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AMERICAN UNIVERSITY OF BEIRUT
Spherically Symmetric Solutions for Five Dimensional
Topological Gravity
by
SAMAR JEAN BECHARA
A thesissubmitted in partial fulfillment of the requirements
for the degree of Master of Science
to the Department of Physicsof the Faculty of Arts and
Sciences
at the American University of Beirut
Beirut, LebanonJanuary 2011
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AMERICAN UNIVERSITY OF BEIRUT
SPHERICALLY SYMMETRIC SOLUTIONS FOR FIVE DIMENSIONALTOPOLOGICAL
GRAVITY
by
SAMAR JEAN BECHARA
Approved by:
Dr.Ali Chamseddine, Professor AdvisorPhysics
Dr.Jihad Touma, Professor Member of CommitteePhysics
Dr.Mounib El Eid, Professor Member of CommitteePhysics
Date of thesis defense: January 17, 2011
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AMERICAN UNIVERSITY OF BEIRUT
THESIS RELEASE FORM
I, SAMAR JEAN BECHARA
authorize the American University of Beirut to supply copies of
mythesis to libraries or individuals upon request.
do not authorize the American University of Beirut to supply
copies of mythesis to libraries or individuals for a period of two
years starting with the dateof the thesis defense.
Signature
Date
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ACKNOWLEDGEMENTS
I would like to express my gratitude to all those who helped me
in completing thisthesis.
Above all, my gratefulness goes to my supervisor, Dr. Ali
Chamseddine, forhis continous help and follow up on the content of
this thesis and to my committeemembers, Dr. Jihad Touma and Dr.
Mounib Eid, for their valuable comments.Moreover, I would like to
thank all the professors who taught me at AUB through-out the
years.
I also offer my special regards and blessings to all my friends
and colleagues,especially Ola Malaeb, for their moral and technical
support. Your friendship andcollaboration means a great deal to
me.
It would not have been possible to complete this thesis without
the supportand encouragement of my family members. Words cannot
express the thanks I oweto each one of you for always being there
for me. I would never be able to pay backall the love and sacrifice
you have provided me with.
v
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AN ABSTRACT OF THE THESIS OF
Samar Bechara for Master of ScienceMajor : Physics
Title : Spherically Symmetric Solutions for Five Dimensional
Topological Gravity
Einstein constructed the theory of general relativity in terms
of a metricand a connection coefficient. After the attempt for
constructing a gauge theoryof gravity by Utiyama and Kibble,
another formulation of general relativity wasintroduced. This
formulation was in terms of the veirbeins and the spin
connections.By gauging the SO(1,5) and the SO(2,4) groups,
Chamseddine then constructeda topological action that contained a
Gauss-Bonnet term, an Einstein term, anda cosmological constant.
Taking a spherically symmetric five-dimensional metricansatz, we
obtained a set of 7 solutions for the derived equations of motion.
Three ofthese solutions were considered physical and were analyzed.
None of them containedsingularities except for coordinate
singularities in the de Sitter space, where a set ofcoordinate
transformations has been applied to eliminate such a singularity.
Finally,
after reducing the metric in five dimensions to a metric in four
dimensions with anadditional scalar field equal toe4
4, we realized that the same set of solutions was still
valid.
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Contents
1 Introduction 2
1.1 Historical Background . . . . . . . . . . . . . . . . . . .
. . . . . . . 21.2 Aim and Organization . . . . . . . . . . . . . .
. . . . . . . . . . . . 7
2 Solutions for the Five-Dimensional Equations of Motion 82.1
Constructing the Five-Dimensional Spherically Symmetric Line
Ele-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 82.2 Equations of Motion in Component Form . . . . . .
. . . . . . . . . . 112.3 Calculating the Curvature tensor
components . . . . . . . . . . . . . 132.4 Equations of Motion . .
. . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Solutions
For Equations of Motion . . . . . . . . . . . . . . . . . . . .
16
3 Analyzing the Physical Solutions 20
3.1 Analyzing the First Physical Solution . . . . . . . . . . .
. . . . . . . 203.1.1 Geodesic Equations . . . . . . . . . . . . .
. . . . . . . . . . . 233.1.2 Coordinate Transformations . . . . .
. . . . . . . . . . . . . . 26
3.2 Analyzing the Second Physical Solution . . . . . . . . . . .
. . . . . . 283.2.1 Geodesic Equations . . . . . . . . . . . . . .
. . . . . . . . . . 323.2.2 Coordinate Transformations . . . . . .
. . . . . . . . . . . . . 34
3.3 Analyzing the Third Physical Solution . . . . . . . . . . .
. . . . . . 363.3.1 Geodesic Equations . . . . . . . . . . . . . .
. . . . . . . . . . 393.3.2 Coordinate Transformations . . . . . .
. . . . . . . . . . . . . 41
4 Dimensionally Reducing the Action 46
5 Conclusion and Future Work 50
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Chapter 1
Introduction
1.1 Historical Background
According to Einsteins theory of general relativity, postulated
at the end of 1915,
gravity is considered as a manifestation of the curvature of
spacetime.[1]
Three main principles lead to Einsteins formulation. These
principles are:
1. The principle of general covariance: laws of physics must
take the same form in
any reference frame.
2. The principle of equivalence: All objects fall at the same
acceleration
irrespective of their masses.
3. Machs principle: the local inertial properties of an object
are determined bythe total mass distribution in the universe.
The first principle lead Einstein to suggest that the equations
should be
written in tensorial form. The second and third principles lead
respectively to the
ideas that gravity should be considered as curvature of
spacetime and that this
curvature is determined by the distribution of matter.
In the theory of relativity, we consider a manifold M, called
Reimannian
manifold, in which the notion of distance is given by:
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ds2 =gdxdx (1.1)
where g is a symmetric tensor called the metric.
On the manifold M, we define coordinates{xa}. Each vector on
themanifold can then be expressed as a linear expansion of a set of
coordinate basis
vectors{ea} such that v=vaea. The same applies for tensors.Since
we are dealing with a curved space, we have to take into
account
that not only the coordinates of a tensor change during a
coordinate
transformation but also the basis vectors themselves.
This is why the notion of covariant derivatives has been
introduced.The
covariant derivative of a vector v, for instance, with respect
to a coordinate is
given by:
v = (ve) = (v)e+ v(e)= (v
+ v)e (1.2)
where
e= e
and
are known as the connection coefficients.
For a torsionless manifold, the connection is symmetric in its
last two
indices and can be written in terms of the metric tensor
according to:
=1
2g(g+ g g) (1.3)
We can define the curvature of the manifold in terms of the
connection
coefficients as:
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R = + (1.4)
If the curvature tensor vanishes, the manifold considered is
flat.
The Einstein field equations for general relativity are given
by
G=R 12
gR = 8T (1.5)
where Tis called the stress-energy tensor.[1]
Another formulation of general relativity is given in terms of
the veilbeins
and spin connections.
