Dynamics of thin walls in space-times with stress-energy

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Dynamics of thin walls in space-times with stress-energy
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vii, 120 leaves : ill. ; 28 cm.
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Vuille, Charles Christopher, 1951-
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Thesis:
Thesis (Ph. D.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references (leaves 118-119).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Charles Christopher Vuille.

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Full Text








DYNAMICS OF THIN WALLS IN SPACE-TIMES WITH STRESS-ENERGY


By

CHARLES CHRISTOPHER VUILLE














A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1989

























To James H. Vuille, Janet Sutor Vuille,
Dianne Kowing, and Kira Vuille-Kowing















ACKNOWLEDGEMENTS








I acknowledge gratefully the help of David Garfinkle and Jim
Ipser in the preparation of this manuscript. I thank also John
Challifour of Indiana University and the members of my committee,

Louis Block, Steve Detweiler, Jim Fry, and Pierre Sikivie, for their
encouragement and support.














TABLE OF CONTENTS



page
ACKNOWLEDGMENTS..................................................................... iii

ABSTRACT..................................................................................................vi

CHAPTERS

1 PRELIMINARIES....................................................... 1

Introduction.................................................................

The Thin Shell Approximation of Israel..............4
2 THIN WALLS IN REGIONS WITH VACUUM
EN ERG Y ........................................................................9

Spherical W alls......................................................... 9

Planar W alls .............................................................49
3 THIN WALLS AND PLANE-SYMMETRIC FLUIDS WITH
PRESSURE EQUAL TO ENERGY DENSITY............77

Introduction............................................................... 77

Taub-Tabensky Space-Times................................ 78

Derivation of the Jump Conditions......................80

Static Space-Time..................................... .......... 83

Dynamic Space-Time.................... .................... 87

Global Structure .....................................................89












4 THIN WALLS IN DUST UNIVERSES..........................94

Introduction......................................................... ...94

Plane-Symmetric Dust Space-Times.................95

Derivation of the Jump Conditions.....................98

Global Structure............................... .............................104

5 CONCLUSIONS.............................................................112

REFERENCES............................................................................118

BIOGRAPHICAL SKETCH... ...................................................120











Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


DYNAMICS OF THIN WALLS IN SPACE-TIMES WITH STRESS-ENERGY

By

Charles Christopher Vuille

May 1989


Chairman: James R. Ipser
Major Department: Physics


The motion of a thin wall is treated in general relativity in the
case where the regions on either side of the wall have non-zero
stress-energy. The space-times studied here are the following: (i)
spherically symmetric with vacuum energy, (ii) plane-symmetric with
vacuum energy, (iii) plane-symmetric fluids with pressure equal to

energy density, and (iv) plane-symmetric dust with vacuum energy.
In connection with the latter case, new classes of exact solutions to
Einstein's equation are presented, corresponding to plane-symmetric
non-homogeneous dust universes. The equation of state of the thin

wall is taken to be T=IF, where z is the tension in the wall, a is the










energy density, and 0< F<1. Particular cases of interest are F=0,

corresponding to a dust wall, and F=1, corresponding to a domain wall.

The dynamics of the thin wall depend on both the stress-energy of
the wall and that of the space-time on either side of the wall.
Oscillation about a point of stable equilibrium is found to be possible
for walls with certain values of the parameters in cases (i)-(iii). In
each such case, the metric on either side of the wall is static. There
are various other possible dynamics, such as expansion followed by
collapse, deflation followed by expansion, simple translation, and
asymptotic approach to points of unstable equilibrium. Finally, the
global structure for each type of space-time is discussed. This permits
the elucidation of the structure of the corresponding wall space-times.













CHAPTER 1
PRELIMINARIES

Introduction


One of the intriguing possible consequences of the hot big bang
model of cosmology is that the universe may have undergone phase
transitions in its early history. These phase transitions can give rise
to various soliton-like structures, such as magnetic monopoles, cosmic
strings, and domain walls.1,2 The dynamics of the latter hypothetical
object, as a special case of a more general class of thin walls, will be
the focus here. The space-time stress-energy on either side of the
wall will be taken to contain, in turn, (i) vacuum energy, (ii) fluid with
pressure equal to energy density, and (iii) dust with vacuum energy.

In connection with the last case, new classes of exact solutions to
Einstein's equation will be presented, corresponding to

non-homogeneous dust universes. Most previous work, with the
exceptions noted below, has dealt simply with vacuum.
Thin walls may arise from contributions by particle fields to the
space-time stress-energy. A simple field theory giving rise to thin

walls contains a real scalar field 4 with a potential V(4), where V(O) is

taken to have two local minima. Let 0+ and 0_ denote the values of 4

corresponding to these minima. During a phase transition, the field 4

1











takes on values near 0+ in one region of space ( the 0+- domain"),

and near 4_ in an adjacent region of space ( the domain "). The

region between the 0+ and 4_ domains has a high energy density, both

because the field takes on values which don't minimize V, and
because there is energy associated with spatial changes in the field.
This high-energy region is designated a domain wall.

To make these ideas more concrete, take a particle field

together with a potential V( ), and the Lagrangian

L=-(1/2)(Va)2--V(O). The contribution to the energy-momentum of

space-time by this field is given by


Tab = (VaO)(VbO) (1/2)gab( VC~Vc + 2V(O)). (1.1)


If 4 is approximately constant throughout a region of space-time, then

the first two terms on the right of (1.1) are negligible in that region,
whereas the third term is a multiple of the metric. Hence in the

0+-domain, say, the contribution to the stress-energy is



Tab+ = -V(+)gab (1.2)


A solution to Einstein's equation with a contribution such as this to
the stress energy is the same as a solution of equations having a











cosmological constant, with A+ = KV(O+) where i is Einstein's

constant. An identical situation holds in the 0_-domain.

Under the conditions described above, the thin wall
approximation developed by Israel3 provides a possible model for a
space-time containing such a wall. To make the calculations tractable,
the wall is assumed to have plane or spherical symmetry, at least,
and often complete reflection symmetry as well. Mathematically, this

latter condition is expressed by Kab+=-Kab-, where Kab is the

extrinsic curvature (defined in the next section). The work done on
planar walls by Vilenkin4, Ipser and Sikivie5, and Ipser6 treated the

case V(O+)=V(_ )=0. Most of the work on spherical domain walls has

been restricted to special values of the four parameters A+ A_, m+ ,

and m3,5,7-17, with some numerical work on the general casel8

Garfinkle and Vuille19 studied the general case V(4+)*V(O_), for both

spherical and plane-symmetric walls. Very little has been done in the
case where there is energy-momentum other than vacuum energy.
Kuchar20 examined special cases involving collapse of a charged shell,
while Maeda21 specialized the method of Israel to space-times
containing an ambient fluid without examining wall dynamics.
To find the motion and the gravitational field of the wall, it is
first necessary to solve Einstein's equation in the presence of the
stress-energy of the wall, the vacuum energy of the domains on











either side of the wall, and the energy-momentum contributed by any
other distributions of matter or fields. This gives the gravity field of

the space-time. Israel's thin shell approximation, discussed in the next
section, gives an equation for the jump in the extrinsic curvature in
terms of the stress-energy of the fluid composing the wall. This leads
to the equation of motion giving the dynamics of the wall in the
space-time. In Chapter Two, the dynamics of thin walls in regions
with vacuum energy will be addressed, for both spherical and planar
symmetry. In Chapters Three and Four the space-times under study
will be plane-symmetric and contain fluid with pressure equal to
energy density, and dust with vacuum energy, respectively.


2. The Thin Shell Approximation of Israel


In the thin shell approximation of Israel3 a thin shell of matter
is idealized as having zero thickness (for a rigorous justification of this

approximation, see reference 22). The history of the thin shell is just a
three-dimensional timelike hypersurface, and thus the space-time

consists of a smooth manifold M, a metric gab and a smooth

three-dimensional timelike hypersurface X. The hypersurface divides

the manifold M into two manifolds with boundary, called M+ and M-.

In M+ and M- separately gab is a smooth solution of Einstein's

equation. Let na be the unit normal to I pointing from M- to M+. Then











in each region M+ and M- the space-time metric gab induces on the

hypersurface I an intrinsic metric hab and an extrinsic curvature Kab

given by


hab= gab nanb (1.3)

Kab = hacVcnb (1.4)


The latter quantity may differ depending on whether it is

calculated in M+ or M-. Denote by 0+ the value of a tensor 0 on I

calculated in M+; define 0_ similarly. The appropriate conditions on

the change in hab and Kab across I are3



hab+ hab_= 0 (1.5)

Kab+ Kab = -K[Sab-(1/2) S hab] (1.6)


where Sab is the so-called surface stress-energy tensor of the shell

and the scalar S is given by S=habSab. Sab may be interpreted as the

limit, as the thickness of the shell goes to zero, of the shell
stress-energy integrated over the thickness of the shell.
Equations (1.5) and (1.6), together with Einstein's equation in
M+ and M- separately, are equivalent to Einstein's equation in all of











M. Thus one can obtain a solution to Einstein's equation in the thin
shell approximation as follows: first choose two smooth solutions to
Einstein's equation on the two manifolds M+ and M-. Then in each

manifold choose a timelike hypersurface (Z+ and E- respectively).

The hypersurface 1+ divides M+ into two regions. Choose one of the

two regions. Then the union of this region with I+ forms a manifold

with boundary which we call M+. In the corresponding way obtain the

manifold with boundary M- from I- and one of the two regions of

M -. Then identify the points of E+ and I- to produce a single

hypersurface E and thus a single manifold M which is essentially M+

and M- "glued together." The surfaces I+ and I- and the map which

identifies them must be chosen so that equations (1.5) and (1.6) are
satisfied. When the surfaces and the map are so chosen, the resulting
space-time M is a solution of Einstein's equation in the thin shell
approximation.

