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DYNAMICS OF THIN WALLS IN SPACETIMES WITH STRESSENERGY 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 VuilleKowing 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 PLANESYMMETRIC FLUIDS WITH PRESSURE EQUAL TO ENERGY DENSITY............77 Introduction............................................................... 77 TaubTabensky SpaceTimes................................ 78 Derivation of the Jump Conditions......................80 Static SpaceTime..................................... .......... 83 Dynamic SpaceTime.................... .................... 87 Global Structure .....................................................89 4 THIN WALLS IN DUST UNIVERSES..........................94 Introduction......................................................... ...94 PlaneSymmetric Dust SpaceTimes.................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 SPACETIMES WITH STRESSENERGY 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 nonzero stressenergy. The spacetimes studied here are the following: (i) spherically symmetric with vacuum energy, (ii) planesymmetric with vacuum energy, (iii) planesymmetric fluids with pressure equal to energy density, and (iv) planesymmetric dust with vacuum energy. In connection with the latter case, new classes of exact solutions to Einstein's equation are presented, corresponding to planesymmetric nonhomogeneous 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 stressenergy of the wall and that of the spacetime 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 spacetime is discussed. This permits the elucidation of the structure of the corresponding wall spacetimes. 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 solitonlike 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 spacetime stressenergy 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 nonhomogeneous 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 spacetime stressenergy. 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 highenergy 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)2V(O). The contribution to the energymomentum of spacetime 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 spacetime, 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 stressenergy 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 spacetime 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 m3,5,717, with some numerical work on the general casel8 Garfinkle and Vuille19 studied the general case V(4+)*V(O_), for both spherical and planesymmetric walls. Very little has been done in the case where there is energymomentum other than vacuum energy. Kuchar20 examined special cases involving collapse of a charged shell, while Maeda21 specialized the method of Israel to spacetimes 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 stressenergy of the wall, the vacuum energy of the domains on either side of the wall, and the energymomentum contributed by any other distributions of matter or fields. This gives the gravity field of the spacetime. Israel's thin shell approximation, discussed in the next section, gives an equation for the jump in the extrinsic curvature in terms of the stressenergy of the fluid composing the wall. This leads to the equation of motion giving the dynamics of the wall in the spacetime. 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 spacetimes under study will be planesymmetric 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 threedimensional timelike hypersurface, and thus the spacetime consists of a smooth manifold M, a metric gab and a smooth threedimensional 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 spacetime 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 socalled surface stressenergy 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 stressenergy 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 spacetime M is a solution of Einstein's equation in the thin shell approximation. The stressenergy tensor of the general (twodimensional) 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 spacetime 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 stressenergy 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 "Fwalls". Domain walls correspond to Fwalls with F=l. Hence the class of Fwalls 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 stressenergy of a Fwall is Sab = o[(lF)vab hab] (1.9) Using equation (1.9) in (1.6) gives the change, for a Fwall, in the extrinsic curvature Kab across : Kab+ Kab = Kc{(lF)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 lefthand side involves components of the metric and its derivatives, and hence depends on the ambient stressenergy 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 spacetime is spherically symmetric if its symmetry group contains the symmetry group of the 2sphere. The general spherically symmetric solution of the vacuum Einstein equation with cosmological constant A is either the Schwarzschildde Sitter metric or the Nariai metric.23 In the Nariai spacetime all the 2spheres 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 Schwarzschildde Sitter metric has the following form:23 ds2 = Fdt2 + Fldr2 + r2(d02 + sin20 d02) (2.1) where the function F is defined by F= 1 2mr1 (/3)Ar2 (2.2) and m is a constant. This spacetime can be regarded as a black hole of mass m in a spacetime which is asymptotically de Sitter. In the case m=0 the spacetime is de Sitter spacetime, while in the case A=0 the spacetime 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 spacetimes 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 spacetime M of the form given in (2.1). Later, these results will be applied to M + and M separately. A threedimensional hypersurface in a fourdimensional spacetime is specified by giving the four spacetime 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 covector 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 F1 (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 righthand 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 = EQ1/2Na (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 + Fldr2 + qabdxadxb (2.8) Thus qab is the "2sphere 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 = Q1/2( F(dT/dr), F1 ) (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 2sphere 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 Q1/2F(dT/dr)qab (2.15) nva = nbabva + vbanb= = (d/dr){EQ 1/2F(dT/dr)}a (2.16) Define the scalar L by L= EQ1/2F(dT/dr) (2.17) Then equations (2.14)(2.17) give Kab = (dL/dr)vavb r1qab (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 + F1 var)2 = = (dR/d)2{ F(dT/dr)2 + F1 (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 righthand 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 spacetime M with the surface I is identical to the spacetime 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 Fwall 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} qabr1 {L+ L_) (2.34) while equations (1.10) and (2.12) give Kab+ Kab= (1/2)Ko{vavb(2F1) 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) r1 (L L_} = (1/2)Ko (2.37) To obtain a first integral of these equations, multiply equation (2.36) by r12F, multiply equation (2.37) by (12F)r12F, and add the resulting two equations to obtain rl2F(d/dr)L, L_} + (12F)rl2F{L+ L_} = 0, (2.38) (d/dr){rl21(L L)} = 0 and (2.39) L L_ = cor21, (2.40) where co is a constant. Note that it follows from equations (2.37) and (2.40) that o= 2co0 1r22. (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 = c0R21. (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 spacetime 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+)co1R12+(1/2)R2F1 (2.43) Squaring equation (2.43) results in (dR/dX)2 = {(F_ F+)/2coR2r}2 + + (1/4)co2R4r2 (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 righthand 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 spacetimes M+ and M, a solution of equation (2.44) determines the motion of the Fwall. Using equations (2.2) and (2.44), obtain (dR/dX)2 = ([(m+ m_) + (1/6)(A+A_)R3]/coR2F 2 + + (1/4)co2R421 + (m+ + m_)R1 + (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) = (aR2F+ bR32F)2 R4F2 cR1 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 Fwall has been reduced to an equivalent problem in onedimensional mechanics. The properties of the onedimensional particle motion, and thus of the rwall 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 nonnegative. 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 1116; 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)} = = 16F2a2Rs42(64F)2b2Rs44 2(34F)2abRsl4F cRs3 4f (4F2)2Rs44. (2.50) Rearranging the terms on the righthand side of (2.50) results in the following equation: Y"(Rs) = 162Rs4r2{ar2+ b(F(3/4))2R3 }2 + + 18F2b2Rs44FIF 3(2 /2)81} ( 3(2 + /2)81} cRs3 4f(4F 2)2Rs4F4 (2.51) All terms on the righthand 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)81} 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 righthand 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 onedimensional 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 timereversal invariant, for any possible motion of the wall, the timereversed 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 timereverse 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 timereverse of the motion in (ii); (v) the timereverse 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+3bR23bR+R2) + 2R3. (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)=6b2a2(1+, 335/8bl/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 onedimensional 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 spacetimes M+ and M are both regions of de Sitter spacetime. From equation (2.48) it follows that V(R) = b2R64 R4F2 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 timereverse of the motion in (ii); (v) the timereverse 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 timereverse 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 timereverse 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 timereverse 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 spacetimes M+ and M are both regions of Schwarzschild spacetime. It then follows from equation (2.48) that V(R)= a2R4FR4F2cRl. (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 timereverse 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 timereverse of the motion in (ii); (v) the timereverse 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 timereverse 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 spacetime. 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 timereverse 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 spacetime consists of two regions of Schwarzschildde Sitter spacetime attached along timelike spherically symmetric surfaces. Thus we first examine the global structure of Schwarzschildde Sitter spacetime. The expression given for the Schwarzschildde 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 spacetime. It is desirable to find a coordinate system which does cover the entire spacetime. To do this, first examine the behavior of radial null geodesics. Consider the twodimensional metric ds2 = Fdt2 + Fldr2. (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 + F1(dr/d,)2 = SF1 {(dr/dX)22}. (2.61) Thus (2.62a) r = tco + ci, and t = c2 F1dr (2.62b) where cl and c2 are constants. If F is nonzero for all values of r then it follows from equations (2.62ab) 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 (2.62ab) that the spacetime 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 Fl(r)dr, (2.63a) v= t + Or Fl(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 spacetime in equation (2.64) is an extension of the spacetime in equation (2.59). The corresponding results hold for the metric in equation (2.65). Using equations (2.63ab), 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 spacetime 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 spacetime. Introducing the appropriate (u,r) and (v,r) coordinate patches and appropriate overlap maps, it is possible to obtain a spacetime with the property that, on every incomplete null geodesic, r approaches 0 as the affine parameter approaches its limiting value. The extended spacetime 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 spacetime. 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 2sphere of symmetry in Schwarzschildde Sitter spacetime. 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 spacetime. 