The Study of Inhomogeneous Cosmologies Through Spacetime Matchings by Dan Giang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright c 2009 by Dan Giang
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The Study of Inhomogeneous Cosmologies Through
Spacetime Matchings
by
Dan Giang
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of PhysicsUniversity of Toronto
The constants a and b are introduced to ensure proper unit bookkeeping. Note that
it is possible to define a consistent time, radial and azimuthal coordinate throughout
many slices, whereas the ‘z’ coordinate is unique in each slice. This inability to define a
consistent ‘z’ coordinate causes some difficulty in defining what is meant by “thickness”.
This will be dealt with in section (3.1.3).
Landry and Dyer have investigated light propagation through the Cheese Slice uni-
verse in detail [68, 69]. Here we follow closely their method.
3.1.1 Null Vectors
To follow the path of the light that reaches the observer we must trace the null geodesics
in our model. A vector tangent to the null geodesic will be denoted ka = dxa/dτ ≡ x
with τ being an affine parameter and the over-dot, ˙ , denotes differentiation with respect
to τ . Let us choose the coordinates such that φ = constant and dφ = 0 and situate the
Chapter 3. Lookback Time and Observational Consequences 32
observer at r = 0. To be null ka is require to satisfy kaka = 0. The geodesic equation,
d2xa
dτ 2+ Γabc
dxb
dτ
dxc
dτ= 0 (3.3)
gives,
t+2
3at1/3(r2 + z2) = 0 (3.4)
tr +4
3tr = 0 (3.5)
tz +4
3tz = 0 (3.6)
in the FLRW regions. Equation (3.5) can be written as,
d
dτ(a2t4/3r) = 0, (3.7)
which can be integrated to give,
r ∝ 1
a2t4/3. (3.8)
Analogously for z we have,
z ∝ 1
a2t4/3. (3.9)
Finally solving for t from equation (3.4) we get,
t ∝ 1
t2/3. (3.10)
Thus in the FLRW regions we have,
kaF =
(
− ξ
t2/3,
ξ√
η2 − 1
ηat2/3, 0, − ξ
ηat2/3
)
, (3.11)
where ξ and η are positive constants yet to be determined. The signs for each component
in equation (3.11) are chosen to describe a time reversed ray (ie. we want ∆t to be
negative). Such a ray originates from the observer and propagates back in time.
Carrying out a similar procedure for the Kasner regions gives,
kaK =
−t−2/3
√
α2 +
(
β
b
)2
t2,α
at4/3, 0, − β
b2t2/3.
, (3.12)
Chapter 3. Lookback Time and Observational Consequences 33
where α and β are constants. Since the geodesics pass through both regions we must
determine how these constants are related. Consider a comoving observer located on the
boundary with normalized 4-velocity ua = (1, 0, 0, 0) as seen from both sides. This is
possible since the coordinates are the same except for z and Z which are both constants.
The 4-velocity must satisfy,
(uaka)F = (uak
a)K and (uAua)F = (uau
a)K = −1. (3.13)
The first is the statement that an observer on the boundary should measure the same
frequency regardless of which side of the coordinates are used. The second ensures that
the observer is timelike. The first condition implies that k0F = k0
K thus,
ξ =
√
α2 +
(
β
b
)2
t2in (3.14)
where tin is the time of entry into the current Kasner slice. Next we can consider a
radially moving observer along the boundary with 4-velocity ua = (√
2, at2/3, 0, 0) again
the same 4-velocity is seen from both sides since the z and Z are constant. This then
implies k1F = k1
K and we can then solve for the constants,
α = ξ
√
η2 − 1
ηand
βtinb
= ξ/η. (3.15)
Equation (3.12) then becomes,
kaK =
− ξ
ηt−2/3
√
(η2 − 1) +
(
t
tin
)2
t2,η√
η2 − 1
ηat4/3, 0, − ξt2/3
ηb2tin.
, (3.16)
3.1.2 Bending Angles
We define ψ to be the angle between the 3-vector tangent to the spatial component of the
null geodesic, kα with α = 1, 2, 3, and a 3-vector normal to the matching surfaces, V α.
We will find the general form of the angle ψ with respects to any surface of z = constant
or Z = constant. This will give us a sense of how the angle evolves as the ray travels
Chapter 3. Lookback Time and Observational Consequences 34
through a region of spacetime. This angle can be calculated from the scalar product in
3-space,
cosψ =V αkα
√
V αVαkβkβ. (3.17)
We have V α = (0, 0, 1) and since ka is null we have kβkβ = gαβkαkβ = k0k0. The angle
can then be simplified to,
cosψ =g33k
3
√
g33(k0k0)=√
|g33|k3
k0. (3.18)
The last equality follows from g33 = |g33| due to our spacetime signature. Since
cos2 ψ + sin2 ψ = 1 we can write,
sinψ = −√
|g11|k1
k0, (3.19)
and also,
tanψ = −( |g11||g33|
)2k1
k3. (3.20)
From equations (3.11) and (3.16) we have
tanψF =√
η2 − 1 = constant and tanψK =tanψF(t/tin)
. (3.21)
The result of ψF being constant is reasonable since the FLRW regions are homogeneous
and isotropic. There should be no preferred direction for the light to bend when it is in
the FLRW region. If we specify the observation angle, ψobs, then ψin can be determined
for each region. This is depicted in Figure (3.1). Using equations (3.21), (3.11) and
(3.16) we arrive at a final form for the null tangent vectors in each region,
kaF =
(
− ξ
t2/3,
ξ sinψinat4/3
, 0, −ξ cosψinat4/3
)
, (3.22)
and
kaK =
−ξ cosψint2/3
√
tan2 ψin +
(
t
tin
)2
,ξ sinψinat4/3
, 0, −ξt2/3 cosψinbtin
, (3.23)
Chapter 3. Lookback Time and Observational Consequences 35
(ψF )out = (ψF )obs
(ψF )obs
(tanψK)out = (tanψF )in
(tout/tin)
(ψK)in = (ψF )out
(ψF )in = (ψK)out
K
F
F
r
z
observer
Figure 3.1: A light ray propagating through different regions is depicted by the thick
line. This ray travels away from the observer. The angles ψ at each point are depicted.
If ψobs is known then all other angles can be calculated.
3.1.3 Calculating the Redshift and Lookback Time
Because each region has its own z-coordinate we must be careful how we choose to
compare relative “thicknesses” of each slice. In each case we can find the distance travelled
by a ray in the z direction as a function of time by integrating dz/dt. In the Einstein de
Sitter regions we have
dz
dt=k3F
k0F
=cosψinat2/3
, (3.24)
which upon integration gives
a(z − zin) = 3t1/3in cosψin
[
(
t
tin
)1/3
− 1
]
. (3.25)
In the Kasner regions we have
dZ
dt=k3K
k0K
=t4/3
btin√
tan2 ψin + (t/tin)2, (3.26)
which gives,
b(Z − Zin) = t4/3in
∫ t/tin
1
(t/tin)4/3d(t/tin)
√
tan2 ψin + (t/tin)2. (3.27)
Chapter 3. Lookback Time and Observational Consequences 36
This integration can be carried out numerically using a Simpson rule integrator. The
relative “thicknesses” will refer to the ratio a(z − zin)/b(Z − Zin). This ratio gives a
sense of distance transversed by the light ray along the z and Z coordinates. This will
not translate to an absolute distance, but will allow us to compare different models using
this ratio as a reference.
For convenience we will work with the redshift factor, x, rather then the redshift
directly,
x ≡ 1 + redshift =(uak
a)observed(uaka)emitted
, (3.28)
where ua is the observer’s 4-velocity. In the Cheese Slice universe, we must treat each
slice separately and therefore replace the observed and emitted times with the time of
entry into the slice, tin and the time of exiting that particular slice, tout. We will assume
a comoving observer with 4-velocity ua = (1, 0, 0, 0). Then from equations (3.23) and
(3.22) we have
xF =
(
tintout
)2/3
. (3.29)
for an Einstein de Sitter region and
xK =cosψin
√
tan2 ψin + (tout/tin)2
(tout/tin)2/3. (3.30)
for the Kasner region. The total redshift factor is the product of the redshift factors from
each slice given by
xtotal =∏
i
xi . (3.31)
From this point forth, we will mean redshift factor when we refer to redshift.
The lookback time, l, is define by
l = temitted − tobserved , (3.32)
but we will work with a normalized lookback time, ℓ, given by
ℓ =temittedtobserved
− 1 . (3.33)
Chapter 3. Lookback Time and Observational Consequences 37
Normalizing the time allows the observer to be at ℓ = 0 and the bang time to be at
ℓ = −1 if the bang time is defined as t = 0.
3.1.4 Numerical Algorithm
The number of slices and relative thicknesses of each slice will always be specified by
the user. Each value entered by the user represents a unique slice and the magnitude of
that value represents the relative thickness. The first slice, the one in which the observer
resides, will always be an FLRW region. The thickness of this slice is defined by the
position of the observer to the first matching surface. Since the ray is travelling in the
positive z-direction, the structure of the model in the negative z-direction (eg. behind
the observer) is irrelevent. The following regions alternate between Kasner and FLRW.
The observation angle is also specified by the user.
Each slice is then divided into one hundred equal points. The lookback time and
total redshift is calculated to each point and plotted. The time spent in each slice is
found from equation (3.25) and by successively integrating equation (3.27) and raising
the upper limit until the required thickness is reached within a preset tolerance. The
integration is carried out using a Simpson rule integrator. Once the time is known the
redshift can be calculated using equations (3.29) and (3.30). Observation angles are
propagated from one slice to the next using equation (3.21).
For Figure 3.5 a slightly different algorithm was used because the goal was to average
over all observation angles. The time between observation and bang-time is divided into
one thousand points. For each point, the distance into the ‘z’ direction is determined
and the number of slices transversed is then known. The redshift is calculated and then
the angle is incremented and calculations repeated for the Simpson rule. The final data
points are then plotted.
Chapter 3. Lookback Time and Observational Consequences 38
Figure 3.2: Lookback time and redshift relation for an Einstein de Sitter model is plotted.
3.2 Results
3.2.1 Lookback Time and Redshift Relations
Recall that the observer must always be in the Einstein de Sitter region because the
Kasner regions are empty. We start with an Einstein de Sitter reference model with no
Kasner regions. The lookback time is plotted in Figure 3.2. Predictably, this relation
does not depend on the observation angle because the Friedmann model is isotropic.
This result could also have been found analytically from equations (3.29) and (3.33).
The normalized lookback time, ℓ, is given by,
ℓ = x−2/3 − 1, (3.34)
since temitted = tout and tobserved = tin.
To see how a Kasner slice would affect this relation, we could build a model with a
thin Friedmann slice and a relatively large Kasner region. We set up a model with the
ratio of thicknesses of FLRW to Kasner to be F : K = (1 : 75), where the ’F’ value refers
Chapter 3. Lookback Time and Observational Consequences 39
Figure 3.3: Lookback time and redshift relation for a thin Einstein de Sitter slice followed
by a large Kasner region with a ratio of F : K = (1 : 75). From left to right, the
observation angle of each curve is ψobs = 0, ψobs = 5, ψobs = 10, followed by increments
of 10 up to ψobs = 80.
to the observer’s local slice and ’K’ the adjacent Kasner region. The thickness of the local
slice is measured from the observer to the first matching surface. The result can be seen
in Figure 3.3. In this case the redshift is highly dependent on the angle of observation.
For an observation angle directly normal to the matching surface we see a blueshift.
This blueshift occurs because the Kasner spacetime contracts in the Z direction while it
expands in the r−φ plane. At about 30 no more blueshift can be observed and at higher
angles the curve begins to resemble that of Figure 3.2. This is expected because as the
observation angle increases the time the ray spends in the FLRW region also increases.
At an angle of ψ = π/2 the ray is entirely in the FLRW region and we recover the result
in Figure 3.2.
Next we try a model with three equal slices. This would give us a sense of how a
Chapter 3. Lookback Time and Observational Consequences 40
Figure 3.4: Lookback time and redshift relation for a three slice model where the Kasner
region is the same thickness as the Einstein de Sitter regions. From left to right the
observation angles are ψobs = 0 followed by increments of 10 up to 80. The curves
overlap in the FLRW regions and are separated as they enter the Kasner region.
distant region of the universe might look if there was a sizable Kasner region between
the source and the observer. The result is plotted in Figure 3.4. We can see that the
Kasner region lowers the redshift for sources at a longer lookback time with the effect
diminishing as the observation angle increases. The result is that for lower angles, it is
possible to have sources with the same redshift but different lookback times.
To get a sense of how the overall redshift would look for the entire sky we can in a
sense “average” out the effect by integrating over the solid angle. We can express this
averaged lookback time and redshift as
L =1
4π
∫ π
0
∫ 2π
0
ℓ sinψdψdθ and X =1
4π
∫ π
0
∫ 2π
0
x sinψdψdθ . (3.35)
If we assume cylindrical and z symmetry about the observer, we can simplify the expres-
Chapter 3. Lookback Time and Observational Consequences 41
Figure 3.5: Lookback time and redshift relation averaged over all observations angles.
The top curve is for an Einstein de Sitter model. The middle curve is for a model with
three equal slices. The bottom curve is for a model with one thin Einstein de Sitter slice
and one large Kasner region.
sions to
L =
∫ π/2
0
ℓ sinψdψ and X =
∫ π/2
0
x sinψdψ . (3.36)
Using a Simpson rule integrator we can numerically plot these values for different models.
The results for the above models (one slice, two slice and three slice models) are plotted
in Figure 3.5. The solid curve in Figure 3.5 represents a model with one slice, which is
equivalent to an Einstein de Sitter universe and serves as an upper bound. As the ratio
of Kasner regions increases the lookback time appears to reach farther into the past for
the same redshift values.
Chapter 3. Lookback Time and Observational Consequences 42
3.2.2 Possible CMB Data
The CMB is often treated as an indication of the isotropy of our universe. If this is true,
then looking at the CMB is a good measure of anisotropy. Could an anisotropic universe
also produce the same degree of isotropy that we see in the CMB? In this section we
attempt to answer that question in relation to the Cheese Slice model. This will allow
us to place a limit on the possible ratio of FLRW regions to Kasner regions. Using our
method to interpret anisotropy in the CMB would mean integrating back to a constant
lookback time, the time of last scattering, and seeing how the redshift varies as one
changes the angle of observation. We will assume that a light ray propagating at 0 will
travel through all the slices in the chosen model. A plot of the results can be seen in
Figure 3.6. If we assume the universe is 13×109 years old [55], and the CMB was formed
when the universe was 4×105 years old, that gives us a normalized time of t = 3.1×10−5
if t = 1 were the present (or ℓ = −0.999969). For an Einstein de Sitter model, we see that
it is isotropic and the redshift is on the order of 1025, which is the same magnitude as
the observed redshift for the CMB [31]. For larger Kasner regions we can see a dramatic
drop in the redshift for lower angles. At higher angles near 90 we see that it always
reaches the Einstein de Sitter limit. This is expected because an observation angle of
90 means the light ray never leaves the observer’s FLRW slice. With this data we see
that a Kasner region of comparable thickness to the Friedman regions would certainly
be noticed in the CMB. However, when the Kasner regions are small in comparison, the
anisotropy is not as noticeable. Thus, we can place a limit on the size of possible Kasner
slices in our universe by imposing a limit of order 10−5 anisotropy that is observed in the
CMB.
To be more rigorous, we can see if the position of a thin Kasner slice would affect
the anisotropy measured in the CMB. We do so by using a three slice model. Using a
relatively thin Kasner slice, we can change the ratio of thicknesses of the two FLRW slices.
In effect we can picture this as “moving” a thin Kasner slice through a predominantly
Chapter 3. Lookback Time and Observational Consequences 43
Figure 3.6: Redshift of the CMB for different models are plotted against observation
angle. A constant lookback time is used to represent the CMB. The top curve represents
the Einstein de Sitter universe, while the dotted curves from top to bottom represent
models with ratios of thicknesses being F : K : F = 0.3 : 0.01 : 2.5, F : K : F : K : F =
0.3 : 0.01 : 0.3 : 0.01 : 2.0 and F : K : F = 1 : 1 : 1. That is, one thin void, two thin
voids and one relatively thick void respectively.