This formulation arises when we try to build gravity as a gauge
theory
that is invariant under a Poincare transformation. The veilbeins
are then the
gauge fields associated with translations and spin connections
are the gauge fields
associated with local lorentz transformations. [2]
The veilbein can be understood as follows: at any point P on the
manifold, we can
define a set of basis vectors{ea} such that this set is
orthonormal. Orthonormalityimplies that
g(ea, eb) =ab
where represents the Minkowski metric.1
This set of orthonormal basis vectors is generally known as the
veilbein. One way
to define this set is by taking:
ea =x
xa (1.6)
where x correspond to the coordinates of a point P on the
manifold and
xa are the coordinates of its mapping in the tangent space of M
at P.[4]
Note that Greek indices are usually used to denote coordinates
in the
1in this paper, the signature of the metric is taken as
(+,-,-,...,-)
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curved space, and Latin indices are used to denote coordinates
on the flat, tangent
space.
In terms of the veilbein, the notion of distance is defined
by:
ds2 = abea eb (1.7)
= abeae
bdx
dx =gdxdx (1.8)
The relation between the metric and the veilbein is then given
by:
g=eaea (1.9)
The spin connection is defined such that the covariant
derivative of the
veilbein vanishes[2]. The covariant derivative of the veilbein
is given by:
Dea=e
a+ w
abe
b
e
a (1.10)
are the Christoffel connections and wab are the spin
conections.
For a torsionless manifold, the spin connection can be expressed
in terms of the
veilbeins [2]. It is given by:
wab =1
2eae
b((e) (e) + (e)) (1.11)
where
= (ec ec)ec (1.12)
The curvature in terms of the spin connection can then be
defined as:
Rab=ab ab + ak bk ak bk (1.13)
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The relation between Rab(w) and R() satisfies:
R() =eaebR
ab(w) (1.14)
The veilbein and the spin connection were actually introduced
when
Utiyama [5], in 1955, followed by Kibble [6] in 1960 , attempted
to construct
gravity as a gauge theory in which the lagrangian is invariant
under general
inhomogeneous lorentz transformations. This has been done by
introducing the
veirbein and the spin connection as gauge fields. It was then
understood [7] that if
we consider the veirbein2 and the spin connection as independent
gauge fields,
then the Einstein action would have a constrained symmetry where
the torsion is
set to zero.
Based on this understanding, Chamseddine was able to construct
[2, 3] a
topological gauge theory of gravity in five dimensions and then
to generalize it to
all odd dimensions. The action is based on the Chern-Simons
five-form given by
5 =
F F A 12
F A A A + 110
A A A A A
(1.15)
where F =dA + A A.
By letting
Aab =ab, Aa5 =ea, a= 1, ..., 5 (1.16)
the action
I5 = kM5
5 (1.17)
takes the form:
2In four dimensions, the veilbein is called veirbein, and in
five dimensions it is referred to asfunfbein.
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I5 = 3 abcde(eaRbc
Rde
2
3
ea
eb
ec
Rde +
1
5
2ea
eb
ec
ed
ee) (1.18)
The first term, which is quadratic in R, is known as a
Gauss-Bonnet term.
The second term, which is linear in R, is the Einstein term, and
the third term is
the cosmological constant term, where is a cosmological constant
that could take
the values:1, 0, 1.3
The equations of motion for this action has been derived by
Chamseddine[2, 3]. They are given by:
abcde(Rab ea eb) (Rcd ec ed) = 0 (1.19)
abcde(Rab ea eb) Tc = 0 (1.20)
whereTa =dea + ab eb (1.21)
is the torsion.
1.2 Aim and Organization
Our aim, in the second chapter, is to construct a spherically
symmetricfive-dimensional space-time configuration for which we
find exact solutions of the
above equations of motion. In the third chapter, we analyze
particular solutions of
physical interest. In the fourth chapter, we reduce our
five-dimensional action to a
four-dimensional action plus a scalar field, where the scalar
field is taken to be e44.
We then check that the obtained solutions are still valid.
3In this paper, we have relabelled in [3] to be.
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Chapter 2
Solutions for the Five-Dimensional
Equations of Motion
2.1 Constructing the Five-Dimensional
Spherically Symmetric Line Element
The general expression for the line element in 5 dimensions is
given by:
ds2 =gdxdx
where = 0, 1, 2, 3, 4.
We assume that the metric is isotropic in three of its spatial
coordinates
x1, x2, and x3.
Denote: xs=(x1, x2, x3) , x4 and x0 t .Because of isotropy, the
metric ds2 depends on x1, x2, and x3 only through
their rotational invariants: xs
xs
r2 , dxs
dxs,and xs
dxs. Thus, the most general
form of the metric must be:
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ds2 = A(t, r)dt2
B(t, r)dt(xs
dxs)
C(t, r)d(xs
dxs)
D(t, r)(xs
dxs)2
E(t, r)(dxs dxs) F(t, r)d2
Transform xs (x1, x2, x3) into spherical polar coordinates.
x1 = r sin cos
x2 = r sin sin
x3 = r cos (2.1)
The rotational invariants are now given by: xs xs=r2 ,xs dxs=rdr
,anddxsdxs=dr2 + r2d2 + r2 sin2 d2.
ds2 = A(t, r)dt2 B(t, r)rdrdt C(t, r)rdrd D(t, r)r2dr2
E(t, r)(dr2 + r2d2 + r2 sin2 d2) F(t, r)d2 (2.2)
Collecting together terms, absorbing factors of r into our
functions, and
redefining A,B,C,D,E, and F, we get:
ds2 = A(t, r)dt2 B(t, r)drdt C(t, r)drd D(t, r)dr2
E(t, r)(d2 + sin2 d2) F(t, r)d2 (2.3)
Define a new radial coordinate by r2 =E(t, r). Again by
redefining A, B, C, D,
and F, we get:
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ds2 = A(t, r)dt2
B(t, r)drdt
C(t, r)drd
D(t, r)dr2
r2(d2 + sin2 d2) F(t, r)d2 (2.4)
Introduce a new time-like coordinate t defined by:
dt= (t, r)
A(t, r)dt 12
B(t, r)dr
(t, r) is an integrating factor that makes the RHS an exact
differential.
Squaring, we get:
dt2 = 2
A2dt2 ABdrdt +14
B2dr2
Thus,
Adt2 Bdrdt = 1A2
dt2 B2
4Adr2
Define new functions:
A= 1
A2
B = B2
4A+ D
The metric now takes the form:
ds2 = A(t, r)dt2 B(t, r)dr2 C(t, r)drd r2(d2 + sin2 d2) F(t,
r)d2 (2.5)
Similarly, introduce a new spacelike coordinate:
d= (t, r)[F(t, r)d+12
C(t, r)dr]
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(t, r) is an integrating factor that makes the RHS an exact
differential.
