Rend. Sem. Mat. Univ. Poi. Torino Voi. 50, 1 (1992)

Singularities in G.R.

G.F.R. Ellis

I N T E R A C T I N G COSMIC S T R I N G S

A b s t r a c t . The pure gravitational aspects of interacting cosmic strings are

investigateci, using an approach introduced by Hellaby (based on use of the

conical approximation for each string). In this picture, the result of the

intersection of two strings is that each of the interacting strings becomes bent

through an angle equal to the deficit angle of the other one, and a new string

is created, joining the places where each of the initial strings has been bent

by the other. Formulae are given for the strength of this new string, and for

the way the angles are perceived when the strings move at relativistic speeds.

This analysis leads to identification of an irreversibility associated with the

interactions of a cloud of strings.

1. Introduction

The gravitational aspects of cosmic strings are far from straightforward; indeed they are somewhat surprising [1-5], despite the fact that at their heart lies a simple description.

Straight strings are known to have a deficit angle structure [6-8], and the deficit angle description is a good approximation to the exterior fìeld of gauge strings, unless the string parameters are very dose to the Planck scale [9]. Furthermore when the energy-momentum of a string is concentrated on a 2-surface (the zero-thickness approximation) and the space-time curvature is sufficiently regular, this kind of conical structure is inevitable [10]. It can be represented by cutting out a wedge from a locally fiat space-time and identifying points [5,6], obtaining a conical-type ("quasi-regular")

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singularity [11] in the fiat space-time; we may cali this the pure gravitational representation of a cosmic string (it makes no attempt to model the field theory aspects of the string, but exactly represents the exterior gravitational field of a straight gauge string, and approximately that of other strings when viewed at appropriate scales [1,10]).

While this conical structure has been emphasized in descriptions of single strings, and indeed is centrai to proposals to search for them observationally by their effect on light propagation in the universe [12], most studies of interacting networks of cosmic strings have concentrated on their field properties, neglecting the implications of the deficit angle structure. The 'pure gravitational' description has been used on the one hand in investigations of pairs of cosmic strings which do not intersect each other [5,13], leading to the discovery of causai violations when the strings are in relative motion; and by Hellaby [14] to investigate the interaction of cosmic strings, when they pass through each other.

Hellaby obtains results that are strikingly different from those obtained by field theory studies of such interactions in fiat space-time, where the strings often break off and reconnect. He finds that they do not break, but rather that new strings form connecting the old ones, which become a bit bent in the process [14].

While his basic approach is very elegant, his presentation is reliant on spherical trigonometry formulae which somewhat obscure the clarity of the basic idea. Here I gì ve a different formalism for doing the calculations, as an alternative way of obtaining his results.

2. The basic idea

The basic idea is illustrated by considering two strings. Starting with fiat space-time, we cut out non-intersecting wedges, one for each string, with the vertex of the wedge defining the position of the string; and then simultaneously identify the pair of faces of each wedge, to create the conical singularities that will represent the gravitational effect of the strings [14]. Thus the procedure is as usuai (see e.g. [5]); the new feature is that the wedges are chosen at right angles to each other and moving towards each other, so that although they are initially apart, after a time they intersect each other.

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When intersection has taken place, some parts of the base space-time will have been (in principle) removed twice. This does not matter; the resulting wedge faces are stili isometric to each other and may be identifìed at each time (Fig. I1) to give the final space (Fig. 22.) In this space there are two strings, joined by a strut (where four lines are identifìed, forming the intersection of the two planes in Figure 2). It will be our aim to identify the nature of this strut; we will find it is another string, linking the first two. Furthermore each of these two strings gets bent by the other once they have intersected, as is clear in figure 2, the points where they are bent being linked by the strut3 .

F i g u r e 1: Two wedges in Minkowski space time that approach and intersect, shown in a spatial section {t=const} after they have intersected each other. The intersecting strings are produced by identifying the opposite pairs of faces of the two wedges. The curve shown does • not intersect either wedge, and so is a regular (closed) curve in fiat space.