Chapter 3. Lookback Time and Observational Consequences 44
Figure 3.7: Redshift of the CMB are plotted against observation angle for a range of
models with a thin Kasner slice between two thick FLRW slices. The ratios of the
FLRW slices are changed incrementally which is equivalent to changing the position of
a thin Kasner slice in a predominantly FLRW model. From top curve to bottom curve
the ratios of thicknesses are F : K : F = (1 : 0.1 : 5), (2 : 0.1 : 4), (3 : 0.1 : 3), (4 : 0.1 :
2), (5 : 0.1 : 1).
FLRW universe. The result is seen in Figure 3.7. The minimum redshift seen in each
curve, that manifests itself as a dramatic dip, is the angle at which the null ray spends
a maximum amount of time in the Kasner slice. At angles higher than this, the bang
time is reached before the ray has entirely reached the end of the Kasner slice. At angles
near 90 the ray spends its entire time in the local FLRW region and we see the usual
Kasner-free redshift we expect. We can see that there is a slight dependence on position
of the Kasner slice. We can see why this is so from equation (3.30). The redshift is very
much dependent on the time of entry and exit from the Kasner slice. The ratio tout/tin
is smaller for slices that are farther away (time reversed ray tracing). This causes an
Chapter 3. Lookback Time and Observational Consequences 45
Figure 3.8: Redshift of the CMB are plotted against observation angle for models with
three slices. The thickness of the Kasner region is changed incrementally. From top the
curve to bottom the ratio of thicknesses are F : K : F = (3 : 0.1 : 3), (3 : 0.2 : 3), (3 :
0.3 : 3), (3 : 0.4 : 3), (3 : 0.5 : 3).
overall lowering of the redshift.
To see how this effect compares to the effect of changing the thickness of the Kasner
slice, we can keep the Kasner slice at the same “distance” from the observer, but change
its thickness. A plot of the effect can be see in Figure 3.8. In this plot we keep the
thickness of the local FLRW slice constant and change the thickness of the Kasner slice.
We can see that the thickness of the Kasner slice causes a drop in the redshift. Again the
feature of a limit is seen for larger angles. The drop in redshift is greater than the one
found in Figure 3.7. Thus we can conclude that the thickness of the slice plays a more
significant role in determining the redshift than the position of the slice.
Finally we can see if the number of slices affects the redshift when the total ratio of
Kasner to FLRW is kept constant. To do so we choose a three slice model with ratio
Chapter 3. Lookback Time and Observational Consequences 46
Figure 3.9: Redshift of the CMB are plotted against observation angle for models with
different number of slices. The total ratio of thicknesses of Kasner to FLRW regions are
kept constant while the number of slices changes from three (top curve) to five (middle
curve) to seven (lower curve).
F : K : F = (1 : 1 : 1) and compare it to a five slice model with ratio F : K : F : K :
F = (4 : 3 : 4 : 3 : 4). In each case the first ’F’ refers to the observer’s local slice and the
consecutive ’K’s and ’F’s refer to subsequent slices. Taking the scheme one step farther,
we have a seven slice model with ratios F : K : F : K : F : K : F = (3 : 2 : 3 : 2 : 3 :
2 : 3). In all three cases the total thickness of the Kasner region is one half of the total
thickness of the FLRW regions. The result is plotted in Figure 3.9. Again we can see
the feature of a limit for higher angles as in the previous Figures 3.7 and 3.8. What we
see is that there is a lower redshift for the model with more slices, but this could also
be due to the fact that we necessarily need to place these slices father away when there
are more slices. Thus it is difficult to separate the change in redshift due to these two
different effects.
Chapter 3. Lookback Time and Observational Consequences 47
3.3 Summary and Discussion
From the above cases, it is clear that a universe with alternating layers of Kasner and
FLRW regions will have profound observational consequences. Depending on the size of
the Kasner region and the angles of observation, it is possible to observe blueshifts where
a redshift might be expected as was shown in Figure 3.3. We could also observe two
sources at different lookback times with the same redshift as in Figure 3.4.
In terms of the construction of the Cheese Slice model, it appears that the thickness
of the Kasner slices seems to have the most effect on the redshift with the position and
number of slices playing a smaller role. Thus the most important parameter we should
consider is the ratio of the thickness.
The observed temperature anisotropies in the CMB are on the order of 10−5 [41].
To achieve an anisotropy of this magnitude in our model, we would require a very thin
Kasner slice. For instance, in a three slice model, we would require a ratio of F : K :
F = (3 : 0.00001 : 2.5), that is a Kasner slice on the order of ten thousand times
thinner than the FLRW regions. While this may seem like an insignificant ratio it does
fundamentally change the spacetime. Inhomogeneity is introduced and the symmetry
of the FLRW spacetime is broken. This could have profound impacts on some features
of the cosmology. We continue to investigate the Cheese Slice “toy model” to discover
the consequences of breaking these symmetries. In Chapter 4 we look at how the initial
singularity might manifest itself and in Chapter 5 we will consider whether this type of
inhomogeneity could be incorporated in braneworld theory.
Chapter 4
The Structure of the Singularity
Perhaps the most compelling aspect of cosmology is the prediction that our universe
began as a singularity. The theorems of Hawking and Ellis [54] have shown, under very
general conditions, that such a singularity is unavoidable. Thus we are forced to confront
a situation in which the theory of General Relativity (GR) appears to predicts its own
downfall. There are two ways in which this breakdown can occur. The first is through
unbounded physical parameters such as infinite densities or infinite spacetime curvature.
The second is through loss of predictability manifesting itself in the existence of a Cauchy
Horizon. It is widely believed that some theory of quantum gravity could resolve such
singularities, but a satisfactory theory of quantum gravity has yet to be produced. In
addition, any theory of quantum gravity would necessarily have to reproduce the results
of GR in the appropriate limit. Thus it is important that we understand the approach
to these singularities.
Investigating the initial singularity is akin to asking what the initial state of our
universe was. The FLRW models have homogeneous and isotropic singularities. These
symmetries hold true regardless of how close to the singularity we choose to examine.
This can be described as a conformal singularity, one that arises from scaling the entire
space. There is no reason to believe that the initial conditions of our universe contains
48
Chapter 4. The Structure of the Singularity 49
such a high degree of symmetry. The question is then, what happens when this symmetry
is broken? This a difficult question to answer in a general sense, but we can make use
of some toy models to break these symmetries and see what kind of behaviour results.
In this chapter we choose to investigate the singularity of the Cheese Slice universe to
see how this singularity might manifest itself when the spacetime is inhomogeneous and
anisotropic. We begin with some general discussion about singularities and then look at
how a matching can affect the singularity structure. In particular we focus on a property
of singularities that is referred to as asymptotically velocity term dominated (AVTD).
4.1 Definition of a Singularity
Many difficulties arise in trying to define the notion of a singularity [103]. There are
certain phenomena that we would like to include in the definition, such as points in
spacetime where the curvature is unbounded or where physical quantities become infinite.
However using physical properties to define a singularity is not sufficient since there are
examples of spacetimes, such as a cone-like spacetimes [37], where there is a point that
cannot be defined and yet the curvature is zero everywhere. In two dimensions this
singular point would be the vertex of a cone.1 Therefore, there must be a definition of
a singularity that encompasses many different spacetimes that we intuitively consider
singular.
The most satisfactory definition of a singularity, one that encompasses the many
different phenomena, is to define it as a point where timelike and null curves cannot be
extended. One also has to add the condition that the spacetime is inextendible. That
is, the spacetime is not isometric to a proper subset of any other spacetime. This avoids
the complications of artificially creating singularities, by removing a point in Minkowski
space for instance.
1Also known as conical singularities. These spacetimes can be envisioned as Minkowski space with awedge removed and remaining boundary planes identified. The vertex of the wedge becomes undefined.
Chapter 4. The Structure of the Singularity 50
Mathematically we describe this end of spacetime as geodesic incompleteness. We
can also add the restriction that it must be either timelike or null incomplete. Since
in these cases we would be considering the motion of force-free particles or observers.
An end of the worldline of a particle would mean that we can no longer describe the
motion of a particle after a finite proper time. Spacelike geodesic incompleteness is not
included in this definition since it doesn’t have clear physical implications. The definition
can be made more robust by including all timelike curves. One can define a generalized
affine parameter on such curves and define the spacetime to be b-complete if there is an
endpoint for every curve of finite length as measured by this parameter [54]. A spacetime
is thus singular if it is b-incomplete.
4.2 Classification of Singularities
With the definition of singularities in hand, we can now go ahead and look at the different
types of singularities that can exist.
4.2.1 Isotropic Singularities
The isotropic singularities are well understood and much work has gone into understand-
ing them [38]. They can also be described as conformal singularities, that is a singularity
that arises from scaling the entire space. For instance, the FLRW singularity can be seen
as arising from a space that has scaled down to the point where the density becomes
infinite. It can also be shown that spacetimes with such singularities must be filled with
matter. Near the singularity, this matter will automatically satisfy the strong and weak
energy conditions. The dominant energy conditions can be satisfied, depending on the
equation of state.
It is the high degree of symmetry that allows isotropic singularities to be examined in
such detail. When it comes to more general singularities, the task becomes much more
Chapter 4. The Structure of the Singularity 51
difficult.
4.2.2 Classification Scheme
Ellis and Schmidt [37] set up a classification scheme to describe different types of singu-
larities.
Quasi-regular singularities are singular points at which there are no curvature ob-
structions to extending the space time. The tensor components Rabcd are well
behaved in a parallel frame along all curves approaching the singularity. The space-
time is well behaved near the singularity. An example of such a singularity is the
vertex of a cone. Such singularities are generally unstable and therefore unlikely to
occur in a physical spacetime.
Curvature singularities are the opposite case where the curvature tensor components
Rabcd are badly behaved. This class of singularities can be further subdivided.
Scalar singularities are singularities where a scalar quantity such as the Kretschmann
scalar, RabcdRabcd, or a physical quantity such as pressure or density, becomes
badly behaved. The Schwarzschild singularity would fit this description as the
Kretschmann scalar diverges.
Non-scalar singularities are singularities in which scalar quantities are well be-
haved yet the curvature is an obstruction to extending the spacetime. One
can perform a conformal transformation on a Taub-NUT spacetime to produce
this type of singularity [94].
These curvature singularities can be further sub divided into divergent singularities
where the relevant components are unbounded, such as the Schwarzschild singular-
ity, or oscillatory singularities where the components are bounded but oscillate
Chapter 4. The Structure of the Singularity 52
as one approaches the singularity such as in the Bianchi IX, “Mixmaster” type
universes [102].
4.2.3 Strength of a Singularity
One may also classify a singularity according to its strength. An early definition arrived
at by Tipler [100] is generally accepted. Under Tipler’s definition, a strong singularity is
one in which a volume element vanishes at the singularity. A volume element is defined
by taking all the linearly independent spacelike Jacobi fields along a timelike geodesic.
If we choose a geodesic such that it terminates at the singularity, we can then find the
volume element and see if it vanishes.
From this definition many physically relevant singularities such as the Schwarzschild,
Reissner-Nordstrom and dust filled FLRW singularities are all strong. Necessary and
sufficient conditions for the occurrence of strong singularities were given by Clarke and
Krolak [23].
Recently this definition has been refined by Ori [84]. To maintain a distinction from
Tipler’s definition, strong singularities according Ori’s definition are referred to as defor-
mationally strong. A singularity is considered to be deformationally strong if any of the
Jacobi fields become unbounded near the singularity. By this definition a Tipler strong
singularity is a subset of the deformationally strong singularity.
4.3 More General Singularity Structures
4.3.1 Properties of the Matching
Before delving directly into the topic of singularities we first look at properties of a
spacetime that results from a matching. We can ask what kind of properties, if any, do
spacetimes constructed from a matching inherit from the spacetimes used in its construc-
tion? Let us call the matched spacetime W and the constituent spacetimes used in its
Chapter 4. The Structure of the Singularity 53
construction V + and V − with the appropriate Σ on each side. The Darmois conditions
are imposed across Σ.
It is clear that even if V + and V − are both homogeneous, W in general is not ho-
mogeneous as we have seen with the Cheese Slice universe. However, such a scenario
is possible. McClure and Dyer [79] have shown that a radiation dominated Einstein-de
Sitter universe can be matched to a matter dominated Einstein-de Sitter universe across
a surface of constant time. In this case the matched spacetime, W , is spatially homo-
geneous at any t = t0, with t0 = constant. It appears that whether or not W inherits
homogeneity from V + and V − depends highly on the nature of the matching.
It would be interesting however if there were some intrinsic properties that W must
inherit from V + and V −. For example, any matching requires the intrinsic metric of Σ
to be identical as viewed from either side obeying equation (2.10). This implies that any
symmetries on Σ seen from V − must also be seen from V +. This is evident in the Swiss
Cheese model where the matching surface is spherically symmetric and in the Cheese
slice where the surface is cylindrically symmetric as seen from either side. However, since
matching conditions are strictly local, it is not known if inherited symmetries on Σ have
any consequences on the global structure of the spacetime.
With regard to singularities, we would like to see what sort of singularity might result
when regions with different singularity structure are matched together. If both V − and
V + have initial singularities and we match them along a timelike surface, what can we
surmise about the singularity that is constructed in W?
4.3.2 BKL Picture of Cosmological Singularities
Belinskii, Khalatnikov and Lifshitz (BKL) tried to construct a picture of the approach
to a generic cosmological singularity [5]. By generic, they refer to a solution of the EFE
that has the most number of arbitrary functions. Initially this was an attempt to see
if singularities were a generic feature of spacetimes or if they were a result of overly
Chapter 4. The Structure of the Singularity 54
simplified FLRW cosmology. The singularity theorems of Penrose and Hawking have
laid this question to rest by showing that singularities can occur given a very general
set of conditions. However these theorems only prove that the singularities exist and do
not indicate the nature of these singularities. Thus the work of BKL remains relevant as
they attempt to describe the general approach to the singularity and shed light on the
behaviour of these singularities.
The BKL approach begins by choosing a frame such that the singularity occurs si-
multaneously for all points in the spacetime. They then found that the terms describing
the time evolution toward the singularity dominated over terms describing the spatial
curvature. By making the assumption that the spatial curvature terms were negligible,
they were able to derive an approach to the singularity such that the spacetime expanded
and contracted in different directions in an oscillatory manner. Furthermore this assump-
tion effectively states that each spatial point evolves independently of any other spatial
point. Any particle’s approach to the singularity is independent of any other particle.
Though they might have influenced each other away from the singularity, these particle
interactions are effectively cut off as one asymptotically approaches the singularity.
In the context of spacetime matchings, this could mean that regions of a matched
spacetime do not effect other as one approaches closer to the singularity.
The BKL approach has not been rigorously proven or refuted [90], but certain space-
times have been demonstrated to satisfy these properties, which came to be known as
“velocity dominated” or “asymptotically velocity term dominated” (AVTD) solutions.
The Gowdy spacetimes [59] and most general Bianchi Types2 [6, 102] were shown to
exhibit these types of singularities. Recent numerical results have also provided some
support for this approach to describing generic singularities [104].
2In particular types VII and IX have oscillatory singularities. The latter is also known as the Mix-master Universe [80].
Chapter 4. The Structure of the Singularity 55
4.3.3 Cauchy Horizon Singularities
Recently evidence for another generic form of singularities have been seen [17]. These
types of singularities occur inside the inner event horizon of charged or rotating black
holes. Due to the structure of these event horizons in-falling radiation from the entire
history of the universe outside the black hole will accumulate at the horizon. An in-falling
observer will see the entire history of the universe in a finite proper time. This in-falling
radiation is infinitely blue shifted. This causes a divergence in the energy momentum
tensor and in turn, from the EFE, results in a curvature singularity.