The stress-energy tensor of the general (two-dimensional)
perfect fluid is given by



Sab = (O'-)vavb thab (1.7)



where a and T are scalars, the energy density and tension of the wall,

respectively, and va is a unit timelike vector, the space-time velocity
of the fluid. The properties of a perfect fluid are given by its equation











of state, a relation which gives t as a function of a A domain wall

has a particularly simple equation of state: t = a. It then follows from

conservation of stress-energy that a is constant and that



Sab = -hab (1.8)



In this work a more general equation of state, t = Fo, will be

assumed, where F is a constant satisfying 0 < F < 1 This equation of

state is also used in references 5,6,7, and 18. Here, only the case a > 0

is considered. Call the class of thin perfect fluid walls with this

equation of state "F-walls". Domain walls correspond to F-walls with

F=l. Hence the class of F-walls is more general than domain walls, but

less general than the whole class of perfect fluid walls with arbitrary
equation of state. It follows from the equation of state that the

stress-energy of a F-wall is



Sab = o[(l-F)vab hab] (1.9)



Using equation (1.9) in (1.6) gives the change, for a F-wall, in the

extrinsic curvature Kab across :


Kab+ Kab = -Kc{(l-F)vavb + (1/2)hab}


(1.10)








8



This is the key equation that will be used to obtain equations of
motion for thin walls under the various conditions. As will be seen in
the next section, the left-hand side involves components of the metric
and its derivatives, and hence depends on the ambient stress-energy
on either side of the wall.













CHAPTER 2
THIN WALLS IN REGIONS WITH VACUUM ENERGY


Spherical Walls


To find the motion and gravitational field of a spherical wall it is
first necessary to find two spherically symmetric solutions of the
vacuum Einstein equation with cosmological constant. A space-time is
spherically symmetric if its symmetry group contains the symmetry
group of the 2-sphere. The general spherically symmetric solution of

the vacuum Einstein equation with cosmological constant A is either
the Schwarzschild-de Sitter metric or the Nariai metric.23 In the
Nariai space-time all the 2-spheres of symmetry have the same area,
and so it is unsuitable for describing an expanding or contracting shell
and will not be considered here. The Schwarzschild-de Sitter metric
has the following form:23


ds2 = -Fdt2 + F-ldr2 + r2(d02 + sin20 d02) (2.1)


where the function F is defined by


F= 1 2mr-1 (/3)Ar2 (2.2)











and m is a constant. This space-time can be regarded as a black hole

of mass m in a space-time which is asymptotically de Sitter. In the

case m=0 the space-time is de Sitter space-time, while in the case A=0

the space-time is Schwarzschild. Note that the expression in equation

(2.1) is badly behaved at points where F=0. This is simply because the
coordinate t is badly behaved at these points. In fact, the metric is
smooth for all values of r greater than zero.
Each of the space-times M+ and M- are taken to have a metric of

the form given in equation (2.1). Let A+ and m+ be the values of the

constants A and m, respectively, for the metric on M +, and

correspondingly define A_ and m_. The cases considered here are

those for which all four of these parameters are greater than or equal
to zero.

Next, it is necessary to find timelike hypersurfaces, I' and I-, in

M+ and M-, and calculate the intrinsic metric and extrinsic curvature

induced on these hypersurfaces. Since the wall is spherically

symmetric, X+ and 1- must be spherically symmetric also. First

consider a general spherically symmetric timelike hypersurface I in

a space-time M of the form given in (2.1). Later, these results will
be applied to M + and M separately. A three-dimensional
hypersurface in a four-dimensional space-time is specified by giving
the four space-time coordinates as functions of any three parameters.

Choose the coordinates r, 0, and 4 as the three parameters. Then the










surface is specified by giving the remaining coordinate t as a function

of the three parameters. Since the surface I is spherically

symmetric, t on E must be a function of r alone. Thus E is
specified by a relation of the form


t = T(r) (2.3)


A normal co-vector Na to the hypersurface I is given by



Na = a(t T(r)) = ( 1 -dT/dr O) (2.4)


Define the scalar Q by Q=NaNa Then equations (2.1) and (2.4) give



Q = F(dT/dr)2 F-1 (2.5)


Since I is timelike, any normal to I must be spacelike. Thus Q must
be greater than zero. Thus the general timelike spherically symmetric
hypersurface in M is given by a relation of the form in equation (2.3)
satisfying the condition that the right-hand side of (2.5) is greater

than zero. Let na be a unit normal to .. Then it follows from the


definition of Q and Na that











na = EQ-1/2Na (2.6)


where = 1. Note that there are two different unit normals

corresponding to the two different possible values of e.

Define the tensor qab by



qabdxadxb = r2(d02 + sin20 d02) (2.7)


Then the metric can be written as


ds2 = -Fdt2 + F-ldr2 + qabdxadxb (2.8)


Thus qab is the "2-sphere part" of gab. A perfect fluid wall has a fluid

velocity va which is a timelike unit vector, tangent to the wall. A
spherically symmetric wall must have spherically symmetric va. Thus
the conditions on va are


vaa = -1 (2.9a)

vana = 0 (2.9b)

vaqab = 0 (2.9c)


Using these conditions and equations (2.4), (2.6), and (2.7), obtain












va = Q-1/2( -F(dT/dr), F-1 ) (2.10)


where 8 =1. It follows, using equations (2.4)-(2.8) and (2.10), that



gab = nanb VaVb + qab (2.11)


Inserting equation (2.11) into (1.3) yields the following expression for
the intrinsic metric:


hab = -Vab + qab (2.12)


Since hab is spherically symmetric, it determines a set of

2-sphere subspaces of I (the spaces of constant r). The metric induced

on a constant r subspace is qab and the area of the subspace is 4X7r2

Thus both qab and r are determined by hab It then follows from

equation (2.12) that vaVb is determined by hab.

Using equations (1.3) and (1.4) and the fact that na is
hypersurface orthogonal, it can be shown that24


Kab = (1/2)fnhab ,


(2.13)










where n denotes the Lie derivative with respect to the vector na

Inserting equation (2.12) results in


Kab = (1/2)(nqab va nvb vb'nva) (2.14)


The Lie derivatives of qab and va are computed by expressing the

Lie derivative in terms of the coordinate derivative operator24 Da

and using the expressions for the coordinate components of qab va

and na. Some straightforward algebra then yields


nqab = nc+cqab + qcbaanc + qac bnc=

= -2er Q-1/2F(dT/dr)qab (2.15)



nva = nbabva + vbanb=

= -(d/dr){EQ 1/2F(dT/dr)}a (2.16)


Define the scalar L by


L= EQ-1/2F(dT/dr) (2.17)


Then equations (2.14)-(2.17) give













Kab = (dL/dr)vavb r-1qab (2.18)


The quantity L can be expressed in two different and useful
ways. First, using equations (2.4), (2.6). (2.8) and (2.17), obtain


na ar = Q- 1/2Na ar = -L = -siLI (2.19)


with the quantity s defined by


s = L/ILI (2.20)


The quantity s can be interpreted as follows: the surface I divides
M into two regions. A region is an "exterior" if r increases along the

direction normal to I and pointing into the region. A region is called

an "interior" if r decreases along the direction normal to I and

pointing into the region. Then divides M into an exterior and an
interior. If s = 1 then na points to the interior. If s = -1, then na
points into the exterior.
The other useful expression for L can be derived using
equations (2.1), (2.11), and (2.19) :


F=gll = (n1)2-(v1)2 =L2 (v1)2.


(2.21)












Then equation (2.20) results in


L = s{F + (vl)2 1/2


(2.22)


Choose a constant r subspace S of I and for each point p E I

and let X(p) be the proper time along the integral curve of va from

S to p. Then there is a function R(X) such that on Z


r = R(?).


(2.23)


It follows that on I


va ar= v1 = dR/dk .


(2.24)


Thus from equation (2.22) there results


L = s(F + (dR/dk)2)1/2


(2.25)


Next it will be shown that the surface is essentially

determined by the function R(X). Given the function R(X) and

equation (2.23), the quantities dR/dX and L2 are determined as

functions of r. Equations (2.3) and (2.24) give












vavat = (dT/dr)(dR/dX) (2.26)


Then equations (2.8), (2.9a), (2.24), and (2.26) result in


-1 = gabvab = -F(va at)2 + F-1 var)2 =

= (dR/d)2{ -F(dT/dr)2 + F-1 (2.27)


Solving for (dT/dr)2 and using equation (3.25) yields


(dT/dr)2 = {L/F(dR/dX)}2. (2.28)


Since the right-hand side of (2.28) is determined as a function of r, it
follows that the function T(r) is determined up to a change of sign and

addition of a constant. Thus if I and .' are two surfaces in M

corresponding to the same R(,), then one can take I to Z' by
changing the coordinate t on M by the addition of a constant and
(possibly) a change of sign. But this coordinate transformation is an

isometry. Thus the space-time M with the surface I is identical to

the space-time M with the surface 1'. So in this sense the surface I

is determined by the function R(,).

These results can now be applied to the surfaces I+ and I-,

which are glued together to form the surface E. Since hab+ hab-= 0,











it follows that if 0 is any tensor defined on YI+ and 1- such that O

is determined by hab, then 0+ = 0. Thus



qab+ qab- = 0, (2.29)

r+ r_ = 0 (2.30)

(Vab)+- (VaVb)- = 0 (2.31)


Choose the origin of X and the direction of va to be the same on I+

and E-. Then it follows from (2.31) that



)+ -? = 0, (2.32)


and from equations (2.23), (2.30), and (2.32) that



R(X+) R(X_) = 0, (2.33)



so that the surfaces Z+ and Y- are described by the same function

R(X). So when the points of I+ are identified with those of Y-, each

point must be identified with a point which has the same value of X in
order to comply with equation (2.30). Similarly each point must be

identified with a point which has the same values of 0 and 0 (up to an

overall rotation) in order to comply with equation (2.29). When the










surfaces are identified in this manner, it follows from equation (2.12)
that the condition hab+-hab_=0 is satisfied.

