SPACE-TIME DEFECTS:OPEN AND CLOSED SHELLS REVISITED
Abstract. Space-times whose Riemann-Christoffel curvature tensors are null, except on a timelike hyper
surface are considered. These geometries with distIibutional curvature tensor can be interpreted as space
time defects with zero associated Newtonian mass. The method to generate axially symmetIic defects is
studied paying special attention to the global aspects of the spacetime that contains the shell. We find shells
that connect the interiors of two spacetimes making a compact space locally isometric to Minkowski space .
Shells connecting two exteriors are also analized . These can be interpreted as examples of wormholes in
Minkowski space . Furthermore we study some new cases of shells with the shapes of cones and hyperbol
oids . We show that these shells can be built with crossed cosmic strings.
I. INTRODUCTION
The formation of structures predicted by theories of the early universe based in thespontaneous symmetry breaking of some unifying group has been the focus of someattention . In particular, cosmic strings are produced in the breaking of an U (1) symmetry. They are good candidates to seed the formation of galaxies and also to modelthe observed anisotropy in the microwave background. Cosmic walls are associatedto the breaking of a discrete symmetry, and may decay later forming cosmic strings(Vilenkin and Shellard 1994).
Geometrically, in the zero width limit, straight cosmic strings and plane symmetricdomain walls are characterized by spacetimes with metrics that have null RiemannChristoffel curvature tensor everywhere except on the lines that represent the stringsand on the planes of the walls (ibid.) . In other words, they have a generalized conicstructure. In general, topological defects are not always characterized in this way,e.g. the spacetime outside a global U(1) loop is not flat.
Motivated by these consideration in the first paper of this series (Letelier and Wang1995), the generation ofspace-times whose curvature is nonzero only on a surface wasstudied in a rather systematic way. Solutions to the Einstein equations that representsshells of matter with spherical, plane, cylindrical , and disk like symmetry were considered in particular. The methodology used was that of the Lichnerowicz-Taub theoryof distribution (Lichnerowicz 1971; Taub 1980), that is mathematically sound for distributions with support in hypersurfaces. In general, the theory of distributions incurved spacetimes presents some drawbacks (Geroch and Traschen 1987). The most
384 R EINALDO 1. GLEISER AND PATRICIO S. L ETELIER
commonly used description of cosmic walls is based in Israel 's theory of thin shell sof matter (Israel 1966, 1967). This theory takes as a departure point the study of theextrinsic geometry of the surfaces that describe the shells of matter via Gauss-Codazziequations. In principle, both approaches give the same result s and are complementary(Taub 1980). In fact , the equivalence of Darmois-Israel and distributi onal meth odsfor thin shells in general relativity has been explicitly demonstrated (Mansouri andKhorrami 1996).
In a second articl e (Letelier 1995a), the same matter was dwelt in, presenting a collection of spacetime defects with different shapes: paraboli c, oblate (prolate) spheroidal, and toroidal. In these two works the geometric interpretation of the solutions aswell as some other global aspect s were almost not touched.
In the present paper we study in some detail the method of generation of axiallysymmetric defects paying special attention to the global aspects of the spacetime thatcontains the shell. In particular, we find shells that connect the interiors of two spacetimes making a compact space locally isometric to Minkowski space. The restrictedcase of two-centered spheres has been previously analyzed, in the context of cosmological models (Lynden-Bell, Katz, and Redmount 1989) , and also , in relation withIsraels's junction conditions (Goldwirth and Katz 1995). Shell s connecting two exteriors are also studied. These can be interpreted as exampl es of wormholes (Morris andThome 1988; Frolov and Novikov 1990; Morri s, Thome, and Yurtsver 1988; Hawking 1992) in Minkowski space. Furthermore we study some new cases of shells withthe shapes of cones and hyperboloids, that can be considered as formed by crossedcosmic strings.