These singularities are distinct from the generic BKL variety because they are null
singularities rather than spacelike. Also, current evidence seems to indicate that such
singularities are deformationally weak [16].
4.4 The AVTD Singularity
Attempts have been made to prescribe some precise definition of what it means to sat-
isfy the BKL properties. The assumption that the evolution towards the singularity is
independent of spatial curvature allows the EFE to be simplified. Eardley, Liang and
Sachs (ELS) [35] formulated a definition of this property that they termed “velocity dom-
inated”. They used a Gaussian normal coordinate system with the timelike coordinate,
t, being orthogonal to all spatial coordinates. The spacetime is then foliated by space-
like hypersurfaces of constant time. These hypersurfaces are three-dimensional manifolds
that have an intrinsic metric, (3)gαβ, extrinsic curvature, Kαβ and an intrinsic Einstein
tensor, (3)Gαβ. By neglecting (3)Gαβ they were able to integrate the four-dimensional
EFE and solve for (3)gαβ. Focusing on dust cosmologies, they defined the singularity
to be at the time t = t0 such that the density ρ→ ∞ as t→ t0. The structure of the
singularity is then given by (3)gαβ∣
∣
t0and its evolution depends only on Kαβ. Since Kαβ
describes how the three-surfaces curve in the ambient spacetime (in this case the ∂/∂t
Chapter 4. The Structure of the Singularity 56
direction), it is often referred to as a “velocity” and thus the term “velocity dominated
singularity” was coined.
This method demonstrates how the structure in the late universe can be directly
attributed to the structure in the singularity itself. However, the ELS approach requires
the use of a time orthogonal coordinate system and is only applicable to dust cosmologies.
One would like to be able to extend this concept of velocity dominated singularity to more
general spacetimes. In that spirit Isenberg and Moncrief has formulated what he refers to
as “Asymptotically Velocity Term Dominated” (AVTD) spacetimes [59]. The definition
of AVTD incorporates more general foliations of spacetimes and matter content. We
begin with a detailed definition of the AVTD property then we define how a spacetime
constructed from a matching can be considered AVTD. Finally, we use the example of
the Cheese Slice universe to demonstrate how this concept can be applied.
Both the FLRW and the Kasner spacetimes have initial singularities that are AVTDS.
We will propose a criterion to determine whether or not a matched spacetime is AVTD
and use this to show that Cheese Slice universe inherits this property and the singularity
is of type AVTDS. We show that this is true in both the matchings with spatially flat
FLRW and open FLRW slices. We conclude by conjecturing that if V − is AVTD and Σ
is a timelike surface, then V + and W are necessarily AVTD as well.
4.4.1 Definitions
Let U be a spacetime with metric gab and coordinates xa. We begin by choosing a
spatial foliation with intrinsic coordinates ξα on each leaf of the foliation as depicted in
Figure (4.1).
Next we identify the intrinsic metric,
γαβ =∂xa
∂ξα∂xb
∂ξβgab, (4.1)
Chapter 4. The Structure of the Singularity 57
∂∂t
coordinates: xaU, metric: gab
Π0,coordinates: ξαmetric: γαβ
na
t0
t1
t2
t3
Figure 4.1: The spacetime U with coordinates xa is foliated with spacelike surfaces, Πi,
with coordinates ξα. Successive leaves of the foliation are labelled by the time coordinate
t. The timelike foliation vector ∂∂t
is in general different from the normal vector, na.
and extrinsic curvature,
Kαβ =∂xa
∂ξα∂xb
∂ξβnb;a, (4.2)
of the spacelike three-surfaces, where na is the normal to the surface. These values are
defined identical to equation (2.8) and (2.9) however we restate them to emphasize that
this foliation is not identical to the matching surface Σ. The leaves of the foliation are
spacelike surfaces whereas Σ, as we have defined it, is a timelike surface. The mean
curvature is then K = Kαα . The timelike foliation vector, ∂/∂t, where t is a timelike
coordinate that labels successive leaves of the foliation, describes the evolution of the
three surface. If we use the normal to define a frame, we can write this vector in terms
of a component normal to the surface and a component tangential to the surface.
∂
∂t= Nna +Ma. (4.3)
These components are called the lapse, N , and shift, Ma, respectively [106].
The matter density, ρ, momentum, Jα, and spatial stress density, Sαβ, must also be
Chapter 4. The Structure of the Singularity 58
considered. These quantities must satisfy the EFE written in the form of constraint
equations [59],
(3)R−KαβKαβ +K2 = 2ρ, (4.4)
(3)∇αKαβ − (3)∇βK = −Jβ (4.5)
and evolution equations,
∂
∂tγαβ = −2NKαβ + LMγαβ, (4.6)
∂
∂tKαβ = N
[
(3)Rαβ +KKα
β + Sαβ +1
2γαβ (ρ− Sνν )
]
− (3)∇α(3)∇βN + LMKαβ , (4.7)
where (3)R and (3)Rab are the spatial Ricci scalar and Ricci tensor respectively. (3)∇ is
the three dimensional covariant derivative and LM is the three-dimensional, spatial Lie
derivative in the direction of Ma. Also, geometrized units have been used where 8πG = 1.
Next, the velocity term dominated solutions (VTD) are defined by neglecting all the
spatial derivatives in the field equations. Equations (4.4)-(4.7) then reduce to the VTD
constraint equations,
KαβKαβ + K2 = 2ρ, (4.8)
(3)∇αKαβ − (3)∇βK = −Jβ, (4.9)
and the VTD evolution equations,
∂
∂tγαβ = −2NKαβ, (4.10)
∂
∂tKαβ = N
[
KKαβ + Saβ +
1
2γαβ
(
ρ− Sνν
)
]
. (4.11)
Note that in general the spatial derivatives in ρ, Jα and Sαβ vanish as well. We use
the over-tilde, ˜, to indicate their distinctiveness from the quantities in the Einstein
equations (4.4)–(4.7).
Solutions of the field equations (4.4)–(4.7) are then defined to be AVTD if in the limit
of large t they approach the solutions to the VTD equations (4.8)–(4.11). That is, as
Chapter 4. The Structure of the Singularity 59
t→ ∞, the values of
γαβ − γαβ
Kαβ − Kαβ
ρ− ρ
Jα − Jα
Sαβ − Sαβ
→ 0. (4.12)
A singularity is said to be an AVTDS if the spacetime is AVTD and the foliation is
chosen such that the singularity is approached as t→ ∞. The AVTDS property is highly
dependent on the choice of foliation. A spacetime that is AVTD in one foliation might
not appear AVTD in another foliation. A spacetime needs only one foliation to satisfy
the AVTD property to be considered AVTD.
4.4.2 AVTD Property of Matched Spacetimes
We will now examine how a spacetime constructed from a matching can be considered
AVTD. The Darmois matching conditions [28], detailed in Chapter 2.1.1 will be used
throughout this section.
Let W be the spacetime constructed from the matching of V − and V + across the
surface Σ. This assumes that the Darmois conditions are satisfied across Σ. Also, let
Π±t± denote leaves of a foliation of V ±, parametrized by t±, such that V ± is AVTD. In
general t− and t+ are different time coordinates. If each leaf of the foliation Π−t− matches
with each leaf of the foliation Π+t+ along the surface Σ, then this constitutes a foliation of
W such that W is AVTD. Note that the corresponding VTD solutions must also match
across the surface Σ in the same manner.
To clarify the matching of the Π−t− with Π+
t+ , let us single out one leaf of the foliation
on each side and call them Π±0 as illustrated in Figure 4.2. Π±
0 are spatial three-surfaces
in V ±. We wish to match Π−0 with Π+
0 across the surface Σ. However, Σ is a timelike
three-surface and the intersection of Π± with Σ is a spatial two-surface. Let us call this
Chapter 4. The Structure of the Singularity 60
t
V +
Π−0
Σ
θ2
θ1Λ
V −
Figure 4.2: A leaf of the foliation on each side is singled out, Π±0 . The intersection of Π±
0
with Σ is what we refer to as the corner, Λ, which is itself a two-surface. Σ is a timelike
surface while Π is a spacelike surface. The corner can also be seen as the intersection of
Π+0 and Π−
0 . The corner is a subspace of all the depicted hypersurfaces.
two surface the “corner” and denote it by Λ with the coordinates ζA. There is also a
two-dimensional space of normals to Λ. Let mAa be an orthonormal basis for this space.
The matching conditions at a corner have already been thoroughly investigated by
Taylor [98] and reviewed in Chapter 2.1.2. In general Π−0 and Π+
0 could meet at any angle;
thus the corner conditions, equations (2.15)–(2.18) are necessary. If the corner conditions
are satisfied on Λ for all time t± then the union of the foliations, Π ≡ Π−0 ∪Π+
0 , constitutes
a foliation of W such that W is AVTD. We will use this method to show that the Cheese
Slice universe is AVTD and the singularity is an AVTDS.
4.5 Singularities in the Cheese Slice Universe
We will use the FLRW line element in the form,
ds2F = −dt2 + a2(t)
[
dr2
1 − kr2+ r2dφ2 + (1 − kr2)dz2
]
, (4.13)
Chapter 4. The Structure of the Singularity 61
where k = −1, 0, 1 and the Kasner line element given by,
ds2K = −dT 2 + T 2p1dX2 + T 2p2dY 2 + T 2p3dZ2, (4.14)
with the restrictions
p1 + p2 + p3 = 1 = p21 + p2
2 + p23. (4.15)
Both the Kasner and the FLRW spacetimes have an initial singularities (t = 0 = T ). We
look at the cases of the spatially flat FLRW matching and the open FLRW matching in
turn.
4.5.1 Case (i) Flat FLRW, k = 0
The coordinates defined in equation (4.13) and (4.14) single out a natural foliation that
we will use to check the AVTD property for the spatially flat case.
The Kasner spacetime satisfies the VTD equation (4.8)–(4.11) directly; therefore it is
trivially AVTD. With the pressure free FLRW spacetime the spatially flat case satisfies
the VTD equation (4.8)–(4.11) as well with the following quantities,
where the brane sits at w = 0 and it is clear that the brane has the same structure as
the Kasner spacetime. The brane must also have a tension and matter-density given by,
σ = ∓ 6
κ25ℓ, and ρ = 0, (5.32)
respectively, which is the same as the Randall-Sundrum tuning condition in equation (5.3).
This means the brane must be empty and has the same fine tuning condition as the
Randall-Sundrum brane. The fact that this brane is a vacuum makes it a poor cosmo-
logical model, but it is important in that it introduces anisotropy into the braneworld
scenario.
Other Authors have investigated anisotropic braneworlds with matter content [1,
4]. Notably, Campos and Sopuerta [20] used dynamical systems techniques to look at
Bianchi-type branes which are homogeneous and anisotropic. 1 However, in these early
studies many assumptions were made about the Weyl term, Eab, due to the lack of an
exact anisotropic bulk solution. This was addressed in [19] for the FLRW and Bianchi I
case and shortly after Campos et al. [18] found a family of exact, anisotropic solutions to
the five-dimensional field equations. Thus they were able to explicitly see the relationship
between the bulk Weyl curvature and the anisotropy on the brane. They found that it is
not possible to have a perfect fluid or scalar field compatible with the anisotropic brane
since the junction conditions require anisotropic stress on the brane. Fabbri et al. [39]
found more exact bulk solutions and agreed that an anisotropic brane cannot support a
perfect fluid if the bulk is static. They found that in some solutions with a non-static
bulk it is possible to have a perfect fluid, but its energy density and pressure is completely
determined by the bulk geometry.
1For a summary of dynamical systems in the context of cosmology, including Bianchi-type cosmolo-gies, refer to [102].
Chapter 5. Cheese Slice Braneworlds 77
Another interesting feature of anisotropic cosmology on the brane is their apparent
tendency to isotropize at the initial singularity. Harko and Mak [53] investigated Bianchi-
type braneworld behaviour near the singularity and at late times and found that they
tend to isotropize for certain matter content. Dunsby et al. [32] used dynamical systems
techniques and confirmed the idea that braneworlds tend to isotropize as the initial
singularity is approached. Furthermore, they used a perturbative analysis of the FLRW
brane and found similar results [51]. It appears that in the braneworld context there is
a natural mechanism for the initial singularity to be isotropic. This is a drastic change
from conventional cosmology where isotropy is taken to be an initial condition.
5.1.4 Brane Collisions
The first Randall-Sundrum braneworld consisted of two branes at some fixed distance
from each other. In general the distance between two branes does not need to be constant.
Thus we can set up a situation in which two branes moving through the bulk could collide
with each other. In fact any number of colliding branes can produce any number of branes
after the collisions. Such a situation has been investigated by Neronov [82] who looked at
Friedmann type branes colliding in an Anti-de Sitter bulk. He was able to derive a simple
relation between the cosmological constants on either side of the colliding branes that
must be satisfied. This was later followed up by Langlois, Maeda and Wands [72] who
showed that this condition is a form of momentum conservation applied to the colliding
branes.
Novel cosmological models have been constructed from the idea of colliding branes.
Khoury et al. produced a model that they termed the ekpyrotic universe [63, 89]. This
model consisted of two vacuum branes which collide resulting in one brane endowed with
stress energy. The collision point is seen as the big bang from an observer in the resultant
brane. This model was motivated by an attempt to construct a cosmology that does not
require inflation. Another alternative was presented by Gen, Ishibashi and Tanaka [45] in
Chapter 5. Cheese Slice Braneworlds 78
which a vacuum bubble forms in a false vacuum bulk. This bubble collides with a brane
resulting in a Friedmann type cosmology on the brane. In these models any bulk effects
on the brane were neglected. These models are currently being studied and contested,
but they serve to show the many possible applications of extra dimensions in cosmology.
5.2 Constructing an Inhomogeneous Brane
We now turn to the problem of constructing new exact braneworld solutions using the
existing solutions. In the spirit of Chapter 2, where exact solutions were pieced together
to form the Cheese Slice universe, we wish to do the same in the braneworld context.
With the existence of Kasner type braneworlds and solutions for cosmological FLRW
type branes, it seems natural to attempt to find a braneworld equivalent of the Cheese
Slice universe. Such a solution would provide an exact inhomogeneous solution to model
inhomogeneities in the braneworld context. Also deviations from the conventional Cheese
Slices might appear that could be used to support or refute the braneworld picture.
Finally, insights gained from attempting a Cheese Slice braneworld matching could be
used to search for new matchings that can generate more exact braneworld solutions.
Some attempts at matching on the brane have been attempted. Germani and Maartens
[50] attempted to match a spherical star of uniform density on the brane to an exterior
solution using the Darmois conditions on the brane. They provide two possible non-
Schwarzschild exterior solutions that were able to match to the same interior solution.
The exterior solution is not unique and also not Schwarzschild due to the Weyl tensor
terms, Eab in equation (5.23). Assumptions were made about this term to arrive at the
solutions. In fact they were found completely on the brane and no extension into the
bulk is known. Despite these difficulties they have attempted to investigate gravitational
collapse on the brane [14]. Without knowing anything about the bulk, they were able to
show that the exterior solution to gravitational collapse cannot be static. That is, the
Chapter 5. Cheese Slice Braneworlds 79
exterior must be non-Schwarzschild due to Weyl tensor terms and a non-zero effective
pressure term on the matching surface.
An attempt at constructing a Swiss Cheese brane has been carried out by Gergely
[49]. In this model, an FLRW brane is punctured by “black string” solutions [21]. The
black strings are characterized by a string-like singularity that extends into the bulk.
The intersection of the black string with the brane forms the familiar Schwarzschild
singularity on the brane. In his treatment Gergely assumes the Weyl tensor term, Eab, is
zero in both the FLRW and the black string regions [48]. Then assuming a perfect fluid
in the FLRW regions and implementing the Darmois matching conditions on a spherical
surface with the Schwarzschild singularity at the centre, he showed that the cosmological
fluid cannot be dust [47, 46]. This is due to the modified Friedmann equations on the
brane. Furthermore, certain values of the cosmological constant required a negative mass
density to ensure a proper matching. Much like the case of a star on the brane, no exact
bulk solution has been found for this construction.