Now the only condition left to be satisfied in order that M be a
solution of the Einstein equation for a F-wall in the thin shell
approximation is equation (1.10). Using equations (2.18) and
(2.29)-(2.31) yields


Kab+- Kab-= vaVb(d/dr){L+- L} qabr-1 {L+- L_) (2.34)


while equations (1.10) and (2.12) give


Kab+- Kab-= (1/2)Ko{vavb(2F-1) qab} (2.35)




Thus Einstein's equation reduces to the following pair of equations:


(d/dr){L+- L_} = K~[F-(1/2)] and (2.36)



r-1 (L- L_} = (1/2)Ko (2.37)


To obtain a first integral of these equations, multiply equation (2.36)
by r1-2F, multiply equation (2.37) by (1-2F)r12F, and add the










resulting two equations to obtain


rl-2F(d/dr)L, L_} + (1-2F)rl-2F{L+- L_} = 0, (2.38)

(d/dr){rl-21(L- L)} = 0 and (2.39)

L- L_ = cor2-1, (2.40)


where co is a constant. Note that it follows from equations (2.37) and

(2.40) that


o= 2co0 -1r2-2. (2.41)


Since the walls treated here have positive energy density, co > 0. Let

F+ and s+ be the values of F and s on IX, with F_ and s_ defined

similarly. Using equations (2.23), (2.25) and (2.40), obtain


s+(F+ + (dR/dX)2)1/2 s_(F_ + (dR/dX)2)1/2 = c0R2-1. (2.42)


Now using the general interpretation of the quantity s, it is

possible to interpret the quantities s+ and s_ as follows: since na

points from M- to M+, the unit normal pointing into M+ is na and the

unit normal pointing into M- is -na. Hence if s+= 1 then M+ is an










interior; if s+=-I then M+ is an exterior. Correspondingly if s_= 1 then

M- is an exterior; if s_=-1 then M- is an interior. Since co> 0, it follows

from (2.42) that the case s+=-l, s_= 1 is forbidden. However, all other

possible combinations are allowed. Thus the thin wall space-time M
can consist of an exterior glued to an interior or two interiors glued
together, but cannot consist of two exteriors glued together.

Next, eliminate s+ and s_ from the equation of motion for R(Q).

Adding s_(F_+ (dR/dX)2)1/2 to both sides of equation (2.42) and

squaring gives


-s_(F_ + (dR/dX)2)1/2 =

= (1/2)(F_- F+)co-1R1-2+(1/2)R2F-1 (2.43)


Squaring equation (2.43) results in


(dR/dX)2 = {(F_- F+)/2coR2r-}2 +

+ (1/4)co2R4r-2 (1/2)(F++F_) (2.44)



A solution R(X) of equation (2.44) determines the constants s+ and s.

as follows: given R(,) that satisfies (2.44), solve equation (2.43) for s.










Then solve (2.42) for s+. The function R(X) determines the surfaces I+

and 1-. The quantity s+ determines which region of M+ (the exterior

or the interior) forms with E+ the manifold M+. The corresponding

result holds for s_ and M-. Note that s_ changes sign when the

right-hand side of equation (2.43) changes sign. This does not

correspond to any pathological behavior of the surface I- or of the

manifold M-, but is simply a point where the vectors va and (3/ar)a

in M- are parallel. The corresponding result holds for s+. Thus given

the space-times M+ and M-, a solution of equation (2.44) determines

the motion of the F-wall.
Using equations (2.2) and (2.44), obtain


(dR/dX)2 = ([(m+- m_) + (1/6)(A+-A_)R3]/coR2F 2 +

+ (1/4)co2R4-2-1 + (m+ + m_)R-1 + (1/6)(A+ + A_)R2 (2.45)


Define the constants a, b, c, f, and E and the variable I by


a 2(m+- m_)/co2 (2.46a)

b= (A+-A_)/3co2 (2.46b)

c =4(m+ + m_)/Co2 (2.46c)

f 2(A+ + A_)/3c2 (2.46d)











E -4/co2 (2.46e)

Sco,/2 (2.46f)


Then equation (2.45) becomes


(dR/d2)2 + V(R) = E (2.47)


where V(R) is given by


Y(R) = -(aR-2F+ bR3-2F)2 R4F-2- cR-1 fR2 (2.48)


Equation (2.47) is just the equation of motion for a particle with
energy E moving in one dimension in a potential V(R). Here R plays

the role of the spatial coordinate, plays the role of time, and the

mass of the "particle" is 2. Note that R takes on only positive values
and that E takes on only negative values. Thus the problem of the

motion of the F-wall has been reduced to an equivalent problem in

one-dimensional mechanics. The properties of the one-dimensional

particle motion, and thus of the r-wall motion, depend on the
properties of the function V(r) and on the value of the constant E. Any
local minimum of V(R) represents a possible point of stable
equilibrium, while any local maximum of V(R) represents a possible
point of unstable equilibrium. Any point where V(R) = E represents a











possible turning point of the motion.

The properties of the function V(R) depend on the values of the

constants a, b, c, f, and F. In this work, certain restrictions have been

placed on the values of these constants. Recall that 0< F <1, and that

A+, A_ m+, and m_ are all non-negative. It then follows that



21al < c and (2.49a)
21bl < f. (2.49b)


Thus the constants c and f are nonnegative. It then follows from
(2.48) that V(R)<0 for all R.

Various special cases of the wall's equation of motion have been
treated previously. These include the case a=b=f=0 treated in 5,7; the

case F=1, a=c=0 treated in 8, 9, 10; the case F=1, a=c/2 treated in

11-16; the case F=b=f=0, a=c/2 treated in 3; and the case F=1, a=0

treated in 17. In addition, a numerical treatment of the general
problem has been done in 18. The method of reference 18 is to choose

various values of the parameters (a,b,c,f,F) and numerically produce

plots of the corresponding potentials. Since the potential has so many
parameters, this is a somewhat unwieldy way to obtain information
about the wall's motion. In this work, analytic methods will be used
to determine, as a function of the parameters, certain features of the
wall's motion: the asymptotic values of R, whether the wall's motion
has a turning point and whether the wall oscillates about stable











equilibrium.
The possible values of the parameters c and f can be divided
into a "generic" case: cP0, f*O, and three "exceptional" cases: (i) c=0,
f*0; (ii) c*O, f=0, and (iii) c=f=0.


Case 1: c*0, f*O


Under these conditions it follows from equation (2.48) that V is
unbounded below both as R- 0 and as R-+*. This means V(R) must
have at least one local maximum. In fact, it is possible to demonstrate

that there is a number Fc such that if r>Fc then V(R) has at least one

local maximum and no other stationary points. Assume that r>0. Let

Rs be a stationary point of V(R), and let a prime denote differentiation

with respect to R. Then V'(Rs) = 0. It follows from equation (2.48)

that


V"(Rs) = {R-(d/dR)(Rd/dR)} =

= -16F2a2Rs-4-2(6-4F)2b2Rs4-4- 2(3-4F)2abRsl-4F

cRs-3 4f (4F-2)2Rs4-4. (2.50)


Rearranging the terms on the right-hand side of (2.50) results in the
following equation:












Y"(Rs) = -16-2Rs-4r-2{ar2+ b(F-(3/4))2R3 }2 +

+ 18F-2b2Rs4-4FIF 3(2- /2)8-1} ( 3(2 + /2)8-1} cRs-3

-4f-(4F- 2)2Rs4F-4 (2.51)


All terms on the right-hand side of (2.51) except for the second are

manifestly negative. In the second term, since F<1 it follows that the

factor F 3(2+/ 2)8-1} is negative. Now define Fc by



c = 3(2-,2)/8 (2.52)


Then if F>Fc it follows that the second term on the right-hand side of

equation (2.51) is manifestly negative. Consider the case >Fc Then

V"(Rs) is negative. Thus V(Rs) is a local maximum, and any stationary

point of V(R) is a local maximum. Since V(R) is smooth it follows that
it can have at most one maximum, and since it has at least one
maximum, it has exactly one maximum.
The one-dimensional motion of a particle in a potential with one
maximum and no other stationary points is fairly simple. The motion
has either one turning point or no turning points depending on
whether the energy is less than the maximum. Since equation (2.47)
is time-reversal invariant, for any possible motion of the wall, the











time-reversed motion is also possible.

Let Vmax be the maximum value of V(R) and let R, be the value

of R which maximizes V(R). Then the possible motions of the wall

depend on Ymax and E. For E> Vmax the possible motions of the wall

are as follows: (i) the wall appears with zero size and expands forever
to unbounded size; (ii) the time-reverse of the motion in (i). For E=

Vmax the possible motions of the wall are as follows: (i) the wall

remains in unstable equilibrium with R=Rs; (ii) the wall appears with

zero size and expands with R asymptotically approaching Rs; (iii) the

wall contracts from unbounded size with R asymptotically

approaching Rs; (iv) the time-reverse of the motion in (ii); (v) the

time-reverse of the motion in (iii). For E< Vmax the possible motions

of the wall are as follows: (i) the wall appears with zero size, expands

to a maximum R less than Rs, contracts to zero size and disappears; (ii)

the wall contracts from unbounded size to a minimum R greater than

Rs and then expands to unbounded size.

An exhaustive description of the possible motions of the wall

has been given, in the generic case, for F> Fc. The F< Fc is more

difficult. One of the features of the motion in the F> Fc case is that

stable equilibrium is not possible. This feature is not always shared

by the F< Fc case. That is, there are some values of the constants a, b,











c, f, E and F, with F< Fc, such that the wall is in stable equilibrium. In

particular, choose r=0 and let b be greater than 4. Choose a,c, and f as

follows: a=-3b2, c=6b2, f=9b3. Then using these choices in equation
(2.48) results in


V'(R) = 6b2(-R5+3bR2-3bR+R-2) + 2R-3. (2.53)


It follows from (2.53) that V'(1)=2, so that in particular


V'(1)>0 (2.54)


Now define a by a=(3b2)-1/8. Note that a
obtain


V'(a)=6b2a-2(1+, 3-35/8bl/4) (2.55)


Since b>4 it follows that V'(a)<0. This means V' changes from negative

to positive as R goes from a to 1, and hence there must be a local
minimum at some value of R in that interval. It follows that for
certain energies the one-dimensional motion in the potential V(R) is
an oscillation about a point of stable equilibrium. The wall, therefore,
can undergo spherical ocillations about a radius of stable equilibrium.

Since y(R) is smooth in the parameters (a, b, c, f, F), any set of











parameters sufficiently near the ones given will also give a V(R)
which has a local minimum and thus give rise to walls with similar
dynamics.