Figure 2a of [14], reproduced with permission.

Figure 3 of [14], reproduced with permission. 3 We do not have space to reproduce ali the relevant diagrams here; and recommend perusing them in the originai paper [14].

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Figure 2: The string space produced by identification of the pairs of faces in Figure 1. The curve shown is topologically equivalent to that in Figure 1; it has been deformed to touch each pair of faces in corresponding pairs of points, but stili lies entirely in the originai fiat space (it does not cross any of the identifìed faces) and so is equivalent, in parallel transfer terms, to the originai curve in Figure 1.

This may seem obvious in Figure 2, however the analysis must be treated with the utmost care. Although the space section shown in each figure is locally fiat, it has non-local curvature because of the conical singularities that result from this construction. As demonstrated by Hellaby, we can fìnd the curvature properties of the space by considering parallel transfer along the loop shown in Figure 1, which lies entirely in the originai fiat space-time and so is unaffected by the identifications macie to create the strings. That is, parallel transfer is integrable and we have parallelism at a distance in the obvious way as long as we do not cross the identifìed sections (e.g. vectors

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parallel to the axes remain parallel to the axes). Thus from this figure we immediately derive parallel transport along any curve that does not cross the identifìed faces. We can also correctly work out what happens to parallel transfer along any curve that does cross the faces as long as we carefully follow the effect of the identifìcation.

Identification gives Figure 2, which gives a correct locai picture of the bending of the strings; but global parallelism does not hold in the obvious sense in this picture. For example, the horizontal string is measured to be bent relative to parallel transfer along a curve which crosses the identifìed face (due to the deficit angle introduced by this identifìcation); thus following the string along its own length, a kink occurs as it crosses this face. However this string is found to be straight relative to parallel transfer along a curve that never crosses the identifìed planes, such as the curve shown in Figure 1. The same holds for the curve in Figure 2 (which is equivalent, in parallel transfer terms, to the curve in Figure 1, for it is obtained from it by deformation in fiat space-time: it is just the same curve but deformed to touch, but not cross, the faces of the wedges that have been cut out).

One should note particularly that the planes shown in Figure 2 are surfaces where the originai coordinates (derived from figure 1) will be discontinuous; but there is no locai problem with the spaice itself at these surfaces, and there exist other locai coordinates (implied in Figure 2 by the orthogonals to the planes) which are perfectly regular across each of these surfaces separately. It is in terms of such coordinates that we "obviously" see that the string is bent. When crossing one of the surfaces, locally there is no problem [as the ghie used to join the faces is perfect and makes the join invisible].

However now there is a deficit angle if we compare parallel transport along a directed curve C\ that crosses the surface relative to a directed curve 6*2 that does not (Figure 3). The point here is that we can compare parallel transport along these curves provided they both go through a point P before and Q after going past the string, for these curves then create a directed loop G = C\ — Ci around the string; parallel transfer around this loop gives the difference between parallel transfer along C\ and Ci- This difference will be a spatial rotation, equal in magnitude to the deficit angle of the string, that can [on using the right-hand rule relative to the direction of the closed loop] be represented by an arrow parallel to the string, with length proportional

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to the deficit angle. If we deform a loop through the edge of the wedge, it will introduce this relative dilTerence in parallel transfer (even though along each curve we simply have locally parallel transport in fiat space time with no rotation, and with nothing special happening on passing through the identified

Figure 3: Deiorming curve C

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string inside. The final curve is a simple loop around the strut connecting the bends in the strings. We can then form the total effect of parallel transfer right around this loop that crosses ali four faces by carefully transporting a vector Xa from an initial point A back to A, obtaining back at A two copies of Xa: the one parallel transported on the outside curves [parallel to the initial value] and the one parallel transported along the inside curves [that has crossed four faces]. We can consider this either in the unidentifìed view [Figure 1] or the identified view [Figure 2]. As the inner curve circles the strut once, it gives the deficit angle of the string that has been constructed along the strut by the intersection of the two initial strings.