The idea of a black hole colliding into the brane or escaping into the bulk has also
been considered by Flachi and Tanaka [42]. To arrive at a result they neglected any self
gravitating effects of the brane and the brane tension as well.
In all these examples, the bulk was not specified or its effects were neglected alto-
gether. Even though Germani and Maartens took bulk effects into consideration, they
did not consider the requirements to match the respective bulks of the interior and the
exterior of the star. In the following we will attempt to construct an inhomogeneous
brane through a Darmois matching and then look explicitly at how the bulk might affect
the matching.
5.2.1 The Cheese Slice Brane
Despite the difficulties encountered so far in braneworld matchings there is reason to
believe that a Cheese Slice brane will be tractable. Both the Kasner and the FLRW
Chapter 5. Cheese Slice Braneworlds 80
branes have natural extensions into the bulk with exact solutions known in each case. In
the vacuum regions we will use the Kasner-AdS metric from equation (5.31), with the
Kasner brane located at w = 0. With this form of the bulk we have Eab = 0. Let us
assume that Eab is zero on the FLRW side as well, allowing us to completely describe
the matching from within the brane. Let us also assume a bulk that is symmetric about
the brane. 2 Thus we only require that our brane solutions satisfy the modified EFE in
equation (5.23). We will use the superscripts, −, and, +, to denote the Kasner side and
the FLRW side respectively. The Kasner metric,
ds2− = −dT 2 + T 2p1dX2 + T 2p2dY 2 + T 2p3dZ2, (5.33)
with the restrictions
p1 + p2 + p3 = 1 = p21 + p2
2 + p23, (5.34)
satisfies the EFE on the brane in the case of a vacuum. This requires,
τ−ab ≡ 0 and Λ−4 = 0. (5.35)
Turning to the flat FLRW metric we have,
ds2+ = −dt2 + a2(t)(
dx2 + dy2 + dz2)
. (5.36)
Assuming a perfect fluid such that,
τ+ab = (ρ+ P )uaub + Pgab, (5.37)
then directly from equation (5.23) we have the modified Friedmann equations,
(a,ta
)2
=Λ+
4
3+κ2
4ρ
3
(
1 +ρ
2σ
)
(5.38)
and
a,tta
=Λ+
4
3− κ2
4P
2
(
1 +ρ
σ
)
− κ24ρ
6
(
1 +2ρ
σ
)
. (5.39)
2The consequences of these assumptions will be explored in Section 5.3
Chapter 5. Cheese Slice Braneworlds 81
From Chapter 2.2 we know that a flat FLRW region and a Kasner region can satisfy
the Darmois conditions along a planar surface z = constant. For this matching to occur
we require,
a(t) = t2
3 p1 = p2 =2
3, and p3 = −1
3. (5.40)
Thus it is possible to have a Cheese Slice matching within the brane, though due to the
modified Friedmann equations (5.38) and (5.39), the matter content is non-trivial.
5.2.2 Energy Conditions
We will now investigate the energy conditions of the cosmological fluid in the FLRW
region. In the case of the Swiss Cheese brane [48, 47] it was shown that the matching
required a non-trivial equation of state, we will see a similar result with the Cheese
Slice brane, and emphasize that a non-trivial matter content is in direct contrast with
the four-dimensional Cheese Slice universe in Chapter 2 where the matter content is
necessarily dust. We will show that even though the equation of state is more complex in
the braneworld case, the energy conditions can be satisfied in certain regimes, allowing
for physically reasonable matter content.
The energy conditions are as follows [103],
The Weak Energy Condition (WEC) is the statement that physically reasonable
matter should have non-negative energy density as seen by any timelike observer.
Specifically,
Tabξaξb ≥ 0 (5.41)
for all timelike ξa.
The Strong Energy Condition (SEC) states that,
Tabξaξb ≥ −1
2T, (5.42)
where T = T aa . This is equivalent to the statement that Rabξaξb ≥ 0, through the
four-dimensional EFE.
Chapter 5. Cheese Slice Braneworlds 82
The Dominant Energy Condition (DEC) states that
−T ab ξb (5.43)
is a future directed timelike or null vector for all future directed timelike ξa. Phys-
ically this condition can be interpreted as stating the speed of energy flow is less
than the speed of light.
The DEC implies the WEC, but otherwise these conditions are mathematically indepen-
dent. These conditions are all assumptions imposed on the spacetime to determine if
they are physically reasonable and are independent from the EFE.
In the case of a perfect fluid these conditions are equivalent to,
WEC
ρ ≥ 0 and ρ+ P ≥ 0, (5.44)
SEC
ρ+ P ≥ 0 and ρ+ 3P ≥ 0, (5.45)
DEC
ρ > |P |. (5.46)
Combining equations (5.38) and (5.39) with equation (5.40) gives the equation of state,
P =
(
Λ+4
κ4
− ρ2
2σ
)
(
1 +ρ
σ
)−1
. (5.47)
We can also solve for the density as a function of cosmological time, t, to get,
ρ
σ= −1 ±
√
1 − 2
κ24σ
(
Λ+4 − 4
3t2
)
. (5.48)
This result is identical to that of Gergely’s black strings cosmology [48]. The positivity
of ρ was investigated in [48] and we quote the result in Table (5.1). The matter density
can always be greater than zero for negative cosmological constant. This can be seen
Chapter 5. Cheese Slice Braneworlds 83
ρ t < t1 t = t1 t1 < t ≤ t2 t > t2
Λ+4 ≤ 0 + + + +
0 < Λ+4 ≤ κ2σ
2+ 0 - -
κ2σ2< Λ+
4 + 0 - no real solution
Table 5.1: Positivity of matter density, ρ, as a function of cosmological time, t, for
different values of the cosmological constant, Λ+4 . The constants are t1 = 2√
3Λ+
4
and
t2 = 2√
23(2Λ+
4−κ2
4σ)
.
−1 0 1 2 3 0
1
2
3
4
−1 0 1 2 3 0
1
2
3
4Positivity of ρ
Λ+
4
κ24σ
no real solution t2κ24σρ > 0
ρ < 0
Figure 5.1: Regions in which ρ is positive. The lower curve represents ρ = 0 or t2 =
23Λ+
4
. The upper curve represents the boundary over which no real solution exists, t2 =
83(2Λ+
4−κ2
4σ)
. The positivity of ρ is also equivalent to the WEC.
Chapter 5. Cheese Slice Braneworlds 84
from the second term in equation (5.48). When Λ+4 is negative, the second term is greater
than 1. With the choice of the + sign ρ will be greater than 0. If Λ+4 is positive, ρ will
only be positive for early times. Now if we consider ρ in the regime which it is positive
and considering the equation of state (5.47) we get,
ρ+ P =
(
ρ+ρ2
2σ+
Λ+4
κ4
)
(
1 +ρ
σ
)−1
=4
3t2κ4
(
1 +ρ
σ
)−1
. (5.49)
In the last equality we have used the modified Friedmann equation (5.38) to simplify the
expression. This right hand side of equation (5.49) is positive for all time as long as ρ is
positive. Thus the WEC is satisfied whenever ρ is positive as outlined in Table (5.1) and
illustrated in Figure (5.1).
The SEC does not necessarily require ρ to be positive. Taking a closer look we see
the first condition of equation (5.45) along with equation (5.49) leads to,
4
3t2κ4
(
1 +ρ
σ
)−1
≥ 0, (5.50)
which is equivalent to the condition,
ρ
σ≥ −1. (5.51)
From equation (5.48) we see that this condition is satisfied whenever the term,√
1 − 2κ24σ
(
Λ+4 − 4
3t2
)
, is real and the positive sign chosen. This requires that,
t ≤ 2
√
2
3(2Λ+4 − κ2
4σ)if Λ+
4 >κ2σ
2(5.52)
and is always true if
Λ+4 ≤ κ2σ
2. (5.53)
The second condition of equation (5.45) leads to,
ρ+4
t2κ4
(
1 +ρ
σ
)−1
− 3ρ ≥ 0. (5.54)
Substituting for ρ we find,√
1 − 2
κ24σ
(
Λ+4 − 4
3t2
)
+2
κ24σ
(
Λ+4 − 1
3t2
)
− 1 ≥ 0. (5.55)
Chapter 5. Cheese Slice Braneworlds 85
−1 0 1 2 3 0
1
2
3
4
−1 0 1 2 3 0
1
2
3
4
Region in which the SEC is satisfied
no real solutionyesno t2κ24σ
Λ+
4
κ24σ
Figure 5.2: The region in which the SEC is satisfied is illustrated. The region includes
both curves on the left and the right, but no points beyond them. The left curve describes
the boundary where equation (5.55) fails and is described byΛ+
4
κ24σ
=−3√y2+y+3y+4
12y, where
y = κ24σt
2. The right curve represents the boundary over which no real solution exists,
t = 2√
23(2Λ+
4−κ2
4σ)
.
As with the WEC we again see a specific range of t and Λ+4 over which the SEC is
satisfied. This region is illustrated in Figure (5.2). Note that equation (5.55) is more
restrictive than either of equations (5.52) and (5.53). The distinctive feature of the SEC
that we can see from Figure (5.2) is that there is no value of Λ+4 for which the SEC is
satisfied for all time. For 0 ≤ Λ+4 ≤ κ2
4σ
2SEC is satisfied for for late times (t → ∞), but
is violated for early times (t→ 0). For Λ+4 >
κ24σ
2the SEC is violated for both early and
late times.
To check the DEC we need to show that ρ ≥ |P |. This necessarily means that ρ is
positive. First we see that the condition,
ρ ≥ −P = ρ− 4
3t2κ4
(
1 +ρ
σ
)−1
(5.56)
Chapter 5. Cheese Slice Braneworlds 86
is always satisfied for positive ρ. The equality stems from substitution of equation (5.49).
Now we check the condition,
ρ ≥ P =4
3t2κ4
(
1 +ρ
σ
)−1
− ρ, (5.57)
which is equivalent to the condition,
2ρ(
1 +ρ
σ
)
≥ 4
3t2κ4
. (5.58)
Substituting from equation (5.48) and choosing the positive sign, we find that the con-
dition becomes,√
1 − 2
κ24σ
(
Λ+4 − 4
3t2
)
+2
κ24σ
(
1
t2− Λ+
4
)
+ 1 ≥ 0. (5.59)
The range over which the DEC is satisfied is illustrated in Figure (5.3). We can see from
the figure that equation (5.59) is not as restrictive as the condition that ρ be positive.
Therefore, the DEC is satisfied in the same region as the WEC. For 0 ≤ Λ+4 ≤ κ2
4σ
2both
the DEC and the WEC are satisfied for all time while for Λ+4 >
κ24σ
2they are only satisfied
for early times.
5.3 Extending the Matching into the Bulk
It appears so far that there is a consistent matching from within the brane, but to have
a consistent braneworld we must be able to embed the brane into a bulk spacetime.
Finding a bulk solution that can induce the given braneworld structure is in general
quite difficult and in some cases such a bulk might not exist. In this section we will
utilize a first order approximation to extend the FLRW and Kasner branes into the bulk.
This approximation will not provide us with a global solution, but will serve to show
whether or not a bulk matching is possible within the vicinity of the brane. We will
show that the bulk of the FLRW and Kasner brane does not match in a straightforward
manner. We then look at the general case of matching any two bulks and show that the
embedding plays a large role in this matching.
Chapter 5. Cheese Slice Braneworlds 87
−1−1−1 0 1 2 3 0
1
2
3
4
0 1 2 3 0
1
2
3
4
0 1 2 3 0
1
2
3
4Region in which DEC is satisfied
ρ > 0 ρ < 0 no real solution
Equation (5.59) is valid.
t2κ24σ
Λ+
4
κ24σ
Figure 5.3: The region in which the DEC is satisfied is illustrated. The lower curve
represents ρ = 0 or t2 = 23Λ+
4
. The right curve represents the boundary over which no
real solution exists, t = 2√
23(2Λ+
4−κ2
4σ)
. Equation (5.59) is valid below the central curve
indicated. This curve is given byΛ+
4
κ24σ
=
√3(3y2+8y)+3y+12
12y, where y = κ2
4σt2.
Chapter 5. Cheese Slice Braneworlds 88
5.3.1 The Bulk of the Cheese Slice Brane
We will use the Darmois matching conditions outlined in Chapter 2 to attempt to match
the bulk. In this case the matching surface will be a four-dimensional surface that
intersects the brane. The procedure must be carried out in two steps. First we have to
show that each bulk can support the respective brane. Then we can check the conditions
to match these two bulks.
We will attempt to keep the bulk as general as possible, assuming only the symmetries
required to produce the FLRW and Kasner branes. On the FLRW side we assume the
same form as in equation (5.5),
ds2+ = −N2(t, y)dt2 + A2(t, y)(
dr2 + r2dφ2 + dz2)
+ dy2 (5.60)
where the fifth coordinate is denoted by y. The brane is located at y = 0 without any
loss in generality. A flat FLRW brane is assumed to ease the calculations.
Let the fifth coordinate on the Kasner side be denoted by w such that the bulk metric
is of the form,
ds2− = −M2(T,w)dT 2 +B2(T,w)(
dR2 +R2dΦ2)
+ E2(T,w)dZ2 + dw2, (5.61)
where the Kasner brane is induced at w = 0. We have assumed a symmetry in the R−Φ
plane as is necessary for the Cheese Slice matching.
For the FLRW bulk we already know the conditions required for a cosmological brane,
namely from equation (5.8) with ρb = ρ+ σ+,
(
N,y
N
)
0+
=κ2
5
6(3P + 2(ρ+ σ+)),
(
A,yA
)
0+
= −κ25
6(ρ+ σ+), (5.62)
where σ+ is the brane tension. We now take the first order expansion, keeping in mind
that the zeroth order terms must be N20 ≡ N2(t, 0) = 1 and A2
0 ≡ A2(t, 0) = t4/3 to
reproduce the flat FLRW cosmology on the brane. The first order expansions are then,
N(t, y) = 1 +N1(t)y and A(t, y) = t2
3 + A1(t)y. (5.63)
Chapter 5. Cheese Slice Braneworlds 89
The conditions in equation (5.62) are then,
N1 =κ2
5
6(3P + 2(ρ+ σ+)),
A1
t2
3
= −κ25
6(ρ+ σ+), (5.64)
On the Kasner side we must work out explicitly what the conditions on the metric
functions should be. Beginning from the matching condition in equation (2.30) and
assuming symmetry about the brane we have,
K−ab = −κ
25
2
(
S−ab −
1
3S−g−ab
)
, (5.65)
where S−ab is the energy-momentum on the brane, g−ab the four-dimensional Kasner metric
and S = Saa . To reproduce the Cheese Slice we need the Kasner brane to be a vacuum
with the possibility of a brane tension. Thus,
S−ab = −σ−g−ab. (5.66)
Equation (5.65) then implies,
(
M,w
M
)
0+
=
(
B,w
B
)
0+
=
(
E,wE
)
0+
= −κ25σ
−
6(5.67)
Similar to the FLRW case, we now take the first order expansion of the metric functions
with the zeroth order terms being M0 ≡ M(T, 0) = 1, B0 ≡ B(T, 0) = T2
3 and E0 ≡
E(T, 0) = T− 1
3 to produce the Kasner brane. The first order expansions are then,
M(T,w) = 1 +M1(T )w,
B(T,w) = T2
3 +B1(T )w (5.68)
and E(T,w) = T− 1
3 + E1(T )w.
With equation (5.67) we can solve for the first order terms giving,
M1 = −µ B1 = −µT 2
3 and E1 = −µT− 1
3 (5.69)
where µ =κ25σ−
6.
