String Rotation

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arXiv:hep-th/9612199v22

5Feb1997

Rigidly Rotating Strings in Stationary

Axisymmetric Spacetimes

V. Frolov1,2,3 , S. Hendy1 and J.P. De Villiers1

February 1, 2008

1Theoretical Physics Institute, Department of Physics, University of Alberta,

Edmonton, Canada T6G 2J12CIAR Cosmology Program3P.N.Lebedev Physics Institute, Leninskii Prospect 53, Moscow 117924, Rus-

sia

1
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Abstract

In this paper we study a motion of a rigidly rotating Nambu-Goto test stringin a stationary axisymmetric background spacetime. As special examples weconsider rigid rotation of strings in a flat spacetime, where explicit analyticsolutions can be obtained, and in the Kerr spacetime where we find an inter-esting new family of test string solutions. We present a detailed classificationof these solutions in the Kerr background.

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1 Introduction

Cosmic strings are cosmologically interesting objects [1] and the motion ofstrings in a curved background is a subject which has recently been inten-sively discussed [2, 3]. If one neglects the gravitational effects of a string andassumes that its thickness is zero, then the string configuration is a time-like minimal surface that is an extremum of the Nambu-Goto action. Oneof the interesting physical applications of the general theory is study of theinteraction of cosmic strings with a black hole. If a (infinitely) long cosmicstring passes nearby a black hole it can be captured [4]. Final stationaryconfigurations of a trapped string were analyzed in [5] and their completeanalytic description was obtained.

In the general case a stationary string in a stationary spacetime is definedas a timelike minimal surface that is tangent to the Killing vector generatingtime translations. In the Kerr-Newman metric the equations describing astationary string allow separation of variables [6, 7, 8] and can be solvedexactly [6]. In this paper we generalize these results to a wider class of stringconfigurations. Namely we study rigidly rotating strings in a stationaryaxisymmetric background spacetime. A rigidly rotating string is a stringwhich at different moments of time has the same form so that its configurationat later moment of time can be obtained by the rigid rotation of the initialconfiguration around the axis of symmetry. Denote by (t) and () Killingvectors that are generators of time translation and rotation. The timelikeminimal worldsheets which represent a stationary rigidly rotating string arecharacterized by the property that the special linear combination (t) + ()is tangent to the worldsheet. Our aim is to study such configurations in astationary axisymmetric spacetime.

The paper is organized as follows. General equations for a stationaryrigidly rotating string in a stationary spacetime are obtained and analyzedin Section 2. As the simplest application we obtain explicit analytical solu-tions describing rotating strings in a flat spacetime (Section 3). One of theinteresting results is the possibility of the rigid rotation of the string with(formally) superluminal velocity, i.e. when r > 1 (r is the distance from

the axis of rotation). A simple explanation of this phenomenon is given inSection 3. Section 4 devoted to rigidly rotating strings in the Kerr spacetime.To conclude Section 4 we present a classification of this new family of teststring solutions in the Kerr spacetime.

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2 General Equations

Consider a stationary axisymmetric spacetime. Such a spacetime possessesat least two commuting Killing vectors: (t) and (). If a spacetime is asymp-totically flat the vector (t) is singled out by the requirement that it is timelikeat infinity. The vector () is spacelike at infinity and it is singled out by theproperty that its integral curves are closed lines. The metric for a stationaryaxisymmetric spacetime can be written in the form

ds2 = V [dt wd]2 + 1V

2d2 + e2(d2 + dz2)

, (1)

where V, w and are functions of the coordinates and z only. This is

the so called the Papapetrou form of the metric for stationary axisymmetricspacetimes (see Ref. [9] for example). In these coordinates (t) = t and

() = .