2. GENERAL METHOD
Since shells correspond to stress - energy - momentum tensors with support on a timelike hypersurface, we may naturally use methods from the theory of distr ibutions, asapplied in the description of distribution valued curvature tensors, to give a generalprescription for the construction of axially symmetric shells in Minkowski spacetime.By axiall y symmetric we mean , as usual , that the spacetime contains a Killing vector8/8¢i, whose orbits are closed. Axially symmetric spacetimes are naturally adaptedto the presence of "cosmic strings" i.e., spacetime defects along the symmetry axis.These "cosmic strings" in tum, are characterized by the existence of a "conic singularity" associated to a "deficit angle" structure in the metric. On this account, we shallassume that the shells (defects) separate two regions of spacetime, which we indi cate respectively by M 1 and M 2, each of whi ch is a portion of Minkowski spacetime(with, possibly, a deficit angle), that can be covered with (or, is contained in) a chartwhere the metric takes the form,
dsZ= - dtZ+ drr + Azrrd¢iz + dzr , i = 1,2 (1)
where the index i refers respectively to M 1 and M 2 . Ai i= 1 corresponds to a nonvanishing deficit angle.
To obtain a spacetime where the distribution method is applicable, we define thelocation of the shell by introducing a hypersurface ~ that is a common boundary for
SPACE-TIME DEFECTS : OPEN AND CLOSED SHELLS REVISITED 385
M I and M2. We also require continuity of the metric , in the sense that the metricinduced on L: from M I should be the same as that induced on L: from M 2.
We consider first the definition of L: as seen from MI . We shall restrict to static ,axially symmetric shells . In this case L: is described by a function j(ZI) , such that L:is the set of points with
(2)
for all <P I and ti ' This introduces a natural parametrization on L: in terms of ZI, <P Iand ti '
Similarly, from the point of view of M 2 , the hypersurface L: is described by afunction h(Z2) , such that on L:
(3)
Again, from the point of view of M2 , a natural parametrization on L: is given in termsof Z2 , <P2and t2·
The corresponding induced metrics on L: are
(4)
anddrY2 = -dt~ + [1+ (h/)2 ] dz~ + A~h2d<p~ (5)
where l' = dj / dZI and h' = dh/ dz2. The continuity condition for the induced metricimplies now ti = t z = t, and we impose <P I = <P2 = <p. We remark now that if theconstruction is at all possible, there should exist a map Z2 = ((zI) , defined on L:.Then, the equality of the induced metrics requires
1 + [1' (ZI)]2
[j(zd ]2
[1+ [h/(() ]2] (::J 2
A2 [h((W
(6)
(7)
where ( stands for ( = (( ZI), and A = A2/ AI . In what follows we shall assume thatAl = 1. The case Al =/= 1 may always be recovered by a redefinition of <p.
Ifwe assume that j(ZI) and A are given, the functions h(() and ((zI) are obtainedas the solutions of the system (6), (7). If they exist and are real the construction ofthe metric is complete.
There are several possibilities and types of solution, depending on the choice of Aand the ranges of r i , as we shall see in the following subsections.
2.1. The interior-interior and exterior-exterior cases
The system (6), (7), considered as a whole , admits a very simple set of solutions thatlead to non trivial shells. Ifwe set A = 1, we obtain a solution by setting ( = ZI andj(zd = h(((ZI)) ' which implies also that r2 = r i on L:. This solution is non-trivialif we impose the condition that the range of r2 on M2 is the same as that of r i onM I, as can be seen by carrying out the procedures for obtaining TJlv that we describein the next section. The spacetime consists of two replicas of a certain region of
386 REINALDO 1. GLEISER AND PATRICIO S. LETELIER
Minkowski spacetime, joined through the "topological defect" Clearly, this works forarbitrary j, although certain restrictions hold if we also impose the condition that thesurface energy density should be positive. The interior-interior case corresponds tothose regions describing the interior ofa shell, while in the exterior-exterior case, theshell provides the inner boundary of two identical open regions, a situation somewhatresemblant of that of a wormhole. Particular examples can be found in (ibid.).
We also remark that, in principle, this case does not require axial symmetry, andthe matching of two identical replicas of Minkowski spacetime can be carried outthrough somewhat arbitrary surfaces, which can be restricted to some extent by thecondition of positivity of the surface energy density.
