Low velocity gravitational capture by long cosmic strings

PHYSICAL REVIEW D, VOLUME 60, 023510

Low velocity gravitational capture by long cosmic strings

Don N. Page*CIAR Cosmology Program, Institute for Theoretical Physics, Department of Physics, University of Alberta,

Edmonton, Alberta, Canada T6G 2J1~Received 8 February 1999; published 14 June 1999!

Coupled ordinary differential equations are derived for the distant gravitational interaction of a compactobject of massM and chargeQ with an initially straight, infinitely long, cosmic string of tensionm!1/G[1, when the relative velocities are very low compared to the speed of lightc[1. @An intermediate result ofthis derivation is that any localized forceF(t) on the string that is confined to a single plane perpendicular tothe initial string configuration gives the intersection of the string with this plane—the point where the force isapplied—the velocityF(t)/(2m).# The coupled equations are then used to calculate the critical impact param-eter bcrit(v0) for marginal gravitational capture as a function of the incident velocityv0. For v0!(12Q2/M2)1/3m2/3, so that the string acts relatively stiffly,bcrit'(p/4)@12m3(12Q2/M2)4#1/5Mv0

27/5

1(p/10)@(54/m)(12Q2/M2)2#1/5Mv021/51O(Mv0 /m). For (12Q2/M2)1/3m2/3!v0!12Q2/M2, so that

the string acts essentially as a test string that stays nearly straight,bcrit'@(p/2)(12Q2/M2)#1/2Mv021/2

1(p/4)(12Q2/M2)mMv0221O(Mm2v0

27/2)1O(M ). Between these two limits the critical impact parameteris found numerically to fit a simple algebraic combination of these two formulas to better than 99.5% accuracy.@S0556-2821~99!01014-0#

PACS number~s!: 98.80.Cq

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I. INTRODUCTION

The gravitational interactions between compact objeand long cosmic strings have been calculated in severacent papers@1–4# in the test-string approximation, in whicone assumes that the dimensionless string tensionm ~usingunits in which Newton’s gravitational constant isG51) ismuch smaller than any other parameter whose ratio withm isrelevant. Then the string obeys the Nambu-Goto equationmotion @5–7# in the gravitational field of the object and hanegligible back reaction on the motion of the object.

It is indeed a good approximation for cosmic strings thm!1, but I shall show that at low initial relative velocitiev0 and near the critical impact parameter, the ratiom/v0

3/2 isimportant, so the test-string approximation is accurate onthe velocity is large in comparison tom2/3. For velocitiesvery low compared with this, the string acts relatively stifflso that the motion of the object responds more than thetion of the string.

Here I shall derive coupled ordinary differential equatioof motion for an initially straight, infinitely long, cosmicstring interacting gravitationally with a compact objectmassM and chargeQ ~i.e., one whose size is negligible icomparison with the distance between the object andstring!, using the approximationsm!1 andv0!1 but allow-ing m/v0

3/2 to be arbitrary. The low velocity allows the part othe string nearest the object to remain nearly straight,then the forceF on the string is a simple function of thstring-object separation and relative velocity. On a mularger scale this force is nearly localized and hence givespoint on the string nearest the object the velocityF/(2m), aswe shall see.

*Email address: [email protected]

0556-2821/99/60~2!/023510~16!/$15.00 60 0235

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For fixedM, Q, andm, after taking out Poincare transformations there is a two-parameter set of solutions that cancharacterized by the initial speedv0 of the object relative tothe string~when they are infinitely far away in the infinitpast! and by the impact parameterb ~how far the objectwould miss the string if both moved uniformly with no interactions!. For eachv0 there is a critical impact parametebcrit , such that ifb,bcrit the object becomes gravitationallcaptured and bound to the string, but ifb.bcrit , the objectand string scatter and asymptotically move freely of eaother. In this paper the coupled equations of motion ofobject and the string will be solved to give the functiobcrit(v0), analytically when the ratiom/v0

3/2 is either verylarge or very small, and numerically otherwise. In the latcase simple algebraic combinations of the analytic resultthe two extremes reproduce the numerical results overentire range to within 1%, or even within 0.43% if two arbtrary exponents are chosen appropriately.

II. GRAVITATIONAL FORCE BETWEEN AN OBJECTAND A STRAIGHT STRING

Strictly speaking, the gravitational interaction betweenobject and a string should be done in terms of curved spatime. However, whenm!1 and when one is in the weakfield regime of the object~which is where the string is, undethe assumption that the object size is much smaller thandistance to the string!, one can do an analysis in termsfictitious forces acting in a fictitious flat background spactime, just as one does for Newtonian gravity~though oneneed not assume that the gravitational field of the stringNewtonian or that the object velocity is low to do this!.

We assume that the string is infinitely long and is neastraight in the region where it is fairly near its minimuseparation from the object, with no oscillations comingalong the string from infinity. Given the assumed initi

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DON N. PAGE PHYSICAL REVIEW D 60 023510

straight, static configuration of the string, this assumptionapproximate straightness of the part of the string nearesobject for all times is justified by the assumption of lotransverse velocities, since the fact that the disturbancethe string propagate up and down it, away from their souat the speed of light, means that the length of string thadisturbed is much greater than the transverse disturbancthat the angular bending of the string is small, even iflinear bending becomes large.

Then we can go to the frame in which the part of tstring nearest the object is at rest along thez-axis and theobject is in thex-y plane with velocity of magnitudev r inthe positivex-direction, the relative velocity of the objecwith respect to the piece of string nearest the object.linear order inM andQ, the gravitational force of the stringon the object is zero whenv r50. However, the string produces a conical spacetime with a deficit angle 2p(4m) @8#,so when this is flattened out in the fictitious flat backgrouspacetime with azimuthal anglew that ranges from 0 to 2p,a moving object appears to be bending toward the strinthe rate of 4m times the rate at whichw is changing, i.e., atthe rate of 4mw54mv ry/r 2, where

r 5Ax21y2 ~1!

is the distance from the object to the string~ignoring relativecorrections of orderm that depend on how the conical spactime is flattened out to produce the fictitious flat spacetim!.

When this bending rate is multiplied by the momentup5Mg rv r of the object, whereg r51/A12v r

2, one gets thepart of the transverse fictitious force on the object thalinear in M, 4mMg rv r

2y/r 2. The longitudinal fictitious force~in the direction of the object velocityvr in the instantaneousframe of the piece of string closest to the object, using boface symbols to denote spatial vectors in the twdimensional plane perpendicular to the string in the fictitioflat spacetime! that is linear inM depends on the flatteninprocedure used to generate the fictitious flat metric. Onechoose it so that the total fictitious force on the object thalinear in M is

Flinear524mMg rv r2r /r 2. ~2!

By Newton’s third law or conservation of momentum, thewill be an equal and opposite fictitious force on the stringdescribe its motion in the fictitious flat spacetime.

To explain and justify this fictitious force in greater detaconsider the geodesic motion of a test massM in the conicalspacetime of a cosmic string@8#:

ds252dt21dz21dr21~124m!2r2dw2

52dt21dz21r 28m~dr21r 2dw2!

52dt21dz21e2f~dr21r 2dw2!, ~3!

where

r 5@~124m!r#1/(124m) ~4!

and

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~not to be confused with the azimuthal anglew).One can go to the frame in which the object moves in

planez50, in which case the motion is in an ultrastatic (11)-dimensional spacetime with conformally flat spatsections, whose general metric form is

ds252dt21e2fd i j dxidxj . ~6!

Timelike geodesics of such metrics have constant

E5Mdt

dt~7!

and constant

P25M2e2fd i j

dxi

dt

dxj

dt5E22M2. ~8!

One can also readily show that the spacetime geodequation implies that the spatial trajectory is itself a geodeof the spatial metrice2fd i j dxidxj and obeys the equation

d2xi

ds25S d i j 2dxi

ds

dxj

ds Df , j , ~9!

wheres is the spatial distance along the trajectory,

ds25d i j dxidxj , ~10!

in the fictitious flat spacetime metric

ds[252dt[

21d i j dxidxj . ~11!