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) SOr F1(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 spacetime 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 F1(r)dr. (2.72) Using the geodesic equation one can show that all null geodesics in this spacetime 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(n1) + } vn= cot{(V)/2). (2.74) Define r in terms of and y as follows: for Ay + =27nn or Vz=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[(4v)/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 spacetime 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 spacetime with 0<9m2A 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 spacetime. Here F=l and there is no curvature singularity. The Penrose diagram for Minkowski spacetime is shown in figure 4, page 45. The case m=0, A*0 is de Sitter spacetime, 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 spacetime, where F=0 at r=2m (figure 6, page 47). There are many possibilities for the global structure of the wall spacetime. Each of the manifolds M+ and M is a region of Schwarzschildde Sitter spacetime 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 16 bounded by a timelike curve. The Penrose diagram of the wall spacetime 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 spacetime. 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 Schwarzschildde Sitter spacetime 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 Schwarzschildde Sitter spacetime 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 Schwarzschildde Sitter spacetime in the case 0<9m2A t=00 r=oo r=O r= oo t=oo Figure 4. The Penrose diagram for Minkowski spacetime (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 spacetime (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 spacetime. 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 spacetime has planar symmetry if its symmetry group contains the symmetry group of the Euclidean 2plane. 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 spacetime has a curvature singularity at r=0, while y=0 corresponds to de Sitter spacetime, and y=A=0 is Minkowski spacetime. There is another type of planar symmetry in Minkowski spacetime. 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+fH1dr. Then the metric in equation (2.76) is given by ds2 = Hdp2 + Hdr2 + 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 spacetimes 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 worldline tangent to the wall and orthogonal to the 2planes of symmetry. The energy density of the wall is given by Y = 2c0 K_1R2F2 (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= CR21. (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(2coR2F1)2 + co2R42/4  (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)co2R42 (1/2)(y + y_)R1 + (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) {aR2F+ bR32)2 R4F2 cR1 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(28F)/3{a+bZ2}2 z(8F2)/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 ) = {ZI[ZW']'}( Zs) = (4/9) Zs(48F)/3{a2(4F1)2+2ab(4F4)2 Zs2+ b2(4F7)2 Z4  (4/9)(4F1)2 Zs(8F8)/3 4f. (2.92) Rearranging terms on the righthand side, arrive at W"( Zs) = (4/9)(4F1)2 Zs(48)/3 { a+b(4F4)2(4'1)2 Zs22 + + 128b2(4F1)2 ZS(88r)/3 {I(8+3/2)81 {F(83/2)81  (4/9)(4F1)2 Zs(8F8)/3 4f. (2.93) All terms on the righthand 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)81 is negative. Now define the number Fc by Fc (83/ 2)/8 (2.94) Then if F>Fc it follows that the second term on the righthand 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(Z22) {(74F)Z2+ (8F2)}  2b2Z + (2/3)(14)Z(8F5)/3 (2.95) From (2.95) it follows that W'(1) = (2/3)b2(2+4F) + (2/3)(14r), (2.96) so W'(1)>0. Define Z1 by Zl~{(28F)/(74F) 1/2. Then Zl<1 and W'( Zp)=2Zl{b2(1/3)(14r)Zl(8F8)/3 }. (2.97) Choose b so that b2>(1/3)(14F) Z(8F8)/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 timereverse 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 timereverse of the motion in (ii); (v) the timereverse 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 timereverse 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 timereverse 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(148r)/3 Z(8F2)/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 timereverse 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 timereverse of the motion in (ii); (v) the timereverse 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 timereverse 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 timereverse 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 spacetime consists of two regions of spacetimes 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 2planes. Consider the two dimensional metric ds2 = 2drd3 Hd32. (2.101) Null geodesics in this metric are the same as planeorthogonal 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 spacetime. In either case, obtain a spacetime, 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 spacetime 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 2plane in the fourdimensional spacetime. 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, A0. 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 spacetime. Define the coordinates 4 and y and the constant .* by Sf0rH1 (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 spacetime 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 spacetime, so new coordinates must be introduced to extend the spacetime. Define the function S(r) by S(r) = (4/3Ac12r)(r2+ clr + c12)3/2 X X expf(3 {(n/6)tan1 [(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)} = AcH1. (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 (2.1092.111), 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 spacetime is the complete extension of the spacetime 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[(N4)/2]cos2[(v+)/2]}1S(d42+ dV2), and (2.115) {(coscosV)/(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 spacetime 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[(N4)/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 spacetime 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 spacetime. In the cases where H>0 introduce the coordinates 5 and y by 3 = tan{(V)/2} (2.121) 2fHldr= sin(y) {cos[(V+4)/2]cos[(y4)/2])1. (2.122) In cases where H<0, introduce the coordinates N and 5 by 0 = tan{((V)/2) (2.123) 2fHldr= 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[(4V)/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 spacetime 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[(v4)/2] }1. (2.128) Solving for r and subtituting into (2.125), obtain the metric ds2={16y1sin(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 spacetime 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= { 16ysin()cos3[(W+)/2]cos3[(vt)/2] }1/2{d2 + dN2).