Case 2: c=0. f*0


Under these conditions, it follows from equation (2.49a) that

a=0. Thus m+=m =0 and the space-times M+ and M- are both regions

of de Sitter space-time. From equation (2.48) it follows that


V(R) = -b2R6-4 R4F-2 fR2. (2.56)


It follows from (2.56) that V(R) is unbounded below as R-+*. It also
follows from (2.56) that the behavior of V(R) as R-*0 depends on the

value of F. For r<1/2, V(R) is unbounded below as R-+0. For F=1/2,

limR,0 V(R)=-I. For F>1/2, limR,0 Y(R)=O. For 1>1/2 it follows from

equation (2.56) that V(R) has no stationary points. For F<1/2 one can

show using (2.56) that Y(R) has exactly one maximum and no other

stationary points. In this latter case, let Vmax and Rs be respectively

the maximum value of V(R) and the value of R at which V(R) attains

this maximum. Then the possible motions of the wall depend on F,

Vmax, and E. For F<1/2, E< Vmax the possible motions of the wall are

as follows: (i) the wall appears with zero size, expands to a maximum











R less than Rs, contracts to zero size and disappears; (ii) the wall

contracts from unbounded size to a minimum R greater than Rs and

then expands to unbounded size. For F<1/2, E= Vmax the possible

motions of the wall are: (i) the wall remains in unstable equilibrium

with R=Rs; (ii) the wall appears with zero size and expands with R

asymptotically approaching Rs; (iii) the wall contracts from

unbounded size with R asymptotically approaching Rs; (iv) the

time-reverse of the motion in (ii); (v) the time-reverse of the motion

in (iii). For r<1/2, E>Vmax the possible motions of the wall are as

follows: (i) the wall appears with zero size and expands to unbounded

size; (ii) the time-reverse of the motion in (i). For F=1/2, E<-l the

motion of the wall is as follows: the wall contracts from unbounded

size to a minimum R and then expands to unbounded size. For F=1/2,
E=-1 the possible motions of the wall are as follows: (i) the wall
contracts from unbounded size with R asymptotically approaching

zero; (ii) the time-reverse of the motion in (i). For r=1/2, E>-1 the

possible motion of the wall is: (i) the wall contracts from unbounded
size to zero size and disappears; (ii) the time-reverse of the motion in

(i). For r>1/2 the motion of the wall is as follows: the wall contracts
from unbounded size to a minimum R and then expands to
unbounded size.











Case 3. c0O. f=0


From equation (2.49b) it follows that b=0. Thus A+=A_=0;

therefore the vacuum energy outside the wall is zero and the
space-times M+ and M- are both regions of Schwarzschild space-time.
It then follows from equation (2.48) that


V(R)= -a2R-4F-R4F-2-cR-l. (2.57)


It follows from (2.57) that V(R) is unbounded below as R-*0. For F>1/2
it follows from equation (2.57) that V(R) is unbounded from below as

R-**. For F=1/2, limR-+* V(R)=-1. For F<1/2, limR* VY(R)=0. For <11/2

it follows from equation (2.57) that V(R) has no stationary points.

Once can show using equation (2.57) that for 1>1/2, V(R) has exactly

one maximum and no other stationary points. For F>1/2 define Rs and

Vmax as before. The possible motions depend on F, Vmax, and E. For

F
size, expands to a maximum R, contracts to zero size and disappears.

For F=1/2, E<-1 the motion of the wall is as follows: the wall appears
with zero size, expands to a maximum R, contracts to zero size and

disappears. For F=1/2, E > -1 the possible motions of the wall are as

follows: (i) the wall appears with zero size and expands to unbounded











size; (ii) the time-reverse of the motion in (i). For F>1/2, E< Vmax the

possible motions of the wall are as follows: (i) the wall appears with

zero size, expands to a maximum R less than Rs, contracts to zero size

and disappears: (ii) the wall contracts from unbounded size to a

minimum R greater than Rs and then expands to unbounded size. For

[>1/2, E=Vmax the possible motions of the wall are as follows: (i) the

wall remains in unstable equilibrium with R=Rs; (ii) the wall appears

with zero size and expands with R asymptotically approaching Rs; (iii)

the wall contracts from unbounded size with R asymptotically

approaching Rs; (iv) the time-reverse of the motion in (ii); (v) the

time-reverse of the motion in (iii). For F>1/2, E> Vmax the possible

motions of the wall are as follows: (i) the wall appears with zero size
and expands to unbounded size; (ii) the time-reverse of the motion in

(i).


Case 4: c=f=0


From equations (2.49a) and (2.49b) it follows that a=b=0. Thus

A+=A_=m+=m =0 and M+ and M- are both regions of Minkowski

space-time. It then follows from equation (2.48) that


(2.58)













For F<1/2 it follows from equation (2.58) that limR- Y(R)=0, V(R) is

unbounded from below as R- 0 and V(R) has no stationary points. For

F=1/2, it follows from (2.58) that V(R)=-I. For F>1/2, limR,0 V(R)=O,

V(R) is unbounded below as R- and V(R) has no stationary points.

The dynamics of the wall depend on F and E. For F>1/2 the motion of

the wall is as follows: the wall contracts from unbounded size to a

minimum radius and then expands to unbounded size. For r=1/2, E=-l

the motion of the wall is as follows: the wall remains with R at a fixed

value. For r=1/2, E >-1 the possible motions of the wall are as follows:

(i) the wall appears with zero size and expands to unbounded size; (ii)

the time-reverse of the motion given in (i). For F<1/2 the motion of

the wall is as follows: the wall appears with zero size, expands to a
maximum radius, contracts to zero size and disappears.


Global Structure


The wall space-time consists of two regions of Schwarzschild-de
Sitter space-time attached along timelike spherically symmetric
surfaces. Thus we first examine the global structure of
Schwarzschild-de Sitter space-time. The expression given for the
Schwarzschild-de Sitter metric in equation (2.1) is badly behaved at

points where F=0; however the curvature scalar aRabcd cd diverges











only as r- 0. Thus the coordinate system in equation (2.1) covers a
region of the manifold where F is nonzero and may not cover the
entire space-time. It is desirable to find a coordinate system which
does cover the entire space-time. To do this, first examine the
behavior of radial null geodesics.
Consider the two-dimensional metric


ds2 = -Fdt2 + F-ldr2. (2.59)


Null geodesics in this metric are the same as radial null geodesics in
the metric of equation (2.1). In the two dimensional metric of
equation (2.59) let ka be the tangent vector to a null geodesic with

affine parameter X. Then there is a constant o such that


o= ka(a/at)a = -F(dt/dX) (2.60)



Since kaka=0 it follows that



0 = -F(dt/d,)2 + F-1(dr/d,)2 =

SF-1 {(dr/dX)2-2}. (2.61)


Thus


(2.62a)


r = tco + ci, and











t = c2 -F-1dr (2.62b)


where cl and c2 are constants. If F is nonzero for all values of r then

it follows from equations (2.62a-b) that for any incomplete null
geodesic, r approaches zero as the affine parameter approaches the

limiting value. If F is zero for some value of r, let ro be the smallest

such value. The it follows from equations (2.62a-b) that the

space-time has incomplete null geodesics for which r approaches ro as

the affine parameter approaches the limiting value.
Introduce the coordinates u and v by


u -t + fOr F-l(r)dr, (2.63a)



v= t + Or F-l(r)dr. (2.63b)


Then the metric in the coordinates (u,r) is


ds2 = 2dudr Fdu2, (2.64)


and in the coordinates (v,r),


ds2 = 2dvdr Fdv2. (2.65)











The metric in equation (2.64) is smooth for < u < and 0< r< *.

For 0< r < ro the metrics in equations (2.64) and (2.59) are isometric.

Thus the space-time in equation (2.64) is an extension of the
space-time in equation (2.59). The corresponding results hold for the
metric in equation (2.65). Using equations (2.63a-b), one finds that

for 0 < r < ro the (u,r) coordinate patch and the (v,r) coordinate patch

overlap. The overlap map is


v= -u + 2f rF -(r)d. (2.66)


This larger space-time contains both the (u,r) coordinate patch and
the (v,r) coordinate patch. In general it takes more than one (u,r) and
one (v,r) coordinate patch to cover the entire space-time. Introducing
the appropriate (u,r) and (v,r) coordinate patches and appropriate
overlap maps, it is possible to obtain a space-time with the property
that, on every incomplete null geodesic, r approaches 0 as the affine
parameter approaches its limiting value. The extended space-time
with this property is called the complete extension.
To display some information about the complete extension in a

simple graphic form, introduce coordinates and such that the

metric has the form ds2= 0 2(-d2 + dN2) for some function Q and

such that all null geodesics approach finite values of and y as their

affine parameters tend to or to a limiting value. Then in the (,Vy)











plane draw the limit points of the null geodesics (including those at
"infinite affine parameter"). These points can be regarded as a
boundary attached to the space-time. Also draw in the points where
F=0. Points representing a curvature singularity are drawn with a
sawtooth line. In the case m=0 there is no curvature singularity and
the points at r=0 are drawn with a dotted line. Each point of the
resulting diagram (called a Penrose diagram) can be regarded as a
2-sphere of symmetry in Schwarzschild-de Sitter space-time. Since
radial null geodesics are lines which make an angle of 450 with the
vertical, it follows that the path of the wall, which is a timelike curve
on this diagram, always makes an angle of less than 450 with the
vertical. In drawing the Penrose diagram, choose a time orientation
for the space-time. Choosing the opposite time orientation simply
results in the same diagram "upside down."
The nature of the complete extension depends on the values of

m and A. Recall that only the cases where A and m are nonnegative

are considered here. There are two "generic" cases: (i) 9m2A > 1, (ii)

0<9m2A<1; and four "exceptional" cases: (i) 9m2A=1, (ii) m=A=0, (iii)

m=0, A*0, (iv) m*0, A=0. The second generic case is covered in

reference 25 and last three exceptional cases are covered in reference
26, so the results for these cases will be only briefly presented. Some
information about the first generic and first exceptional cases is
presented in reference 16.

In the generic case 9m2A>l, it follows from equation (2.2) that










F<0 for all r. Thus the (r,t) coordinate patch covers the complete

extension. Define the coordinates and xV and the constant ,. by



S= t (2.67)

S-Or F-1(r)dr (2.68)

= limr-+*. (2.69)


Then it follows from equation (2.59) that


ds2 = -F(-dA2 + dy2). (2.70)


The points at 4=0 correspond to r=0; those at = 00. correspond to r=-.