The effect is equivalent to vector addition at one point (the axes for this addition can be obtained by parallel transport of a vector basis along the outside curve). The result is a non-zero change because the inside vector gets a change A l [relative to the outside vector] at face 1; thus at face 2, where it gets a change A2, it is no longer orthogonal to the string. Thus we need to calculate the total change to the vector that crosses ali the faces, relative to a parallel propagated basis that does hot cross any of the faces (and so is integrable). We can do so in practice by starting witli an outer loop Oi, along which parallel propagation is integrable (it is contained in a single Minkowski patch). We shrink it through four edges of excised wedges, to give the inner loop O2 that circles the strut once and gives the deficit angle we require: the deficit angle of the newly formed string. Each time we move the loop through an edge (i.e. through the location of a string, by passing through the piane rather tlian around it) we obtain a deficit angle for parallel propagation along the inner loop relative to the outer loop. The total deficit angle is obtained by adding tlies.e contributions (relative to a basis of vectors parallel propagated along the outer loop).

In the remainder of the paper, we calculate this effect by using an orthonormal basis in the originai coordinates [the Minkowski space, unidentifìed, but with wedges removed]. Each deficit angle corresponds to a spacelike rotation with axis the edge of the wedge. We first look at the situa.tion in a 3-space [the rest space òf an observer moving with 4-velocity ua], and then at the Lorentz transformation properties of strings.

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3. 3-space description

3 .1 . Single deficit angle

A string appears as a straight line with unit tangent vector Fa. The spatial deficit angle holonomy [relative to the basis mentioned above] is a rotation generated by the bivector

(1) Fab = eabcFc

where Faf, is a unit bivector (so the components form a skew matrix with elements -fi» 0, or -1). We write the (finite) rotation as

(2) Rab = exp(/3F)a

6 = 6* + 0Fab + ~P2F2a

b + ^/33F3ab + ...

where F2ab = Fa

cFcb, F3a

b = FacFc

dFdb, etc, and (3 is the angle of rotation

[it can easily be checked for example that this gives the correct form for spatial rotations about the x-axis, when the components of F^ are ali zero except F23 = 1 = — JF32]. We can alternatively express (2) in terms of the bivector fai, generating the rotation:

(3) Rah = ( exp / ) .* , /„» = 0Fab ^/3

2 = \fahfab.

3.2. Intersecting strings

Now we consider two wedges each with a deficit angle as above; we use subscripts 1 and 2 to denote them respéctively, so the rotation axes are F\a, F2 a , the bivectors -Fi^, F

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where the signs of the rotation have to be taken in accordance with the loop direction and the right hand rule (this is implied by the use of the esymbol to relate the bivectors to the vectors).

3.2 .1 . Power series calculation

Now we use the power series expansion (2), keeping terms up to the second order. The result (basically a standard result in exponentials) is

(5) Tah = Sa

b + A / ^ F , , F2]ah + 0(f)

where the commutator [,] is defìned by

(6) [F1,F2}ab = Fla

cF2cb - F2a

cFlcb

showing how the effect comes directly from the non-commutativity of the rotation group. Substituting from (1), we can re-expresn this as

(7) [Fi,F2]ab = FlaF2b-F2aFlb = (F1AF2)ab,

Equation (5) gives the deficit angle generated by the strut, to second order; this agrees with the calculation by Hellaby. Naturally (4) gives a more correct result for large angles. However in the cosmological context of cosmic strings, the deficit angle will always be very small and the result (5) will be adequately accurate.

The conclusion is that when two strings intersect, the result is that each is bent by the deficit angle of the other, their bends being joined by a new string with deficit angle (4) with (5) a good approximation in the usuai context.

4. 4-dimensional formulation

4 .1 . Single string

Now a string is a 2-dimensional world-sheet. It can be characterised by uni't orthogonal timelike and spacelike vectors that span it, say va, Fa; then

(8) Fab = VabcdFcvd, vava .=' - 1 , F

aFa = 1, vaFa = 0

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where these two vectors are arbitrary by a boost:

(9) v,a = cosh ip va + sinh ip Fa, F,a = cosh i/> Fa + sinh ip va.