Squaring, we get:
d2 = 2[F2d2 + CF drd+1
4C2dr2]
Thus,
F d2 + Cddr= 1
F2d2 C
2
4Fdr2
Define new functions:
F = 1
F2
C= B C2
4F
The metric now becomes diagonal and takes the form:
ds2 = A(t, r)dt2 B(t, r)dr2 r2(d2 + sin2 d2) C(t, r)d2 (2.6)
We take as an additional assumption the metric to be static,
i.e. A,B, and
C are now independent of t. Redefining the functions A,B, and C
and removing
the bars from the coordinates, the metric takes the form:
ds2 =A2(r)dt2 B2(r)dr2 r2(d2 + sin2 d2) C2(r)d2 (2.7)
2.2 Equations of Motion in Component Form
As mentioned before, the equations of motion derived from the
five-dimensional
action are given by:
abcde(Rab
ea
eb)
(Rcd
ec
ed) = 0 (2.8)
abcde(Rab ea eb) Tc = 0 (2.9)
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where
Ta =dea + ab eb (2.10)
is the torsion, and the curvature R is given by:
Rab =dab + ac bc (2.11)
where is the spin connection and e is the funfbein.
Considering both the funfbein and the spin connection as gauge
fields, the
Einstein action becomes invariant only under a constrained
symmetry for which the
torsion is set to zero[3]. Therefore, we are only interested in
the case where Ta = 0
Writing the first set of equations of motion in component form
we obtain:
abcde(Rab eaeb)(Rcd eced) = 0 (2.12)
Of course, this is a set of 25 different equations of motion
where
e= 1, 2, 3, 4, 5 and = 1, 2, 3, 4, 5.
Our aim is to find a solution for the equations of motion using
the metric
ansatz derived above.
Relabeling dummy indices in the above equation and using the
fact that:
abcde= e[ae
be
ce
de
]e (2.13)
The equations of motion take the form:
abcdeRabR
cd 2(2!)e[cede]eRcd+ 4!2eedet(e1) (2.14)
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2.3 Calculating the Curvature tensor
components
Remember that our static, spherically symmetric metric is given
by:
ds2 =A2(r)dt2 B2(r)dr2 r2(d2 + sin2 d2) C2(r)d2 (2.15)
In terms of the funfbeins, the line element is given by:
ds2 =abeae
bdx
dx (2.16)
where in our case, the funfbein has the form:
A(r) 0 0 0 0
0 B(r) 0 0 00 0 r 0 0
0 0 0 r sin() 0
0 0 0 0 C(r)
(2.17)
ab is the five dimensional flat space-time metric with signature
(+,-,-,-,-).
Since we are only interested in the case where torsion is set to
zero, we can
derive the spin connections from the funfbeins using equation
1.11.
The non-zero s turn out to be:
010 = 100 = A(r)
B(r)
122 = 212 = 1
B(r)
133 = 313 = sin B(r)
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233 = 323 = cos
144 = 414 = C(r)
B(r) (2.18)
(2.19)
where prime denotes derivatives with respect to r, and dotted
indices denote
curved indices.
The curvatures are then calculated from the spin connections
using
equation 1.13.
Rab=ab ab + ak bk ak bk (2.20)
After calculating the curvature tensor components, we discover
that the
only non-zero components are:
R0101 =
A(r)B(r)
B(r)2 A(r)
B(r)
R0202 = A(r)
B(r)2
R0303 = A(r)
B(r)2sin
R0404 = A(r)C(r)
B(r)2
R1212 = B(r)
B(r)2
R1313 = B(r)B(r)2
sin
R1414 = C(r)B(r)
B(r)2 +
C(r)
B(r)
R2323 = sin
1
B(r)2 1
R2424 = C(r)
B(r)2
R3434 = C(r)
B(r)2
sin
(2.21)
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where here also prime denotes derivatives with respect to r.
2.4 Equations of Motion
Plugging the curvature tensor components obtained into the 25
equations of
motion, we notice that there are only 5 non-trivial equations,
those for which
e= r, from which we obtain 3 independent equations (e=r=0,1,5)
for the three
unknowns A(r), B(r), and C(r).
The simplified equations obtained are the following:
242B(r)r2C(r)sin 16rC(r)sin
B(r) 8C(r)sin
B(r) + 8B(r)C(r)sin
+8r2C(r)B(r)sin
B(r)2 8r
2C(r)sin
B(r) +
16rC(r)B(r)sin
B(r)2
24B(r)C(r)sin
B(r)4 +
8B(r)C(r)sin
B(r)2 8C
(r)sin
B(r)3 +
8C(r)sin
B(r) = 0
242r2A(r)C(r)sin 16rA(r)C(r)sin
B(r)2 8A(r)C(r)sin
B(r)2
+8A(r)C(r)sin 8r2A(r)C(r)sin
B(r)2 16rA
(r)C(r)sin
B(r)2
+24A(r)C(r) sin
B(r)4 8A(r)C(r)sin
B(r)2 = 0
242r2A(r)B(r)sin 8A(r)sin B(r)
+ 8A(r)B(r)sin +16rA(r)B(r)sin
B(r)2
16rA(r)sin B(r)
+8r2A(r)B(r)sin B(r)2
8r2A(r)sin B(r)
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24A(r)B(r)sin
B(r)4 +
8A(r)B(r)sin
B(r)2 +
8 sin A(r)
B(r)3 8sin A
(r)
B(r) = 0
2.5 Solutions For Equations of Motion
After inserting the above three equations into MAPLE, we
obtained 7 different
solutions.
First let us state the solutions obtained:
solution 1:
A(r) = 0
B(r) = 33r2 + 3
C(r) =C1LegendreP
10 1
2, 52
,r
+ C2LegendreQ
10 1
2, 52
,r
(r2 + 1)1/4
(2.22)
solution 2:
A(r) = C1
r2 + 1
B(r) = 1r2 + 1
C(r) = C(r)
(2.23)
solution 3:
A(r) = C3r
B(r) = 3
9r2
+ 3C(r) = C1r+ C2
3r2 + 1
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(2.24)
solution 4:
A(r) = 0
B(r) = B(r)
C(r) = Y(r)
(2.25)
such that Y(r) is the solution for
1B(r)(B(r)2r2 + 1 B(r)2) (Y
(r)B(r)3r2 Y(r)B(r) + Y(r)B(r)3
2Y(r)rB(r)3 Y(r)B(r)2r2B(r) + 3Y(r)B(r) + Y(r)B(r)2B(r)
+3B(r)52Y(r)r2 + B(r)3Y(r) B(r)5Y(r) + 2B(r)2Y(r)B(r)r) = 0
(2.26)
solution 5:
A(r) = A(r)
B(r) = 1r2 + 1
C(r) = C1
r2 + 1
(2.27)
solution 6:
B(r) = B(r)
C(r) = 0
A(r) = Y(r)
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(2.28)
such that Y(r) is the solution for
1B(r)(B(r)2r2 + 1 B(r)2) (Y
(r)B(r)3r2 Y(r)B(r) + Y(r)B(r)3
2Y(r)rB(r)3 Y(r)B(r)2r2B(r) + 3Y(r)B(r) + Y(r)B(r)2B(r)
+3B(r)52Y(r)r2 + B(r)3Y(r) B(r)5Y(r) + 2B(r)2Y(r)B(r)r) = 0
(2.29)
solution 7:
A(r) = r
C2
exp
2 + B(r)2r2
r dr
dr+ C3
B(r) = 1r2 + C4
C(r) = C1exp
B(r)2(3A(r)r2B(r)2 + 2rA(r) + A(r)
B(r)2A(r))
2B(r)2A(r)r+ B(r)2r2A(r) + 3A(r) A(r)B(r)2 dr(2.30)
Note that in all solutions C1, C2, C3, and C4 are constants of
integration.