The Penrose diagram of this space-time is shown in figure 1, page 42.
The lines are labelled with the corresponding values of r.

In the exceptional case 9m2A=1, it follows from equation (2.2)

that F and (dF/dr) vanish at r=ro where ro=3m. For all r*ro, F(r)<0.

The complete extension is covered with coordinate patches (un, r) and

(vn, r), one for each integer n. The metric on each (un, r) coordinate

patch has the form given in equation (2.64) and in each (vn, r)

coordinate patch the metric has the form given in equation (2.65).

Each (un, r) coordinate patch overlaps with the (vn, r) coordinate











patch for r>ro and with the (vn+1, r) coordinate patch for r< ro. The

overlap maps are, for r>ro


Vn =Un 2f r** FI(r)dr,


(2.71)


and for r< r


Vn+1 = -un + 2f r F-1(r)dr.


(2.72)


Using the geodesic equation one can show that all null geodesics in

this space-time either have r=ro and are complete or have r take on

all values between 0 and and are incomplete only as r 0.

Next, introduce the coordinates 5 and y as follows: for 0< i< r

define un and vn in terms of 5 and y as follows: for

(27n-)

un = cot{((+V)/2.


(2.73)


For {27(n-1) + }

vn= cot{(--V)/2).


(2.74)












Define r in terms of and y as follows: for Ay + =27nn or V-z=27n, r=ro.

For other values of y and t (with 0<
(2.74) and either (2.71) or (2.72). Then it follows that un and vn are

C" functions of 4 and y and that r is a C1 function of 4 and V. The

metric in the and y coordinates is


ds2= -F(-d42 + dv2){4sin2[(4+V)/2]sin2[(4-v)/2] }-1, (2.75)


where r is regarded as a function of 4 and y and the metric

coefficients are defined by continuity at those points where r=ro. The

Penrose diagram for this space-time is shown in figure 2, page 43.

The geodesic limit points with r=0 occur at 4=0, W*27in; those with r=*

occur at =Tc, yV(2n+l)t; those with r=ro occur at =0, y=2nn and =Tn,

V=(2n+l)7r.

The Penrose diagram for the space-time with 0<9m2A shown in figure 3, page 44.25 In this case, F(r) vanishes at two values

of r, designated r+ and r.

Finally the results for the last three exceptional cases will be

presented.26 The case m=A=0 is Minkowski space-time. Here F=l and
there is no curvature singularity. The Penrose diagram for Minkowski

space-time is shown in figure 4, page 45. The case m=0, A*0 is de











Sitter space-time, with F=0 at r=(3/A)1/2 and no curvature

singularity (figure 5, page 46). The case m*O, A=0 is Schwarzschild

space-time, where F=0 at r=2m (figure 6, page 47).

There are many possibilities for the global structure of the wall
space-time. Each of the manifolds M+ and M- is a region of
Schwarzschild-de Sitter space-time with a boundary which is a
timelike spherically symmetric surface. Thus the Penrose diagram of
M+ or M- consists of a piece of one of the Penrose diagrams of figures
1-6 bounded by a timelike curve. The Penrose diagram of the wall
space-time consists of the Penrose diagrams for M+ and M- with the
boundaries identified at corresponding values of r. Figure 7, page 48,
is the Penrose diagram for one possible wall space-time. In this case

9m+ 2A>l and 0<9m 2A_<1. The curved dotted lines represent the

paths of the wall in M+ and M- and are identified.























r=co












r=O


Figure 1. The Penrose diagram for Schwarzschild-de Sitter
space-time in the case 9m2A>1.






















r=oo r=oo r= oo




ro0 o ro ro






r=0 r=0

Figure 2. The Penrose diagram for Schwarzschild-de Sitter
space-time in the case 9m A=l.









44










r=0 r=O
r=oo T=oo









rT=oo r=o
r=O r=0

Figure 3. The Penrose diagram for Schwarzschild-de Sitter
space-time in the case 0<9m2A



















t=00


r=oo







r=O





r= oo

t=-oo








Figure 4. The Penrose diagram for Minkowski space-time
(m= A = 0).














r=00






ro ro

r=O
rr=0

















(m=O, A* 0). Here, ro-(3/A)1/2






















r=0


r=oo "r=oo




r=oo r=oo


r=0



Figure 6. The Penrose diagram for Schwarzschild space-time
(m*0, A=0). r = 2m and t = O on the inner diagonals.





















r=O r=oo r=oo










r=0 r=0


Figure 7. The Penrose diagram for a possible spherical
wall space-time. 9m+2A>l, 0<9m 2A<1. The dotted lines
are identified.










Planar Walls


To find the motion of a planar wall one must first find two
solutions of the vacuum Einstein equation with cosmological constant
with planar symmetry. A space-time has planar symmetry if its
symmetry group contains the symmetry group of the Euclidean
2-plane. The general planar symmetric solution of the vacuum Einstein

equation with cosmological constant A has the form23


ds2 = -Hdp2 + 2drdp + r2(dx2 + dy2) (2.76)


where the function H is defined by


H = y/r Ar2/3 (2.77)


and y is a constant. The surfaces of constant r and p are surfaces of

planar symmetry. In the case y*0 this space-time has a curvature

singularity at r=0, while y=0 corresponds to de Sitter space-time, and

y=A=0 is Minkowski space-time.

There is another type of planar symmetry in Minkowski
space-time. Writing the Minkowski metric in the usual Cartesian
coordinates,


ds2 = -dt2 + dz2 + dx2 + dy2,


(2.78)













it is manifest that the surfaces of constant t and z are surfaces of
planar symmetry. However, the volume element on these surfaces is
independent of t and z. This form of planar symmetry is therefore not
suitable for describing an expanding or contracting wall and will not
be considered here.

In the case where y 0 or A*0 one can rewrite the metric in

equation (2.76) by introducing the coordinate p defined by

p=-13+fH-1dr. Then the metric in equation (2.76) is given by


ds2 = -Hdp2 + H-dr2 + r2(dx2 + dy2) (2.79)


This is similar to the expression in equation (2.1) for the spherically
symmetric case.
Choose M+ and M- to be space-times with metrics of the form

given in equation (4.1). Let A+ and y+ be the values of the constants A

and y in M+, respectively, and in the same way define A_ and y_. Here,

only the cases where both A+ and A_ are greater than or equal to zero

will be considered.
The procedure for treating planar walls and deriving their
equation of motion is completely analogous to the method used in the
last section for spherical walls. Therefore we omit the derivation and
simply present the results. The wall hypersurface is specified by a











relation of the form


r = R() ,


(2.80)


where X is proper time on a world-line tangent to the wall and

orthogonal to the 2-planes of symmetry. The energy density of the
wall is given by


Y = 2c0 K_1R2F-2


(2.81)


where co is a constant. The equation of motion of the wall is



s+{H, + (dR/dX)2 }1/2 s_ H_ + (dR/dX)2} 1/2= CR2-1. (2.82)



The quantities H+ and H_ denote the values of H on I+ and I-

respectively. The constants s+ and s_ are defined as in the spherical

case. From equation (2.82) obtain


(dR/dX)2 = (H_- H+)2(2coR2F-1)-2 + co2R4-2/4 -


-(H+ + H_)/2 .


(2.83)


Then, using equation (2.77), obtain












(dR/dk)2= {(7_- y+)+(1/3)(A+- A_)R3 2(2coR2F)-2 + (1/4)co2R4-2


-(1/2)(y + y_)R-1 + (1/6) (A + A_)R2 .


Define the constants a, b, c, and f and the variable .k by


a (y_- y+)/co2

b (A+- A_)/3co2,

2
c -2(y++y_)/co2,

f- 2(A++ A_)/3co2,

2= cok/2


Then equation (2.84) becomes


(dR/d.)2 + V(R) = 0 ,


where V(R) is given by


Y(R) -{aR-2F+ bR3-2)2 R4F-2 cR-1 fR2.


Equation (2.86) is the equation of motion for a particle with zero


(2.84)


(2.85a)

(2.85b)

(2.85c)

(2.85d)

(2.85e)


(2.86)


(2.87)











energy moving in one dimension in a potential V(R). The variable R
takes on values in the open interval (0, *). The special case a=b=f=0
has been treated in references 4,5,6, and 7.

Since A+ and A_ are nonnegative, it follows that



21b < f. (2.88)


Thus the constant f is nonnegative also. Note, however, that there is no
restriction on the sign of the constant c. Consider the case c >0. It then
follows from equation (2.87) that V(R)<0 for all R. The motion of a
particle of zero energy has no turning points in a potential that is

everywhere negative. If a, c, and F satisfy the special conditions a=c=0,

F>1/2, then limR 0 V(R)=0 and the possible motions of the wall are as

follows: (i) the wall contracts from unbounded values of R with R

asymptotically approaching zero; (ii) the time reverse of the motion in

(i). If these special conditions on a, c, and F are not satisfied then the

possible motions of the wall are as follows: (i) the wall appears with
R=0 and expands forever to unbounded values of R; (ii) the time
reverse of the motion in (i).
Next consider the case c<0. Introduce the variable Z by


Z= R32. (2.89)











Note that Z is a monotonic function of R; thus dZ/dL, has the same sign

as dR/d&.. Rewriting equation (2.86) in terms of the variable Z results
in


(4/9)(dZ/dk)2 + W(Z) = c (2.90)


where the function W(Z) is defined by


W(Z) -Z(2-8F)/3{a+bZ2}2 z(8F-2)/3- fZ2. (2.91)


Equation (2.90) is the equation of motion of a particle of mass 8/9 and
energy c moving in a potential W(Z).
Next, examine the properties of the function W(Z). Consider a
"generic" case: F*l/4, a*0 and three "exceptional" cases: (i) F=l/4, a*0;

(ii) 1/4, a=0 and (iii) F=1/4, a=0.