Denote by a two-dimensional timelike minimal surface representingthe motion of a string in this spacetime and denote by St the spatial slicet = const. The intersection of with the surface St is a one-dimensionalline t representing the string configuration at the time t. We define a rigidcosmic string as one whose shape and extent (but not necessarily position)are independent of the coordinate time t. If xi are spatial coordinates (formetric (1) (,z,)) then t is given by the equations x

i = xi(, t), where isa parameter along the string. Since () is tangent to St it is a generator of

symmetry transformations (spatial rotations) acting on St. It is evident thatthis transformation preserves the form and the shape of the string t. Ourassumption that the string at the moment t is obtained by a rigid rotationfrom the string t0 can be written as

(, t) = (, t0) , z(, t) = z(, t0) , (, t) = (, t0) + f(t, t0) . (2)

Moreover we assume uniform rotation, so that f(t, t0) = (t t0), where is a constant angular velocity. It is evident that the following combination = (t)+

() of the Killing vectors

(t) and

() is tangent to the worldsheet

of a uniformly rotating string.

In a region where

is timelike one can define a set of Killing observerswhose four-velocities are u = / |2|1/2. This set of observers form a rigidlyrotating reference frame that is the frame moving with a constant angularvelocity . One could choose to define a rigidly rotating string as a stringwhich was fixed in form and position in the frame of some Killing observer

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with angular velocity . It can be shown that if all the string is located

in the region where

is timelike this definition is equivalent to that givenabove. But, as we shall demonstrate later, situations are possible when arigidly rotating string lies in a region where the Killing vector is spacelikeyet its world sheet surface remains timelike. With this possibility in mindwe will use the former definition of the rigid string rotation.

We begin by performing the following coordinate transformation:

= t, (3)where is a constant. Metric (1) now takes the form

ds2 =

V[(1

w)dt

wd]2+

1

V2(d + dt)2 + e2(d2 + dz2)(4)

and the Killing vector has components = (1, 0, 0, 0). The Killing tra-jectories of (that might be timelike or spacelike) are: ,z, =const .

A configuration of a test cosmic string in a given gravitational backgroundis represented by a timelike two-surface (a world sheet) that satisfies theNambu-Goto equations of motion. A world sheet can be described inthe parametric form x = x(A), where x are spacetime coordinates, andA (A = 0, 1) are the coordinates on the world sheet. For these coordinateswe shall also use the standard notation (0, 1) = (, ). The Nambu-Gotoaction is given by

S[x] = d2G, (5)where G is the determinant of the induced metric on the world sheet GAB =gx

,Ax

,B and is the string tension.

For a stationary world sheet configuration one can choose parameters(, ) in such a way that

x(A) = ( + f(), (), z(), ()) , (6)

where f is some function of . The determinant of the induced metric GABon the world sheet is

G = e22V

2+ z

2

2 2. (7)

where

2 = V + 2wV + 2(2/V w2V) , (8)

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Note that neither the function f nor its derivative f appear in the action so

they may be specified freely. It is convenient to choose f so that the inducedmetric is diagonal, i.e. G = gx,x

, = 0. In this case we find f must be

chosen to satisfy the condition

f = V w + (2/V w2V)2

. (9)

A stationary string configuration (6) provides an extremum for the reducedNambu-Goto action

E =

d

e2(2)

V 2 + z2

+ 2 2. (10)

Hence a stationary string configuration xi = ((), z(), ()) is a geodesicline in a three-dimensional space with the metric

dh2 =e2(2)

V

d2 + dz2

+ 2d2 . (11)

The Nambu-Goto equations for a stationary rigidly rotating string are

2e2GV

=

1G

1

2

2e2

V

2+ z 2

2

, (12)

2

G

= 0, (13)

2e2GV z

=

1

2G

z

2e2

V

2+ z 2

, (14)

where G is given by (7). Equation (13) can be integrated immediately togive

2 =2

V

L2e2

2(2 L2)

(

2+ z 2). (15)

Here L is a constant of the integration. The constant L is associated with the-independence of the Lagrangian and is related to the angular momentumof the string. In what follows we choose L to be non-negative.