Notice that in the first solution, we have A(r) = 0, which in
other words
means that the time component in our line element vanishes. This
makes the
solution non-physical. We will thus discard this solution.
For the same reason, we discard solution 4 that contains A(r) =
0.
Solution 6 which contains C(r) = 0 will also be discarded
because it is not
five-dimensional.
We are left out with 4 solutions: solutions 2, 3, 5,and 7.
solution 7 turns out to be mathematically complicated and will
not be
discussed in this paper.
The remaining three solutions will be analyzed throughout the
following
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chapter.
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Chapter 3
Analyzing the Physical Solutions
In this chapter, we analyze the three physical solutions that we
have obtained.
3.1 Analyzing the First Physical Solution
The first physical solution to be analyzed is solution 2. Recall
that it is given by:
A(r) = C1
r2 + 1
B(r) = 1r2 + 1
C(r) = C(r)
(3.1)
where C1 is a constant of integration.
Plugging the solutions for A,B, and C into the metric ansatz, we
obtain:
ds2 =C21(r2 + 1)dt2 1
r2 + 1dr2 r2d2 r2 sin2 d2 C(r)2d2 (3.2)
We absorb the constants of integration by redefining our
coordinates such that:
t C1t
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The metric finally becomes:
ds2 = (r2 + 1)dt2
1
r2
+ 1
dr2
r2d2
r2 sin2 d2
C(r)2d2 (3.3)
for =0 the metric takes the form:
ds2 =dt2 dr2 r2d2 r2 sin2 d2 C(r)2d2 (3.4)
for =1 the metric takes the form:
ds2 = (r2 + 1)dt2 1r2 + 1
dr2 r2d2 r2 sin2 d2 C(r)2d2 (3.5)
for =-1 the metric takes the form:
ds2 = (r2 + 1)dt2 1r2 + 1 dr2 r2d2 r2 sin2 d2 C(r)2d2 (3.6)
It is clear that no singularities exist for = 0, 1. However, for
= 1, wehave:
B(r) for r= 1
To check whether this singularity is a coordinate singularity or
a real
singularity, we need to calculate RR for this metric.
To calculate this term, we start by calculating the Christoffel
symbols.
These are given by equation (1.3). The non-zero components turn
out to be:
001 = 010=
r
r
2
+ 1100 = (r
2 + 1)r
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111 = r
r2 + 1
122 = (r2 + 1)r
133 = (r2 + 1)r sin2
144 = (r2 + 1)C(r)C(r)
212 = 212=
1
r
233 = sin cos
313 = 331=
1
r
3
32
= 3
23
=cos
sin
414 = 441=
C(r)
C(r) (3.7)
From the Christoffel symbols we can calculate the components of
the
Reimann curvature tensor, given by equation (1.4).
The non-zero components are then given by:
R0101 =
R0202 = (r2 + 1)r2
R0303 = (r2 + 1)r2 sin2
R0404 = (r2 + 1)rC(r)C(r)
R1212 = r2
r2 + 1
R1313 = r2
r2 + 1sin2
R1414 = C(r)(C(r)r2 + C(r) + rC(r))
r2 + 1
R2323 = r4 sin2
R2424 = (r2 + 1)rC(r)C(r)
R3434 = (r2 + 1)rC(r)C(r)sin2
(3.8)
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Taking= 1, We can now calculate RR. The result obtained is:
RR =
16r2C(r)2
C(r)2 16C
(r)2
C(r)2 +
8C(r)2
C(r)2r2+
4r4C(r)2
C(r)2
8r2C(r)2
C(r)2 +
8r3C(r)C(r)
C(r)2 +
4C(r)2
C(r)2 8rC
(r)C(r)
C(r)2 + 24
We notice that the above expression does not blow up for r= 1,
as long as
C(r) is a well-behaved function having no roots. Therefore, we
conclude that thisis a coordinate singularity.
3.1.1 Geodesic Equations
Now let us derive the geodesic equations for this metric.
We can now derive the geodesic equations for the coordinates:
t,r,,, from the
following formula:
x + xx = 0
where dot represents the derivative with respect to an affine
parameter .
However,we choose to derive the geodesic equaitons by varying a
Lagrangian. The
Lagrangian, L= gxx, for this metric is given by:
L= (r2 + 1)t2 1r2 + 1
r2 r22 r2 sin2 2 C(r)22 (3.9)
The geodesic equations are then obtained from the
Euler-Lagrange
equations:
dd
Lx
Lx
= 0 (3.10)
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d
d2(r2 + 1)t= 0
r
r2 + 1+ rt2 r
(r2 + 1)2 r(2 + sin2 2) C(r)C(r)2 = 0
+2r
r sin cos 2 = 0d
d
2r2 sin2
= 0
d
d
2C(r)2
= 0
(3.11)
This set of equations can be easily simplified to:
(r2 + 1)t= k
r
r2 + 1+ rt2 r
(r2 + 1)2 r(2 + sin2 2) C(r)C(r)2 = 0
+2r
r sin cos 2 = 0
r2= h
C(r)2=l
(3.12)
where k has the significance of energy and h and l have the
significance of angularmomenta along the and axes respectively.
Note that = 2
is a solution for the third equation. Since our metric is
spherically symmetric, we can, without loss of generality,
confine our attention to
particles moving in the equatorial plane, = 2
.
The 4 remaining equations now become:
(r2 + 1)t= k
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r
r2 + 1+ rt2 r
(r2 + 1)2 r2 C(r)C(r)2 = 0
r2= h
C(r)2=l
(3.13)
This set of equations is valid for both null and non-null
geodesics.
We now replace the complicated r-equation by a first integral of
the
geodesic equations.
For a non-null geodesic, this is given by:
gxx = 1 (3.14)
where we have taken c2 = 1.
For a null geodesic, it is given by:
gxx = 0 (3.15)
We are interested in trajectories of photons. Therefore, the set
of null geodesic
equations becomes:
(r2 + 1)t= k
(r2 + 1)t2 1r2 + 1
r2 r22 C(r)22 = 0
r2= h
C(r)2=l
(3.16)
To perform a coordinate transformation that will get rid of the
coordinate
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singularity, we use the analogous to Eddington-Finkelstein
coordinates. Therefore,
we are interested in the radial motion of photons. For radial
motion, we have
= 0 and = 0.