If an observer moves with 4-velocity va, (8) will reduce to (1) above, but is now fully covariant. As this is a simple bivector (Fabr}

a°cdFC(i = 0), an alternative representation is: there exist unit orthogonal spacelike vectors Xaì Ya in the rest-space of v

a such that

(10) Fab = XaYb - YaXb, X.X = Y.Y = 1, X.Y = 0, X.v = Y.v = 0 .

These vectors are arbitrary by a rotation in their piane.

If an observer moves with 4-velocity ua, he will measure as a deficit angle the rotation generated by the projection of fab = fiFab orthogonal to his 4-velocity [for that defìnes the rest- space within which the rotation takes place; the holonomy circuit willbe in this rest space, and will pick out only these components of the full tensor]. Thus he will measure the effective rotation basis to be

(11) F±ab = ha'h^Fsu hab = gab + uaH

which is no longer a unit bivector. It can be rewritten as

(12) FLah = Fab -UarìhscdusvcFd + uwàtcrfvOF*.

The last two terms vanish if ua and v are coplanar with the string, i.e. if the string is "at rest" relative to the observer (or if the observer's 4-velocity is chosen to move with the string); note that a velocity along the string is not measurable (if ua lies in the piane of the string we can boost va to lie along ua, and these relations will ali remain valid).

The exponential formula

(13) Rj = (exp f)ab, fab = PF±ab

will give the deficit angle experienced by an arbitrarily moving observer (or equivalently, experienced by a "stationary" observer due to an arbitrarily moving string); it reduces correctly to the formulae above when the string and observer are at rest relative each other. However it is useful to know more

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explicitly the transformation properties of the deficit angles under change of the observer's 4-velocity.

4.2. Velocity dependence

It is convenient to re-express F_\_ai in other ways. One convenient option is to project (10) in the following way:

(14) F±ab = X X o y X 6 - Y±aX±b, XI = habX

b, Yl = habYb

(note that these vectors are orthogonal to each other and to ua but are in general not unit vectors). This is again a simple bivector, and we can express it relative to ua analogously to (8):

(15) F±ab = r,abcdGcud, uaua = - 1 , G

aGa = 1, uaGa = 0 .

Now if we substitute in here from (8) and (11), and multiply by riabstut, we find the rotation basis vector Ga in the rest-frame of ua to be given by

(16) Ga = (-u.v)Fa +(u.F)va

(which correctly obeys the relation u.G = 0 and reduces to Ga = Fa whenever ua = va). Thus this vector gives the axis of the rotation as seen by the arbitrarily moving observer ua. The magnitude /3' of the rotation will (using (8), (13), (15), (16)) be given by

(17) /?'2 = l-fahfah = P2GaGa = f {(-u.v)

2- (u.F)2)

which shows explicitly the velocity dependence of the effective deficit angle /3' on the observers' 4-velocity ua.

Because of the freedom (9) in choice of va and F in the piane of the string, if we select one observer with 4-velocity ua and let him watch arbitrarily moving strings go by (so that ua is fìxed but each string has different va and Fa), then we can set the vectors Fa to ali lie in the observers' rest-frame, without loss of generality, by choosing

(18) tanhV> = (u.F)/(-u.v)

in each case, so that finally (dropping the primes)

(19) u.F = 0.

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Equations (18), (19) give a unique choice of va and Fa in (8), so defìning uniquely the velocity of the string relative to the observer [by resolving the arbitrariness (9) of va in the piane of the string).

With this choice, from (17) the eiTective deficit angle becomes

(20) /?' = fi(-u.v) = h where 7 = (—u.v) is the Lorentz transformation factor between ua and v a , determining the speed of radiai motion of the string relative to the observer. As the choice (19) uniquely fixes the velocity relation along the string, f3' is in fact independent of the motion parallel to the string.