2.2. The interior-exterior case
This is the case where A -=I=- 1. Assuming that j and A -=I=- 1 are given, we may use (7)to eliminate h in (6). We have,
.u; _ V 1- A2 I 2dZ
l- 1 - jf2[J (zdJ . (8)
Assuming that d(/dz; -=I=- 0, solving (8) we may also find Zl as a function of (. Then,h is given by
1h(() = A j(Zl(()). (9)
It can be seen that for this construction to work, it is necessary that the argument ofthe square root in (8) be non negative. This implies the restriction on j
(10)
Therefore, for any given A < 1, we have an upper bound on 1f' 1. In particular, animmediate consequence of (10) is that the inside of a sphere cannot be matched toa flat exterior metric, for in this case, for a sphere of radius R, we have j(zd =JR2 - z;, and f' is not bounded for IZl l near R.
In the next sections we give some examples where the construction is possible, butwe first discuss the construction of the energy-momentum tensor TJ-lv.
2.3. The distributional stress-energy-momentum tensor
The results of the previous section give the relation between the functions j(Zl) andh(() that characterize the surface where the spacetime defect is located . The function((Zl) (and its inverse), provide the necessary connection between points on this surface as seen from each of its sides. Although this proves the existence of a continuousmetric, in the application of distributional methods we still need an explicit form forthe overall metric, in a chart where its components are continuous functions of thecoordinates, in the neighbourhood of all points, even those on 2:: . We may use theprevious results on continuity to construct this new chart. We consider first Ail and
SPACE-TIME DEFECTS: OPEN AND CLOSED SHELLS REVISITED 387
define new coordinates n ,w by the relations
(11)
where n :S 0, and the range of w is, in princip le, unrestricted. The hypersurface 2:corresponds to n = 0, but it is easily seen that a/an is not, in general, orthogonal to2:. A simple choice that makes a/an orthogonal to 2: is B, = El, with El = ± 1, andB2 = -Ed ' . E = 1 corresponds to the case where r l decreases as we move awayfrom 2:, while if E = - 1, rl increases as we recede from this hypersurface . The metricin M 1 is then given by
dsi = - dt2+ (1+ f'2)dn2+ 2nf'1"dndw (12)
+(J + El )2d¢? + [f' 2+ (1 - Eln1")2]dw2
where f and its first and second derivatives are given as functions of w.Similarly, we consider in M 2 a coordinate transformation of the form
(13)
where, n ;::: 0, and, in accordance with Equations (6) and (7), the functions h(( (w)),and ((w) satisfy the equations
1 + [f' (wW
[j(w) ]2
[1+ [h' (( (w)W] ( :~) 2= A2[h (( (w))J2.
(14)
(15)
We may again check that the conditions ofcontinuity of the metric , and orthogonality of a/an to 2: are satisfied if we choose
(16)
where E2 = ±l. Again, we remark that rz increases with increasing n for E = 1, whilewe have the opposite situation for E = -l.
With these choices, and taking into account (14) and (15), we finally obtain
and ( " = d(' / dw. It is clear now that if we consider a chart where n takes values ina neighbourhood of n = 0, and the metric is given by (12) for n :S°and by (17)for n ;::: 0, the overall metric is contin uous, with discontinuous first derivatives. We
388 REINALDO 1. GLEISER AND PATRICIO S. LETELIER
may then apply the distribution theory for the curvature tensor and obtain the Einsteintensor. We find that the non zero components of the Einstein tensor are given by
c.:
(19)
(20)
(21)
The physical interpretation of the corresponding components of the energy - momentum tensor is made simpler by replacing n by proper distance along a/an,whichwe indicate by s, and by referring the components of T/Lv to an orthonormal basis .
This amounts to replacing 8(n) by VI + f' z8(s), and taking out a factor pin G¢¢and a factor 1 + i" in Gw w . We also take 8KI>: = 1. The final result is
Tww
(22)
(23)
(24)
where the "hats" over ¢ and w indicate components with respect to an orthonormalbasis on E . The last equation shows, as expected, that the "Newtonian mass" of theshell vanishes (Vilenkin and Shellard 1994) .
We remark that in this construction, both the geometry of the shell, and the surfacestress-energy-momentum are completely specified once the function [ , and the constants A, and t i are given. We need to solve equations (6) and (7) only if we want toknow what the hypersurface E looks like as seen from Mz .