The first integral reduces to:
(r2 + 1)t2 1r2 + 1
r2 = 0 (3.17)
which upon simplification gives:
dr
dt = (r2
+ 1) (3.18)
for = 1 we have:dt= dr
1 r2 (3.19)
upon integration, this gives:
t= 1
2ln1
r
1 + r + constant (outgoing photons)
t= +1
2ln
1 r1 + r + constant (incoming photons) (3.20)
3.1.2 Coordinate Transformations
We will now perform a series of coordinate transformations that
will get rid of the
coordinate singularities.
First, denote the above integration constants by u and v and
take them as new
coordinates.
We have:
u= t +1
2ln
1 r1 +r
(outgoing photons)
v=t 12
ln1 r1 + r
(incoming photons) (3.21)25
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differentiating we get:
du= dt 11 r2 dr
dv= dt + 1
1 r2 dr (3.22)
Multiplying du and dv together by (1 r2) we get:
(1 r2)dudv= (1 r2)dt2 11 r2 dr2 (3.23)
Substituting into the line element, we get:
ds2 = (1 r2)dudv r2d2 r2 sin2 d2 C(r)2d2 (3.24)
r is now viewed as a function ofu and v defined implicitly
by:
ln
1 r1 + r = u v2 (3.25)
From the above equation we can rewrite equation (3.24) as
ds2 = (1 +r)2 exp
v u2
dudv r2d2 r2 sin2 d2 C(r)2d2 (3.26)
Now let
V= expv
2U= exp
u2
(3.27)
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Differentiating, we get:
dV =1
2
exp v
2 dv
dU=1
2exp
u2
du (3.28)
We then multiply dU by dV to get:
dUdV =1
4exp
v u
2
dudv (3.29)
Substituting back into the line element, we get:
ds2 = 4(1 + r)2dUdV r2d2 r2 sin2 d2 C(r)2d2 (3.30)
Notice that the line element has become well behaved at r= 1. We
have
got rid of the singularity at r= 1 by performing this series of
coordinate
transformations.
3.2 Analyzing the Second Physical Solution
The second physical solution to be analyzed is solution 3.
Recall that it is given by:
A(r) = C3r
B(r) = 3
9r2 + 3
C(r) = C1r+ C2
3r2 + 1
(3.31)
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where C1, C2, and C3 are constants of integration.
Plugging the solutions for A,B, and C into the metric ansatz, we
obtain:
ds2 =C23r2dt2 3
3r2 + 1dr2r2d2r2 sin2 d2(C1r+C23r2 + 1)2d2 (3.32)
We absorb the constants of integration by redefining our
coordinates such that:
t C3t and C1.We also denote C2
C1by C, where C is some integration constant that can be fixed
by
boundary conditions.
The metric finally becomes:
ds2 =r2dt2 33r2 + 1
dr2 r2d2 r2 sin2 d2 (r+ C
3r2 + 1)2d2 (3.33)
for =0 the metric takes the form:
ds2
=r2
dt2
3dr2
r2
d2
r2
sin2
d2
(r+ C)2
d2
(3.34)
for =1 the metric takes the form:
ds2 =r2dt2 33r2 + 1
dr2 r2d2 r2 sin2 d2 (r+ C
3r2 + 1)2d2 (3.35)
for =-1 the metric takes the form:
ds2 =r2dt2 33r2 + 1 dr2 r2d2 r2 sin2 d2 (r+ C
3r2 + 1)2d2 (3.36)
It is clear that no singularities exist for = 0, 1. However, for
= 1, wehave:
B(r) for r= 13
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Again, to check whether this singularity is a coordinate or real
singularity,
we need to calculate RR for this metric.
We start by calculating the Christoffel symbols using equation
(1.3). The non-zero
components turn out to be:
001 = 010=
1
r
100 = 1
3(3r2 + 1)r
111 = 3r
3r2 + 1
122 = 1
3(3r2 + 1)r
133 = 1
3(3r2 + 1)r sin2
144 = 1
3
3r2 + 1
r+ C
3r2 + 1
3r2 + 1 + 3Cr
212 = 212=
1
r
233 =
sin cos
313 = 331=
1
r
332 = 323=
cos
sin
414 = 441=
3r2 + 1 + 3Cr
(r+ C
3r2 + 1)
3r2 + 1(3.37)
(3.38)
The components of the Reimann curvature tensor are calculated
from
equation (1.4). The non-zero components are given by:
R0101 = 3r2
3r2 + 1
R0202 = 13
(3r2 + 1)r2
R0303 = 13
(3r2 + 1)r2 sin2
R0404 = 133r2 + 1r
r+ C3r2 + 1
3r2 + 1 + 3Cr
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R1212 = 3r2
3r2 + 1
R1313 = 3r2
3r2 + 1sin2
R1414 =3
3r2 + 1r2 + 6r3C+ 2Cr+ 33r2 + 1C2r2 + 3r2 + 1C2(3r2 + 1)3/2
R2323 = 1
3r2 sin2 (2 + 3r2)
R2424 = 1
3
3r2 + 1r
r+ C
3r2 + 1
3r2 + 1 + 3Cr
R3434 = 1
3
3r2 + 1r
r+ C
3r2 + 1
3r2 + 1 + 3Cr
sin2
(3.39)
Taking= 1, We can now calculate RR. The result obtained is:
RR = 720r4
C2
(r+ C3r2 + 1)4 + 360r
4
C4
(r+ C3r2 + 1)4 + 40r
4
(r+ C3r2 + 1)4
480C3r33r2 + 1
(r+ C3r2 + 1)4 +
160Cr33r2 + 1
(r+ C3r2 + 1)4
312C2r2
(r+ C3r2 + 1)4
240C4r2
(r+ C3r2 + 1)4
8r2
(r+ C3r2 + 1)4 +
184C3r3r2 + 1
(r+ C3r2 + 1)4
24Cr3r2 + 1
(r+ C3r2 + 1)4
76C2
(r+ C3r2 + 1)4 +
4
(r+ C3r2 + 1)4
+
64C4
(r+ C3r2 + 1)4 40C3
3r2 + 1
(r+ C3r2 + 1)4r + 40C
3r2 + 1
3(r+ C3r2 + 1)4r+
52
3
C2
(r+ C3r2 + 1)4r2
16C4
(r+ C3r2 + 1)4r2 +
8
3
C4
(r+ C3r2 + 1)4r4
+32
3
C33r2 + 1
(r+ C3r2 + 1)4r3
We notice that the above expression does not blow up for r=
13
. Therefore, we
conclude that this is a coordinate singularity.