One can reproduce the spatial geodesic ofconformally-flat spatial metrice2fd i j dxidxj by a fictitiousspatial force

F5dp

dt[5Mg rv r

2“f1 f vr ~12!

in the fictitious flat spacetime, where

p5Mg rvr5Mg rdr /dt[ ~13!

is the fictitious flat spacetime momentum of the particlemassM, of flat spacetime position vectorr ~with componentsxi that are raised and lowered by the fictitious flat spametric d i j dxidxj in this fictitious force analysis!, of spatialvelocity vr5dr /dt[ with magnitudev r5Avr•vr5Ad i j v r

i v rj ,

and ofg r51/A12v r2.

In the expression above for the fictitious forceF, one canhave an arbitrary functionf multiplying the velocityvr if onejust wants to reproduce the spatial trajectory of the geodof the actual metric~6!. If one also wants to reproduce thtime dependence of the trajectory, soxi(t[) under the ficti-tious force in the fictitious metric has the same dependeon t[ asxi(t) does ont for the geodesic in the actual metricthen one needs

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LOW VELOCITY GRAVITATIONAL CAPTURE BY LONG . . . PHYSICAL REVIEW D 60 023510

f 52Mg r~g r211!vr•“f. ~14!

However, for the purposes of this paper, the precise tdependence of the trajectory of an object passing a cosstring is not so important as the trajectory itself, so I shallf 50. Then if one uses the form off given by Eq.~5! for thecosmic string, one gets the fictitious force given by Eq.~2!.

To quadratic order inM andQ, Smith@9# calculated fromthe effect of the deficit angle on the three-dimensionalplacian for the Newtonian gravitational and electrostafields of the object that the string exerts an attractive force~ifM2.Q2) of magnitudepm(M22Q2)/(4r 2) on the objectwhen the relative velocityv r can be neglected.~Whenv r isnot negligible in comparison with unity, the quadratic foris overwhelmed by the linear force, so one does not needhigh-velocity corrections to the quadratic force.! In two-dimensional vectorial form,

Fquadratic52pm~M22Q2!r /~4r 3!. ~15!

The conservation of momentum implies that the objexerts an equal but opposite force on the string, andindeed gives precisely the bending of a distant static strinthe Reissner-Nordstrom metric@10#.

When we add the fictitious forces that are linear and qdratic inM andQ, we get a total fictitious force on the objethat, in the limit of a small dimensionless string tensionm!1 and in the limit of a large separationr @M , is approxi-mately

FM524mMg rv r

2r

r 22

pm~M22Q2!r

4r 3. ~16!

The first term is valid for all velocitiesvr but neglects addi-tional terms in the direction ofvr , and the second termwhich is only important if the velocity is very small, negleccorrections whenv r is not small.

III. MOTION OF A LONG STRINGWITH A LOCALIZED FORCE

In the fictitious flat spacetime, the response of the copact object to the fictitious force is simply the relativistversion of Newton’s second law, the first equality of E~12!. However, the response of the string to the equalopposite fictitious force needs to be worked out.

As noted above, the low-transverse-velocity assumpmeans that the length of string that is bent~which develops atthe speed of light! is much greater than the transverse being ~which develops at the transverse velocity of the strimuch less than the speed of light by assumption!. Hence, onecan look at the bending on a scale much larger than thathe transverse separation between the string and the oand yet much smaller than that of the length of string thabent. On this scale, the force on the string appears tolocalized at the position on the string nearest the objectis given to good approximation by Eq.~16! in terms of thestring-object transverse separationr and the relative velocityv r .

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Let us do a general analysis of the effect of such a locized force on an initially static, straight cosmic string, witout, in the general analysis, assuming that the transverselocity of the string is small, even though in our particulapplication we need this assumption to justify our assumtion that the force is effectively local and is given by E~16!.

Suppose a time-dependent horizontal force in thex-yplane is applied atz50 to an infinite string initially lying atrest along thez-axis ~vertically!, with no other forces on it.This force causes equal horizontal disturbances to go up~forz.0) and down~for z,0) the string at the speed of ligh~propagating away from where they are generated bytime-dependent force, atz50; assuming that the infinitestring was initially at rest and has no forces on it atzÞ0implies that no disturbances are coming in toz50):

r s~ t,z!5u~z!r'~ t2z!1u~2z!r'~ t1z!, ~17!

whereu(z) is the Heaviside step function ofz and each ofthe two occurrences ofr' represents the same function of thargument inside the following bracket@either t2z for thefirst term, which is nonzero forz.0 whereu(z)51 andu(2z)50, or t1z for the second term, which is nonzero foz,0 whereu(2z)51 andu(z)50#.

Focus on the infinitesimal piece of string just abovez50 at some arbitrary timet. Its tilt or slope away fromvertical is

]r s~ t,z!

]z52

]r s~ t,z!

]t52 r' , ~18!

where the overdot in the last term denotes a derivative wrespect to the unlisted argumentt2z evaluated atz50. De-note the magnitude of this slope by

vs[u r'u[Ar'• r'5tanc, ~19!

wherec is the instantaneous angle of the tilt from the vercal.

This piece of string can be considered to be movstraight upward at the speed of light~unity!, the velocity atwhich the pattern moves forz.0, but locally any componenof the velocity parallel to the string has no effect. To calclate the energy and momentum carried by a piece of strwhat is relevant is the transverse velocity, the componperpendicular to the string. Here the infinitesimal piecestring is tilted at an anglec from the vertical, so the transverse velocity is tilted at an anglep/22c from the verticaldirection along which the pattern speed is unity. Therefothe magnitude of the transverse velocity is

v t5sinc5vs

A11vs2

, ~20!

and the associated relativisticg factor is

g t51

A12v t2

5secc5A11vs2. ~21!

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The length of this infinitesimal piece of string is

dl5~secc!dz5A11vs2dz, ~22!

and its energy is

dE5mg tdl5m~sec2c!dz5m~11vs2!dz. ~23!

The transverse momentum~at the anglec from the horizon-tal x-y plane! is the transverse velocity times the energwith magnitude

dpt5v tdE5m~sinc sec2c!dz5mvsA11vs2dz. ~24!

The vertical component of this momentum is

dpz5sincdpt5m~ tan2c!dz5mvs2dz, ~25!

and the horizontal component has magnitude

dph5coscdpt5m~ tanc!dz5mvsdz ~26!

and is in the direction ofr' .This infinitesimal piece of the displaced-string pattern t

is moving upward at the speed of light from the force az50 takes a timedt5dz to be generated. A mirror-imagpattern is moving downward at the speed of light from tforce atz50, and during the same infinitesimal time an ifinitesimal piece of string with the same magnitude ofdz isgenerated moving downward just belowz50. The verticalcomponents of these two pieces are opposite and hencecel, but the horizontal components are in the same direcand so add to make the two infinitesimal pieces of strgenerated during the timedt have a total momentum of

dp52m r'dt52mdr'52mdr s , ~27!

where here and henceforth I shall user s for r s(t,0) atz50,the horizontal position of the string in thex-y plane wherethe force is applied.

Since the string was assumed to start off from rest althe z-axis, at r s(t,z)50 for large negativet, the total mo-mentum of the string is simply

ps52mr s , ~28!

and the force on the string is

Fs5dps

dt52m r s . ~29!

The string will react back with an equal and opposforce on whatever is forcing it to move horizontally. Itinteresting that this reaction force is precisely an idealizfriction force directly proportional to velocity. The fact thamomentum is carried away along the string at the speelight, never to return~since the string is assumed to be innite!, is apparently what allows an initially straight, infinitelong string to act as a perfect source of friction.

One application of Eq.~28! is to give a very simple deri-vation of the net displacement of an initially straight stri

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by a gravitating object of massM that passes by with largeimpact parameterb@M /v0

2, so that the fictitious force termlinear in M, the first term of Eq.~16!, dominates over theforce term quadratic inM andQ, the second term of Eq.~16!.Then the object trajectory bends by half the deficit angle, iby an angle 4pm!1, so its momentum changes by

Dp54pmp54pmMg0v0 . ~30!