In the next section we give some examples where the construction is carried out.
3. SOME EXAMPLES
3.1. The general interior-interior case
As indicated this is the case where t l = 1 and t Z = - 1. Ifwe further requ ire that M zis a regular interior, we must choose A = 1. We then have
T¢¢41" (25)(1+ f' Z)3/Z 8(s)
Tww jVl~ f'z 8(s)(26)
T££ - T¢¢ -Tww' (27)
SPACE-TIME DEFECTS: OPEN AND CLOSED SHELLS REVISITED 389
Therefore, we have only surface tension and a positive energy density if I" ::::: O. Inparticular this holds for a large class of smooth regular closed shells, including spheres(this case is considered below) and ellip soids with axial symmetry.
We may also have pure tension and positive energy density if f" is essentiallyequal to zero , except on a bounded region. An exampl e would the case of two oppositely oriented cones connected through a smooth "throat."
There are, clearly many other possib ilities, which we shall not discuss here.
3.2. Spheres
Spherical shells can only be constructed in the interior - interior or exterior - exteriorcases. Here we have
11 = f (zd = j '5 - z? (28)
where 1 0 is the radius of the sphere. A regular interior implies A = 1 and it is easy tocheck that
4T -- -8(s) (29)¢¢ 10
4Tww - 8(s) (30)
1 0
8TU - - 8(s). (3 1)
10
Namely, we obtain, as espected, a constant positive energy density, and a constantisotropic surface tension , see also (Letelier and Wang 1995).
3.3. Cylinders
This is the general case where 11 = 1 0, with 1 0 a constant. In this case f = 10 andf' = f" = O. This implies that we may take ( = Z I , and h(() = ' o/A . We also haveT¢¢ = O.
This still leaves open the choic e of t l and t 2 . We consider first t l = 1. In this caseM 1 is the region inside a cylinder of radiu s 1 0 . We have
r.; 21 -/2A 8(s)
-r; «.
(32)
(33)
We have now two choices for t2. If t2 = - 1, the region M 2 corresponds to (part of)the interior of a cylinder of radius 10 / A. The surface energy density is positive for allvalues of A, but if A > 1, we end up with a negative "deficit angle" on the symmetryaxis. For A < 1, the region M 2 corre sponds to a "co smic string" spacetime thatextends from the symmetry axis up to radiu s 1 0 /A. Finally, if A = 1, what we obtainis the matching of the interi ors of two identical cylinders of radiu s 1 0 , both with regularsymmetry axes. The surface energy density is equal to 2(1 + A) /,O, and is also equalto the tension along the generatrixes .
390 R EINALDO 1. GLEISER AND PATRICIO S. L ETELIER
If we choose 1:2 = 1, the region M 2 is that external to a cylinder of radius rooPositivity of the energy density requires A < 1. As result we have a deficit angle8 = (1-A)21f in the external region. Therefore, what we have obtained is a matchingof the region inside of a cylinder of radius ro with part of the region exterior to acylinder of rad ius ro/ A. (We recall that 8 f. °impli es that a "wedge" has been cutout from the external region). This result is, of course, the same as that obtained in(Letelier and Wang 1995; Tsoubelis 1989).
In the case 1:1 = - 1, the region M 1 is that external to a cylinder of radiu s rooThe only new case is that where 1:2 = 1. This is the external - external case. It iseasily seen that in this case the energy density is negative. We have a wormhole typeof spacetime in this case, we shall be back to this point later.
We shall not consider further cases with 1:1 = - 1, so that in the remaining examples M 1 will correspond to a regular region, interior to the shell.
3.4. Cones
We define a cone byj(zr) = alzll
where a is a real constant. We may then choose 1:2 = 1, and
(34)
(35)
(36)
(37)
J A2 - (1 - A2)a2( = A Z I
provided that A2 2': (1 - A2)a2, and, therefore,
ah(() = J A2 - (1 _ A2)a21( 1.