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These equations are invariant under reparameterization = ().In the region where

= 0 one can use this ambiguity to put = . For thischoice equations (12)-(14) reduce to

d

d

2=

2L2e22(2 L2)V

1 +

dz

d

2 , (16)

d

d

2e2

GVdz

d

= 1

2

G

z

2e2

V

1 +

dz

d

2 , (17)

where

G = G 2

= 22e2

V(L2 2)1 + dz

d

2 . (18)The solutions represent a timelike two-surface provided the determinant

G is negative definite. Thus we see that the rigidly rotating strings areconfined to regions (for V > 0) where

I L2 2

2> 0. (19)

When L2 = 2 the world sheet has a turning point in as a function of .

In general, in order to ensure rigid rotation of a string, an external forcemust act on it. For example, one could assume that a string has heavymonopoles at the end and that a magnetic field is applied to force them tomove along a circle. In this case a solution of equations (16)-(17) describesthe motion of the string interior. In order to escape a discussion of the detailsof the motion of the end points we shall use the maximal extensions of thestring solutions, continuing them until they meet the surface where 2 = 0.Since the invariant I changes its sign at this surface, the minimal surfacedescribing the rigidly rotating string ceases to be timelike here. The endpoints of such a maximally extended string move with the velocity of lightalong this surface. In this paper, for brevity, we call such solutions open

strings. In what follows we restrict ourselves to the study of such openstrings and do not discuss how these solutions can arise when real forces areacting on the string or when a part of the string is involved in a rigid rotation.

Our assumption of rigidity implies that the coordinates and zof the endpoints of the string remain fixed. Under these conditions the end points of an

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open string are located on the surfaces where 2 = 0. In a flat spacetime

this timelike surface is a cylinder located at the radial distance 1

from theaxis of symmetry. In the general case we shall refer to the surfaces where2 = 0 as null cylinders. Note that if L2 2 vanishes at the same pointas 2 it is possible for the world sheet to pass through the surface 2 = 0and remain regular and timelike.

In order to find a rigidly rotating string configuration one needs to fixfunctions V(, z), (, z), and w(, z) that specify geometry. It is quiteinteresting (as was remarked by de Vega and Egusquiza [10]) that if themetric (1) allows a discrete symmetry z z, then equations (16) and (17)always have a special solution, namely a string configuration described bythe relations z = 0 and = const. De Vega and Egusquiza called these

straight rigidly rotating strings in axially-symmetric stationary spacetimesplanetoid solutions.

3 Rotating Strings in Flat Spacetime

Our main goal is a study of rigidly rotating strings in a spacetime of arotating black hole. But before considering this problem we make a fewremarks concerning rigidly rotating strings in a flat spacetime. We recoverthe Minkowski metric

ds

2

= dt2

+ d

2

+ dz

2

+

2

d (20)from (1) by setting the metric functions V = 1 and w = = 0. We also have2 = 22 1. Since the metric is independent of z one can integrate (17)once to reduce the equations of motion to the form

d

d= L (1

2 2)

(1 p2 + L2 2) 2 2 4 L2, (21)

dz

d= p

(1 p2 + L2 2) 2 2 4 L2, (22)

where p is a constant of integration. These equations can be solved analyti-

cally.In order for (21) and (22) to be real-valued, is constrained to lie in the

interval 0 < < + where the upper and lower bounds are given by, =

1

(B C)/2 , (23)

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where,

B = 1 p2 + L2 2, and C = B2 4 2 L2 . (24)The equations for z() and () can be integrated readily (the substitutionu = 2 reduces these to standard integrals) with solutions,

z() = p2

arcsin

B 2 2 2C

2

, (25)

() = 12

arcsin

B 2 2 L2C 2

+ L arcsinB 2 L2 2

C+

2(1

L) (26)

where for convenience we have chosen the initial conditions z() = () =0.