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3.2.1 Geodesic Equations
Let us derive the geodesic equations for this metric. Again, we
derive the geodesic
equaitons by varying a Lagrangian. The Lagrangian L= gxx for
this metric is
given by:
L= r2t2 33r2 + 1
r2 r22 r2 sin2 2 (r+ C
3r2 + 1)22 (3.40)
The geodesic equations are then obtained from the
Euler-Lagrange
equations. They are given by:
d
d
2r2t
= 0
3r
3r2 + 1+ rt2 9r
(3r2 + 1)2r2 r(2 + sin2 2)
+(r+ C3r2 + 1)
1 + 3Cr3r2 + 1
2 = 0
+2r
r sin cos 2 = 0d
d
2r2 sin2
= 0
d
d
2(r+ C
3r2 + 1)2
= 0
(3.41)
This set of equations can be easily simplified to:
r2t= k
3r
3r2 + 1+ rt2 9r
(3r2 + 1)2r2 r(2 + sin2 2)
+(r+ C3r2 + 1) 1 + 3Cr3r2 + 1
2 = 0
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+2r
r sin cos 2 = 0
r2 sin2 = h
(r+ C3r2 + 1)2=l
(3.42)
where k has the significance of energy and hand l have the
significance of angular
momenta along the and axes respectively.
Because of spherical symmetry, we can, without loss of
generality, confine
our attention to particles moving in the equatorial plane, = 2
.
The remaining equations now become:
r2t= k
3r
3r2 + 1+ rt2 9r
(3r2 + 1)2r2 r2
+(r+ C
3r2 + 1) 1 + 3Cr
3r2
+ 1
2 = 0
r2= h
(r+ C
3r2 + 1)2=l
(3.43)
This set of equations is valid for both null and non-null
geodesics.
We are interested in trajectories of photons. Therefore, we
replace the
r-equation by gxx = 0.
r2t= k
r2t2 33r2 + 1
r2 r22 r2 sin2 2 (r+ C
3r2 + 1)22 = 0
r2= h
(r+ C
3r2 + 1)2=l
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(3.44)
Again, we are interested in the radial motion of photons. For
radial
motion, we have = 0 and = 0.
The first integral reduces to:
r2t2 33r2 + 1
r2 = 0 (3.45)
which upon simplification gives:
dt=
3r2(3r2 + 1)
dr (3.46)
for = 1 we have:dt=
3
r2(3r2 + 1)dr (3.47)
After integrating, we get:
t=
3tanh1(
1 3r2) + constant (outgoing photons)
t=
3tanh1(
1 3r2) +constant (incoming photons) (3.48)
3.2.2 Coordinate Transformations
We now perform a series of coordinate transformations to get rid
of the coordinate
singularity.
First, denote the above integration constants by u and v and
take them as
new coordinates. We have:
u= t +
3tanh1(
1 3r2)
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v= t
3tanh1(
1 3r2)
(3.49)
differentiating we get:
du= dt +
3
r2(3r2 + 1)dr
dv= dt
3
r2(3r2 + 1)
dr
(3.50)
Multiplying du and dv together by r2 we get:
r2dudv =r2dt2 31 3r2 dr
2 (3.51)
Substituting into the line element, we get:
ds2 =r2dudv r2d2 r2 sin2 d2 (r+ C
3r2 + 1)2d2 (3.52)
Now let
T =u+ v
2
X= u
v
2 (3.53)
which gives dT2 dX2 =dudv.In terms of the new coordinates T and
X, the line element takes the form
ds2 =r2dT2 r2dX2 r2d2 r2 sin2 d2 (r+ C
3r2 + 1)2d2 (3.54)
Notice that the line element has become well behaved at r=
13
.
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The relation between the new coordinates T and X and the old
coordinates t and r is now given implicitly by:
T= U+ V
2 =t
X= U V
2 =
3tanh1(
1 3r2) (3.55)
(3.56)
We have got rid of the singularity at r= 13
by performing this series of
coordinate transformations.
3.3 Analyzing the Third Physical Solution
The third physical solution to be analyzed is solution 5. Recall
that it is given by:
A(r) = A(r)
B(r) = 1r2 + 1
C(r) = C1
r2 + 1
(3.57)
where C1 is a constant of integration.
Plugging the solutions for A,B, and C into the metric ansatz, we
obtain:
ds2 =A(r)2dt2 1r2 + 1
dr2 r2d2 r2 sin2 d2 C21 (r2 + 1)d2 (3.58)
We absorb the constants of integration by redefining our
coordinates such that:
C1.
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The metric finally becomes:
ds2 =A(r)2dt2
1
r2
+ 1
dr2
r2d2
r2 sin2 d2
(r2 + 1)d2 (3.59)
for =0 the metric takes the form:
ds2 =A(r)2dt2 dr2 r2d2 r2 sin2 d2 d2 (3.60)
for =1 the metric takes the form:
ds2 =A(r)2dt2 1r2 + 1
dr2 r2d2 r2 sin2 d2 (r2 + 1)d2 (3.61)
for =-1 the metric takes the form:
ds2 =A(r)2dt2 11 r2 dr
2 r2d2 r2 sin2 d2 (1 r2)d2 (3.62)
It is clear that no singularities exist for = 0, 1. However, for
= 1, wehave:
B(r) for r= 1
To check whether this singularity is real or coordinate, we need
to
calculate RR for this metric.
We start by calculating the Christoffel symbols from equation
(1.3). The
non-zero components turn out to be:
001 = 010 =
A(r)
A(r)100 = (r
2 + 1)A(r)A(r)
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111 = r
r2 + 1
122 = (r2 + 1)r
133 = (r2 + 1)r sin2
144 = (r2 + 1)r
212 = 212 =
1
r
233 = sin cos
313 = 331 =
1
r
3
32
= 3
23
=cos
sin
414 = 441 =
r
r2 + 1
(3.63)
We now calculate the components of the Reimann curvature tensor
using
equation (1.4). The non-zero components are then given by:
R0101 = A(r) (A(r)r2 + A(r) + A(r)r)
r2 + 1
R0202 = (r2 + 1)A(r)A(r)r
R0303 = (r2 + 1)A(r)A(r)r sin2
R0404 = (r2 + 1)A(r)A(r)r
R1212 =
r2
r2 + 1
R1313 = r2
r2 + 1sin2
R1414 =
R2323 = r4 sin2
R2424 = (r2 + 1)r2
R3434 = (r2 + 1)r2 sin2
(3.64)
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Taking= 1, We can calculate RR. The result obtained is:
RR = 24 +
16r2A(r)2
A(r)2 16A
(r)2
A(r)2 +
8A(r)2
A(r)2r2+
4r4A(r)2
A(r)2
8r2A(r)2
A(r)2 +
8r3A(r)A(r)
A(r)2 +
4A(r)2
A(r)2 8rA
(r)A(r)
A(r)2
We notice that the above expression does not blow up for r= 1 if
A(r) is a
well-behaved function that has no roots. Therefore, we conclude
that thissingularity is a coordinate singularity.