Case 1: r 1/4. a*0


From equation (2.91) it follows that W(Z) is unbounded from
below both as Z 0 and as Z-+*-. Thus W(Z) must have at least one local

maximum. There is a number Fc such that if F>Fc then W(Z) has

exactly one local maximum and no other stationary points. To see this,

let Zs be a stationary point of W(Z) and let a prime denote










differentiation with respect to Z. Then W'(Zs)=0. It then follows from

equation (2.91) that


W"( Z ) = {Z-I[ZW']'}( Zs)

= -(4/9) Zs(-4-8F)/3{a2(4F-1)2+2ab(4F-4)2 Zs2+ b2(4F-7)2 Z4 -


-(4/9)(4F-1)2 Zs(8F-8)/3 4f.


(2.92)


Rearranging terms on the right-hand side, arrive at


W"( Zs) = -(4/9)(4F-1)2 Zs(-4-8)/3 { a+b(4F-4)2(4'-1)-2 Zs22 +

+ 128b2(4F-1)-2 ZS(8-8r)/3 {I-(8+3/2)8-1 {F-(8-3/2)8-1 -


-(4/9)(4F-1)2 Zs(8F-8)/3 4f.


(2.93)


All terms on the right-hand side of (2.93) are manifestly negative
except for the second term. In the second term, since F<1 it follows

that the factor r-(8+3v2)8-1 is negative. Now define the number Fc by


Fc (8-3/ 2)/8


(2.94)


Then if F>Fc it follows that the second term on the right-hand side of

(2.93) is manifestly negative. Consider the case F>Fc. Then W"( Zs) is











negative. Thus W( Zs) is a local maximum. This means any stationary

point of W(Z) is a local maximum. Since W(Z) is smooth it follows that
it can have at most one maximum, and since it has at least one
maximum, it has exactly one maximum.

Let Wmax be the maximum value of W(Z) and let Zs be the

value of Z at which W(Z) attains this maximum. Define Rs by Rs- Zs2/3

For c>Wmax the possible motions of the wall are as follows: (i) the

wall appears with R=0 and expands to unbounded values of R; (ii) the

time reverse of the motion in (i). For c= Wmax the possible motions of

the wall are as follows: (i) the wall is in unstable equilibrium with

R=Rs; (ii) the wall appears with R=0 and expands with R asymptotically

approaching Rs; (iii) the wall contracts from unbounded values of R

with R asymptotically approaching Rs; (iv) the time reverse of the

motion in (ii); (v) the time reverse of the motion in (iii). For c< Wmax

the possible motions of the wall are as follows: (i) the wall appears

with R=0, expands to a maximum R less than Rs, contracts to R=0 and

disappears; (ii) the wall contracts from unbounded values of R to a

minimum R greater than Rs and then expands to unbounded values of

R.

The generic case when F<.Fc is more difficult, but, as in the

spherical case, it is possible to demonstrate that for some choices of










the parameters a, b, c, f, and F the wall can undergo oscillations about
a point of stable equilibrium. Let F be any number satisfying 0<.<1/4.
Choose a=-2b and f=b2. Note that for this choice of f to be compatible
with equation (2.88) one must have b2>4. Then, using equation (2.81),
find


W'(Z)=-(2/3)b2Z-(1+8F)/3(Z2-2) {(7-4F)Z2+ (8F-2)} -


-2b2Z + (2/3)(1-4)Z(8F-5)/3 (2.95)


From (2.95) it follows that


W'(1) = (2/3)b2(2+4F) + (2/3)(1-4r), (2.96)


so W'(1)>0. Define Z1 by Zl~{(2-8F)/(7-4F) 1/2. Then Zl<1 and



W'( Zp)=-2Zl{b2-(1/3)(1-4r)Zl(8F-8)/3 }. (2.97)


Choose b so that b2>(1/3)(1-4F) Z(8F-8)/3. Then from (2.97) it follows

that W'( Z1) < 0. Therefore it follows that W(Z) has a local minimum at

some Z between Z1 and 1. Hence for some values of the "energy" c and

initial value of R the wall undergoes oscillations about a point of stable











equilibrium.


Case 2: F=1/4. a*0


From equation (2.91) it follows that


W(Z) = -(a+bZ2)2 1 fZ2. (2.98)


From equation (2.98) the following properties hold for the function
W(Z): (i) if (2ab+f)<0 then W(Z) has exactly one local maximum and no
other stationary points; (ii) if (2ab+f)>0 then W(Z) has neither local
maxima nor local minima; (iii) limZ,0W(Z)=-(l+a2); (iv) if f=O then

(from equation (2.88)) b=0 and W(Z)=-(1+a2). Recall that Z takes on
values only in the open interval (0, *), so one must take limits as Z- 0
and Z=0 cannot considered as a possible location for a minimum or

maximum of W(Z). For (2ab+f)<0 let Wmax be the maximum value of

W(Z), let Zs be the value of Z at which this maximum is achieved, and

let Rs be given by Rs=Zs2/3. The dynamics of the wall depend on 2ab+f

and c. For (2ab+f)<0, c>Wmax the possible motions of the wall are as

follows: (i) the wall appears with R=0 and expands to unbounded
values of R; (ii) the time-reverse of the motion in (i). For (2ab+f)<0,

c=Wmax the possible motions of the wall are as follows: (i) the wall is











in unstable equilibrium with R=Rs; (ii) the wall appears with R=0 and

expands with R asymptotically approaching Rs; (iii) the wall contracts

from unbounded values of R with R asymptotically approaching Rs;

(iv) the time-reverse of the motion in (ii); (v) the time-reverse of the

motion in (iii). For (2ab+f)<0, -(1+a2)< c
the wall are as follows: (i) the wall appears with R=0, expands to a
maximum value of R less than Rs, contracts to R=0 and disappears; (ii)

the wall contracts from unbounded values of R to a minimum value of

R greater than Rs and then expands to unbounded values of R. For

(2ab+f) <0, c <-(1+a2) the wall contracts from unbounded values of R

to a minimum value of R greater than Rs and then expands to

unbounded values of R. For (2ab+f) >0, c >-(l+a2) the possible motions
of the wall are: (i) the wall appears with R=0 and expands forever to
unbounded values of R; (ii) the time-reverse of the motion in (i). For
(2ab+f)>0, c<-(l+a2) the wall contracts from unbounded values of R to a
minimum value of R greater than Rs and then expands to unbounded

values of R. For (2ab+f)>0, f*0, c=-(l+a2) the possible motions of the
wall are as follows: (i) the wall contracts from unbounded values of R
with R asymptotically approaching zero; (ii) the time-reverse of the
motion in (i). For f=0, c=-(l+a2) the wall remains with R at any
constant value.











Case 3: r*1/4, a=0


From equation (2.91) it follows that


W(Z) = -b2Z(14-8r)/3 Z(8F-2)/3 fZ2. (2.99)


From (2.99), the following properties hold for the function W(Z): (i) if

r>1/4 then limZ,0W(Z)=0 and W(Z) has no stationary points; (ii) if

F<1/4 and f=0 then b=0, limZ,*.W(Z)=0 and W(Z) has no stationary

points; (iii) if F<1/4 and f*0 then W(Z) has exactly one maximum and

no other stationary points. For r<1/4, f*O define Wmax, Zs, and Rs as

before. The dynamics of the wall depend on the values of F, f, and c.

For F>1/4 the wall contracts from unbounded values of R to a
minimum value of R and then expands to unbounded values of R. For

r
R, contracts to R=0 and disappears. For F<1/4, f*0, c>Wmax the

possible motions of the wall are: (i) the wall appears with R=0 and
expands forever to unbounded values of R; (ii) the time-reverse of the

motion in (i). For F<1/4, f*0, c=Wmax the possible motions of the wall

are:(i) the wall remains in unstable equilibrium with R=Rs; (ii) the wall

appears with R=0 and expands with R asymptotically approaching Rs;











(iii) the wall contracts from unbounded values of R with R

asymptotically approaching Rs; (iv) the time-reverse of the motion in

(ii); (v) the time-reverse of the motion in (iii). For r<1/4, f*0, c
the possible motions of the wall are: (i) the wall appears with R=0,

expands to a maximum value of R less than Rs, contracts to R=0 and

disappears; (ii) the wall contracts from unbounded values of R to a

minimum value of R greater than Rs and then expands to unbounded

values of of R.


Case 4: F=1/4, a=0


From equation (2.91) it follows that


W(Z)= -b2Z4 1 fZ2 (2.100)


From (2.100) the following properties hold for the function W(Z): (i)

limZOW(Z)=-l; (ii) if f=0 then W(Z)=-1; (iii) if f*0 then W(Z) has no

stationary points. The dynamics of the wall depend on the values of f
and c. For c>-1 the possible motions of the wall are: (i) the wall
appears with R=0 and expands forever to unbounded values of R; (ii)
the time-reverse of the motion in (i). For c=-l, f 0 the possible
motions of the wall are: (i) the wall contracts from unbounded values
of R with R asymptotically approaching zero; (ii) the time-reverse of











the motion in (i). For c=-l, f=0 the wall remain at any constant value of
R. For c<-l, f must be nonzero and the wall contracts from unbounded
values of R to a minimum value of R and then expands to unbounded
values of R.


Global Structure


The global structure of the wall space-time consists of two
regions of space-times of the form given in equation (2.76) attached
along timelike plane symmetric surfaces. Begin by considering the
properties of plane orthogonal null geodesics, that is, null geodesics
which are orthogonal to the symmetry 2-planes. Consider the two
dimensional metric


ds2 = 2drd3 Hd32. (2.101)


Null geodesics in this metric are the same as plane-orthogonal null
geodesics in the metric of equation (2.76). If H is nonzero for all values

of r then one can show that for any incomplete null geodesic in the
metric of equation (2.101) r approaches zero as the affine parameter
approaches its limiting value. If H vanishes at some value of r,
introduce new coordinates to extend the space-time. In either case,
obtain a space-time, the complete extension, in which any incomplete
null geodesic has the property that r approaches zero as the affine











parameter approaches its limiting value.
To illustrate the properties of the complete extension, introduce

coordinates 4 and y such that the metric has the form

ds2=Q2(-d42+dy2) for some function C1 and all null geodesics approach

finite values of 4 and y as their affine parameters tend to + or to a
limiting value. Then produce the Penrose diagram of the space-time

by drawing, in the 4,y plane, the limit points of the null geodesics and
the places where H vanishes. Each point in the Penrose diagram can be
regarded as a 2-plane in the four-dimensional space-time. Curvature

singularities are again represented by sawtooth lines; if y=0 then there
is no curvature singularity and the places where r=0 are represented
by broken lines.