By Newton’s third law, the string’s momentum changesthe opposite amount, so Eq.~28! implies that the string getsdisplaced by

Dr 5Dp/~2m!52pMg0v0 , ~31!

just as De Villiers and Frolov found@2#.If one inverts Eq.~29! to get

r s5Fs

2m, ~32!

one notices the curious fact that apparently a force larthan twice the string tensionm can move a string faster thathe speed of light, seemingly a violation of special relatividespite the fact that a completely~special! relativistic analy-sis was used in the derivation of the relationship betwehorizontal force and velocity. However, one can make tcomments about this paradox:

First, for all finite horizontal speedsvs of the string, even‘‘superluminal’’ vs.1, the physical transverse speedsv t ofthe string, given by Eq.~20! above, are less than the speedlight. The horizontal speedvs of the kink in the string atz50 where the force is applied is somewhat analogous tospeed of the point where the two blades of a pair of scissintersect. Even though the two blades are constrainedmove slower than the speed of light, there is nothing in scial relativity that constrains the intersection point to moslower than light.

One might object that the scissors are more nearly angous to a pair of intersecting strings both moving in the saplane, whose intersection point can indeed move faster tlight ~at least if one assumes no interactions at the interstion, so that the two strings each move freely!. In this case itseems to be the prior existence of each of the two stringsboth sides of the intersection that allows the intersectionmove faster than light while neither string does so. Howevfor the single string being considered in this paper, whokink at z50 moves horizontally faster than light if the forcis greater than twice the string tension, there is no pexisting string beyond the kink, so it seems that somethphysical must be moving faster than light.

The partial answer to this worry is that for a horizontforce greater than 2m, string is being created faster thalight, but it is not being moved faster than light~in the trans-verse direction, the only motion that has a physical meanfor a longitudinal-boost-invariant string of the type beinconsidered here!. This is somewhat analogous to the possibnearly-instantaneous freezing of a lake, in which the waice boundary can in principle move faster than the speedlight as ice is created~but not moved! faster than light.

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Another analogy would be electron-positron pair creatin a hypothetical imploding cylindrical electromagnetic wathat builds up a huge longitudinal electric field, say betwetwo capacitor plates at the ends of the cylinder. The pcreation can in principle discharge the plates much fathan the time taken for light to go from one plate to the othIf one draws the Feynman diagrams of the pair creationannihilation that accomplish this process, with positrons ptured as electrons moving backward in time, one canelectron lines that zigzag forward and backward betweencreations and annihilations and have a net spatial mofaster than light between the two plates. One might call ta tachyonic lightning bolt. And yet there would be no violtion of special relativity with signals traveling faster thalight in this hypothetical process, since it would be the iploding electromagnetic wave that locally causes the pcess, with no causes traveling faster than the speed of l

Considering the creation of string faster than the speelight leads on to the second comment, which is that althoa string with a kink having a horizontal velocityvs.1 doesnot seem to violate the principles of causality in special retivity ~signals not traveling faster than light!, it does seem tobe associated with an instability analogous to tachyonichavior. The example of the nearly-instantaneous freezinga lake illustrates the instability of the liquid phase of waterlow temperatures, and the example of the pair creation inimploding electromagnetic field illustrates the instabilitythe zero-current ‘‘vacuum’’ of the Dirac electron-positrofield in the presence of a strong electric field. Similarly,chyonic negative mass-squared terms in relativistic wequations, such as the Klein-Gordon equation, do not leaviolations of causality~since disturbances do not propagafaster than light whatever the sign of the mass-squared te!but instead lead to instabilities of exponential growth.

In the case of applying a strong force to a string, this clead to an instability in the production of strings. For eample, suppose the force is that of a gauge field on a mless charged particle attached to the string at the kinkz50. ~If the particle were massive, its inertia would prevethe kink from moving faster than light, so I shall assumecontrary to continue the argument.! But then if the forcewere greater thanm, the energy that a pair of oppositecharged massless particles extract from the gauge fielbeing created and separated would be greater than the enneeded to create the string joining them. This would leadthe rapid pair production of pieces of string with these opositely charged gauge particles at the ends in the strgauge field, an instability that would presumably rapidly dcharge the gauge field until the force on a charged partwould be no greater thanm.

Therefore, if the force is that of a stable gauge field aplied to a massless charged particle on the string, it appthat the maximum force is the string tensionm, and thereforethe maximum horizontal velocity of the string kink where tparticle is attached is one-half the speed of light~relative tothe distant parts of the string that are still at rest alongz-axis according to the assumption that the entire stringoriginally at rest along thez-axis and that the force is applieonly at z50).

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IV. COUPLED MOTION OF A SLOWLY MOVINGCOMPACT OBJECT AND A NEARLY

STRAIGHT INFINITELY LONG STRING

Now let us combine the expression~16! for the force be-tween a compact object and a distant nearly straight lstring with Newton’s second law for the response of the oject and with Eq.~32! for the response of the string, when threlative velocities are all small compared with the speedlight, so that Eq.~16! indeed applies for giving the force anso that the force is effectively localized~relative to the lengthof string that is bent! in order for Eq.~32! to apply as well.Use a fictitious flat spacetime metric with Cartesian coornates chosen so that the string is initially at rest alongz-axis and the compact object moves in thex-y plane (z50), and use boldface letters to denote vectors in this pla~The same letters when not bold will denote the magnituof the corresponding boldface vectors.! In particular, letr Mdenote the position of the compact object~with massM andchargeQ) relative to the origin (x,y)5(0,0), let r s denotethe position of the string atz50 relative to the origin, and le

r5r M2r s ~33!

denote the position of the compact object relative tostring.

Then the nonrelativistic coupled equations of motionthe compact object and the string become

M r M522m r s52pm~M22Q2!r

4r 32

4mM ~ r• r !r

r 2.

~34!

It is now convenient to use the unit of time

Tu[M

2m~35!

and the unit of length

Lu[F p

16mS 12Q2

M2D G1/3

M , ~36!

which we henceforth define to be unity~unless otherwisespecified!. In terms of these units the speed of light is nlonger unity but is

c5Fp2 m2S 12Q2

M2D G21/3Lu

Tu5Fp2 m2S 12

Q2

M2D G21/3

@1.

~37!

SettingTu5Lu51, Eqs.~34! become

r M52 r s52r

r 32

4m~ r• r !r

r 2. ~38!

At an arbitrary timet i , the freely specifiable initial con-ditions arer s(t i), r M(t i), and r M(t i) in the two-dimensionalx-y plane, a total of six parameters. If one uses the Euclidinvariance of thex-y plane to take out translations and rot

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tions, one is left with three Euclidean-invariant initial condtions, for example the object-string separation distancer, theobject speed

v[u r Mu[Ar M• r M, ~39!

and the anglec between the string-object separation vectorand the object velocity vector

v[ r M , ~40!

defined so that

r•v5rv cosc. ~41!

Instead ofc, it is sometimes convenient below to use

C[cosc5r•v

rv~42!

or

X[sinc

25A12C

25Arv2r•v

2rv. ~43!

Note that thev given by Eq.~39! as the speed of theobject in the frame in which the string was initially at resand used with this meaning henceforth in this paper, isthe same as thev r used in Sec. II, e.g., in Eq.~16!, whichdenotedu r u, the instantaneous relative speed of the objectthe string, and it is also not the same as thevs used in Sec.III, which by Eq. ~19! denoted what we are now callingu r su,the speed of the point on the string nearest the object, inframe in which the string was initially at rest.

If one takes out the time-translation invariance, one iswith only two initial conditions, which can be the initiaspeedv0 at t52` and the impact parameterb5r sinc att52`, wherer is infinite andc is p. There Eqs.~38! implythat r s50, and one can choose the initial string position tor s50 ~the string initially at rest atxs5ys50) and the objectto be coming in fromxM52` with yM5b and with veloc-ity v0 in the positive x-direction initially at t52`. These initial conditions and the first of Eqs.~38!imply that

r s5v02 r M5v02v. ~44!

Inserting this into the second of Eqs.~38! gives a singlesecond-order differential equation for the object vector potion r M(t), although the second term in the rightmost exprsion of Eqs.~38! makes this single equation nonlinear inr M .