It is instructive to examine the special case where p = 0 further wherethe solutions are confined to the z = const plane. The solution (26) can berewritten in the form

() =

arctan k k1 arctan

, (27)

where k = (L)1 and = (2 L2)/(1 22).

In order for solutions to exist, the invariant I (L2

2

)/2

must benon-negative. Thus there are a number of cases to resolve. We know that thestring can end only on the null cylinder where 2 vanishes, i.e. = 1/. Wealso see that the string may have a turning point at = L. When L = 0 werecover the rigidly rotating straight strings of De Vega and Egusquiza [10].When L > 0 there are two cases;

1. L < 1/: the string lies in the region L < < 1/, has end-points at = 1/ and a turning point at = L (see Figure 1),

2. L > 1/: the string lies in the region L > > 1/. It has end-points

at = 1/ and a turning point at = L (see Figure 2).(The case L = 1/ is excluded since I < 0 and hence no solution exists.)

In the latter case the Killing vector is spacelike. Nonetheless the world-sheet is timelike; in fact the tangent vector x, is timelike in this region.

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However the solution lies in the region L > > 1/ and appears, by com-

paring t = const slices in non-rotating coordinates, to move at superluminalvelocities (except at the end points which move at the speed of light). Thisis in apparent contradiction with the observation that the world sheet istimelike.

The puzzle is clarified if we note that the apparent velocity of the stringin the surface t = const is not the physical velocity of the string. Recall thatthe Nambu-Goto action is invariant under world-sheet reparameterizations.This reparameterization can be used to generate a motion of the stringalong itself, which evidently is physically irrelevant. In other words onlyvelocity normal to the string world-sheet has physical meaning (see e.g. [1]).

Hence, on the t = const hypersurfaces, we must to consider the compo-

nent of the apparent string velocity normal to the string configuration. Thenormal component of the velocity is the physical component. The appar-ent three velocity, vi (i = 1, 2, 3), of the string in (,z,) coordinates asmeasured by a static observer at infinity is

vi = (0, 0, 1) (28)

and has magnitude v = .Its projection on the normal to the string ui in the (t = const) plane is

ui =

1 + 2

2

(

2, 0, 1), (29)

and the magnitude of this normal velocity is

u2 =22

1 + 2 2=

2(L2 2)2L2 1 . (30)

Thus we see that if 1/2 > 2 > L2 (case 1) then u2 < 1 as expected.Furthermore if L2 > 2 > 1/2 (the apparently superluminal case 2) wesee that u2 < 1 also. The physical velocity of the string is subluminal in allcases where the solution exists.

This phenomenon is evidently of a quite general nature. In order to

separate these two different types of rigid rotation of strings we will call themotion superluminal if they are tangent to with 2 > 0 and subluminalif the world-sheet is tangent to with 2 < 0.

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4 Rigidly Rotating Strings in the Kerr Space-

time

4.1 Equations of Motion

In Boyer-Lindquist coordinates the Kerr metric is given by:

ds2 =

dt a sin2 d

2+

dr2 + d2

+sin2

(r2 + a2)d adt

2, (31)

where = r2

2Mr +a2 and = r2+a2 cos2 . This spacetime is stationary

and axisymmetric with two Killing vectors: (t) = t and () =

.

The relationship between the Boyer-Linquist coordinate functions andthe Papapetrou coordinate functions is straightforward; the time coordinatet and the angular coordinate that appear in both the metric (1) and themetric (31) are simply identified and

2 = sin2 , z = (r M)cos . (32)In terms of the Boyer-Lindquist coordinates the Papapetrou metric functionsfor the Kerr metric are

V = a2

sin

2

, (33)

w =2Mra sin2

a2 sin2 , (34)

e2 = a2 sin2

cos2 + (r M)2 sin2 . (35)

We can now write down the string equations of motion for the Kerr space-time in the Boyer-Linquist coordinates. The string configuration is deter-mined by two functions (r) and (r) which satisfy

ddr

2=

GL

2

2 sin4 (36)