Chapter 5. Cheese Slice Braneworlds 90
With these conditions on the first order terms in mind, we now turn to the bulk
matching. Let the matching surface be denoted by z − Σ(t, y) = 0. Within the brane
the matching surface must be z = constant, and thus Σ0 ≡ Σ(t, 0) = constant. Let the
parametrization on Σ be such that,
t = u, T = T (u), (5.70)
φ = θ = Φ, (5.71)
r = v = R, (5.72)
y = x, w = w(u, x), (5.73)
z = Σ(u, x) and Z = Z(u, x). (5.74)
Z must be constant in the brane as well, ie. Z(t, 0) = constant. With this parametriza-
tion we can find the first fundamental forms one either side of Σ. Keeping only first order
terms in y and w, we get,
Υ+00 = (u
2
3 + 2A1x)u2
3 Σ2,u − (1 + 2N1x), (5.75)
Υ+03 = (u
2
3 + 2A1x)u2
3 Σ,uΣ,x, (5.76)
Υ+11 = (u
2
3 + 2A1x)u2
3 , (5.77)
Υ+22 = (u
2
3 + 2A1x)u2
3v2, (5.78)
Υ+33 = (u
2
3 + 2A1x)u2
3 Σ2,x + 1 (5.79)
and
Υ−00 = T− 2
3 (1 − 2µw)Z2,u − (1 − 2µw)T 2
,u + w2,u, (5.80)
Υ−03 = T− 2
3 (1 − 2µw)Z,uZ,x + w,uw,x, (5.81)
Υ−11 = T
4
3 (1 − 2µw) , (5.82)
Υ−22 = T
4
3 (1 − 2µw) v2, (5.83)
Υ−33 = T− 2
3 (1 − 2µw)Z2,x + w,x, (5.84)
(5.85)
Chapter 5. Cheese Slice Braneworlds 91
in the FLRW and Kasner sides respectively.
From the Darmois matching conditions we must have Υ+ab = Υ−
ab. Equating equa-
tions (5.77) and (5.82) we can solve for w to get,
µw =1
2− (u
2
3 + 2A1x)u2
3
2T4
3
. (5.86)
The equality of equations (5.78) and (5.83), Υ+22 = Υ−
22, is now automatically satisfied.
Equating Υ+03 = Υ−
03 and taking the zeroth order term gives the condition,
0 =2A1u(uT,u − T )
3µ2T11
3
. (5.87)
Thus we must have, on integration with respect to u,
T = Cu, (5.88)
where C is a constant of integration. Using equations (5.88) and (5.86) the condition
Υ+00 = Υ−
00 leads to,
N1 = −(
C
u
)2/3
A1. (5.89)
If we recall the matching conditions for the bulk to support an FLRW brane from equa-
tion (5.64) we can now arrive at an equation of state,
0 = 3P + (2 − C2
3 )ρ− (1 − C2
3 )σ+. (5.90)
However, this equation of state contradicts with the one we arrived at in equation (5.47)
by assuming a matching within the brane. Thus it is not possible to match both the bulk
and the brane simultaneously.
The assumptions we made along the way included the symmetry of the bulk as well
as the form of the matching surface z − Σ(t, y) = 0. One could now try to relax these
assumptions to find a bulk that does match, but such a prescription would be mostly trial
and error and there is no guarantee that a solution can be found. Rather than taking
that route we opt to investigate matched branes in general to see what insights can be
gained from the bulk matching.
Chapter 5. Cheese Slice Braneworlds 92
5.4 General Embedding of Matched Branes
It is clear from section 5.3.1 that a symmetric embedding of the Cheese Slice brane into a
bulk with Eab = 0 is not possible. In this section we investigate what class of matchings
are possible to embed into a bulk and see if there are any restrictions as to what types
of branes are possible.
There are theorems that state an analytic spacetime can be locally embedded into a
higher dimensional Cauchy development given appropriate initial data [27]. Initial data
in this sense consists of a manifold Σ, with an intrinsic metric hab and extrinsic curvature
Kab, all of which are analytic. However, in an inhomogeneous model created from a
matching, there is no guarantee that the spacetime is analytic at the matching point.
Thus these theorems cannot be invoked. Furthermore, we are not only interested in the
existence of the bulk. We would also like to see how the bulk embedding would affect
the brane.
In the following we use the concept of a brane constructed from a matching to inves-
tigate what restrictions the bulk imposes on the brane. By constructing a brane through
a matching we allow for discontinuities in the matter across the matching surface. As
we saw in section 5.2.1 this is entirely consistent within the brane. However we find that
embedding into the bulk severely restricts the brane configurations that are possible. In
particular if the embedding contains no corners, there cannot be discrete jumps in the
matter content on the brane. Conversely this means that if we are to have jumps in the
matter content on the brane, the embedding must allow for a corner to appear at that
point.
5.4.1 Set-up
Let M+4 and M−
4 be two branes with the respective metrics given by g+ab and g−ab. We will
assume that M+4 and M−
4 can be matched along some hypersurface Σ+3 = Σ−
3 = Σ3 using
Chapter 5. Cheese Slice Braneworlds 93
the Darmois matching conditions. We wish to see the consequences of embedding this
brane into some bulk. More precisely, we embed M+4 and M−
4 into respective bulks M+5
and M−5 and match the bulks along a hypersurface Σ±
4 , which we wish to be an extension
of Σ3. Thus we choose Σ±4 such that, Σ3 = M±
4 ∩ Σ±4 .
Expressions with + or − refer to the respective quantity in either M+5 or M−
5 . In
the following the superscripts will be left off general expressions that apply to both sides
of the matching.
Assuming the Darmois conditions across Σ3 implies
h+AB = h−AB where hAB ≡ gAB −mAmB. (5.91)
and also,
Ω+AB = −Ω−
AB where ΩAB ≡ hDAmB‖D =1
2LmhAB, (5.92)
where mA is the normal to Σ3 in M4.The minus sign arises from our choice that the unit
normals be pointing “inward” in both M+4 and M−
4 . Note that first fundamental form of
Σ3, hAB, and the second fundamental form, ΩAB, are calculated on the three-dimensional
subspace. The brane metric gAB is a four-dimensional quantity. hAB is also the intrinsic
metric and projection operator that projects quantities in M4 onto Σ3. The ‖ denotes
covariant differentiation on M4 associated with gAB and L the Lie derivative.
Let the unit normal to M4 be denoted nA, such that nAnA = 1, when embedded in
M5. Then we can write the bulk metric in the form,
gAB = gAB + nAnB, ds2 = gab(xc, w)dxadxb + dw2. (5.93)
The “5th coordinate”, w is defined normal to the brane such that nAdxA = dw. The
brane is located at w = 0 without loss of generality.
5.4.2 An Embedding With no Corners
We first investigate the consequences of an embedding with no corners. This gives us a
unique normal at all points p ∈ Σ3 ⊂ M4. Let us denote the normal to Σ3 by mA such
Chapter 5. Cheese Slice Braneworlds 94
that
mAmA = 1 and mA ∈M4. (5.94)
It follows from equation (5.92) that we have
m+A = −m−
A, (5.95)
and also nAmA = 0 by the definition of nA.
We have not yet defined how M+4 and M−
4 meet when they are embedded in M5. For
instance, they could meet at some angle θ measured in M5 forming a corner when viewed
from the bulk, similar to the situation in Figure 2.2 with M±4 in place of Πi and M5 in
place of V . However, in this section we will consider what happens when there is no
corner and insist that
θ = π, (5.96)
which is equivalent to stating,
n+A
∣
∣
p= n−
A
∣
∣
pfor all p ∈ Σ3. (5.97)
5.4.3 The Bulk Matching Surface
The matching surface in the bulk, Σ4, has yet to be determined. However we know that
at w = 0 this surface must coincide with Σ3. This allows us to state some properties of
the normal to Σ4 that will be important when investigating the bulk matching.
Let there be a unique normal to Σ4 at each point, denoted mA, such that mAmA = 1
and
nA∣
∣
p6= mA
∣
∣
pfor all p ∈ Σ3. (5.98)
The normalized projection of mA onto M4 at point p is then equivalent to mA at point p,
gBAmB
‖gBAmB‖
∣
∣
∣
∣
p
= mA
∣
∣
pfor all p ∈ Σ3. (5.99)
Chapter 5. Cheese Slice Braneworlds 95
To see this we can choose an orthonormal basis, e(B)A , at point p such that e
(α)A lies in Σ3
for α = (0, 1, 2). We can choose e(3)A = mA
∣
∣
pand we are left with e
(4)A = nA
∣
∣
p. Since mA
is perpendicular to Σ3 ⊂ Σ4, the only non-zero components are,
mA
∣
∣
p= m(3)e
(3)A + m(4)e
(4)A . (5.100)
The projection is then,
gBAmB
∣
∣
p= (gBA − nAn
B)mB
∣
∣
p
= (gBA − e(4)A eB(4))(m(3)e
(3)B + m(4)e
(4)B )
= m(3)e(3)A + m(4)e
(4)A − e
(4)A m(4)
= m(3)e(3)A , (5.101)
and upon normalization we have
gBAmB
‖gBAmB‖
∣
∣
∣
∣
p
= e(3)A = mA
∣
∣
pfor all p ∈ Σ3, (5.102)
thus confirming equation (5.99). Let the angle between nA and mA be ψ. Then nAmA = cosψ,
allowing us to simplifying the normalized projection,
gBAmB
‖gBAmB‖=
gBAmB√
gEF mEmF
=mA − nAn
BmB√1 − nEnF mEmF
=mA − nA cosψ√
1 − cos2 ψ
= mA cscψ − nA cotψ (5.103)
Thus at any point p ∈ Σ3 we have,
mA
∣
∣
p= (mA cscψ − nA cotψ)
∣
∣
p(5.104)
or
mA
∣
∣
p= (mA sinψ + nA cosψ)
∣
∣
p. (5.105)
Chapter 5. Cheese Slice Braneworlds 96
5.4.4 Approximation of the Bulk
The general expression for the Taylor expansion of a tensor is given by,
T (q) = T (p) + (Lξ1T )∣
∣
pλ+ (Lξ2 + L
2ξ1
)T∣
∣
pλ2 +O(λ3), (5.106)
where ξA1 and ξA2 are the generators of the diffeomorphism that are free to be specified
depending on the desired direction of the Taylor expansion. Refer to Appendix A for a
derivation of equation (5.106).
To generate an approximation for the metric tensor of the bulk, gAB, we let nA
generate the first order flow into the bulk and the natural parameterization of this flow
is the coordinate w. With the brane located at w = 0, we have to first order in w,
gAB(w) = gAB∣
∣
p+ (LngAB)
∣
∣
pw +O(w2), (5.107)
where p is a point on Σ3 (ie. w=0). Now we can use the metric form from equation (5.93)
to get,
gAB(w) = (gAB + nAnB)∣
∣
p+ Ln(gAB + nAnB)
∣
∣
pw +O(w2)
= (gAB + nAnB)∣
∣
p+ 2KAB
∣
∣
pw +O(w2), (5.108)
where the last equality follows from the extrinsic curvature of the brane,
KAB ≡ 1
2LngAB and LnnA = 0 (5.109)
Similarly we can also expand the normal to Σ4 in the same way,
mA(w) = mA
∣
∣
p+ (LnmA)
∣
∣
pw +O(w2). (5.110)
Substituting from equation (5.105) we get,
mA(w) = (mA sinψ + nA cosψ)∣
∣
p+ sinψLnmA
∣
∣
pw +O(w2).
Chapter 5. Cheese Slice Braneworlds 97
5.4.5 Matching the Bulk
We are now prepared to examine the matching conditions in the bulk. The intrinsic
metrics of Σ+4 and Σ−
4 must match, that is,
h+AB = h−AB where hAB ≡ gAB − mAmB. (5.111)
From the first order Taylor expansion of equation (5.108) and equation (5.111) we have,
hAB = (gAB + nAnB) + 2KABw
− [(mA sinψ + nA cosψ) + sinψLnmAw] [(mB sinψ + nB cosψ) + sinψLnmBw]
+O(w2), (5.112)
where it is understood that all the coefficients are evaluated at p. We can rearrange
equation (5.112) to get,
hAB = hAB − (mA cosψ − nA sinψ)(mB cosψ + nB sinψ)
+ 2[
KAB − sinψLnm(A(mB) sinψ + nB) cosψ)]
w +O(w2), (5.113)
where parenthesis, (· · · ), on the indices denote symmetrization. We can now examine
the matching to each order using equation (5.113).
Zeroth Order Matching
The first terms automatically match since
hAB∣
∣
+
p= hAB
∣
∣
−p
(5.114)
from equation (5.91).
For an embedding with no corners we must have,
ψ+ + ψ− = π, (5.115)
as illustrated in Figure (5.4). This follows from equation (5.96) since θ ≡ ψ+ +ψ−. Thus
Chapter 5. Cheese Slice Braneworlds 98
M+4
M−4
Σ+4
Σ−4
ψ−
nA
m+A
m−A
m−A
m+A
p
ψ+
Figure 5.4: Matching of two branes extended into the bulk. The solid curve represents
the branes M+4 and M−
4 . The bulks are matched across the surface Σ4 represented by
the dashed curve. The angles ψ± are measured between the normals to Σ±4 and M±
4 and
satisfy ψ+ + ψ− = π to avoid a canonical singularity at the point p ∈ Σ3.
Chapter 5. Cheese Slice Braneworlds 99
we have
cosψ+ = − cosψ− and sinψ+ = sinψ−. (5.116)
Together with equation (5.95) and equation (5.97) we have,
(m+A cosψ+ − n+
A sinψ+)∣
∣
p= (m−
A cosψ− − n−A sinψ−)
∣
∣
p. (5.117)
Thus the second term in equation (5.113) can be matched across Σ4 and the zeroth order
term of the condition in equation (5.111) is satisfied. This zeroth order matching is
essentially the matching of the brane across Σ3.
First Order Matching
For convenience in the first order matching we scale w such that w+ = w−. The first
order condition of equations (5.111) and (5.113) can be written out as,
K+AB−sinψ+
Lnm+(A(m+
B) sinψ++n+B) cosψ+) = K−
AB−sinψ−Lnm
−(A(m−
B) sinψ−+n−B) cosψ−).
(5.118)
Due to equations (5.116) and (5.95), the condition can be simplified into,
K+AB = K−
AB. (5.119)
Since KAB is the second fundamental form of M4, it is related to the matter content on
the brane by,
[Kab] = −κ25
(
Sab +1
3Sgab
)
. (5.120)
The [. . .] denotes the jump in that quantity across the brane and Sab is the energy
momentum tensor on the brane. To describe the jump in KAB across the brane we must
now consider the bulk on the other side of the brane. Let us denote these respective
quantities with a bar, ¯. This gives
[KAB]+ = K+AB −K+
AB (5.121)
Chapter 5. Cheese Slice Braneworlds 100
and
[KAB]− = K−AB −K−
AB. (5.122)
We expect the barred quantities to obey the same conditions for the matching to be valid
on the other side of the brane as well. This does not imply that the structure of the bulk
is symmetric, it only relies on the matching conditions being the same on either side.
Due to the condition of equation (5.119) we have
[Kab]+p = [Kab]
−p (5.123)
at point p. Combined with equation (5.120) we have
S+ab
∣
∣
p= S−
ab
∣
∣
p, (5.124)
since K+ = K− implies S+ = S−. The stress-energy on the brane must be the same at p
in both branes M+4 and M−
4 . That is, observers in M+4 and M−
4 must agree on the value
of the energy momentum at the matching surface.
5.4.6 Consequences of Assuming No Corner
Since we have assumed that there is no corner, the matching surface Σ3, is not unique.
Any surface that separates the brane into two distinct regions of M+4 and M−
4 can be be
defined as Σ3. We can conclude that the condition of equation (5.124) must hold true
throughout the brane. This imposes a strong constraint on the matter content of the
brane.