As the object comes in from infinity, it will be pulledtoward the string, so that at first it will speed up and bedownward, andyM will decrease below its initial value ofb.Eventually the object will either be captured by the string~sothat r andv tend to zero!, or it scatter so thatr increases toinfinity again, depending on the initial velocityv0 and theimpact parameterb. For eachv0, the boundary between thestwo qualitatively different asymptotic behaviors is given

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d

the critical impact parameterbcrit(v0): for b,bcrit(v0) theobject is captured, and forb.bcrit(v0) the object scatters tor 5` with v asymptotically greater than zero. Forb5bcrit(v0), the marginal capture case,r reaches a minimumand then slowly increases back to, but with v asymptoti-cally decreasing toward zero, so that the object essenthas precisely the escape velocity from the string.

The main goal of this paper, besides deriving the coupequations of motion~34! or ~38! given above, is to calculatethe critical impact parameterbcrit(v0) as a function of theinitial velocity v0 when this velocity is low compared withthe speed of light. I shall derive one approximate formuwhen Lu /Tu!v0!c, the regime in which the string actessentially as a test string at the critical impact paramete~atleast untilr reaches its minimum value; during the late stagwhen r slowly increases back to , the back reaction of thestring on the object becomes important!, and another ap-proximate formula whenv0!Lu /Tu!c, the regime in whichthe string acts stiffly and is only slightly perturbed by thforce from the object~though this small perturbation is essential for causing the object to lose enough energy tocome bound!.

In this paper I shall focus on the case in which the ratiothe second term to the first term on the extreme right hside of Eqs.~38!, 4mr ( r• r ), is much less than unity, so wshall omit the last term of Eqs.~38!. For the dominant part ofthe scattering,r is of the order of magnitude of the impacparameterb, and r• r is of the order of magnitude of thesquare of the initial velocityv0, so this ratio is of the order omagnitude of mbv0

2 ~in units with Tu5Lu51). For v0

!Lu /Tu , we shall find below thatbcrit}v027/5, so for very

small v0, certainly mbv02!1 for the case of marginal cap

ture. Forv0 of order unity (Lu /Tu), the truncated Eqs.~38!~with the last term omitted! have no small or large parameters, so we expect that thenbcrit is also of order unity (Lu),giving mbcritv0

2;m!1. For v0@1, bcrit'2v021/2, so

mbcritv02;mv0

3/2. This is also small compared to unity whev0!m22/3;(12Q2/M2)1/3c.

Therefore, for calculating the critical impact paramebcrit(v0), it appears to be a good approximation to neglectlast term of Eqs.~38! for velocities low compared with thespeed of light times the cube root of (12Q2/M2). Unlessthe object is significantly charged~e.g., nearly extremelycharged if the object is a black hole!, the limitation of con-sidering only velocities low compared with the speed of ligis sufficient for dropping the last term of Eqs.~38! when onecalculatesbcrit(v0). It is apparently not sufficient when (12Q2/M2)1/3c!v0!c, but the calculation ofbcrit(v0) in thisregime for nearly extremely charged objects will be left asexercise for the reader.

When we drop the last term of Eqs.~38!, use Eq.~44! tosolve the first part of these equations, and define

R5r M2v0 , ~45!

then what remains of Eqs.~38! becomes the single secondorder vector equation

0-6

r

s

n

a

bitr

-

icity

get

ut

ua-s


R52R1R

uR1Ru3. ~46!

In terms ofR, one has

r M5v01R, ~47!

r s5v02R, ~48!

r5R1R, ~49!

v[ r M5R. ~50!

Alternatively, Eq.~46! can be written as two first-ordevector equations:

r5v2r

r 3, ~51!

v52r

r 3. ~52!

If we instead go to the three Euclidean-invariant quantitier,v, and either C5cosc5r•v/(rv) or X5sin(c/2)5A(12C)/2, one gets the three first-order scalar equatio

r 5Cv21

r 2 , ~53!

v52C

r 2 , ~54!

and either

C5~12C2!S vr

21

r 2v D ~55!

or

X52X~12X2!S vr

21

r 2v D . ~56!

Another choice of variables is motivated by the fact thin the stiff-string regime, wherer 2v@1, the equations areapproximately those of a unit-mass Keplerian particle oring a stationary string that exerts a unit inverse-square foon the particle,r'v andv52r /r 3. For such a particle, conserved quantities are the energy,

E[1

2v22

1

r, ~57!

the angular momentum,

L[rv sinc5rvA12C252rvXA12X2, ~58!

one-eighth the cube of the angular momentum,

02351

s

t

-ce

h[1

8L35r 3v3~X22X4!3/2, ~59!

the eccentricity of the orbit,

e[A112EL2, ~60!

and one-fourth the excess of the square of the eccentrover unity,

g[1

4~e221!

51

2EL25~r 2v422rv2!X2~12X2!

5@~rv221!221#~X22X4!. ~61!

Only two of these quantities are independent, and tofairly simple differential equations below, I shall focus ongandh. In terms of them andX22X45(1/4)sin2 c, one mayreadily solve algebraically forr andv:

r 5h2/3

AX22X4~AX22X41AX22X41g!, ~62!

v5h21/3~AX22X41AX22X41g!. ~63!

Because the string is in actuality not completely stiff bmoves toward the object, the quantitiesE, L, h, e, andg arenot precisely conserved in the coupled object-string eqtions but instead are all decreasing with time, at the rate

E521

r 4 , ~64!

L52L

r 3 , ~65!

h523h

r 3 523h21~X22X4!3/2~AX22X4

1AX22X41g!3, ~66!

e52L2

er4 ~rv221!52v3

eC, ~67!

g52L2

2r 4 ~rv221!521

2v3C52v3XX

522h22~X22X4!3/2AX22X41g~AX22X4

1AX22X41g!4. ~68!

Now we can insert Eq.~62! and Eq.~63! into Eq. ~56! toget

X52~X22X4!3/2

hXAX22X41g~AX22X41AX22X41g!.

~69!

0-7

rc

n

-o

r

a

l,

se

neis

e

e

-

er

sessthe

e-

he

alt


Then dividing g and h by X gives two coupled first-ordedifferential equations for the coupled motion of the objeand the string:

dg

dX5

2

hX~AX22X41AX22X41g!3, ~70!

dh

dX5

3X~AX22X41AX22X41g!2

AX22X41g. ~71!

At t52`, one has the object approaching the string, aso the anglec between the object velocityv and theinfinitely-long string-object separation vectorr is p, givingX5sin(c/2)51 initially. The energy is thenE05v0

2/2, andthe angular momentum is thenL05bv0, so

g~X51!51

2E0L0

251

4b2v0

4 , ~72!

h~X51!51

8L0

351

8b3v0

3 . ~73!

Then asX decreases, so dog andh. If the object scattersand remains unbound, the energy and henceg remain posi-tive as the object returns to infinitely larger, but now withthe object moving away from the string, sov and r asymp-totically approach the same direction, givingc50 and henceX50. If the object becomes bound to the string,E5v2/221/r becomes negative, and hence alsog. Equation ~71!becomes singular whereAX22X41g goes to zero, but onecan switch back to Eqs.~53!–~55! or ~56! to continue thecalculation, asX andAX22X41g reverse sign and the evolution continues, withX oscillating as the object spirals in tjoin the string after a finite time.

In the intermediate case in which the impact parametebhas the critical valuebcrit(v0), E and g tend to zero as theobject slowly moves to infinite separationr with essentiallyjust the escape velocity. In this case, like the scattering cin which g stays positive, the object velocityv and the string-object separation vectorr asymptotically become parallegiving X50 at t5`.

It is now easy to calculate a one-parameter family ofv0’sand their correspondingbcrit(v0)’s using Eqs.~70! and ~71!,without requiring a shooting method from initial conditionat t52` or X51. Instead, one evolves backward in timfrom t51` or X50, where one setsg(X50)50 but uses

h[h~X50!51

8L3~ t5`! ~74!

as the single free parameter for determining the oparameter family of critical solutions. Each solutionevolved to X51, where one evaluatesg(h,X51) andh(h,X51) and then inverts Eqs.~72! and ~73! to get

v0~h!5g~h,X51!1/2h~h,X51!21/3, ~75!

bcrit„v0~h!…52g~h,X51!21/2h~h,X51!2/3. ~76!