3.3.1 Geodesic Equations
The Lagrangian L= gxx for this metric is given by:
L= A(r)2t2 1r2 + 1
r2 r22 r2 sin2 2 (r2 + 1)2 (3.65)
The geodesic equations are then obtained from the
Euler-Lagrange
equations:
d
d
2A(r)2t
= 0
r
r2 + 1A(r)A(r)t2 r
(r2 + 1)2r2 r(2 sin2 2) r2 = 0
+2r
r sin cos 2 = 0d
d
2r2 sin2
= 0
d
d2(r
2 + 1)= 0(3.66)
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This set of equations can be easily simplified to:
A(r)2t= k
r
r2 + 1A(r)A(r)t2 r
(r2 + 1)2r2 r(2 sin2 2) r2 = 0
+2r
r sin cos 2 = 0
r2 sin2 = h
(r2 + 1)=l
(3.67)
where k has the significance of energy, and h and l have the
significance of angular
momenta along the and axes respectively.
As in the previous two cases, we can confine our attention to
particles
moving in the equatorial plane, = 2
The 4 remaining equations now become:
A(r)2t= k
r
r2 + 1A(r)A(r)t2 r
(r2 + 1)2r2 r2 r2 = 0
r2= h
(r2 + 1)=l
(3.68)
This set of equations is valid for both null and non-null
geodesics.
We are interested in trajectories of photons. Therefore, we
replace the
r-equation by the first integral for null geodesics.
A(r)2t= k
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A(r)2t2 1r2 + 1
r2 r22 r22 (r2 + 1)2 = 0
r2= h
(r2 + 1)=l
(3.69)
For radial motion, we have = 0 and = 0. The first integral
reduces to:
A(r)2t2 1r2 + 1
r2 = 0 (3.70)
which upon simplification gives:
dt= drA(r)2(r2 + 1)
(3.71)
for = 1 we have:dt= dr
A(r)2(r2 + 1)
(3.72)
t=
drA(r)2(r2 + 1)
(3.73)
which of course cannot be solved unless A(r) is given.
However, let dr
A(r)2(r2+1)=F(r) + constant , then:
t= F(r) + constant (outgoing photons)
t= F(r) + constant (incoming photons) (3.74)
3.3.2 Coordinate Transformations
Denote the above integration constants by u and v and take them
as new
coordinates. We have:
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u= t + F(r)
v= t F(r)
(3.75)
Differentiating we get:
du= dt + dr
A(r)2(3r2 + 1)
dv= dt drA(r)2(3r2 + 1)
(3.76)
(3.77)
Multiplying du and dv together by A(r)2 we get:
A(r)2dudv =A(r)2dt2 11 r2 dr
2 (3.78)
Substituting into the line element, we get:
ds2 =A(r)2dudv
r2d2
r2 sin2 d2
(1
r2)d2 (3.79)
Now let
T =u + v
2
X= u v
2
which gives dT2 dX2 =dudv.
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In terms of the new coordinates T and X, the line element takes
the form
ds2 =A(r)2dT2
A(r)2dX2
r2d2
r2 sin2 d2
(1
r2)d2 (3.80)
The relation between the new coordinates T and X and the old
coordinates t and r is now given implicitly by:
T =u+ v
2
=t
X= u v
2 =F(r)
(3.81)
Notice that the above line element still has a singularity at r=
1. This
singularity appears in g44.
To get rid of this singularity, we consider motion of photons in
the-direction. Let us first rewrite the first integral of motion
for photons in terms of
the new coordinates. This is given by:
A(r)2 T2 A(r)2 X2 r22 r2 sin2 2 (r2 + 1)2 = 0 (3.82)
We now consider motion in the equatorial plane and, in
particular, motion along
the -direction. Therefore we have: = 2
, X= 0, and = 0.
Now, for = 1, the first integral becomes:
A(r)2 T2 (1 r2)2 = 0 (3.83)
which upon solving gives
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T =
1 r2|A(r)| + constant (3.84)
Denote the integration constants by xand y respectively and take
them as the new
coordinates.
x = T
1 r2|A(r)|
y = T+
1 r2
|A(r)|
(3.85)
Derive w.r.t to get:
dx
d
= dT
d
1 r2
|A(r)|dy
d =
dT
d +
1 r2|A(r)|
(3.86)
Multiplying the above equations together by A(r)2d2 and
replacing inside the line
element, it becomes:
ds2 =A(r)2dxdy A(r)2dX2 r2d2 r2 sin 2d2 (3.87)
Now, let
U = y+ x
2
V = y x
2(3.88)
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The line element then becomes:
ds2 =A(r)2dU2
A(r)2dX2
r2d2
r2 sin 2d2
A(r)2dV2 (3.89)
The relation between the new coordinate U and Vand the old
coordinates T and
is given by:
U = x+ y
2 =T
V = y x2
= 1 r2|A(r)|(3.90)
We have got rid of the singularity at r= 1 by performing this
series of
coordinate transformations. Note, however, that A(r) should be a
well-behaved
function that has no roots.
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Chapter 4
Dimensionally Reducing the
Action
In this chapter, we will reduce the action in 5-dimensional
spacetime of part 1 into
an action in four dimensional spacetime plus a scalar field. We
check that the
same set of solutions is still valid.
Recall that the action in five-dimensional spacetime is given
by:
I5= 3
abcde(eaRbcRde 2
3eaebecRde +1
52eaebeced ee) (4.1)
which in component form becomes:
I5 = 3
abcde(eaR
bcR
de
2
3eae
be
cR
de+
1
52eae
be
ce
de
e) (4.2)
where of course the Greek indices are curved indices, and Latin
indices are tangent
space indices.
This action is dimensionally reduced by Chamseddine [3] as
follows:
First, let = 4, then
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I4= 3k M4
d4xabcde(eaR
bcR
de4+ e
a4R
bcR
de
2eae
be
c4R
de
43
eaebe
cR
de4+
2eaebe
ce
de
e4) (4.3)
Second, rewrite the above expression in terms of the
four-dimensional
fields. The action is then given by:
I4 = 3kM
4
d4xabcd(e4R
abR
cd4+ 2e
aR
bcR
d44+ 2e
aR
b4R
cd4
+e44RabR
cd+ 4e
a4R
bcR
d4
8
3eae
be
cR
d44
4eaebe44Rcd 4eaebe4Rcd4 4e4eaeb4Rcd 4eaebec4Rd4+2eae
be
crhoe
de
44+ 4
2eaebe
ce
4e
d4) (4.4)
(4.5)
where
Rab=ab ab + ak bk ak bk+ a4 b4 a4 b4 (4.6)
We use this formula to calculate the curvature tensor components
from
the veilbeins and spin connections in chapter 1. For our choice
of metric ansatz,
the curvature tensor components turn out to remain the same.