Consider a situation in which condition (5.124) would fail. For example the matching
surface of any Cheese Slice or Swiss Cheese model will have vacuum on one side ρ− = 0
and some uniform mass density on the other ρ+ = ρ0 6= 0. If this matching surface is
given by y = 0, then,
ρ+y→0+ 6= ρ−y→0− . (5.125)
Chapter 5. Cheese Slice Braneworlds 101
This type of discontinuity in the energy-momentum is not permitted in our embedding
of the brane. This confirms the result from section 5.3.1 where we failed to find a bulk
solution for the Cheese Slice brane. In that case we implicitly chose an embedding with
no corner when we assumed symmetry across the brane. When a corner does exist the
bulk is necessarily asymmetric in that the angle of the corner is different when viewed
from either side, θ 6= θ.
Conversely if we wish to construct an inhomogeneous brane with jumps in the energy-
momentum tensor or have any object on the brane, such as star, equation (5.96) must be
broken and θ 6= π. This would need a corner or some form of conical singularity about
Σ3.
5.5 The 3+1+1 Decomposition
The result of section 5.4.6 is a clear indication that we need to allow for a corner at
a point where we wish to have a discrete jump in the matter content. However, using
the method of a Taylor approximation into the bulk, it is not clear how a corner could
be incorporated. This difficulty arises because the corner is required to subtend some
angle, φ 6= π. This adds an additional variable to the condition of equation (5.111).
Furthermore, a matching surface in the bulk might not be needed.
The solution we found to this was to carry out a 3 + 1 + 1 decomposition of the
spacetime. This is inspired by the ADM approach [106], commonly called a 3 + 1 de-
composition, which views a four-space as the timelike evolution of a three-dimensional
hypersurface. This 3 + 1 approach was adapted to the braneworld context by Aliev and
Gumrukcuoglu [2]. They performed a 4 + 1 decomposition, thereby expressing the bulk
spacetime as the spacelike evolution of the brane. We take this one step further and
express the bulk as the evolution of a three-dimensional hypersurface in two spacelike
directions. This three-surface is then taken to be the corner. As we will show, this gives
Chapter 5. Cheese Slice Braneworlds 102
us a straightforward way to describe the brane with a corner and allows us to draw some
conclusions.
In the following we will focus on the bulk as a continuous region (ie. no bulk matching
surface). We will call the boundaries to the bulk Σ4 and M4. These two boundaries will
later be interpreted as the brane.
5.5.1 Defining the Normals, Bases and Metrics
We begin with a five-dimensional spacetime (M5, gAB) that is covered by the coordinates
xA and define two arbitrary but distinct scalar functionsW (xA) and Z(xA) such thatW =
constant describes a family of non-intersecting timelike four-dimensional hypersurfaces
that cover M5 and likewise for Z = constant. We focus on the W = 0 and Z = 0 surfaces
and insist that they intersect along a three-dimensional timelike surface which we call Σ3
such that Σ3 divides each surface into two distinct regions. We choose one of the W = 0
and one of the Z = 0 regions, calling them (M4, gab) and (Σ4, hij) respectively. The brane
can now be defined as the union of M4 and Σ4 with Σ3 ≡M4 ∩Σ4 being the corner. We
take the brane as the boundary to a region of M5 that we call the bulk. Figure (5.5)
visually depicts this situation.
The initial decomposition
This follows closely the 4+1 decomposition of the bulk described in [2]. We can introduce
two unit spacelike normals to M4 and Σ4,
nA = N∂AZ and mA = M∂AW, (5.126)
which must satisfy,
gABnAnB = 1 = gABm
AmB. (5.127)
The two functions,
N = |gAB∂AZ∂BZ|−1/2 M = |gAB∂AW∂BW |−1/2 (5.128)
Chapter 5. Cheese Slice Braneworlds 103
Z
M5
Σ3
Σ4, W = 0
M4, Z = 0 W
Figure 5.5: The Z-W plane is defined. The five-dimensional bulk is called M5. Within M5
are two timelike hypersurfaces defined by two functions Z(xA) = 0 and W (xA) = 0. We
call them M4 and Σ4 respectively. The intersection of these two hypersurfaces is a three-
dimensional, timelike hypersurface denoted by Σ3. The highlighted four-dimensional
hypersurfaces are the regions that define the brane.
are defined as the lapses associated with each part of the brane. To realize a corner these
two normals must not be collinear,
gABnAmB = cos θ 6= ±1, (5.129)
where θ is the angle between the two normals. For the later purpose of matching space-
times we insist that both normals be pointing “inward” into the bulk, see Figure 5.6 for
clarity.
The parametric equation of the brane can be defined separately for each side, xA = xA(ya),
xA = xA(ηi) and for the corner, xA = xA(ξα), as well. This allows us to find local frames
for each region given by,
eAa =∂xA
∂ya∈M4, fAi =
∂xA
∂ηi∈ Σ4, and ζAα =
∂xA
∂ξα∈ Σ3. (5.130)
These vectors are thus orthogonal to the normals,
nAeAa = 0 = mAf
Ai , and nAζ
Aα = 0 = mAζ
Aα . (5.131)
Chapter 5. Cheese Slice Braneworlds 104
(Σ3, γ)
(M−5 , g
−)
φ− φ+
(Σ−4 , h
−) (M+4 , g
+)
(M−4 , g
−) (Σ+4 , h
+)
u+
n+
v+
m+
v−
u−
m−
n−
(M+5 , g
+)
Figure 5.6: Illustration of the matching conventions that are being used. The − brane
uses the same conventions as the + brane to make the notation symmetrical. (M−5 , g
−)
and (M+5 , g
+) are the two bulks to be matched with the respective branes acting as the
matching surface.
Chapter 5. Cheese Slice Braneworlds 105
They also satisfy the completeness relations,
ebAeAa = δba f jAf
Ai = δji and ζβAζ
Aα = δβα. (5.132)
The induced metrics on the hypersurfaces are then,
gab = gABeAa e
Bb in M4, hij = gABf
Ai f
Bj in Σ4, and γαβ = gABζ
Aα ζ
Bβ in Σ3.
(5.133)
This allows us to write the bulk metric as either,
gAB = gabeaAe
bB + nAnB (5.134)
or
gAB = hijfiAf
jB +mAmB. (5.135)
Four-dimensional indices of tensors on the brane are raised and lowered with gab and
hij depending on which part of the brane the tensor is evaluated in. Three-dimensional
tensors on the corner have the indices raised and lowered by γαβ.
A Recursive Decomposition
Up to this point we have essentially carried out two distinct 4 + 1 decompositions of M5.
To complete the decomposition we make use of Σ3 to perform a recursive decomposition
that could be understood as a (3 + 1) + 1 decomposition. Since Σ3 is defined as the
intersection of M4 and Σ4, it must be a member of a family of surfaces that intrinsically
cover the brane. We introduce normals to Σ3 such that,
ua = U∂aW ∈M4 and vi = V ∂iZ ∈ Σ4. (5.136)
Here U and V are lapse functions within M4 and Σ4 respectively. Analogous to the above
procedure we insist that they be unit normals satisfying,
gabuaub = 1 = hijv
ivj, (5.137)
Chapter 5. Cheese Slice Braneworlds 106
We can define local frames orthogonal to these normals by,
εaα =∂ya
∂ξαand ǫiα =
∂ηi
∂ξα(5.138)
satisfying,
εaαεβa = δβα = ǫiαǫ
βi and uaε
aα = 0 = viǫ
iα, (5.139)
which allows us to carry out the (3 + 1) decomposition in M4 and Σ4 respectively with
the resulting metrics,
gab = γαβεαaε
βb + uaub (5.140)
and
hij = γαβǫαj ǫβj + vivj. (5.141)
The three-bases εaα, ǫjα and ζAα all span the corner, Σ3, and are related by
εaαeAa = ǫiαf
Ai = ζAα . (5.142)
We choose to use ζAα as the preferred basis for Σ3.
Now since ua is in M4, we can write it in terms of the M4 frame which is orthogonal
to nA such that,
(uaeAa )nA ≡ uAnA = 0. (5.143)
Similarly in Σ4 we have,
(vifAi )mA ≡ vAmA = 0. (5.144)
From our initial assumption that nA and mA are not collinear in equation (5.129), it
follows that uA and vA are also not collinear.
Combining equations (5.134), (5.135) and (5.140) we can express the bulk metric in
its desired form,
gAB = γαβζαAζ
βB + uAuB + nAnB. (5.145)
In Σ4 we can perform the same procedure to get an alternate, but equivalent decompo-
sition,
gAB = γαβζαAζ
βB + vAvA +mAmB. (5.146)
Chapter 5. Cheese Slice Braneworlds 107
A Note on Sign Conventions
As alluded to after equation (5.129), we must state some sign conventions before we
continue. Our initial construction of the brane leaves some ambiguity in determining the
angle between M4 and Σ4 that we will now clarify.
Let φ be the angle of the corner as measured in the two-space normal to Σ3. This
space is spanned by nA and mA with the angle between them being given by θ,
gABnAmB = cos θ. (5.147)
Being orthogonal to Σ3, uA and vA are also in this two-space and since they are in M4
and Σ4 respectively, the angle between them will give us the angle of the corner,
gABuAvB = cosφ. (5.148)
The ambiguity lies in the relation between θ and φ which is determined by whether the
normals point “outwards” or “inwards”. We will adopt the convention that all normals
point “inward” into their respective spaces as dipicted in Figure 5.6. This results in the
relation,
θ = π − φ with
0 < φ < 2π and φ 6= π
−π < θ < π and θ 6= 0. (5.149)
Since φ is the angle between uA and vA we have,
gABuAvB = cosφ = − cos θ. (5.150)
To further clarify the relation between the various normal vectors we note that mA, vA
forms an orthonormal basis of the two-space. Thus we can express nA as a linear combi-
nation of the basis vectors,
nA = C1vA + C2m
A, (5.151)
where C1 and C2 are constants. Now if we project nA onto Σ4 using gab eAa e
bB as the
Chapter 5. Cheese Slice Braneworlds 108
projection operator we get,
(gab eAa e
bB)nB = (gAB −mAmB)nB (5.152)
= nA −mA cos θ (5.153)
= C1vA. (5.154)
Only the vA component remains since mA is by definition orthogonal to Σ4. Contracting
equations (5.153) and (5.154) with gABnB gives,
1 − cos2 θ = C1 gABnBvA (5.155)
and contracting equation (5.151) with gABvB gives,
gABnAvB = C1. (5.156)
Therefore C1 = sin θ. Applying the same method with the projection of mA onto M4
allows us to find that 3,
nA −mA cos θ = vA sin θ. (5.157)
Likewise,
mA − nA cos θ = uA sin θ, (5.158)
and it follows that,
gABnAvB = sin θ = gABm
AuB. (5.159)
The relation between uA, νA, mA and nA will become important when we examine
the bulk matching in section 5.6, particularly in Figure 5.6, but first we must fix the
coordinates and find the form of the metrics.
5.5.2 Fixing the Coordinates
Next we define the spacelike vectors WA and ZA such that,
WA∂AW = 1 = ZA∂AZ, (5.160)
3Note the similarity between these definitions and those of section 5.4.3
Chapter 5. Cheese Slice Braneworlds 109
which can be thought of as the “evolution vectors” off the brane into the bulk. These
vectors are tangent to the congruence of curves intersecting the hypersurfaces that cover
the bulk. In general these vectors are not orthogonal to the brane. However, we can
decompose them using the bases defined in section 5.5.1 giving the components orthogonal
and tangential to the corner,
WA = NnA +NaeAa ZA = MmA +M ifAi
= NnA + (Uua + ναεaα)eAa = MmA + (V vi + µαǫiα)f
Ai
= NnA + UuA + ναζAα = MmA + V vA + µαζAα
(5.161)
Here the 4-vectors Na and M i are known as the shift vectors in the 4+1 decompositions.
M,N,U, and V are the same as those defined in equations (5.126) and (5.136). We make
use of our recursive strategy to further decompose these 4-vectors into “3 + 1-vectors”,
thereby giving the final line of equation (5.161). The 3-vectors, να and µα are the shift
vectors of Σ3 associated with the evolution of WA and ZA respectively. Also, U and V
are the respective lapses of Σ3 within M4 and Σ4.
We now use these vectors to fix the coordinates of M5 such that,
xA ≡ (ξα, Z,W ), (5.162)
ya ≡ (ξα, Z), (5.163)
ηi ≡ (ξα,W ). (5.164)
Then by equations (5.160) and (5.130) we have,
ZA =
(
∂xA
∂Z
)
W=0
= δAZ , (5.165)
WA =
(
∂xA
∂W
)
Z=0
= δAW , (5.166)
ζAα =
(
∂xA
∂ξα
)
Z=W=0
= δAα . (5.167)
Chapter 5. Cheese Slice Braneworlds 110
5.5.3 Finding the Metrics
We begin with M4, which in the coordinates of xA, can now be defined as the W = 0
hypersurface. In this case we have,
dya =∂ya
∂ξαdξα +
∂ya
∂ZdZ (5.168)
= εaαdξα + eaAZ
AdZ (5.169)
since
∂ya
∂Z=∂ya
∂xA∂xA
∂Z= eaAZ
A. (5.170)
From equation (5.168) we find that,
dyadyb = εaαεbβdξ
αdξβ + εaαdξα(ebBZ
B)dZ + εbβdξβ(eaAZ
A)dZ + eaAebBZ
AZBdZ2. (5.171)
Now we can use equations (5.140) and (5.171)to find the line element.
ds2 = gabdyadyb (5.172)
= (γα′β′εα′
a εβ′
b + uaub)dyadyb (5.173)
= γαβdξαdξβ + 2γαβ′ζβ
′
A ZAdξαdZ
+[
γαβ′ζαAζβ′
B ZAZB + uAuBZ
AZB]
dZ2 (5.174)
= γαβdξαdξβ + 2µαdZdξ
α +[
µαµα + (M sin θ − V cos θ)2
]
dZ2 (5.175)
Equations (5.142) and (5.161) were used, as well as,
uAZA = M sin θ − V cos θ, (5.176)
which results from equation (5.159). We can then read off the brane metric,
gab =
γαβ µα
µβ µαµα + M2
(5.177)
where M ≡ (M sin θ − V cos θ). The inverse is then,
gba =
γαβ +µαµβ
M2− µβ
M2
− µα
M2
1
M2
, (5.178)
Chapter 5. Cheese Slice Braneworlds 111
Recall that the indices are such that a, b, . . . = 0, 1, 2, 3 and α, β, . . . = 0, 1, 2. γαβ is
therefore three-dimensional. The metric of Σ4 can be found in a similar manner,
hij =
γαβ νβ
νβ νανα + N2
(5.179)
where N ≡ (N sin θ − U cos θ). With the inverse,
hij =
γαβ +νανβ
N2− νβ
N2
− να
N2
1
N2
. (5.180)
Now in these coordinates we have,
nA = (0, 0, 0, 0, N) and mA = (0, 0, 0,M, 0) (5.181)
From equations (5.158) and (5.157) we have,
uA =mA − nA cos θ
sin θ= (0, 0, 0,M csc θ,−N cot θ) (5.182)
and
vA =nA −mA cos θ
sin θ= (0, 0, 0,−M cot θ,N csc θ). (5.183)
Thus,
ua = (0, 0, 0,M csc θ) (5.184)
and
va = (0, 0, 0, N csc θ) (5.185)
Therefore the lapses introduced in equation (5.161) are,
U ≡ M
sin θand V ≡ N
sin θ. (5.186)
Thus we can eliminate U and V .
Chapter 5. Cheese Slice Braneworlds 112
5.5.4 The Bulk Metric
Now we carry out the same procedure for the bulk spacetime. We have,
dxA = ζAα dηα + ZAdZ +WAdW. (5.187)
The metric can be taken in either form,
gAB = γαβζαAζ
βB + uAuB + nAnB (5.188)
(5.189)
= γαβζαAζ
βB + vAvB +mAmB.