Ifwe choose the range of r l as °:::; r l :::; lazl l, and r2 2': alz21/ J A2 - (1 - A2)a2,we have the matching of the region inside of a cone of aperture angle 0: = tan-1 (a),with part of the region external to a cone with aperture angle0: ' = tan " ! (a/ J A2 - (1 - A2)a2). We notice that 0:' approaches 1f / 2 as the limiting condition A2 = (1 - A2 )a2 is approached.
From the general expre ssions for T/Lv we obtain T ¢¢ = 0, and, therefore, T£i =- Tww, which can be interpreted as indicating that the conical shell is generated bya continuous distribution of cosmic strings, distributed along the generatrices of thecone. The surface energy density is given by
Although, in general, we cannot write ((z), (or its inverse) in closed form , it isstraightforward to obtain a power series expansion for z(( ), around ( = 0 assuming that z(O) = O. The first non vanishing terms are
-r l- c2r3 (c2 -1)(13c2 -1) r 5
z - ,, + 6b2 " + 120b4 ,, + ...
Replacing this expansion in j, we obtain an expansion for h,
h(r) = ab ~r2 _ (4c2
- l)a r4 (88c4
- 44c2 + l)a (6
" A + 2Ab" 24Ab3" + 760Ab5 + .. .which should be compared with
The external shape of the shell is not , therefore, that of (a part of) a hyperboloid. Itreduces to a hyperboloid only in the trivial limit A = 1 (where we also have c2 = 1).
A particular case where we obtain closed expressions, is the limiting case c2 = O.Here we have
and
( = bln(z jb + v z2 jb2 + 1)
z = bsinh(( jb) .
(48)
(49)
392 REI NALDO 1. GLEISER AND PATRICIO S. L ETELIER
(50)
Replacing in h we find
h(() = h COSh (~)
which clearly shows the different shapes of the surfaces Tl = j (Zl ) and T2 = h(Z2).These results have an interesting interpretation. The hyperboloids belong in the
family of ruled surfaces , i.e., surfaces that can be generated by the motion of straightlines. In particular, for hyperb oloids, there are two families of straight lines , tangentto the hyperboloid in all their points, that cross each other in "wire wastepaper basket" fashion. We may therefore think of the hyperboloids in these examples as beinggenerated by a continuous distribution of straight cosmic strings, that cross each otherin such a way that the result ing tension has components in both the 0/o¢ and 0/owdirections. It is well known , however, that when we have two cosmic strings that crosseach other at an angle , if we adapt our coordinate system to one of them , so that itlooks straight, the other appears to be "bent" on account of the "cone" singularitynature of the spacetime associated to a cosmic string . In our case, this "bending" iscontinuous, and gives as a result that if from the inside of the shell the strings lookstraight, they appear to be "bent" from the outside, which corresponds to the fact thecurvature of the external surface is larger than that of the internal one.
4. OPEN SH E LLS
So far we have restricted our treatment to the cases where the condition
(51)
was satisfied. In this Section we analyze what happens if this condition is violated.Consider first the case when [j '( ZlW -. A2/ (1 - A2) as Z l -. 00 . In this case wemay write
(52)
where g(Zl) > 0, and g(Zl ) -. °for large Zl . We then have
(53)
Here we have two possibilities, depending on the behaviour of g(zd for large Zl . Ifg(zd is such that the integral in (53) diverges , the external side of the shell extendsto all values of Z2, and, since for large Zl we have
(54)
from the relation
(55)
SPACE-TIME DEFECTS: OPEN AND CLOSED SHELLS REVISITED 393
we find that, for large ZI , r2 is also divergent, and the shell approaches a conical shapeon both sides .
On the other hand , if g(zd goes to zero sufficiently fast, the integral in (53)is convergent, and (, (and therefore Z2 ) , approaches a finite value ( 00 = Z2 - 00, asZI ----+ 00 . We notice, however, that , since (54) holds also in this case, r2 is unboundedas Z2 approac hes Z2 - 00 . Thus, the shell approaches a conical shape from inside, butan infinite disk-like shape from outside.
Simi lar considerations hold in the limit ZI ----+ - 00 . Thus, this cases are essentia llysimilar to that where [II(zdF< A2 / (1- A2 ) for all ZI , in the sense that the resultingspacetime is covered by the union of two charts , M I and lV12, and the hypersurface2:.