2G

d

dr

2

G

d

dr

=

cos

sin3 Z , (37)

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where

Z = L2 + b2

1 + (1 a22 sin2 )2

(38)

b2 = L2 sin2 , G = 2 sin2

b2

1 +

d

dr

2 . (39)

2 =sin2

a2 + 4Mra + 2((r2 + a2)2 a2 sin2 )

(40)

We were not able to solve this system analytically. Moreover one can

show (see Appendix) that the only case when the variables in the problemunder consideration can be separated by the Hamilton-Jacobi method (atleast in these coordinates) is when = 0, so that the problem reduces to theone considered in [6]. In what follows we consider the case when a rotatingstring is located in the equatorial plane, which allows more detailed analysis.

4.2 Rigidly Rotating Strings in the Equatorial Plane

For the motion of the string in the equatorial plane = /2 equation (37) issatisfied identically (both the left and right hand sides vanish) and equation(36) takes the form

d

dr

2=

L22

2(L2 ) . (41)

Here

2 =F

r, F = 2r3 + (a22 1)r + 2M(a 1)2 . (42)

Solutions exist only if the right-hand side of (41) is non-negative and hence(for L = 0)

I L2

2

0. (43)

The positivity of the invariant I also guarantees that the world-sheet of thestring is a regular timelike surface (cf. (19)).

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To simplify the analysis of the structure of the null cylinder surfaces

instead of r, a, L, and F we introduce the dimensionless variables

=r

M, =

a

M, =

L

M, = M , F=Mf. (44)

In these variables

2 =f

, f = 23 + (22 1) + 2( 1)2 . (45)

The surface where L2 vanishes corresponds, in general, to turningpoints of the string in r. Now L2 has only one zero outside the horizon,namely r0 = M0 with

0 = 1 + 1 + 2 2 . (46)For r > r0, L

2 < 0, and for r < r0, L2 > 0.Firstly we note that when L = 0 then we obtain the planetoid solution

of De Vega and Egusquiza [10]. In this case the solution is a rigidly rotatingstraight string with end points on the null cylinders r = r1 and r = r2 wherer2 > r1 are the zeroes of

2. Note that for a given if these zeroes donot exist (so that is spacelike everywhere) then there are no such rigidlyrotating straightstrings.

In the general case the endpoints of an open string must be locatedon the null cylinders where 2 = 0. Note one can think of the equation

f(,,) = 0 as a quadratic in for fixed and . The zeroes of (45), thatis the solutions of the equation f(,,) = 0, are

=2 2 2 + 2

3 + 2 + 22. (47)

They bound the interval () < < +() where is timelike at a givenradius r = M . We note that inside the horizon where 22 + 2 < 0, it isnot possible for to be timelike or null except within or on the inner Cauchyhorizon. Since we are interested in the motion of the strings in the black holeexterior from now on we restrict ourselves by solutions in the region > +

where+ = 1 +

1 2. (48)

Equation (47) shows that |r+ = BH /(2+ + 2) (BH/M is theangular velocity of the black hole). At large distances = 1/ reproduces

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flat space behavior. For fixed the function +(, ) has a maximum, and

the function (, ) has a minimum. The points of the extrema can bedefined as joint solutions of the equation f(,,) = 0 and the equationf/|(,) = 0. The latter equation implies that

= m , m =

1 223|| . (49)

A simultaneous solution of this relation and equation (47) defines a max-imal value max() of + and a minimal value min() of . We concludethat for a given value of the rotation parameter the Killing vector canbe timelike only if min() < < max(). If there exists a region where

the Killing vector is timelike, this region is inside interval 1 < < 2,and one has min() < 1 and max() > 2 (where 1(, ) < 2(, ) arethe zeroes of the polynomial f). Here min() and max() are given by (49)with = min() and = max(), respectively.