Relation (20) shows how, if the sky is filled with a random pattern of strings moving at high speeds in ali directions, those moving either towards us or away from us at high speed will appear to havé greatly increased deficit angles, which would be observed in lensing of more distant objects by these strings.

4.3. Interacting strings

When there are two interacting wedges, the deficit angle of the strut can be calculated as before, by the same formulae (4) as above (applied to the projected bivectors). The Lorentz transformation results of the previous subsection will apply independently to each string in a collision. The commutator formula (5) above will apply to the projected bivectors, giving the strut angle. The analogous formula to (6) and (7) in terms of the projected bivector and the effective vector Ga will give the observer's measurements.

5. Issues

When a network of cosmic strings continuously intersect each other, we can use the same approach as used above for the case of two strings. The conclusion is that each time they intersect, they do not break; rather a new string is generated that links the other two between the points where they have been bent by the collision. The new string is weaker than the other two (if they have small deficit angles and are slowly moving) but must be present for consistency of the solution (otherwise the initial two strings could not be bent by the collision, and this must happen because of their deficit angle

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nature). The new string can have a large deficit angle if the two colliding strings are in rapid relative motion (even if their intrinsic deficit angles, that is the angles evaluated in their respective rest frames, is small).

An interesting point that then emerges is that there appears to be an effect of increasing entropy associated with the string collisions. This is indeed so: after each collision there will be more strings present than before. Why is this so? The point is that in principle we can undo the effect of a collision by reversing the velocities after the collision has taken place, so undoing the bends in each string and de-creating the strut (we run the collision backwards in time, obtaining the previous initial state: two separated strings). However to do so we have to get the reversal of velocities exactly right. If we make any error whatever, instead of undoing the strut we miss, and create yet another new one. Thus the network of strings will in practice never succeed in undoing any of the struts once they are formed: there will be an ever increasing density of strings (admittedly of ever decreasing deficit angles, but there ali the same, for without them space-time singularities must occur). The existence of the new strings is a consequence of the Regge calculus forni of the Bianchi identities for conical singularities.

As has been emphasized, this is a pure gravitational view of the interactions. Presumably (cf. the discussion given in [14]) real strings have an interaction with some elements from this picture, and some from the more usuai field theory calculations in a fiat background, that ignore the conical singularities and the consistency requirements that they impose.

I thank Charles Hellaby for useful comments, and the MURST for finali ci al support.

REFERENCES

[1] V P . FROLOV, W. ISRAEL, W.G. UNRUH, Phys. Rev. D39 (1989), 1084. [2] C.J.S. CLARKE, G.F.R. ELLIS, J.A.G. VICKERS, Chss. Qu. Grav. 7(1990),

1-14. [3] W.G. UNRUH, G. HAYWARD, W. ISRAEL, D. MeMANUS, Phys. Rev. Lett.

62 (1989), 2897. [4] W. ISRAEL, Annales de Physique 14 Coli. 1 (1989), 109-114. [5] J.R. GOTT, Phys. Rev. Lett. 66 (1991), 1126. [6] A. VILENKIN, Phys. Rev. D 23 (1981), 852.

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[7] J.R. GOTT, Astrophys. Journ. 288 (1985), 422. [8] W. HISCOCK, Phys. Rev. D 31 (1985), 3288. [9] D.N. VOLLICK, W.G. UNRUH, Phys. Rev. D 42 (1990), 2621.

[10] J.A.G. VICKERS, Class. Qu. Grav. 7 (1990), 731. [11] G.F.R. ELLIS, B.G. SCHMIDT, Gen. Rei. Grav. 8 (1983), 915. [12] A. VILENKIN, Phys. Rep. 121 (1985), 263. [13] C. CUTLER, Phys. Rev. D 45 (1992), 487. [14] C. HELLABY, Gen. Rei. Grav. 23 (1991), 767.

George F.R. ELLIS, SISSA, Trieste, Italy,

and Department of Applied Mathematics, University of Cape Town, South Africa.

Lavoro pervenuto in redazione il 28.2.1992.