The nature of the complete extension depends on the values of y

and A. There are two "generic" cases: (i) y<0,A>0; (ii) y>O, A>0; and four

"exceptional" cases: (i) y=A=-; (ii) y=0, A>0; (iii) y>0, A=-; (iv) y<0, A-0.

In the generic case y<0, A>0, it follows from equation (2.77) that

H<0 for all r. Thus the (P,r) coordinate patch covers the entire

space-time. Define the coordinates 4 and y and the constant .* by



S-f0rH-1 (r)dr (2.102)

V 3 + (2.103)

*= limr-*.4 (2.104)












It then follows from equation (2.101) that


ds2 = -H(-d2 + dy2) (2.105)


The points at t=0 correspond to r=0; those at S=*. correspond to r= *.

The Penrose diagram of this space-time is shown in figure 8, page 70.

In the generic case y>0, A>0, define the constant cl by



cl {33y/A}1/3. (2.106)


Then it follows from equation (2.77) that


H= (A/3r)(c13 -r3). (2.107)


Thus H(cl)=0. For r< cl, H>0; for r>cl, H<0. The coordinates in equation

(2.101) only cover part of the space-time, so new coordinates must be
introduced to extend the space-time. Define the function S(r) by


S(r) = (4/3Ac12r)(r2+ clr + c12)3/2 X

X expf(3 {(n/6)-tan-1 [(2r+c1)/cl,/3] } (2.108)











Then for r< cl it follows from equations (2.107) and (2.108) that



(d/dr){ln(S/H)} = AcH-1. (2.109)


For r< c1 define the coordinates A and B by



A= exp{-Acclp/2}, (2.110)

AB= (2/Acc)2H/S (2.111)


Then, using equations (2.101) and (2.109-2.111), obtain


ds2 = SdAdB (2.112)


Here r is a function of A and B given by equation (2.111). The metric
in (2.112) is smooth for all values of A and B. The region A>0, B>0 is

isometric to the (p, r) coordinate patch. This space-time is the complete
extension of the space-time of metric (2.101).

Define the coordinates 5 and y by


A =tan{((v-)/2} (2.113)

B = tan{(v+t)/2) (2.114)











Then, using equations (2.111) and (2.112), obtain


ds2 = {4cos2[(N-4)/2]cos2[(v+)/2]}-1S(-d42+ dV2), and (2.115)

{(cos-cosV)/(cos4+cosV)} = (2/Ac1)2H/S (2.116)


The points at r=0 correspond to y=-n/2 or V=n/2; those at r=*

correspond to cos4=-cos(V)tanh(7t/2I 3) The Penrose diagram for this
space-time is shown in figure 9, page 71.

In the exceptional case y=A=0, H=0 and thus


ds2 = 2drd3 (2.117)


All null geodesics in this metric either have r=constant and P varying

linearly with the affine parameter, or P=constant and r varying
linearly with the affine parameter. Thus on all incomplete null
geodesics r approaches zero as the affine parameter tends to its
limiting value. Introduce the coordinates 5 and V by


r = tan{(v+t)/2} (2.118)

S= tan ((V-)/2} (2.119)


Substituting into equation (2.117) results in










ds2= {2cos2[(V+4)/2]cos2[(N-4)/2] -1 {-d2 + dV2) (2.120)


The points at Ny+=0 correspond to r=0; those at N+4=nC correspond to
r=*. The Penrose diagram for this space-time is in figure 10, page 72.
In the last three exceptional cases, H is nowhere zero and thus
the (3, r) coordinate patch covers the entire space-time. In the cases

where H>0 introduce the coordinates 5 and y by


3 = tan{(V-)/2} (2.121)
2fH-ldr= sin(y) {cos[(V+4)/2]cos[(y-4)/2])-1. (2.122)


In cases where H<0, introduce the coordinates N and 5 by


0 = tan{((-V)/2) (2.123)
2fH-ldr= sin(v) {cos[(t+v)/2]cos[(-V)/2] }-1. (2.124)


In both cases the metric (2.101) takes the form


ds2= {4cos2[(4+/)/2]cos2[(4-V)/2] }-1HI {-dd2 + dy2} (2.125)


In the exceptional case y=0, A>0, H= -Ar2/3<0. Using equation
(2.124) obtain











(6/Ar) = sin(v) cos[L( +V)/2]cos[((-v)/2] -1. (2.126)


Solving for r and substituting into (2.125) results in


ds2 = (3/A) {-d[ 2 + dy2} /sin2S (2.127)


The points at 4+y=Tt or -V=7tt correspond to r=0; those at =0 correspond
to r=*-. The Penrose diagram for this space-time is figure 11, page 73.

In the exceptional case y>0, A=0, H=y/r>0. Using equation (2.122)
results in


r2/y = sin(v) {cos[(v+)/2]cos[(v-4)/2] }-1. (2.128)


Solving for r and subtituting into (2.125), obtain the metric


ds2={16y-1sin(y)cos3[(yV+)/2]cos3 [(V-_)/2] }-1/2{ d42 + dy2}. (2.129)


The points at V=0 correspond to r=0; those at yV+=n or Vy-=t correspond
to r=*. The Penrose diagram for this space-time is figure 12, page 74.

Finally, in the exceptional case y<0, A=0, H=y/r<0. From equation
(2.124) arrive at


r2/y = sin(v){cocos[(+N-)/2]cos[(-V)/2]}-1


(2.130)













As before, use this and equation (2.125) to get


ds2= { 16y-sin()cos3[(W+)/2]cos3[(v-t)/2] }-1/2{-d2 + dN2).(2.131)


The points at =0 correspond to r=0; those at t+y=-xT or 5-y=-7

correspond to r=*. The Penrose diagram for this space-time is figure
13, page 75.
There are many possibilities for the global structure of the
planar wall space-time. The Penrose diagram for each of the manifolds
M+ and M- consists of a piece of one of the Penrose diagrams of figures
8-13 bounded by a timelike curve. The Penrose diagram of the wall
space-time consists of the Penrose diagrams of M+ and M- with the
boundaries identified at corresponding values of r. Figure 14, page 76,
is the Penrose diagram for one possible wall space-time. In this case

A+>0, y+<0, A_> 0, and y_> 0. The paths of the wall in M+ and M- are

identified.




















r= o


r=0



Figure 8. The Penrose diagram for the plane-symmetric
space-time in the case y<0, A>0.



















r=oo







r=O r=O








r=oo


Figure 9. The Penrose diagram for the plane-symmetric
space-time with y> 0, A > 0. The inner diagonals correspond
to r= c1.























r= oo


r=O


Figure 10. The Penrose diagram for the planar
symmetric space-time with y=A=O.





















N


r=O


r=O


r=00


Figure 11. The Penrose diagram for the plane-symmetric
space-time with y=0, A=O.
































r=O





rI'= 00











Figure 12. The Penrose diagram for the plane-symmetric
space-time in the case y > 0, A > 0.























r = 0
r=0













r=oo r= c00





Figure 13. The Penrose diagram for the plane-symmetric
space-time in the case y<0, A=0.

























r= OO








*


r=0
r=O



Figure 14. The Penrose diagram for a possible planar
symmetric wall space-time. A+>0, y+< 0, A_> 0, y_> 0.

The dotted lines are identified.














CHAPTER 3
THIN WALLS AND PLANE-SYMMETRIC FLUIDS
WITH PRESSURE EQUAL TO ENERGY DENSITY


Introduction


In this chapter the motion of a thin wall is treated in general
relativity for the case where the regions on either side of the wall are
filled with fluid. The fluid is taken to be irrotational and

plane-symmetric, with the pressure p equal to the energy density p.

As before, the wall is assumed to have the equation of state T = o ,

where 0 < F < 1 being the tension in the wall and a its

energy-density. Unlike the treatment in chapter two, the wall
space-times are taken to be completely reflection symmetric (as in

references 5,6, and 7) in order to make the equations tractable.

Mathematically, this means Kab+=-Kab-. A class of static space-times

is found to permit oscillatory solutions whenever F>l/4. In a

separate class, which includes a Robertson-Walker space-time, the
walls simply translate through space without turning points. The
treatment extends to fluid space-times previous results involving thin
walls in vacuum, as studied in references 5 and 6. The global structure
of the wall space-time is discussed in the final section of the chapter.
77











Taub-Tabensky Space-Times


To derive the metric, start from Einstein's equation, which is


Rab (1/2) Rgab = KTab ,with (3.1)



Tab = (P + P)uab + Pgab (3.2)



where p, p and ua are the energy density, pressure, and

four-velocity of the ambient fluid, respectively. Outside the wall the

scalar field 0 makes no contribution to Tab since 4 is approximately

constant and V(O) is taken to be vanishingly small.
Tabensky and Taub27 solved equation (3.1) under the

assumption that the energy density p equaled the pressure p, and that
the fluid was irrotational. With these conditions it follows from the

conservation of Tab that there exists a function F such that



ua = VaF/(-VcF VF)(1/2) (3.3)


and that equation (3.1) becomes


Rab = 2cpuaub = VaF VbF


(3.4)











The pressure and energy density of the fluid is given by


2Kp = 2Kp = -VcF VCF,


(3.5)


while the equation of motion is


VCVcF = O.


(3.6)


With the above equations and the assumption
symmetry, one can proceed as in Tabensky and Taub, and
solution


ds2 = + t-1/2exp()( -dt2 + dz2 ) + t( dx2 + dy2)


of plane
obtain the


(3.7)


where Q is an arbitrary line integral in the t-z plane,


Q = ft{( F,t2 + F,z2)dt + 2F,tF,zdz) ,


(3.8)


with comma designating partial derivative. In these coordinates F
satisfies


F,zz = F,tt + t1 F,t .


(3.9)











Note that either t or z can act as a time coordinate, depending
on the choice of sign in equation (3.7). The choice of solution in (3.9)
determines the metric.