We can arrive at the same conclusion by slightly different reasoning whichwill allow us to make further simplifications. For fixed and the functionf(,,) is a cubic polynomial in . It tends to as andf(0) 0 (f(0) = 0 only if = 1). For || 1 it is monotonic, and hencealways positive at > 0. For || < 1 the function f() has a minimumat = m and a maximum at = m, where m is given by (49). At theminimum point, f takes the value

fm = f(m, ) = 2( 1)21 1

3

3||

(1 + )3

1

. (50)

The minimum value fm vanishes if the following equation is satisfied

(2 + 27)3 + (32 27)2 + 3 + 1 = 0 . (51)Solutions of this equations () are also solutions of the two equations f = 0and rf = 0, and hence they coincide with min() and max().

The numerical solution of equation (51) is shown in Figure 3. Line a

represents solution max and line b represents solution min. These linesbegin at = 0 at their Schwarzschild values 33/2 and reach values 1/2and 1/7 respectively for the extremely rotating black hole. Figure 4 showsthe corresponding radii max (curve a) and min (curve b) as the functions ofthe rotation parameter .

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The third branch c in Figure 3, which intersects a at = 1, corresponds

to the minimum values of inside the Cauchy horizon. Line d is the solutionof the equation = 1. For the values of the parameters in the () planelying in the region outside two shaded strips function f is positive for any > 0. For the values of the parameters inside two shaded strips function fhas two roots, 0 < 1 < 2, and f is negative for lying between the roots.The upper shaded region corresponds to the situation where the roots of flie within the inner Cauchy horizon. The lower shaded region is the areawhere the two roots of f lie outside the event horizon.

Having obtained this information on the structure of the null cylindersurfaces we now discuss the different types of motion of a rigidly rotatingstring in the equatorial plane of the Kerr spacetime. The simplest situation

clearly occurs when has no zeroes in the region + (where + isdefined by equation (48)). This happens for values of the parameters whichlie outside the shaded region restricted by lines a and b in the ( )-plane(see Figure 3). In this case there is only one allowed type of solution 0 > +corresponding to solutions in the region + 0. These configurationsbegin and end in the black hole and have a turning point at = 0 in itsexterior (the form of these solutions is qualitatively similar to that of thesolution shown in figure 6). Such a solution may describe a closed loop-likestring, part of which has been swallowed by a black hole. Centrifugal forcesconnected with the rotation allow the other part of the string to remain in

the black hole exterior.For the values of the parameters lying inside the shaded region restrictedby lines a and b in the ( )-plane the situation is somewhat more com-plicated. In this case 2 has two zeroes that we denote by 1 and 2(+ < 1 < 2). The Killing vector is timelike in the region 1 < < 2.Since the invariant I defined by (43) must be positive there are then a numberof different possibilities depending upon the choice of the angular momentumparameter 0.

1. 0 < 1. The invariant I is positive either if (a) 1 < < 2 or(b) < 0. In the former case f < 0 and the motion of the string is

subluminal, with the ends of the string at 1 and 2 (Figure 5). Inthe latter case the motion of the string is superluminal, the stringbegins and ends on the black hole and has a radial turning point 0 inthe black hole exterior (Figure 6).

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2. 1 < 0 < 2. The invariant I is positive either if (a) 0 < < 2

or (b) < 1. In the former case f < 0 and the motion of the stringis subluminal, with both ends of the string at 2 and 0 being radialturning points (two examples (with = 0.5 and = 2.0) are shown inFigure 7). In the latter case the motion of the string is superluminaland string begins at the horizon + and ends at 1 (Figure 8).

3. 2 < 0. The invariant I is positive either if (a) 2 < < 0 or (b) < 1. In both cases f > 0 and the motion is superluminal. In theformer case the ends of the string are at 2 and a radial turning pointis at 0 (an example of such a configuration with = 4.0 is given inFigure 7). In the latter case the string configurations are similar to the

one shown in Figure 8.