02351

t

d

se

-

V. THE CRITICAL IMPACT PARAMETERIN THE STIFF-STRING REGIME, v0!L u /Tu

Now we shall find approximate analytic solutions of thevolution Eqs.~70! and ~71! in the marginally bound case(g(X50)50 or E(t5`)50) in the two limiting cases inwhich h5h(X50)5L3(t5`)/8 is either very large or verysmall.

For h@1, the factor ofh in the denominator of Eq.~70!means that to lowest orderg(X) stays very small and can bneglected on the right hand sides of Eqs.~70! and~71!. Thenone gets

g~X!'1

64h~12c28 sin 2c1sin 4c!

51

8h@3 sin21X1~2322X2

124X4216X6!AX22X4#, ~77!

h~X!'h13

8~2c2sin 2c!

5h13

2@sin21X1~2112X2!AX22X4#. ~78!

This is sufficiently accurate to give

h~h,X51!5hF113p

4h1OS 1

h2D G . ~79!

However, to getg(h,X51) with a relative error of onlyO(h22) rather thanO(h21), one needs to insert the approximate expressions~77! and~78! into the right hand sideof Eq. ~70! and evaluate it through the next-to-lowest ordin h21. This gives

g~h,X51!53p

16hF110

h1OS 1

h2D G . ~80!

If we insert these expressions into Eqs.~75! and ~76! toget v0(h) andbcrit(h) and then eliminateh to getbcrit(v0),we find that

bcrit5~6p!1/5v027/51

1

5~6p!3/5v0

21/51O~v0!. ~81!

Large h gives smallv0'A3p/(4h5/6), so Eq.~81! appliesfor v0!Lu /Tu[1. At the critical impact parameter, this ithe stiff-string regime, where the string responds much lthan the object to the force between them. If one restoresunitsTu andLu given by Eqs.~35! and~36!, one obtains theformula given in the Abstract above for the stiff-string rgime.

One can give an order-of-magnitude derivation of tcritical impact parameter to show that it indeed goes asv0

27/5

for v0!1. What happens is that the object comes in onslightly hyperbolic, but nearly parabolic, orbit with initiaenergyE05v0

2/2 just slightly positive. Then as the objec

0-8

tojeb

y

c

n

ra

he

it

e.

e

re

b

ingtu

t-thm

allin

e-bedonhe

lly

onely

t iser

theimeinm-

hatob-y.nga

theentn-estthelyinge itwerelynd

sed

topob-

se

fi-the

p-

rm


swings around the string, the small motion of the stringward the object drains just enough energy from the obthat it goes back out on an asymptotically parabolic orwith E approaching zero.

In this analysis, use' for approximate numerical equalitunder the limits given~i.e., x!1 implies that 11x'1), anduse; for order-of-magnitude equality that has the correpowers of small quantities~e.g., 2x;x).

With impact parameterb@1/v0, the angular momentumLstays nearly constant at its initial valueL05bv0@1 ~as Ishall show momentarily!, so at the perispaggon, the point othe object orbit nearest the string, the minimal separationr mand the object speedvm are related byr mvm'L'L05bv0.Assuming thatE5v2/221/r !1/b ~which for E;E0 re-quires b!1/v0

2, so we need 1/v0!b!1/v02 and hencev0

!1), we getvm2 /2'1/r m@v0

2/2. Combining this withr mvm

'L0 then gives

r m'1

2L0

2 , ~82!

vm'2

L0. ~83!

The time spent near perispaggon istm;r m /vm;L03. Dur-

ing this time, the angular momentum decreases at thegiven by Eqs.~65!, L52L/r 3;2L0 /r m

3 ;21/L05 , and thus

it decreases by a total amount roughlytm as large, orDL;21/L0

2. The magnitude of this decrease is much less thanL0

for L0@1 or b@1/v0, as claimed above, so in this regime tangular momentum is approximately constant.

The energy decreases at the rate given by Eqs.~74!, E521/r 4;21/r m

4 ;1/L08 , and multiplying this by the timetm

gives a total decrease2DE;1/L05'1/(bv0)5. For the final

energy to be zero, this decrease needs to equal the inenergy, giving v0

2;E0;2DE;1/(bv0)5 or b5bcrit

;v027/5. For v0!1, this is indeed within the allowed rang

1/v0!bcrit;v027/5!1/v0

2 of the approximations used aboveOne can easily get the correct coefficient (6p)1/5 of the

v027/5 term for bcrit by calculating the time integral of th

energy loss rateE521/r 4 for a parabolic orbit of angulamomentumL05bv0, but that confirmation of the precisform of the first term of the right hand side of Eq.~81! willbe left as an exercise for the reader. Similarly, one shouldable to get the second term on the right hand side of Eq.~81!from first-order perturbations due to the motion of the strand the resultant loss of the energy and angular momenof the object.

VI. THE CRITICAL IMPACT PARAMETERIN THE TEST-STRING REGIME, v0@L u /Tu

Now I shall consider the opposite case,h[h(X50)5L3(t5`)/8!1, which leads tov0@Lu /Tu[1 and bcrit!Lu[1. This is the test-string limit, for which the lowesorder approximation is to ignore the string tension andeffect of the string on the object. The critical impact para

02351

-ctit

t

te

ial

e

m

e-

eter in this limit was found in@3# to be bcrit.2v021/2 after

converting to the units used in this paper. Here I shrederive this result and give the lowest-order correctionthe small quantity 1/v0.

In this test-string regimev0@1, the qualitative picture isthat initially the object moves with very nearly constant vlocity, and the string gets bent toward the object as descriin @3#. In that paper, where the back reaction of the stringthe object was totally ignored, the final situation, when timpact parameterb has the critical valuebcrit for marginalcapture, is for the string to develop a kink that asymptoticatrails the object at a distance ofv0

21/25bcrit/2 behind it, anunstable equilibrium point where the force of the objectthe string causes the string to follow the object at precisthe same speed as the object, by Eq.~32!. In the completetest-string approximation, the deceleration of the objecignored, so it continues moving forever, pulling an evlonger kink of string behind it.

When one does include the back reaction effect ofstring tension on the object but stays in the test-string regof v0@1, the picture during the first stage of the evolutionthe marginal capture case is very similar to that of the coplete test-string approximation used in@3#, so the position onthe string nearest the object approaches quite near to wwould be the unstable equilibrium separation behind theject if the object continued moving with constant velocitBut then in actuality the object slows down as the striweakly pulls back on it~somewhat analogous to the wayfish slows down when attached to a weak fishing line@12#!.If it had been at the unstable equilibrium separation forvelocity of the object~i.e., the separation that would be thequilibrium one if the object continued moving with constavelocity!, then the slowing down of the object would evetually lead to its falling into the string. For the object to bjust marginally not falling into the string, the separation mube greater, by a precise tiny amount, than what would beequilibrium separation at constant velocity. At this slightgreater separation, the force on the kink of the string trailthe object would be just slightly less than needed to movat the speed of the object, so the string moves slightly slothan the object and hence trails behind it at an indefinitincreasing separation, gradually slowing down more amore.

If the object had had an impact parameterb slightlygreater thanbcrit(v0), it would maintain a velocity boundedaway from zero as its separation from the string increaindefinitely. On the other hand, if the object hadb slightlysmaller thanbcrit(v0), the object would eventually slowdown more than the string, so that the separation would sincreasing and would henceforth decrease to zero as theject falls into the string. But in the marginal capture cawith impact parameterb5bcrit(v0), the separationr betweenthe string and the object, although continuing to grow indenitely, grows slowly enough that the object, as well asstring, continues to slow down just fast enough that it asymtotically approaches, but never reaches, zero velocity.