A different situation results if we have [j' (zd F = (A2 /(1 - A2 ) for some finitevalue of ZI . Suppose this equality holds for ZI = z~ . Then, if we assume that j(ZI) isdifferentiable for ZI = z~, we have
where B is some constant. Then, a simple computation shows that
(57)
where 0 1 and O2 are some constants , for ZI < z~ , near ZI = z~ .
This result impl ies that the hypersurface 2: is bounded both in M I and in M 2 .
Therefore, we may naturally extend, say M I , so that we go around the border of 2:,and see the "other side of the shell" but staying all the while in M 1, and not goingthrough 2:. If, instead, we do move through the shell, but start ing on its inner sideon M I , we end up in M 2. Similarly, we may start in M 2, and either go throu ghthe shell, or around it, and end up in different subspaces. We may reconcile all theseobservations by adding a third chart M 3, that is reached when we cross the shell fromthe "outside" in M I . The shape of the shell in M 3 is the same as that in M 2 , sothat the matching conditions are satisfied, but it is easy to check that surface energydensity is now negative. Cross ing the shell from the "outside" in MI, we reach the"inside" in M 3 . Now, if we cross the shell from the "outside" in M 3 , we reach a copyof M l' which we indicate with M 4, in such a way that the "outside" of the shell inM 3 corresponds to the "inside" of the shell in M 4. This corresponds to a shell withpositive energy density. We may finally complete the picture by imposing that whenwe cross the shell from the "outside" in M 4,we reach the "inside" of the shell in M2.This time the shell appears with negative energy density.
In this process, we have actually added not only the two charts M 3 and M4 , butalso three more matching hypersurfaces. Namely, besides 2:, commo n to M I andM 2, we have 2:13 , common to lV1 1and M 3, 2:34 , common to M 3 and M 4, and 2:42 ,
common to M4 and M 2. The resulting spacetime has, therefore, a rather non trivialtopology, besides containing shells with negative surface energy density.
394 REINALDO 1. GLEISER AN D PATRICIO S. LETELIER
5. FINAL COMMENTS
In the exterior-exterior case limited by a cylindrical surface, we have that this surfaceacts as the throat of a wormhole, i.e., its connects to asymptotically flat spacetimes.The fact that the energy density is negative in this case is a well known fact of staticwormholes (Morris and Thome 1988). We have that nonstatic wormholes do notalways need to violate the energy conditions, see for instance (Wang and Letelier1995). One can also do, in a similar way, constructions with toroid al and spheroidalshells like in (Letelier 1995a).
The conical shells as well as the hyperbolic shell (ruled surface) that can be considered as formed by usual straight cosmics strings, i.e., in other words as continuous limits of the metric that represents several crossed cosmic strings (Letelier andGal 't sov 1993). Of course the cylindrical shell considered here can be also thoughtas the continuous limit of the metric that represent s several parallel strings that is aspecial case of the one above mentioned.
In this work we have constructed some new static shells and mainly made a geometric interpretion of a variety of closed and open shells. The case of nonstatic shellswill be treated in a separate paper.
In this series of papers the defects we dealt with are curvature defects, i.e., spacetimes whose curvature tensor is a Dirac type of distribution with support on a hypersurface. One can likewise consider other type of defects in which the torsion is not nullas in a usual Riemannian manifold, but it is also a distribution. This is the analog of thescrew dislocat ion defect in solids, see for instance (Letelier 1995b). Also defects inpure torsion geometries, Weizembock spaces, has been considered (Letelier 1995c).Global properties of defect space-times can also be studied using loop variables orholonomy transformations (Bezerra and Letelier 1996).
ACKNOWLEDGEMENTS
This work was supported in part by research grant s from CONICET, CONICOR,SECYT - UNC, FAPESP and CNPq. R. 1. G. is grateful for the hospitality extended to him while visiting the Departamento de Matematica Aplicada - IMECC, wherepart of this work was done. R. 1. G. is a member ofCONICET.
Universidad Nacional de CordobaUniversidade Estadual de Campaninas
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