For subluminal motion the apparent velocity is less than the velocityof light, for superluminal motion the apparent velocity is greater thanthe velocity of light. In both cases the physical (orthogonal to the string)velocity is less than the velocity of light. We described the origin of thisphenomenon in Section 3. We recall that in the above analysis we restrictedourselves to the case of rotating strings. In the absence of rotation stationarystring configurations (both equatorial and off-equatorial) allow a completedescription (see [6]).

Besides these main categories of motion there are possible different bound-

ary cases when 0 coincides either with 1 or with 2. These cases requirespecial analysis. For special values of the parameters one might expect thata string passes through (and beyond) these points remaining regular andtimelike (see Figure 9 for example). A similar situation was analysed in [5]for special configurations of strings which pass through the event horizon.

Acknowledgements

The work of V.F. and J.P.D. was supported by NSERC, while the work byS.H. was supported by the International Council for Canadian Studies. The

authors would like to thank Arne Larsen for useful discussions.

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Appendix

In this appendix we analyze separability of the Hamilton-Jacobi equationsdescribing rigidly rotating strings in the Kerr spacetime. Since a rigidlyrotating string in the axisymmetric stationary spacetime provides the min-imum of the energy functional (10) and its configuration is a geodesic in athree-dimensional space with metric (11) one can use the following form ofthe Hamilton-Jacobi equation for defining such configurations

S

+

1

2hij

S

xiS

xj= 0 . (52)

Here hij is the metric inverse to hij given by (11).

For the special case of Minkowski spacetime, where V = 1, w = = 0,(and setting m = 1), the Hamilton-Jacobi equations are trivially separable

S = 12

m2 + L + pz+ A() , (53)

and function A() obeys the equation

dA()

d

2+ p2 +

L

2 1

1 2 2

= 0 . (54)

By solving these equations and varying the action S with respect to separa-

tion constants and m one gets the complete set of equations that is equivalentto (21) and (22).Turning now to the case of Kerr spacetime, it is easier to apply trans-

formation (3) directly to the Kerr metric (31) and repeat the steps outlinedabove. It is straightforward, again, to show that the components of thespatial metric are,

hrr = 1h , h = T

2 sin2 R2 , h = sin2 (55)where,

T() = 1

a sin2 , R(r) = a

(r2 + a2) . (56)

Assuming the separation of variables in this metric and working with anaction of the form,

S = 12

m2 + L + A(r) + B() (57)

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the Hamilton-Jacobi equation yields

dA(r)

dr

2

+

dB()

d

2

+L2 T2

sin2

L2 R2

m2

T2 R2 sin2

= 0. (58)

The L2 terms are effectively separated. Only the m2 term requires con-sideration. In order to complete the separation of variables, it is requiredthat,

m2 T2 R2 sin2

= K1(r) + K2(). (59)

Expanding the bracket, T2 R2 sin2

= a2 sin2 (60)

+sin2

a2 sin2

r2 + a22

+ 2 a

2 M r Q2

.

The first two terms are separated. Simple analysis shows that the last onecannot be. That is why in order to provide separability one must put = 0.The separation of variables for this case was studied earlier [6]. In the specialcase where the string is confined to the equatorial plane, the Hamilton-Jacobiequations are trivially separable since the function B() is eliminated from

the outset. It is then straightforward to obtain the results of section 4.2.

References

[1] A.Vilenkin and P.Shellard, Cosmic Strings and Other Topological De-fects, Cambridge University Press (1994).

[2] H.J. de Vega and N. Sanchez, Talk given at the 3rd Colloque Cos-mologique, (Paris, France 1995), hep-th/9512074.

[3] N.Sanchez and G.Veneziano, Nucl. Phys. B333, 253, (1990).

[4] S.Lonsdale and I.Moss, Nucl. Phys., B298, 693, (1988).

[5] V.P.Frolov, S.C.Hendy and A.L.Larsen, Phys. Rev. D54, 5093, (1996).

