braneworlds with Wrapped Extra Dimensions. This phe-
nomenon of localizing gravity was discovered by Randall
and Sundrum [45] in 1999. RS braneworlds do not rely
on compactification to localize gravity on the brane, but
on the curvature of the bulk, sometimes called “warped
compactification.” What prevents gravity from leaking into
the extra dimension at low energies is a negative bulk
cosmological constant. There are two popular models. The
first, is called RS-1 and has a finite size for the extra
dimension with two branes, one at each end. The second,
RS-2, is similar to the first, but one brane has been placed
infinitely far away, so that there is only one brane left in the
model. They also used their model to explain the hierarchy
problem [46] in particle physics.
For simplicity, we consider RS-2 model which has a single
brane embedded in five-dimensional bulk with negative
cosmological constant. The action for this model is given by
S5=M3
∗
2d4x+∞
−∞ dyG(R5−2Λ)+d4xg(T+LSM),
(29)
where Λdenotes the negative cosmological constant and Tis
the brane tension. The equation of motion derived from this
action is given by
M∗GRAB −1
2GABR=−
M3
∗ΛGGAB
+Tggμνδμ
Aδν
Bδy.
(30)
In this convention the brane is located in extra space at y=
0.Theaboveequationshaveasolutioninfour-dimensional
world volume as
ds2=e−|y|/Rημνdxμdxν+dy2.(31)
8ISRN Astronomy and Astrophysics
It is important to emphasize that the five-dimensional
action is integrable with respect to yfor the zero mode. That
is,
M3
∗
2d4x+∞
−∞ dyGR5−→ M3
∗(2R)
2d4xgR. (32)
The result of this integration is a conventional four-
dimensional action. Hence we find the relation between four-
dimensional Planck mass and M∗
M2
pl =M3
∗(2R).(33)
This looks similar to that in ADD model with one extra
dimension. The similarity is due to the fact that the effective
size of the extra dimension that is felt by the zero-mode
graviton is finite and is of the order of Rin both the models.
Besides the zero-mode, there are an infinite number of
KK modes. Since the extra dimension is not compactified,
the KK modes have no mass gap. In the zero-mode approx-
imation, these states are neglected. However at distances
smaller than the size of the extra dimension, the effects of
these modes become important.
The static potential between two sources on the brane is
given by
V(r)=−GNm1m2
r1+ (2R)2
r2.(34)
The first term is the conventional four-dimensional law of
Newtonian dynamics, whereas the second term is due to
exchange of KK modes which becomes dominant when r
R.
4.3. Braneworlds with Infinite Volume Extra Dimensions. This
mechanism of obtaining (3 + 1) gravity on the brane is
different from the earlier two as it allows the volume of the
extra dimension to be infinite. This model was proposed
in 2000 by Dvali et al. [47]. In the first model the four-
dimensional gravity could be reproduced at large distances
due to finite volume of extra space. This is usually done by
compactifying the extra space. Alternatively, this is done by
warping the extra dimensions in the second model where still
the volume of extra space is finite. But in this scenario the
size of the extra dimensions does not need to be stabilized
since the extra dimensions are neither compactified nor
wrapped because of the presence of infinite-volume extra
dimensions and hence gravity is modified at large distances.
This gives rise to new solutions for late-time cosmology
and acceleration of the Universe which comes from type-Ia
supernovae observations. This can also explain dark energy
problem and Cosmic Microwave Background.
The action in five dimensions with one infinite volume
extra dimension is given by
S5=M3
∗
2d4x+∞
−∞ dyGR5+d4xg⎛
⎝M2
pl
2T+LSM⎞
⎠.
(35)
To study the gravity described by this model, we introduce
the quantity
rc∼M2
pl/M3
∗.(36)
When rc→∞the four-dimensional term dominates but
in the opposite limit rc→0, the five-dimensional term
dominates. Therefore we expect that for rrcto recover
the four-dimensional laws on the brane while for rrcfive-
dimensional laws.