We can put this discussion into a more quantitative fo

0-9

u

e

pd

y-

are

ly

of

ro


in the following way: Forg!X2!1, Eqs.~70! and~71! be-come

dg

dX5

16X4112X2g1O~X6!1O~g2!

h, ~84!

dh

dX512X21O~X4!1O~g2!, ~85!

with the critical solutions (g50 andh5h.0 at X50)

g~X!516

5 S X5

h2

X8

h2D1OS X7

h D1OS X10

h2 D 1OS X11

h3 D ,

~86!

h~X!5h14X31O~X5!1O~h21!. ~87!

To get the time dependence of these solutions, we canEq. ~69!, which for g!X2!1 becomes

X54X413X2g1O~X6!1O~g2!

2h. ~88!

Insertingg(X) and h(X) from Eqs.~86! and ~87! into thisand integrating gives

t5h

6X3 24

5ln X1OS X3

h D , ~89!

dropping the irrelevant constant of integration. For largtthis can be inverted to give

X5S h

6t D1/3F11

4

45tln

6t

h1OS ln2t

t2 D G , ~90!

Then from Eqs.~86!, ~87!, ~62!, and~63!, one may get

g~ t !'16

5h2/3~6t !25/3S 11

4

9tln

6t

h D , ~91!

h~ t !'hS 113

2t1

8

45t2 ln6t

h D , ~92!

r ~ t !'1

2~6t !2/3S 12

8

45tln

6t

h D , ~93!

v~ t !'2~6t !21/3S 114

45tln

6t

h D . ~94!

Alternatively, one can divide Eq.~53! by Eq. ~54! to get

dr

dv5secc2r 2v'12r 2v ~95!

for X[sinc/2!1, where one may recall thatc is the anglebetween the velocity of the object and the string-object seration, soc goes fromp initially ~as the object heads towarthe string! to 0 finally for marginal capture~as the object

02351

se

a-

heads away from the string with slower and slower velocit!.WhenX can be neglected, Eq.~95! leads to the series solution

1

r'

1

2v22

1

20v51

1

160v82

7

8800v111

1

9856v141O~v17!

~96!

or

v~r !'A2r 21/211

5r 22

1A2

25r 27/21

24

1375r 251O~r 213/2!. ~97!

All of the series expansions given so far in this sectiongood only whent@1, r @1, andv!1, in addition toX!1~i.e., when the object is moving very slowly nearly directaway from the string at very late times!. Whenh!1, as oneintegrates backward in time from the final conditiong50and h5h at X50 ~where t5`, r 5`, andv50), the ap-proximate second equality of Eq.~95! remains valid so longas X!1 and gives, whenv becomes very large andr be-comes very small,

1

r'v1/22

1

4v212

5

32v25/22

15

64v24

21105

2048v211/21O~v27!, ~98!

up to terms that decrease roughly as the exponential2(4/3)v3/2, so long asX remains negligible. Inverting thisgives

v~r !'1

r 2 11

2r 1

1

8r 41

5

32r 71O~r 10! ~99!

for small r, again neglecting all contributions from nonzeX and terms exponential in24/(3r 3).

To get theX-dependence of the critical solution whenX!1 and to prepare the way for the corrections whenX is nolonger small, divide Eq.~56! by Eq. ~54! to get

dX

dv5

X~12X2!

122X2 S 121

rv2D rv'XS 121

rv2D rv, ~100!

with the approximate equality applying whenX!1.When v!1 so that Eq.~96! is applicable, Eq.~100! be-

comes

dX

dv'

X

v, ~101!

which, when matched to the boundary condition thatL52rvXA12X2 is equal to 2h1/3 at X50, gives

X'1

2h1/3v'

h1/3

A2r~102!

0-10

e

io

ofas-

imc

sts

a

ifn

ctgst

e

r,

ndhe

.

q.

.

lly

es,

o


for v!1, r @1, or X!h1/3.At X;h1/3!1, v surpasses unity and thereafter becom

large, andr becomes small~until X gets near 1 andr be-comes large again in this evolution backward in time!. Whenv@1 so that Eq.~98! is applicable, withrv2'v3/2@1, Eq.~100! becomes

dX

dv'

X~12X2!rv122X2 'Xrv, ~103!

where the first approximate equality applies for allX@h1/3

and the second approximate equality applies forh1/3!X!1.

Equation~103!, along with Eq.~98!, implies that oncevgets large,X grows roughly exponentially with (2/3)v3/2. Af-ter X passesv21/2, dv/dX becomes small, and the largevchanges very little during the rest of the backward evolutto the initial conditions atX50 (t52`,v5v0). Since thisexponential growth ofX with v starts atX;h1/3 where v;1 and ends atX;1 wherev;v0, one gets

v0;S 21

2lnh D 2/3

~104!

for very small h. This very slow growth ofv0 with 1/himplies that when one solves Eqs.~70! and~71! numericallywith the numerical initial conditionsg(X50)50 andh(X50)5h ~final conditions in time for marginal capture!, onemust chooseh to be extremely tiny to get a large valuev0, and for the approximations of this section, one mustsume byh!1 not just thath is more than an order of magnitude ~say! smaller than unity, but that2 ln h is more thanan order of magnitude~say! larger than unity.

Once the marginal capture solution forh!1 has beencarried pastX;(2 ln h)21/3 wheredv/dX becomes small,vstays nearly constant and one is in the test-string regtreated in@3#, where one can to first approximation neglethe back reaction of the string on the object~the object’sacceleration toward the string!. In this paper, however, wewant to treat this back reaction perturbatively in this testring regime and get the first-order correction to the testring critical impact parameter formulabcrit.2v0

21/2 givenin @3#.

For this purpose it helps to change the dependent vables of integration from theg(X) and h(X) used in Eqs.~70! and~71! to two other variables that would be constantthe test-string approximation were completely accurate. Osuch variable isv, the magnitude of the velocity of the obje~in the frame in which the string was initially at rest alonthe z-axis!. A second such variable is motivated by the testring Eq. ~67! of @3#, where the polar coordinate angleudefined just before Eq.~54! of that paper is essentially thsame asc here, wherer there is the same asr here, wherezgiven there just before Eq.~54! as r sinu is hence the sameas r sinc52rXA12X2 in the notation of the present papeand where the constantL given by Eq.~50! ~not to be con-fused with the angular momentumL of the present paper! is

02351

s

n

-

et

-t-

ri-

e

-

the same asv021/2 in our present unitsTu[1 and Lu[1.

Then Eq.~67! of @3#, plus the equationbcrit.2L three linesbelow, imply that

b22bcrit2 .4r 2X2~12X2!24v0

21X2 ~105!

is a constant of motion in the test-string approximation, afurthermore that it is zero for marginal capture. If we use tfact that v.v0 to replacev0 by v and then multiply theexpression on the right hand side byv/4, we get the quantity

w[r 2v~X22X4!2X25L2

4v2X2

5h

AX22X41AX22X41g2X2. ~106!

Equations~70! and ~71! now imply that the alternativedependent variablesv andw obey the differential equations

dvdX

5X~122X2!

Av~X22X4!~X21w!F12A X22X4

v3~X21w!G21

,

~107!

dw

dX5

X@X22~122X2!w#

Av3~X22X4!~X21w!F12A X22X4

v3~X21w!G21

.

~108!

We can see from Eq.~106! that w is not a convenientvariable to integrate all the way fromX50, since sufficientlynear there thath'h and g!1, the last expression of Eqs~106! implies that we havew'h/(2X), which diverges atX50. Alternatively, in the second-to-last expression of E~106!, one gets a divergence from the fact thatL goes to thenonzero constant 2h1/3 at X50, but v goes to zero thereHowever, from Eq.~98! for h1/3!X!1, wherev@1 andr'v21/2!1, one can see that thenw'(1/2)v23/2X2, which isnegligibly small for small enoughX. For example, forX

;h1/3;e2(2/3)v03/2

, where v;1 and r;1, w;e2(2/3)v03/2

isexponentially small for largev0. Therefore, with only anexponentially small error, we can setw50 andv@1 as ini-tial conditions in integrating Eqs.~107! and~108! from somevery smallX, such ash1/3, which for practical purposes wecan replace with 0 with only an additional exponentiasmall error.