Computing the line element allows us to express the metric in matrix form. With
A,B = 0, 1, 2, 3, 4 and α, β = 0, 1, 2, we arrive at,
gAB =
γαβ µα να
µβ µαµα + M 2 Ψ
νβ Ψ νανα + N 2
, (5.190)
where,
M2 ≡ M2 +N2 sin2 θ (5.191)
N2 ≡ N2 +M2 sin2 θ (5.192)
Ψ ≡ M2 +N2 −MN cot2 θ cos θ . (5.193)
Now we can see that θ, M , N , µα, να and γαβ constitute the 15 arbitrary functions that
we would expect in a general five- dimensional metric. Equivalently one could use Ψ, M ,
N in place of θ, M and N . They are also related to M and N by,
M |W=0 = M and N |Z=0 = N . (5.194)
To find the inverse we also make use of a recursive strategy, we first write the bulk
metric as a 4 + 1 decomposition,
gAB =
gab Pa
Pb PaPa +B2
, (5.195)
Chapter 5. Cheese Slice Braneworlds 113
where Pa and B we treat as unknown functions. In this form it is easy to read off the
inverse as,
gBA =
gba +P bP a
B2
−P b
B2
−P a
B2
1
B2
, (5.196)
which is identical in block form to equation (5.178). We can in fact explicitly write out gab
from equation (5.178), but we must be careful to use M as the lapse function to ensure
we do not loose the W -dependence of gAB. We must now decompose Pa and find the
contravariant component P a ≡ gabPa. We identify equation (5.196) with equation (5.190)
and find that Pa is given by,
Pa ≡
να
Ψ
. (5.197)
Explicitly performing the calculation for P b gives:
P b =
γαβ +µαµβ
M 2− µβ
M 2
− µα
M 2
1
M 2
να
Ψ
=
νβ −(
Ψ − µαναM 2
)
µβ
Ψ − µαναM 2
(5.198)
=
νβ − Ψµβ
M 2
Ψ
M 2
(5.199)
≡
P β
P 3
,
Chapter 5. Cheese Slice Braneworlds 114
where we have defined Ψ ≡ Ψ − µανα. Expanding equation (5.196) gives,
gBA =
γαβ +µαµβ
M 2+P βPα
B2− µβ
M 2+P 3P β
B2−P
β
B2
− µα
M 2+P 3Pα
B2
1
M 2+P 3P 3
B2−P
3
B2
−Pα
B2−P
3
B2
1
B2
. (5.200)
To carry on we identify equation (5.190) with equation (5.195) to give,
νανα + N
2 = PaPa +B2 (5.201)
= PαPα + P3P
3 +B2
= νανα − ναµ
αΨ
M 2+
ΨΨ
M 2+B2
= νανα +
Ψ2
M 2+B2 (5.202)
and solve for B2 giving,
B2 =N 2M 2 − Ψ2
M 2. (5.203)
Now all that remains is to substitute equations (5.203) and (5.199) into the expanded
form of gAB (equation (5.200)). After some simplification we arrive at the final form of
the inverse metric,
gBA =
γβα +N 2µβµα + M 2νβνα − 2Ψµ(βνα)
Φ
−N 2µβ + Ψνβ
Φ
−M 2νβ + Ψµβ
Φ
−N 2µα + Ψνα
Φ
N 2
Φ
−Ψ
Φ
−M 2να + Ψµα
Φ
−Ψ
Φ
M 2
Φ
,
(5.204)
Chapter 5. Cheese Slice Braneworlds 115
where we have defined Φ ≡ M 2N 2 − (Ψ − µγνγ)2 and recall Ψ ≡ (Ψ − µγν
γ). It is
straightforward to verify that equation (5.204) is indeed the inverse by checking that
gAB gBA = I. We could have equally well used hij in the initial 4 + 1 decomposition to
arrive at the same result.
5.6 The Matching of the Bulk
The theorems of Clarke and Dray [22] state that the minimum condition for two space-
times to match is that the intrinsic metric on the matching surface agrees. In addition,
Taylor’s corner conditions [98] insist that the matching at a corner also requires that
observers on either side of the matching agree on the angle of the corner.
From section 5.4.6 we know that jumps in the matter content on the brane are not
possible without a corner in the embedding. The 3 + 1 + 1 decomposition above forms a
natural environment to investigate these corners. When a corner exists, we have an angle
between the branes such that φ 6= π. Thus the coordinate system used in equation (5.162)
are well defined.
Let M4 be part of the brane with some matter content and Σ4 be the other part of
the brane with different matter content, possibly a vacuum. Let the different sides of
the bulk be denoted by, +, and, −, superscripts. The metric of the bulks are then g+AB
and g−AB. This construction is depicted in Figure 5.6. We can now see the advantage of
this method over the method of section 5.4 as it does not require a superfluous matching
surface in the bulk.
5.6.1 The Matching Conditions
We would like to keep the same conventions on either side of the brane when it comes to
defining angles and norms which is discussed in detail in section 5.5.1. To ensure that we
can use the method on both sides we must match Σ−4 to M+
4 and M−4 to Σ+
4 . This ensures
Chapter 5. Cheese Slice Braneworlds 116
that the sign conventions are the same on either side, making the − bulk completely
analogous to the + bulk, and allowing for an intuitive definition of φ±. Figure 5.6 shows
how the φ+ and φ− are defined.
The matching of the bulk along Σ−4 and M+
4 requires the first fundamental forms to
be identical when calculated on either side. Thus
h−ab = g+ab (5.205)
and the matching along Σ+4 and M−
4 requires that
g−ij = h+ij. (5.206)
From equations (5.177) and (5.179) it follows that,
γ+αβ = γ−αβ µ+
α = ν−α ν+α = µ−
α (M+)2 = (N−)2 and (N+)2 = (M−)2
(5.207)
The final condition required at the corner is that,
φ+ + φ− = 2π, (5.208)
or equivalently,
θ+ = −θ− (5.209)
From the definitions of M , N , equation (5.207) and equation (5.209) we have,
M+ = ςN− (5.210)
M+ sin θ+ − V + cos θ+ = ςN− sin θ− − ςU− cos θ− (5.211)
M+ sin θ+ −N+ cot θ+ = ςN− sin θ− − ςM− cot θ− (5.212)
M+ sin θ+ −N+ cot θ+ = −ςN− sin θ+ + ςM− cot θ+ (5.213)
M+ sin θ+ + ςN− sin θ+ = N+ cot θ+ + ςM− cot θ+ (5.214)
(M+ + ςN−) sin θ+ = (N+ + ςM−) cot θ+ (5.215)
(M+ + ςN−) = (N+ + ςM−) cot θ+ csc θ+, (5.216)
Chapter 5. Cheese Slice Braneworlds 117
where ς = ±1. Now from the other relation in equation (5.207) involving N+ and M−
we have analogously,
(M− + N+) = (N− + M+) cot θ+ csc θ+ (5.217)
where = ±1.
We can consider two cases:
Case 1; ς = :
In this case we have,
1 = cot2 θ+ csc2 θ+ (5.218)
or equivalently
cos2 θ+ ± cos θ+ − 1 = 0, (5.219)
which, surprisingly, is the equation for the golden ratio. It is interesting that our matching
conditions lead to this specific value. The cosine is then equal to the golden ratio with
the possibility of sign differences,
cos θ+ =1 ±
√5
2or
±1 +√
5
2. (5.220)
In our predefined range given in equation (5.149) we have the possibilities of
θ+ = ±51.8,±128.2 (5.221)
and
φ+ = 51.8, 128.2, 231.8, 308.2 (5.222)
We list the φ values, as they give a more intuitive picture of the matching angle. We can
also solve for M± and N± since cot θ+ csc θ+ = ±1,
(M+ + ςN−) = q(N+ + ςM−) (5.223)
(M− + ςN+) = p(N− + ςM+) (5.224)
Chapter 5. Cheese Slice Braneworlds 118
where p, q = ±1. If p = q there is no new information. In the case that p 6= q we are
left with the result,
M+ = −ςN− and N+ = −ςM− (5.225)
Case 2; ς = −:
In this case we have,
(M+ + ςN−) = (N+ + ςM−) cot θ+ csc θ+ (5.226)
(M− + N+) = (N− + M+) cot θ+ csc θ+, (5.227)
which is equivalent to
(M+ + ςN−) = (N+ + ςM−) cot θ+ csc θ+ (5.228)
(ςM− −N+) = (ςN− −M+) cot θ+ csc θ+. (5.229)
Unlike case 1, θ can take on any value and we are left with,
(M+)2 − (N−)2 = (N+)2 − (M−)2. (5.230)
The jumps in the lapses squared is equal on either side of the corner.
5.6.2 The Second Fundamental Form and Matter Content
The matching conditions stated in section 5.6 are the minimal conditions to ensure that
a spacetime exists. We have yet to consider what stress energy is on the brane. To do
so we must look at the jump in extrinsic curvature or the second fundamental form.
We define the extrinsic curvature of M4 and Σ4 respectively as,
Kab = nA||BeAa e
Bb ∈M4 (5.231)
and
Ωij = mA||BfAi f
Bj ∈ Σ4, (5.232)
Chapter 5. Cheese Slice Braneworlds 119
where || denotes covariant differentiation with respect to gAB The extrinsic curvature of Σ3
has two components due to the two-space of normals. We define the second fundamental
form of this space through any two vectors that span this two-space. For example, we
have the two choices,
nωαβ = nA||BζAα ζ
Bβ (5.233)
mωαβ = mA||BζAα ζ
Bβ , (5.234)
where we use the left-superscript to denote which normal is being used to compute the
component of the extrinsic curvature.
In M4 the normal to Σ3 is ua and in Σ4 the normal is vi. Thus the extrinsic curvature
of Σ3 as measured within the brane is,
uωαβ = ua;bεaαε
bβ ∈ Σ3 ⊂M4 (5.235)
vωαβ = vi;jǫiαǫjβ ∈ Σ3 ⊂ Σ4 (5.236)
where ; denotes the covariant derivative with respect to the brane metric (either gab ∈M4
or hij ∈ Σ4). Now we observe that,
ua;b = uA||BeAa e
Bb (5.237)
vi;j = vA||BfAi f
Bj (5.238)
since ua and vi are defined to lie in constant Z and W surfaces respectively. Using
equation (5.142), we can write equations (5.235) and (5.236) as,
uωαβ = uA||BζAα ζ
Bβ (5.239)
vωαβ = vA||BζAα ζ
Bβ (5.240)
Finally, using equations (5.182) and (5.183) we have,
uωαβ =(
mA||B csc θ − nA||B cot θ)
ζAα ζBβ (5.241)
vωαβ =(
nA||B csc θ −mA||B cot θ)
ζAα ζBβ (5.242)
Chapter 5. Cheese Slice Braneworlds 120
or in terms of Kab and Ωij,
uωαβ = Ωijǫiαǫjβ csc θ −Kabε
aαε
bβ cot θ (5.243)
vωαβ = Kabεaαε
bβ csc θ − Ωijǫ
iαǫjβ cot θ, (5.244)
If the observer on the brane is to see no stress energy on the surface Σ3, then the jump
in extrinsic curvature, as measured intrinsic to the brane, must be zero. This means,
uωαβ = vωαβ. (5.245)
Therefore from equations (5.243) and (5.244)
Kabεaαε
bβ = Ωijǫ
iαǫjβ. (5.246)
Taking the trace of the left-hand side of equation (5.246) gives,
(Kabεaαε
bβ)γ
αβ = Kabεaα(ε
αc g
cb) (5.247)
= Kabgab (5.248)
= K (5.249)
The trace of the right hand side is,
(Ωijǫiαǫiβ)γ
αβ = Ωijǫjα(ǫ
αi h
ij) (5.250)
= Ωijgij (5.251)
= Ω (5.252)
Therefore
K = Ω. (5.253)
From the definitions of Kab and Ωij (equations (5.231) and (5.232)) we also have,
nA||BζAα ζ
Bβ = mA||Bζ
Aα ζ
Bβ . (5.254)
Chapter 5. Cheese Slice Braneworlds 121
The consequences of this condition on the bulk matching can now be examined. Let
the matter content on the brane be denoted Tab ∈M4 and Sij ∈ Σ4 to distinguish between
either region of the brane. Using the matching situation depicted in Figure (5.6) we have,
K+ab + Ω−
ab = κ25(Tab −
1
3g+abT ) (5.255)
and
Ω+ij +K−
ij = κ25(Sij −
1
3h+ijS). (5.256)
Here we add the extrinsic curvature because they point in different directions. If we now
look only at the components on Σ3, applying equation (5.246), we have,
Tabεaαε
bβ −
1
3γαβT = Sijǫ
iαǫjβ −
1
3γαβS (5.257)
Then the traces of equations (5.255) and (5.256) along with equation (5.253) give,
K+ + Ω− =κ25
3T
Ω+ +K− =κ25
3S
=⇒ T = S, (5.258)
and finally equation (5.257) takes the form,
Tabεaαε
bβ = Sijǫ
iαǫjβ (5.259)
Thus the matter content in each region of the brane, projected onto the matching surface,
Σ3, must be equal. If we have dust on one side, Tab = ρgab, and vacuum on the other,
Sab = 0, we arrive at an inconsistency.
This is in direct contrast to the case of a star on the brane discussed by Germani
and Maartens [50] where they assumed a perfect fluid, constant density star ρ = const
matched to an external vacuum ρ = 0 = P . They then go on to investigate the non-local
effects that the bulk embedding imposes on brane. However, their initial assumption
appears to contradict our result. The same case arises in [47] where a perfect fluid brane
is punctured with Schwarzschild voids.
Chapter 5. Cheese Slice Braneworlds 122
One difference in our results from those of Germani and Maartens is that they assumed
the non-local bulk effects on the brane were different inside the star and outside the star.
In essence this assumes that the bulk itself is inhomogeneous and has a different structure
in either region. In comparison, our decomposition assumed the bulk (on either side of
the brane) to be one continuous region. This suggests that a combination of the methods
in section 5.4 and section 5.5 is required for a full treatment of inhomogeneous branes.
In contrast, the Swiss Cheese type brane of Gergely assumed that the non-local bulk
effects were zero in either region. This makes a strong assumption about the bulk and
is similar to what we did in section 5.3, where we found this type of assumption to be
too restrictive. This assumption is also stronger than that of Germani and Maartens.
Our results suggest that the initial assumptions of the Swiss Cheese brane need to be
reconsidered.
5.6.3 Matching Four Bulks
The condition of equation (5.257) is rather restrictive. It is possible to have more general
brane constructions using our formalism by considering matching several bulks together.
If we take four different bulk spacetimes, as depicted in Figure (5.7), and apply our
decomposition in each bulk region M i5, where i = 1, 2, 3, 4. We will use integer super-
scripts to distinguish between the four different regions with the variables defined as in
Figure (5.7) The matching conditions can be applied to each region requiring,
h1 = g2, h2 = g3, h3 = g4, h4 = g1, (5.260)
where the tensor indices have been left off for clarity, and the angle condition,
φ1 + φ2 + φ3 + φ4 = 2π. (5.261)
If we choose M14 and M3
4 (equivalently Σ44 and Σ2
4) to define the brane, then the
three-dimensional matching requires,
u1
ωαβ = u3
ωαβ. (5.262)
Chapter 5. Cheese Slice Braneworlds 123
φ1
n1
m1
u1
γαβ v1
(Σ24, h
2)
n2
m2
u2
φ4
φ3 φ2
v3
u3v2
m3
n3
(Σ44, h
4)
(M34 , g
3)
(Σ34, h
3)
m4
n4v4
u4
M 45
M 35 M 2
5
M 15
(Σ14, h
1)
(M24 , g
2)
(M14 , g
1)
(M44 , g
4)
Figure 5.7: The matching of four different bulks is depicted. The solid lines represent the
brane and the dotted lines represent the matching surface. The 3 + 1 + 1 decomposition
can be applied to each region. The four corners are identified with γαβ being the metric
on the corner.
Chapter 5. Cheese Slice Braneworlds 124
In terms of the extrinsic curvature of the brane this gives,
Ω1ijǫ
iαǫjβ csc θ1 −K1
abεaαε
bβ cot θ1 = K3
abεaαε
bβ csc θ3 − Ω3
ijǫiαǫjβ cot θ3, (5.263)
This condition is not as restrictive as equation (5.245) and it is evident that breaking up
the bulk allows for more general matchings to occur.