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[6] V.Frolov, V. Skarzhinski, A. Zelnikov and O.Heinrich, Phys. Lett.,

B224, 255, (1989).[7] B.Carter and V.Frolov, Class. Quant. Grav., 6:569 (1989).

[8] B.Carter, V.Frolov and O.Heinrich, Class. Quant. Grav., 8:135 (1991).

[9] R.M. Wald, General Relativity, University of Chicago Press, Chicago,(1984).

[10] H.J. de Vega and I.L. Egusquiza, Phys. Rev. D54, 7513 (1996).

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Figure Captions

Figure 1. String configurations in flat spacetime for p = 0 with L < 1/.Solid lines represent strings for 4 different values L = 0.05, 0.25, 0.5, and 0.9of the angular momentum. A dashed line is a null cylinder = 1/ (here = 1). The arrow in this, and subsequent, figures indicates the direction ofrotation of the strings.

Figure 2. String configurations in flat spacetime for p = 0 with L > 1/.Solid lines represent strings for 4 different values L = 1.1, 1.5, 1.75, and 1.95of the angular momentum. A dashed line is a null cylinder = 1/ (here = 1).

Figure 3. The roots of equation (51) are plotted in the plane(curves a, b and c) along with the curve = 1/ (curve d). The shadedregions correspond to parameter values where f has two positive roots; in theupper region between c and d these roots lie inside the inner Cauchy horizonof the black hole and in the lower region between a and b they lie outside theevent horizon of the black hole.

Figure 4. The functions max() (curve a) and min() (curve b) areplotted for 0 < < 1.

Figure 5. This and next figures illustrate qualitatively different typesof motion of rigidly rotating strings in the Kerr spacetime. In all figures theinner solid circular line is the event horizon +. The nearest to the horizondashed circle is = 1, and the outer dashed circle (if shown) is = 2.String configurations at the given moment of time are shown by solid lineswith the indication of the corresponding value of the angular momentumparameter. The present figure illustrates Case 1(a) and shows a typical pairof string configurations ( = 0.25, = 0.45) in the region 1 < < 2with no turning points. The strings have end-points at = 1 and = 2( = 0.5, = 0.05).

Figure 6. Case 1(b): A string configuration (t = const slice) in theregion < 0. The string has a turning point at = 0 ( = 0.5, = 0.06).

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Figure 7. Case 2(a): String configurations inside the region 0 < 2( = 0.5, = 2.0) with turning points at 0. Case 3(b): A typical stringconfiguration in the region 2 < 0 ( = 4.0) with a turning point at 0( = 0.5, = 0.05).

Figure 8. Case 2(b): A typical string configuration where < 1. Thestring is seen to spiral into the horizon = + from = 1 ( = 0.43, = 0.04).

Figure 9. A string configuration where 0 = 1. The string is seen topass through 1 and to spiral into the horizon = + ( = 0.43, = 0.04).

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0 0.2 0.4 0.6 0.8 1

X

-1

-0.5

0

0.5

1

Y

L=.05

L=0.25

L=0.5

L=0.9

Figure 1


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0 0.5 1 1.5

X

-1

-0.5

0

0.5

1

Y

L=1.95

L=1.75

L=1.5

L=1.1

Figure 2


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a

w

-1/7

1/2

1

0.2 0.4 0.6 0.8 1-1

0

1

2

3

a

b

Figure 3

c

d


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0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

a

b

Figure 4


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1 2 3 4 5 6 7

X

-3

-2

-1

0

1

2

3

Y

Figure 5

=0.45

=0.25


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-3 -2 -1 0 1 2 3

X

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

Y

Figure 6

= 0.5


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-4 -2 0 2 4 6 8

X

-4

-2

0

2

4

6

Y

Figure 7

=4.0

=0.5

=2.0


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-3 -2 -1 0 1 2 3

X

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Y

Figure 8

=1.0


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-4 -2 0 2 4

X

-6

-5

-4

-3

-2

-1

0

Y

Figure 9

=0.113

String Rotation

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