Because Eqs.~107! and ~108! havev1/2@1 andv3/2@1,respectively, in the denominators of the right hand sidwhereas the other factors are of order unity~except for harm-less square-root singularities in 12X2 at X51), v and wchange by only small amounts as one integrates toX51 toget the temporally initial conditions for marginal capture. Tlowest order in 1/v0, one can thus neglectw on the righthand sides and rewrite Eqs.~107! and ~108! as

dvdX

'122X2

Av~X22X4!'

122X2

Av0~X22X4!, ~109!

0-11

ct

t

ith

is

bngalt p

-o

or

os

nc

ol-bu

rge

utof

ly

wond

ofqs.

led

y

be

-

,.-


dw

dX'

X2

Av3~X22X4!'

X2

Av03~X22X4!

. ~110!

Then Eq.~110! integrates to give

w~X51!5L0

4v0215

1

4bcrit

2 v0215v023/21O~v0

23!.

~111!

From this we can readily solve for the critical impaparameter, in the units ofTu and of Lu given by Eqs.~35!and ~36!, for largev0@Lu /Tu[1:

bcrit~v0!52v021/21v0

221O~v027/2!. ~112!

When the units are restored, one gets the result given inAbstract.

An order-of-magnitude argument for thev021/2 power law

of the leading term was given in@3#. Summarizing it verybriefly in the units used here, if the object is incident wvelocity v0@1 and impact parameterb such that it is notcaptured, the distance of closest approach will ber;b, thevelocity at that point will bev;v0, the time when the stringis roughly that close will be;r /v;b/v0, the length of stringthat bends significantly will be the speed of light times thor ;cb/v0, the bending angle will be;1/(cr2);1/(cb2),and hence the total transverse bending of the string willthe length of string significantly bent times this bendiangle, or;1/(bv0). Marginal capture requires that this tottransverse bending be of the same order as the impacrameter, givingb5bcrit;1/Av0.

VII. THE CRITICAL IMPACT PARAMETEROVER THE ENTIRE RANGE

OF NONRELATIVISTIC VELOCITIES

Now that we have found analytically Eq.~81! for thecritical impact parameter at very low initial velocities,v0!Lu /Tu!c, and Eq.~112! for bcrit(v0) at high ~but stillnonrelativistic! velocities,Lu /Tu!v0!c, we would like tofind it numerically for all nonrelativistic velocities, and ideally find a fairly simple formula that gives a good fit tbcrit(v0) for all v0.

It was straightforward to program Mathematica 2.0 on mancient NeXT to integrate Eqs.~70! and ~71! numericallyfrom the temporally final conditionsg50 and h5h at X50 back to the temporally initial conditions atX50 andthen use Eqs.~75! and~76! to solve forv0 andbcrit(v0). It isslightly awkward that by this method one cannot straightfwardly choose the values ofv0 for which one getsbcrit(v0),but this is a small price to pay for not having to use a shoing method to find the critical or marginally bound solution@One could use a shooting method to get any particularv0desired, but for determining the characteristics of the fution bcrit(v0), this is hardly necessary.#

Another slight awkwardness is that to get large valuesthe initial velocityv0, one must use exponentially small vaues ofh, and this can lead to some numerical inaccuracy,

02351

he

e

a-

y

-

t-.

-

f

t

this was not a serious problem for at least moderately lavalues ofv0.

I calculated over 50 pairs of@v0 ,bcrit(v0)#, ranging from(0.00104198, 26910) through~for example! (0.834385,3.48671) to (357.89, 0.105727). Larger values ofv0 requiresuch extraordinarily small values ofh that it would be some-what difficult by my numerical method to obtain them, bwhat I could easily attain extends well into the domainvalidity of both Eq.~81! and Eq.~112!.

The results from the numerical integrations fit extremewell to the truncated series expansions of Eq.~81! at verysmall v0 and of Eq.~112! at largev0, confirming both thetwo coefficients and the two exponents of each of the texpressions~eight parameters calculated analytically aconfirmed numerically!.

One can combine Eqs.~81! and ~112! into the interpola-tion formula

bcrit~v0!.B(a,b)~v0!

[F S 6p

v07 D a/5

1S 4

v0D a/2G1/a

1F S 55v0

63p3D b/5

1v02bG21/b

, ~113!

which depends not only on the eight parameters~four coef-ficients and four exponents! calculated analytically and givenin Eqs. ~81! and ~112!, but also on two new parameters,aandb. Let us see how well this formula works for suitableaandb.

In order that the first two terms of the right hand sideEq. ~81! be the dominant terms of the right hand side of E~113! at v0!1, one needsa.4/3 andb.0, and in orderthat the first two terms of the right hand side of Eq.~112! bethe dominant terms of the right hand side of Eqs.~113! atv0@1, one needsa.5/3 and b.0. The combination ofthese two pairs of inequalities gives what might be calcriterion A, a.5/3 andb.0. If one wants to avoid anydeviations larger than theO(v0) term of Eq.~81! at smallv0,one needsa>8/3 andb>2/3, and if one wants to avoid andeviations larger than theO(v0

27/2) term of Eq.~112! at largev0, one needsa>10/3 and b>5/6. The combination ofthese latter two pairs of inequalities gives what mightcalled Criterion B,a>10/3 andb>5/6.

If one wants a simple choice of the pair of fitting parameters (a,b) ~e.g., a pair of integers!, criterion A motivatedmy first choice, (a,b)5(2,1). As a crude fit, this is not toobad, giving a maximum relative error of slightly over 10%e.g. 10.382% error at@v0 ,bcrit(v0)#5(0.940172,3.10142)My second choice, (a,b)5(3,1), is the smallest pair of integers that~almost! satisfies criterion B. This gives

bcrit~v0!.B(3,1)~v0!

[@~6pv027!3/518v0

23/2#1/3

1F S 3125v0

216p3 D 1/5

1v02G21

. ~114!

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aar

gixthlaix

to

.

4thtiv

anoselywe0

ineue

exi

rrehet

-x

.snf-

icalan

t

e-, I

the

ial

,n-

heoc-

ctele,

eryter

ofcu-sn

low


This approximate formula gives a much better bit, withmaximum relative error of only about 0.85%, ne@v0 ,bcrit(v0)#5(1.79814, 1.78528).@The position of themaximum relative error was not calculated to the six-difigures given for„v0 ,bcrit(v0)… here, but they are given to sidigits so that one can see how accurate this, or some oapproximation is at this sample point, a point where the retive error was about as large as I could find given the sdigit apparent accuracy of the Mathematica calculation.#

I then used the FindMinimum function of Mathematicaminimize, as a function of the pair (a,b), the sum of the100th powers of the relative errors ofB(a,b)(v0) over thefirst 45 pairs of values of@v0 ,bcrit(v0)# that I had calculatedThis sum is an analytic function of (a,b) whose minimumshould be at a value of (a,b) which is near the point wherethe maximum absolute value of the relative error for thepoints is minimized, since the sum of the 100th power ofrelatives errors should be dominated by the largest relaerror.

Mathematica gave the minimization as occurring(a,b)5(4.00448, 1.36884), though the result woulddoubt have been slightly different if I had used a differentof @v0 ,bcrit(v0)# and a different power of the errors. Idealone should use an infinite set of points and an infinite poof the errors, but I considered 45 points and a power of 1as being high enough to give a good feel for the true mmum, over (a,b), of the maximum of the magnitude of threlative error. The largest relative error I found for this valof (a,b) was 0.427328%.

By comparison, minimizing the sum of the squares~2ndpowers! of the relative errors led to (a,b)5(3.98412,1.37903), which is not that much different evfor this relatively small value of the power, though the mamum error I found for this choice of (a,b), 0.47019%, isabout 10% higher.

A fairly simple rational approximation for the (a,b) thatminimizes the maximum relative error ofB(a,b)(v0) is(4,15/11), which then leads to

bcrit~v0!.B(4,15/11)~v0![F S 6p

v07 D 4/5

1S 2

Av0D 4G 1/4

1F S 3125v0

216p3 D 3/11

1v030/11G211/15

. ~115!