This construction is very similar to the idea of brane collisions which we will discuss
in detail in section 5.7
5.6.4 Special Cases: Breaking the Angle Condition
An interesting case occurs if we are not required to satisfy equation (5.208). This forms
a conical singularity in the bulk; however the brane can still be well defined as we will
show.
Let us examine the trivial case where we assume that one part of the brane is a
vacuum. In this case, there is no jump in extrinsic curvature, Ωij or Kab, across the
brane in that region. Let us take the brane defined by the matching of M−4 and Σ+
4 to
be the vacuum. The normals are co-linear,
nA− = −mA−, (5.264)
and the bulk would appear continuous at that point with no jump in the stress energy.
Thus it no longer matters how we define this part of the brane. We can use the remaining
normals, which are not co-linear, to span the two-space that defines our Z −W plane.
In essence, rather than matching two bulks, we can view this construction as a removal
of a wedge in the bulk and then identifying the resulting boundaries. See Figure 5.8 for
an illustration of this construction. It is then straightforward to apply the 3 + 1 + 1
decomposition by using the normals nA+ ≡ nA and mA− ≡ mA.
The matching conditions, equation (5.205),then require that,
gab = hab. (5.265)
Chapter 5. Cheese Slice Braneworlds 125
m−
(M+
4 , g+)
φ
φ−
(M−
4 , g−)
u−
n−
m−
v−
(Σ−
4 , h−)
φ+
(Σ+
4 , h+)
u+
n+
v+
m+
(M+
4 , g+)
n+
u+
v−
(Σ−
4 , h−)
Figure 5.8: If we assume that M−4 and Σ+
4 are vacuums then there is no jump in extrinsic
curvature across that region of the brane. We can then apply the 3 + 1 + 1 approach of
section 5.6 to M+4 and Σ−
4 allowing m− and n+ to span the Z−W plane. In essence, this
is identical to removing one wedge out of the bulk and matching the resulting boundaries
to each other. There is a conical singularity at the corner, however Σ3 and the brane
itself is well defined.
Chapter 5. Cheese Slice Braneworlds 126
It follows that µα = να and N2 = M2. The bulk metric must take the form,
gAB =
γαβ µα µα
µβ µαµα + M 2 Ψ
µβ Ψ µαµα + M 2
. (5.266)
5.7 Summary and Discussion
We have shown that it is possible to match the FLRW and the Kasner regions within
the branes. The solution is similar to the four dimensional Cheese Slice universe with
the exception of the equation of state that had changed through the modified Friedmann
equation. In general the pressure in the FLRW region is non-zero in the braneworld
context, while in the four-dimensional case zero pressure was required for the match-
ing. Through a detailed investigation of the energy conditions, we have shown that the
energy conditions can be satisfied in certain regimes and thus the matter content can
be considered physically reasonable. However we were unable to find a straightforward
bulk for the Cheese Slice brane. It appears that failure to find the bulk arises from the
assumption of bulk symmetry, which necessitates an embedding with no corners.
Through the investigation of general brane matchings we have shown that if there are
no corners in the embedding, the matter content in the brane must be continuous at all
points. This result was arrived at when attempting to match the first fundamental form
of the bulk to first order. It is likely that higher order matchings would impose even
greater constraints on the brane configurations that are possible.
Finally the 3 + 1 + 1 decomposition of the bulk seems to be the most useful and
applicable method. It allowed us to quite easily find the matching conditions that are
required at a corner and draw some conclusions about the matter content of the brane.
It appears that in an inhomogeneous brane, the matter content in each region must agree
when projected onto the matching surface. This is unintuitive as it rules out physically
reasonable spacetimes, like the Swiss Cheese models. We then found that allowing the
Chapter 5. Cheese Slice Braneworlds 127
bulk to have more structure gave more freedom to the brane configuration. For instance,
the matching of four bulks did not have the same restriction as matching two bulks.
This matching of four bulks relates closely to the idea of brane collisions as each
matching surface can be considered a brane in itself. Our investigation decomposed the
bulk spacetime along two spacelike directions. If one were to repeat our decomposition
along a timelike direction and one spacelike direction then our four step matching can be
interpreted as two branes colliding, producing two other branes. This type of collision
was investigated by Neronov [82] who looked at Friedmann type branes colliding in an
Anti-de Sitter bulk. He was able to derive a conservation law which was later generalized
by Khoury et al. [63] who showed that the conservation law amounted to momentum
conservation. In these cases they all assumed the bulk to be vacuum Schwarzschild-Anti-
de Sitter spacetimes. Though our method was developed to investigate inhomogeneities
on the brane, it could also be used to investigate brane collisions. Our method has the
advantage of not assuming any specific form of the bulk, thereby allowing a more general
way to describe these collisions.
Returning to our matching conditions, our results suggest that inhomogeneities on
the brane require non-trivial embeddings. We emphasize the need to take the bulk into
consideration and carefully consider its implications. As in the case of Germani and
Maartens [50] some sort of structure must be allowed to exist in the bulk to differentiate
the matter filled regions and the vacuum regions of the brane. This structure in the
bulk is likely to induce some sort of non-local effect on the brane as well. Perhaps the
most vital lesson is that we cannot assume an arbitrary matter content on the brane.
Any inhomogeneous brane must be viewed in the context of its bulk embedding and any
non-local effects must be considered as well.
Chapter 6
Summary and Conclusions
Throughout this thesis we have utilized the concept of spacetime matchings in different
contexts to investigate inhomogeneous cosmologies. We first used the Darmois matching
conditions to construct the Cheese Slice universe. This toy model served as the example
of choice in our investigations since it has the advantage of being a mathematically simple
model which can incorporate inhomogeneities.
Through investigating the lookback time verses the redshift relation we were able
to demonstrate that an observer in the Cheese Slice universe will see anisotropies that
depend on the angle of observation relative to the matching plane. The relative thickness
of the Kasner vacuum layers and the matter field FLRW regions plays a dominant role in
determining this anisotropy with the number of layers and distribution of layers playing
a smaller role. When comparing such results with the CMB data we find that the Kasner
regions must be on the order of ten thousand times thinner than the FLRW regions to
fall within limits of the observed CMB anisotropies. Though this might seem like a small
vacuum region, the very existence of such an inhomogeneity breaks the symmetries of
the FLRW cosmology and will affect some properties of the spacetime.
The approach to the singularity is one situation in which even a small inhomogeneity
should be carefully examined. Conventional cosmologies assume an isotropic homoge-
128
Chapter 6. Summary and Conclusions 129
neous singularity and attempt to explain the formation of the large scale structure as
the model evolves. This does not need to be the case as the initial singularity itself
can be inhomogeneous. The concept of an AVTD singularity describes an approach to
the singularity that is independent of the spatial curvature. Thus, the three-space could
have any structure, including an inhomogeneous one, without affecting the evolution to-
wards the singularity. We were able to devise a criterion for a matched spacetime to be
AVTD based on the ability to match the foliations used in the definition of AVTD. This
demonstrates another application of matching conditions, in this case applied to each
hypersurface that constitutes a leaf of the foliation. Using this criterion the Cheese Slice
universe, both with flat and open FLRW regions, were shown to be AVTD. Thus the
structure of the inhomogeneities, the size and distribution of the slices, arises directly
from the singularity itself. This means that structure in the universe could be an initial
condition in addition to any structure that forms during the cosmological evolution.
We conjecture that any spacetime that can be matched to an AVTD spacetime would
also be AVTD. This is based on the fact that the Darmois conditions, which are used
to match the foliations, are equivalent to the Lichnerowicz conditions. The Lichnerowicz
conditions guarantee a coordinate system that is continuous through the matching sur-
faces. In this coordinate system it is likely that a reasonable foliation could be found to
prove the AVTD property. A rigorous proof of this conjecture could be the subject of fu-
ture investigations. These AVTD singularities are of much interest because the approach
to the singularity plays a fundamental role in any cosmology. Our study has offered a
new perspective on these singularities in terms of matching foliations and demonstrates
that an inhomogeneous universe can be AVTD.
In higher dimensional models, matching conditions play a pivotal role. The braneworld
models are essentially the matching of two five-dimensional spacetimes across a brane rep-
resenting the four-dimensional universe. The problem then becomes one of embedding of
the brane within a bulk. Focusing on the Cheese Slice model we were able to show that
Chapter 6. Summary and Conclusions 130
it is possible to construct an inhomogeneous brane with matter content that satisfies the
energy conditions in certain regimes. This brane and matter content obeys the modi-
fied Friedmann equations that arise from projecting the five-dimensional EFE onto the
brane. However, we came across difficulties when attempting to embed this brane into a
bulk. We showed that it is not possible to find approximations to a symmetric bulk that
could support such a brane. This serves as a cautionary example that shows one cannot
arbitrarily construct branes, such as a Swiss cheese brane or stars on the brane, without
taking the bulk matching into consideration.
In the more general case, we looked at the conditions for any brane constructed
from a matching to be embedded in a bulk. Using a Taylor expansion of the bulk
in a neighbourhood of the brane we found that if the embedding had no corners, the
brane cannot have discrete jumps in the energy-momentum tensor. This is makes sense
intuitively as the extrinsic curvature is directly related to matter content. Any discrete
jump in energy-momentum should be accompanied by jumps in extrinsic curvature along
the brane. It is clear that if the brane were to be inhomogeneous there must be corners
in the embedding. Thus it would be prudent for those studying braneworld models to
consider the consequences of assuming symmetry about the brane. It is possible to have
corners and also symmetry, but this would require addition structure in the bulk. There
is then no advantage to assuming symmetry in the context of an inhomogeneous brane.
This leads us to the 3 + 1 + 1 decomposition of the bulk inspired by the ADM
decomposition. We were able to find a coordinate system that was adapted to suit the
brane with a corner that can be applied to either side of the bulk. The coordinates were
such that W = 0 represented one region of the brane and Z = 0 defined the other. In fact
this type of coordinates would only be well defined if there is a corner making the W and
Z coordinates distinct. Through this construction we found that the matter content of
each brane had to be equal when projected onto the matching surface. This proved to be
very restrictive and rules out reasonable matter content on the brane, such as the Cheese
Chapter 6. Summary and Conclusions 131
Slice brane. It lead us to conclude that an inhomogeneous brane must be embedded into
an inhomogeneous bulk.
It is still possible to apply the 3+1+1 decomposition to an inhomogeneous bulk. This
would require decomposing more regions of the bulk, which lead us to consider a model
consisting of matching four separate bulk regions. The similarity between this type of
model and braneworld collisions became apparent. Our method can be applied to the
collisions by decomposing the spacetime along one timelike and one spacelike direction.
This can allow us to describe brane collisions in more general bulk spacetimes rather then
the Anti-de Sitter bulks that are commonly assumed. Though this wasn’t the initial aim
of our study, it is interesting to see that our method can apply to other braneworld
constructions.
An interesting case occurred when we broke the angle condition when matching
around the corner. We showed that it was possible for a corner to manifest itself as
a conical singularity in the bulk. The brane remains well behaved at the corner.
Ideally one would like to be able to find exact solutions for these inhomogeneous
branes. It is perhaps possible to do so by utilizing the decomposition technique. One
possible method would be to decompose the five-dimensional field equations in this man-
ner. This will provide a natural environment to search for exact solutions with corners.
Exact bulk solutions will allow us to calculate the non-local effects of the bulk on the
perceived matter content of the brane, something that is still elusive in the current un-
derstanding of braneworlds.
Our study has highlighted some of the intricacies of inhomogeneous universes and
the important role of matching conditions in cosmology. We hope that this serves as
another step towards describing our universe using more comprehensive models that do
not require the assumption of homogeneity from the onset.
Appendix A
Taylor Expansion of a Tensor Field
Some authors [15, 95, 81] have formulated a generalized expression for the Taylor expan-
sion of a tensor field on a manifold. This comes in useful as we pursue the matching in
the bulk. Within a neighbourhood of the brane the Taylor approximation of the bulk
metric can give us an impression of which matchings are possible. This will not generate
a global solution, but it will be able to rule out matchings that are not possible since the
first order matching must be satisfied before any higher order matchings can be consid-
ered. We therefore treat the brane as the zeroth order of the Taylor series and expand
the metric into the bulk. We begin with a look at how the Taylor expansion of a tensor
field can be defined. Then we apply it to our brane construction.
Let T (p) be the value of a tensor field at point p ∈ M . The Taylor expansion is an
approximation of the tensor field T (q) at a point q ∈M where q in the neighbourhood of
p. To define it we must introduce a one parameter family of diffeomorphisms φ : D →M
such that, φ(0, p) = p and D = R ×M . We will denote the diffeomorphism by φλ(p) ≡
φ(λ, p) where λ acts as the parameter label.
First we assume that φ is a flow1 generated by a vector field ξ. Then the pullback of
1A flow is defined by φσ+λ = φσ φλ.
132
Appendix A. Taylor Expansion of a Tensor Field 133
a tensor field T can be expanded as a Taylor series,
φ∗λT = T +
d
dλφ∗λT
∣
∣
∣
∣
λ=0
λ+1
2
d2
dλ2φ∗λT
∣
∣
∣
∣
λ=0
λ2 + . . . , (A.1)
where φ∗0T = T was used in the first term. In general φ∗
λT is a tensor-valued function of
the parameter λ. By definition of the derivative we have,
d
dλφ∗λT ≡ lim
λ→0
(
φ∗λT − T
λ
)
= LξT, (A.2)
which is just the Lie derivative in the direction of the flow. Thus the Taylor series can
be more compactly expressed as,
φ∗λT =
n∑
i=0
λi
i!L
iξT, (A.3)
up to order n.
However, not all diffeomorphisms can be expressed as the flow of a vector field. To
incorporate a more general diffeomorphism into this scheme Sonego and Bruni [95] have
formulated the concept of a knight diffeomorphism. They prove that any one parameter
family of diffeomorphisms, denoted by ψλ, can be approximated by a combination of
flows,
ψλ = φ(n)λn/n! · · · φ
(2)
λ2/2 φ(1)λ (A.4)
up to any desired order n. A diffeomorphism defined in this way is termed a knight
diffeomorohism or knight. Each flow, φ(i)
λi/i!is generated by a respective vector field ξi.
This can be understood as a displacement along the integral curves of ξi to a parameter
distance of λ, followed by a displacement along the integral curve of ξ2 to a parameter
distance of λ2/2, and so on.2 Each successive flow, φ(n), can be seen as a correction
to φ(n−1). Thus we can express the Taylor series of any diffeomorphism by combining
equations (A.4) and (A.3),
ψ∗λT =
n−1∑
l=0
λl
l!
∑
Jl
l!
2!j2 · · ·n!jnj1! · · · jn!L
j1ξ1
· · ·L jnξnT + λnR
(n)λ T (A.5)
2Taking the first two orders into consideration, we see a parameter displacement in the direction ofξ1 followed by a smaller parameter displacement in a different direction ξ2. This is reminiscent of themovement of a knight piece in the game of chess, thus inspiring the terminology.
Appendix A. Taylor Expansion of a Tensor Field 134
where R(n)λ T is a remainder term with a finite limit as λ→ 0 and
Jl ≡
(j1, · · · , jn) ∈ Nn∣
∣
n∑
i=1
iji = l
(A.6)
are summation indexes.
The tensor field at point q can be expressed in terms of the knight to the desired
order,
T (q) = T (ψλ(p)) = (ψ∗λT )(p), (A.7)
and thus be expressed as a Taylor expansion. Applying equation (A.5) explicitly to
second order we have,
T (q) = T (p) + (Lξ1T )∣
∣
pλ+ (Lξ2 + L2
ξ1)T∣
∣
pλ2 +O(λ3), (A.8)
where ξA1 and ξA2 are the generators of the diffeomorphism that are free to be specified
depending on the desired direction of the Taylor expansion.
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