The static gravitational potential between the sources in
the four-dimensional world volume of the brane is given by
V(r)=− 1
8π2M2
pl
1
rsinr
rcCir
rc
+1
2cosr
rcπ−2Sir
rc,
(37)
where Ci(z)≡γ+Len(z)+z
0(cos(t)−1)dt/t,Si(z)≡
z
0(sin(t)dt/t and γ0.77 is the Euler-Mascheroni constant,
and the distance rcis defined as follows:
rcM2
pl
2M3
∗
.(38)
In this model rcis assumed to be of the order of the
present Hubble size, which is equivalent to the choice M∗∼
10–100 MeV. It is useful to study the short distance and the
long distance behavior of this expression. At short distance,
when rrcwe get
V(r)=− 1
8π2M2
pl
1
rπ
2+−1+γ+ln
r
rcr
rc
+Or2.
(39)
Therefore, at short distances the potential has the correct
four-dimensional Newtonian 1/r scaling. This is subse-
quently modified by the logarithmic “repulsion” term in the
above expression. At large distances rrc, the potential
takes the form
V(r)=− 1
8π2M2
pl
1
rrc
r+Or2.(40)
Thus, the long distance potential scales as 1/r2in
accordance with laws of five-dimensional theory.
4.4. Braneworlds with Universal Extra Dimensions. Universal
Extra Dimensions model was proposed by Appelquist et
al. [48] in 2001. In this model the extra dimensions are
accessible to all the standard model fields, referred to here
as universal dimensions which may be significantly larger.
The key element is the conservation of momentum in the
universal dimensions. In the equivalent four-dimensional
theory, this implies KK number conservation. In particular
there are no vertices involving only one nonzero KK mode,
and consequently there are no tree-level contributions to the
electroweak observables. Furthermore, nonzero KK modes
may be produced at colliders only in groups of two or
ISRN Astronomy and Astrophysics 9
more. Thus, none of the known bounds on extra dimensions
from single KK production at colliders or from electroweak
constraints applies for universal extra dimensions.
The full Lagrangian of this model includes both the
bulk and the boundary Lagrangian. The bulk Lagrangian
is determined by the SM parameters after an appropriate
rescaling. The very important property of this model is the
conservation of KK parity that implies the absence of tree
level KK contributions to low energy processes taking place
at scales very much less than 1/R In the effective four-
dimensional theory, in addition to the ordinary particles of
the SM, denoted as zero modes, there are infinite towers of
the KK modes. There is one such tower for each SM boson
and two for each SM fermion, while there also exist physical
neutral and churched scalars with (n≥1) that do not have
any zero mode partners.
5. Conclusion
Many of the major developments in fundamental physics of
the past century arose from identifying and overcoming con-
tradictions between existing ideas. For example, the incom-
patibility of Maxwell’s equations and Galilean invariance led
Einstein to propose the special theory of relativity. Similarly,
the inconsistency of special relativity with Newtonian gravity
led him to develop the general theory of relativity. More
recently, the reconciliation of special relativity with quantum
mechanics led to the development of quantum field theory.
We are now facing another crisis of the same character.
Namely, general relativity appears to be incompatible with
quantum field theory. Any straight forward attempt to
“quantize” general relativity leads to a nonrenormalizable
theory. This has led to theories like superstrings and
braneworlds. Even though these theories look rather exotic,
at least for the moment. Yet they lead to important insights
and also provide a framework for addressing a number of
phenomenological issues. Furthermore, new ideas emerge
in approaching fundamental problems which have been
puzzling physicists over the centuries. All this makes the
subject interesting and lively. The question is whether the
mother nature follows any of these routes being explored in
this context.
Acknowledgments
The first author (V. H. Satheeshkumar) would like to
thank Gerald Cleaver and Anzhong Wang for many useful
discussions on the topics mentioned in this paper.
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