Derivation of the Jump Conditions


The Tabensky-Taub space-time metric takes the form


ds2 = eH(t,z)(-dt2 + dz2) + t(dx2 + dy2) ,


(3.10)


where e= + 1 and H = -(1/2)exp(Q), with Q as defined in the previous

section. Specify the hypersurface I by z=P(t). Then in the coordinates

( t, z, x, y) a normal covector to I is given by


Na = (-P,t, 0, 0)


(3.11)


Thus the unit covector na and the F-fluid four-velocity va are given


na =jQ-(1/2)( -P't 1 ,0,0), and

Va Q-(1/2)( 1 ,-P,t,0,0 ),


(3.12)

(3.13)










where Q = eH-_[-(P,t)2 + 1] the norm of Na and 8 and j are +1 ,

corresponding to the two possible directions of these covectors. Using
(3.12) and (3.13) one can write


gab = -Vab + nanb + qab (3.14)


just as in equation (2.11). Here, qabdxadxb = t( dx2 + dy2) The

intrinsic metric of the wall will then be hab= -VaVb + qab With this

expression the extrinsic curvature can again be computed from
equation (2.13). A straightforward computation yields


n(va) = jeH-1I(Q-1/2 Pt),t + (Q-1/2)z}va (3.15)


n(qab) = jeHlt-l(Q-(1/2)Pt)qab (3.16)


and hence


Kab = -VabeH-1( L,t + j(Q-1/2),z) + (1/2)qabEH-lt-1L (3.17)


where L = jQ-1/2pt Equations (1.10), (3.14), and (3.17) give two

equations for Kab+- Kab- one in terms of the stress-energy of the

wall, and the other in terms of the purely geometric quantities of











equation (3.17). Equating the parts tangential and orthogonal to va
results in the jump conditions:


(1/2)eH + t- L+ (1/2)EH_-1 t-L_= -(1/2)ic ,


(3.18)


-eH -( L+,t + j+(Q-1/2),) + eH_-( L_,t + j_(Q_-1/2),) =


= (r- 1/2) ica .


(3.19)


From vaa = -1 and the above definitions a useful expression

for L can be found:


L = sH( (va at )2 -EH-1)1/2


(3.20)


Here s=+l, and will be interpreted in the following sections.
Substituting into equation (2.10) results in


(1/2)et-s((vav t)2 EH+-1)1/2 -(1/2)t-1 s_((a at)2 EH_-1)1/2


= -(1/2)K .


(3.21)


In the next two sections, the equations developed thus far will be used
to study the motion of thin walls in two particular Tabensky-Taub
space-times.











Static Space-Time


Choose F= az a a constant, for the solution of equation (3.9),

and let E =-1 in the metric (3.10), so that the four-velocity of the

ambient fluid will be timelike and at rest with respect to these
coordinates. Using equations (3.7) and (3.8) results in the metric:


ds2 = t-1/2exp(a2t2/2)(-dz2 + dt2) + t( dx2 + dy2) (3.22)


Note that z is the time coordinate here, and t a space
coordinate, so the metric is static. The energy density and pressure of
the fluid is


p = p = (1/2)K-la2tl/2exp(-a2t2) (3.23)


which is smooth for all t > 0. Before invoking the assumption of
reflection symmetry, it will be necessary to interpret the quantity s

in (3.20), and then examine (3.21) to determine what values of s+ and

s_ might be allowed. For e=-1 it follows that



naVat = n = -slLIH-1, (3.24)


which says that, in the direction of na, t decreases when s=l, and











increases when s=-l. Looking then at equation (3.21), it is clear that if

the energy density of the wall, a is to be positive, either s+=l and

s_=-l, or s+=s_=l with the additional proviso that H+ > H-1 The

assumption of reflection symmetry leaves only the former possibility.
Hence in a direction perpendicular to the wall and pointing into M+ t
is decreasing, and area elements are, from (3.22), shrinking. M+, as in
chapter two, is designated an "interior." Note that na points from M- to
M +, so the same calculation in M- must be done using -na This

means that s_=-1 also corresponds to an interior. Thus the wall

space-time will consist of two interiors glued together at the wall.
In view of the above considerations, it is now possible to

specialize equations (3.18) and (3.19) to the present case, Kab+=-Kab_:



-H -1 t-lL = -Ko/2 (3.25)

2H -IL+,t = (2F 1)Ko/2 (3.26)



Equations (8.4) and (8.5) can be integrated, yielding


L+ = Cot(2F-1)/2, (3.27)


and so the energy density of the r-fluid, from (3.18), is











o = 2ic-1CotF-lexp(-a2t2/2) .


(3.28)


From (3.18) (3.20) and (3.23) comes the equation of motion


I (v av a2 + H1 1/2 = Cotrexp(-a2t2/2).


(3.29)


Squaring both sides of this equation leads to the pseudo-Newtonian
energy equation


(vaVat)2 + V(t) = 0,


(3.30)


where


V(t) = tl/2exp(-a2t2/2)f 1 Co2t(4-1)/2exp(-a2t2/2) (3.31)


The effective energy of the F-wall is zero in these coordinates. The

motion of the wall can now be completely analyzed for every F.


Case 1: F< (1/4)


Write (3.31) as V(t)=t2Fexp(-a2t2){t(1-4F)/2- Co2exp(-2t2/2)}.

Note that the first multiplicative factor is positive definite for t > 0,











hence acts only as an amplitude. Focus the attention, therefore, on the

second term, (t(1-4F)/2- Co2exp(-a2t2/2)}. For small enough t, this

term is strictly negative, as long as CO is non-zero (as it must be, by

(3.23)). As t increases, t(1-4F)/2 increases monotonically, while the
exponential term similarly decreases. Therefore there will be a point

tmax at which V(t) ceases to be negative and becomes positive for all

t > tmax The possible motion of the wall is simple: it emerges from

t=0, expands to tmax and returns.


Case 2: F= (1/4)


In this case, (3.31) becomes


V(t) = tl/2exp(-a2t2){ 1 Co2exp(-a2t2/2)}. (3.32)


If ICol < 1, no motion is possible. If ICol > 1 then V will be negative

on (0, tmax) for some tmax and positive on (tmax ), as before. The

wall expands to tmax and then contracts.











Case 3: (1/4) < F


This case includes F= 1, corresponding to domain walls. Set

B(t)= { 1-Co2t(4F-l)/2exp(-c2t2/2)}, the second multiplicative factor on

the right-hand side of equation (3.31). Note that B(t) has one critical
point for t > 0, which clearly corresponds to a minimum. It then

follows that if the expression {[(4F-1)/4]-lln(Co2) + ln[(4r-1)/2a2]} is

less than one, no motion is possible, while if it is greater than or equal
to one there is a point of stability around which the wall may oscillate,
or remain at rest. Hence in this case the motion of the wall depends

critically on the choice of the parameters Co, a, and F.


Dynamic Space-Time


Choose F=b In(t), b a constant, for the solution of equation

(3.9), and let e = +1 in the metric (3.10). This gives


ds2 = to(-dt2+ dz2) + t(dx2 + dy2) (3.33)


where co=b2-1/2. The energy density and pressure of the fluid are


p = p = (1/2)&-lb2t-(o+2)


(3.34)











Again, the ambient fluid is at rest relative these coordinates.
The metric is singular at t = 0, and the wall space-time consists of two
interiors glued together, just as in the previous case. Proceeding in the

same way as before yields L+=-CotF-1/2. The energy density of the

F-fluid is given by


o = 2i-1CotF-co-3/2. (3.35)


An explicit expression for the ordinary velocity, dz/dt, can be easily
found:


dz/dt = +.Cot( C2t21 + 1 )-1/2 (3.36)



where p=F-(b2/2) (1/4) .While this expression can be exactly

integrated only for special values of the parameters, it is nonetheless
clear by inspection what the possible motions of the wall must be.


Case 1: > 0



Here the wall starts from rest at some zo and accelerates to the

speed of light in the limit as t goes to infinity.















In this case the wall proceeds at some uniform velocity in either
the positive or negative z-direction.


Case 3: D < 0


The wall moves initially at the speed of light and slows to zero
in the limit as t goes to infinity.


Case 4. b=(3/2)1/2


This corresponds to a Robertson-Walker metric, as can be seen
by setting T=(2/3)t3/2, which puts the metric in the usual form,

ds2=-dT2+a(T)(dx2+dy2+ dz2 ). Here (3 = r-1, which means that domain

walls will translate at uniform velocity through the space-time. As the
wall expands, space is opened up and fluid is created at the expense
of the energy density of the wall.


Global Structure


As discussed previously, for the space-times considered here
one must glue together two interiors. In both cases, there is a real
singularity at t=0, time-like for the static space-times and space-like











for the dynamic space-times. These space-times are topologically R4
To draw the Penrose diagrams, consider the two dimensional metric


ds2= eH(t,z)[-dt2+dz2] (3.37)


Each point of the diagram will correspond to a 2-plane. The null
geodesics issue from the singularity at t=0 and extend to arbitrarily
large parameter values. Since t is strictly positive in these space-times,
the Penrose diagrams will be very similar to the diagram for
Minkowski space in spherical coordinates26, with the exception that
t=0 will correspond to a real singularity, rather than simply a
coordinate singularity. The desired coordinate transformation is


t= tan((t*+z*)/2) + tan((t*-z*)/2) (3.38)
z = tan((t*+z*)/2) tan((t*-z*)/2) (3.39)


This transformation yields the two-metric


ds2= EHsec2((t*+z*)/2)sec2((t*-z*)/2) -dt*2+dz*2} (3.40)


Figures 15 and 16, pages 92 and 93, are Penrose diagrams for the
static and dynamic cases, respectively, each with a sample trajectory.

To construct the Penrose diagram for the F-wall space-times studied

here, cut along the trajectory of the wall, taking the piece in which the








91


area elements are shrinking in the direction normal to the wall. This
piece corresponds to the interior. Gluing this piece to its mirror image
gives the desired diagram.


































t= 0 o




Z=-00









Figure 15. The Penrose diagram for static Tabensky-Taub
space-time, with sample trajectory.








93


















t= 00 t= c


Z=- 00 Z= 00








t=0


Figure 16. The Penrose diagram for dynamic Tabensky-Taub
space-time, with sample trajectory.