(2.131) The points at =0 correspond to r=0; those at t+y=xT or 5y=7 correspond to r=*. The Penrose diagram for this spacetime is figure 13, page 75. There are many possibilities for the global structure of the planar wall spacetime. The Penrose diagram for each of the manifolds M+ and M consists of a piece of one of the Penrose diagrams of figures 813 bounded by a timelike curve. The Penrose diagram of the wall spacetime 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 spacetime. 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 planesymmetric spacetime in the case y<0, A>0. r=oo r=O r=O r=oo Figure 9. The Penrose diagram for the planesymmetric spacetime 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 spacetime with y=A=O. N r=O r=O r=00 Figure 11. The Penrose diagram for the planesymmetric spacetime with y=0, A=O. r=O rI'= 00 Figure 12. The Penrose diagram for the planesymmetric spacetime in the case y > 0, A > 0. r = 0 r=0 r=oo r= c00 Figure 13. The Penrose diagram for the planesymmetric spacetime in the case y<0, A=0. r= OO * r=0 r=O Figure 14. The Penrose diagram for a possible planar symmetric wall spacetime. A+>0, y+< 0, A_> 0, y_> 0. The dotted lines are identified. CHAPTER 3 THIN WALLS AND PLANESYMMETRIC 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 planesymmetric, 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 energydensity. Unlike the treatment in chapter two, the wall spacetimes 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 spacetimes is found to permit oscillatory solutions whenever F>l/4. In a separate class, which includes a RobertsonWalker spacetime, the walls simply translate through space without turning points. The treatment extends to fluid spacetimes previous results involving thin walls in vacuum, as studied in references 5 and 6. The global structure of the wall spacetime is discussed in the final section of the chapter. 77 TaubTabensky SpaceTimes 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 fourvelocity 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 = + t1/2exp()( dt2 + dz2 ) + t( dx2 + dy2) of plane obtain the (3.7) where Q is an arbitrary line integral in the tz 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 TabenskyTaub spacetime 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 Ffluid fourvelocity 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) = jeH1I(Q1/2 Pt),t + (Q1/2)z}va (3.15) n(qab) = jeHltl(Q(1/2)Pt)qab (3.16) and hence Kab = VabeH1( L,t + j(Q1/2),z) + (1/2)qabEHlt1L (3.17) where L = jQ1/2pt Equations (1.10), (3.14), and (3.17) give two equations for Kab+ Kab one in terms of the stressenergy 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 tL_= (1/2)ic , (3.18) eH ( L+,t + j+(Q1/2),) + 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 EH1)1/2 (3.20) Here s=+l, and will be interpreted in the following sections. Substituting into equation (2.10) results in (1/2)ets((vav t)2 EH+1)1/2 (1/2)t1 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 TabenskyTaub spacetimes. Static SpaceTime 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 fourvelocity 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 = t1/2exp(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)Kla2tl/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 = slLIH1, (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+ > H1 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 spacetime 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 tlL = 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(2F1)/2, (3.27) and so the energy density of the rfluid, from (3.18), is o = 2ic1CotFlexp(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 pseudoNewtonian energy equation (vaVat)2 + V(t) = 0, (3.30) where V(t) = tl/2exp(a2t2/2)f 1 Co2t(41)/2exp(a2t2/2) (3.31) The effective energy of the Fwall 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(14F)/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(14F)/2 Co2exp(a2t2/2)}. For small enough t, this term is strictly negative, as long as CO is nonzero (as it must be, by (3.23)). As t increases, t(14F)/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)= { 1Co2t(4Fl)/2exp(c2t2/2)}, the second multiplicative factor on the righthand 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 {[(4F1)/4]lln(Co2) + ln[(4r1)/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 SpaceTime 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=b21/2. 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 spacetime consists of two interiors glued together, just as in the previous case. Proceeding in the same way as before yields L+=CotF1/2. The energy density of the Ffluid is given by o = 2i1CotFco3/2. (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 zdirection. 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 RobertsonWalker 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 = r1, which means that domain walls will translate at uniform velocity through the spacetime. 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 spacetimes considered here one must glue together two interiors. In both cases, there is a real singularity at t=0, timelike for the static spacetimes and spacelike for the dynamic spacetimes. These spacetimes 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 2plane. The null geodesics issue from the singularity at t=0 and extend to arbitrarily large parameter values. Since t is strictly positive in these spacetimes, 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 twometric 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 Fwall spacetimes 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 TabenskyTaub spacetime, with sample trajectory. 93 t= 00 t= c Z= 00 Z= 00 t=0 Figure 16. The Penrose diagram for dynamic TabenskyTaub spacetime, with sample trajectory. 