Replacing the values of the curvature tensors and veirbeins in
the action,
we get an action in terms of the fields A(r), B(r), and
C(r).
This system corresponds to gravity coupled to a vector and a
scalar, where
C(r) =e44
is the scalar field and is called a dilaton. The vector field
corresponds to
non-diagonal components in the metric and turns out to be zero
in this case.
The Lagrangian for this action is then given by:
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L=
8sin C(r)r2A(r)
B(r)
+12A(r)C(r)sin
B(r)3
12A(r)B(r)C(r)sin
B(r)4
+8sin rC(r)A(r)
B(r)3 +
4sin A(r)C(r)B(r)
B(r)2 + 8 sin B(r)C(r)A(r)
8sin C(r)A(r)B(r)
+8sin C(r)A(r)B(r)
B(r)2 +
8sin rA (r)C(r)
B(r)3
4sin A(r)C(r)
B(r) 4sin A(r)C
(r)
B(r) +
4sin A(r)C(r)
B(r)3
+8sin C(r)A(r)
B(r)3 24sin C(r)A
(r)B(r)
B(r)4 +
16 sin C(r)A(r)rB (r)
B(r)2
+24A(r)B(r)r2 sin 2C(r) +16 sin r2
A(r)C(r)B(r)B(r)2 8sin C(r)A(r)B(r)+
8sin C(r)r2A(r)B(r)
B(r)2 24rB
(r)sin A(r)C(r)
B(r)4 16sin C(r)rA
(r)
B(r)
16sin r2A(r)C(r)
B(r) 16sin r
2A(r)C(r)
B(r) 32 sin rA(r)C
(r)
B(r)
(4.7)
The equations of motion are then obtained by using the
Euler-Lagrange
equations for A(r), B(r), and C(r).
Since the Lagrangian contains second derivatives, the
generalized
Euler-Lagrange equations are given by:
L
r
L
(r)
+
2
r2
L
(2r)
= 0 (4.8)
where (A(r), B(r), C(r)).The equations of motion for A,B, and C
respectively are then given by:
24B(r)r2 sin 2C(r) 8sin r2C(r)
B(r) 16 sin rC
(r)
B(r) 24B
(r)C(r)sin
B(r)4
+8sin C(r)B(r)
B(r)2 + 8 sin B(r)C(r) 8sin C(r)
B(r) +
8sin r2C(r)B(r)
B(r)2
+16sin C(r)rB(r)
B(r)2 +8sin C
(r)B(r)3
8sin C(r)B(r)
= 0
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(4.9)
8sin r2A(r)C(r)
B(r)2 + 8 sin C(r)A(r) 8 sin A
(r)C(r)
B(r)2
16sin C(r)rA(r)
B(r)2 +
24A(r)C(r)sin
B(r)4 16 sin rA(r)C
(r)
B(r)2
8sin C(r)A(r)B(r)2
+ 24A(r)r2 sin 2C(r) = 0
(4.10)
8sin r2A(r)
B(r) + 8 sin B(r)A(r) 8sin A(r)
B(r) +
8sin A(r)B(r)
B(r)2
8sin A(r)
B(r) +
8 sin A(r)
B(r)3 24A
(r)B(r)sin
B(r)4 + 24A(r)B(r)r2 sin 2
+16sin A(r)rB (r)
B(r)2 +
8 sin r2A(r)B(r)
B(r)2 16sin rA
(r)
B(r) = 0
(4.11)
After solving this set of differential using MAPLE, we found out
that the
solutions turn out to be exactly the same as those obtained for
the action in
five-dimensional spacetime. As a double check, the solutions,
mentioned in chapter
2 above, were inserted in the above equations of motion in
equation 4.9, 4.10, and
4.11. It was verified that all solutions satisfy the above
equations.
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Chapter 5
Conclusion and Future Work
In the problem above, we considered a topological action for
gravity in five
dimensions from which equations of motion are derived. The
action contained a
quadratic term in R, the Gauss-Bonnet term; a linear term in R,
an Einstein term;
and a cosmological constant. We then took as an ansatz a static,
spherically
symmetric metric in five dimensions. The metric contained three
unknown
functions A(r), B(r), and C(r). After calculating the curvatures
from the spin
connections, we plugged into the equations of motion to get
three coupled,
independent differential equations in terms of A(r), B(r), and
C(r). This system is
completely solvable and turns out to have seven independent
solutions that were
found using MAPLE. Three of these solutions were analyzed for
singularities.
Neither of the analyzed solutions contained a singularity except
for a coordinatesingularity in the deSitter space for each. After
deriving the geodesic equations
that describe the radial motion of photons, we perform
coordinate transformations
to get rid of the coordinate singularities. These
transformations are analogous to
the Eddington-Finkelstein coordinate transformations applied
Schwarzschild
geometry. Next, we dimensionally reduced the action in five
dimensions to an
action in four dimensions that contained, in addition to the
four dimensional
metric components, a scalar field. The scalar field was taken to
be C(r). The
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scalar curvatures were then plugged in to obtain a lagrangian in
terms of the three
unknown functions A, B, and C. The equations of motion were then
obtained by
considering the generalized Euler-Lagrange equations for these
three fields. After
plugging the equations into MAPLE, they turned out to have the
same set of
solutions as the equations considered in five-dimensions.
As future work, we might work on finding solutions for these
equations of
motion by taking a non-static spherical metric rather than a
static one.
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Bibliography
[1] M. Alcubierre, Introduction to 3+1 Numerical Relativity,
International Series of
Monographs on Physics, Oxford Science Publications, pp. 1
32.
[2] A.H. Chamseddine, Applications of the Gauge Principle to
Gravitational
Interactions, Int.J.Geom.Meth.Mod.Phys.3:149-176, 2006.
[3] A.H. Chamseddine, Topological Gravity and Supergravity in
Various
Dimensions,Nuclear Physics B, Volume 346, Issue 1, p. 213-234,
1990.
[4] A.E. Blechman, A Mathematics Primer for Physics Graduate
Students , 2007.
[5] R. Utiyama, Invariant Theoretical Interpretation of
Interaction, Phys. Rev.
101, 15971607, 1956.
[6] T.W.B. Kibble Lorentz Invariance and the Gravitational
Field, Journal of
Mathematical Physics, Vol. 2, p.212-221, 1960.
[7] A.H. Chamseddine, Topological Gauge Theory of Gravity in
Five and All Odd
Dimensions, Physics Letters B, Volume 233, Issue 3-4, p.
291-294,1989.
[8] R.M. Wald,General RelativityThe University of Chicago Press,
Chicago and
London, 1984, pp. 148 160.
[9] M.P. Hobson, G. Efstathiou, and A.N. Lasenby, General
Relativity, An
Introduction for Physicists, Cambridge University Press, New
York, 2006.
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[10] C.W Misner, K.S. Thorne, and J.A. Wheeler,Gravitation, W.H.
Freeman and
Company, U.S.A, 1973.
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