Here the number 4 in the exponents of the first term cosponds toa, and the number 11 in the exponents of tsecond term corresponds to 15/b, so if one wanted differenvalues of (a,b), one could simply change these two numbers in this expression. This approximation gives a mamum relative error of just under 0.43%~I found a maximumof 0.429991%!, near @v0 ,bcrit(v0)#5(0.820853, 3.54436)Thus this B(4,15/11)(v0) is over 99% as good aB(4.00448,1.36884)(v0) @has a maximum relative error less tha1.01 times the maximum relative error oB(4.00448,1.36884)(v0)# and is roughly twice as good an approximation ~i.e., with roughly half the maximum relativeerror! as the simplerB(3,1)(v0) given by Eq. ~114! that I

02351

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er,--

5ee

t

t

r0i-

n-

-

i-

guessed on my second attempt without doing any numererror minimization. To get a relative accuracy better throughly one part in 234 forbcrit(v0), one would need a dif-ferent functional form thanB(a,b)(v0) given by Eq.~113!,but I will not pursue such refinements here.

Since the critical impact parameterbcrit goes asv027/5 for

small velocityv0 and asv021/2 for largev0, one can see tha

if one multipliesbcrit by v0 to get the critical initial angularmomentumLcrit(v0) ~per unit object massM, and in units ofLu

2/Tu after dividing out by thisM ), this rises with both lowand highv0 and hence must have some minimum in btween. By calculating several points near the minimumfound that the minimum value is

Lmin'2.9026, ~116!

which occurs near the sample point@v0 ,bcrit(v0)#5(0.834385, 3.48671) given above. When one restoresunits and multiplies by the object massM to get the trueminimum angular momentum of the object about the initstatic, straight string configuration, one gets

Lmin'2.9026Fp2

32mS 12

Q2

M2D 2G1/3

M2

'1.9611m1/3~12Q2/M2!2/3M2. ~117!

A compact gravitating~and possibly electrically charged! ob-ject approaching from infinity toward an initially staticstraight, infinitely long cosmic string with angular mometum ~about the initial string axis! less than this would inevi-tably be captured by the string.

VIII. CRUDE FORMULA FOR THE CRITICAL IMPACTPARAMETER FOR ALL VELOCITIES

In the test-string approximation@3#, I calculated the criti-cal impact parameter for velocities much lower than tspeed of light and then a correction term linear in the velity divided by the speed of light, getting Eq.~88! of thatpaper. In the case of a Schwarzschild black hole (Q50), Iarbitrarily added a~relatively small! constant term to makethe resulting formula agree with the known critical impaparameter at the speed of light@1#, which is the same as thcritical impact parameter for a massless point particA27M . The resulting formula, Eq.~89! of @3#, though just aguess by no means rigorously derived, is apparently a vgood approximation to the actual critical impact paramefor a test string at all calculated velocities~i.e., up to a rela-tivistic gamma-factor ofg0;10) @4,11#, though it misses aninfinite series of discontinuities that I predicted@3# occur atlarge values ofg0 ~none seen yet in the analyses of@2,4# upto g0;10). At velocities at all comparable to the speedlight, the critical impact parameter apparently must be callated numerically by solving partial differential equation@2,4#, which is a much more difficult numerical problem thausing the ordinary differential equations of@3# and of thepresent paper that are only accurate at velocities verycompared with the speed of light.

Here I would like to give an analogue of Eq.~89! of @3#

0-13

n-

nt

f

i-rom


that incorporates~a! Eq. ~115! as an excellent approximatioto my numerical calculations at all velocities very low compared with the speed of light,~b! Eq. ~88! of @3# for the termlinear in velocity that is independent of the string tensioand also~c! the right limit for the critical impact parameter athe speed of light for a Reissner-Nordstrom black hole~ar-bitrary Q2<M2). This last value is the minimum value oA2guu /g00 in the Reissner-Nordstrom metric, which is

ve

tio

anq.

tio

ird

mne

ly

eorthns

er

02351

,

bcrit~v051!5A~3M1A9M228Q2!3

2M12A9M228Q2. ~118!

The resulting crude formula for all velocities for the critcal impact parameter for the capture of a Reissner-Nordstblack hole of massM and chargeQ by an initially straight,static cosmic string with dimensionless tensionm[Gm!1is thatbcrit(v0) is roughly the same asbguess(v0), with

bguess~v0!

M5S p

2D 1/2S 12

Q2

M2D 1/2

v021/2~12v0

1/2!F11p2

16S 9

2D 2/5S 12

Q2

M2D 6/5

m12/5v0218/5G1/4

1p

4S 12

Q2

M2Dmv022~12v0

2!F11S 3125

1728D 3/11S 12

Q2

M2D 9/11

m18/11v0227/11G211/15

264

15~12v0!1A~31A928Q2/M2!3

212A928Q2/M2. ~119!

a-

erht

-lget

me

The first two terms dominate forv0!1 ~where we havenow returned to units in which the speed of light isc51).For this highly nonrelativistic velocity these two terms giessentially the same as Eq.~115!, which I found numericallyis accurate to greater than 99.5% accuracy for all nonrelaistic velocities whenm!1. The terms which were added tEq. ~115! to get these two terms are essentially constants~upto correction factors that go to unity whenm is taken to zero!at v0@m2/3 and were designed to make these two terms vish at v051. The third term on the right hand side of E~119! is the contribution of the term linear inv0 calculatedfor the test string in@3# ~though with the recognition thathere might be other such terms missed in that calculatand perhaps other terms that are independent ofv0 and thatgo asv0

1/2), corrected with a constant term to make this thterm also vanish atv051. The fourth~and last! term on theright hand side of Eq.~119! is independent ofv0 and givesthe critical impact parameter at the speed of light, the saas that of a massless particle impinging upon the ReissNordstrom black hole.

Although I have no real justification of Eq.~119! for rela-tivistic velocities, the fact that it was found to be a fairgood approximation for all velocities calculated so far~i.e.,at least for velocities not too ultrarelativistic! for Schwarzs-child black holes@11# leads me to conjecture that it may baccurate to within several percent for all velocities fReissner-Nordstrom black holes as well, assuming onlym!1. However, apparently only numerical calculatiocould confirm this.

Following Eq.~90! of @3#, one might note that this crudguess, Eq.~119!, gives a minimum critical impact parameteof

v-

-

n,

er-

at

bguess min

M'6S p

15D1/3S 12

Q2

M2D 1/3

2S p

2 D 1/2S 12Q2

M2D 1/2

264

151A~31A928Q2/M2!3

212A928Q2/M2~120!

at

v0'~225p!1/3

32 S 12Q2

M2D 1/3

, ~121!

ignoring correction terms of orderm that form!1 would benegligible in comparison with theO(1) errors I would ex-pect to be in Eq.~119!. It would be interesting to learn fromnumerical calculations how the minimum critical impact prameter~say divided byM to make it dimensionless!, and thevelocity at which it occurs, varies as a function ofQ2/M2 fora Reissner-Nordstrom black hole.

IX. EFFECTS OF A LONG STRING ON THE SUNOR EARTH

It may be amusing to conclude this hypothetical papwith a discussion of what would happen if a long straigstring were impinging upon the Sun, Earth, or Moon.

Approximate these objects as spherical, with massM,chargeQ50, and radiusR, not to be confused with the magnitude of R defined in Eq.~45!. Then the mathematicaanalysis above applies so long as the string does notinside the object, so the separationr between the string andthe center of mass of the object, the magnitude ofr definedin Eq. ~33!, must remain greater thanR. We shall find thatfor solar system objects, this puts us in the stiff-string regiunlessm is extremely tiny, much smaller thanm;1026 ex-pected for cosmic strings.

In the critical capture case, inserting the unitsTu andLu

0-14

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c-iu

b

s

e

e

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trthhetion-

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n

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from Eqs. ~35! and ~36! into Eqs. ~82! and ~83!, with L05bcritv0 and with bcrit(v0) being given by Eq.~81! in thisstiff-string regime, gives the minimum separationr m and therelative velocityvm at this separation being given by

r m'M S 9p5

2048D1/5S 12

Q2

M2D 3/5

m1/5v024/5, ~122!

vm'F8

3 S 12Q2

M2Dm2v02G1/5

'Fpm

2 S 12Q2

M2D M

r mG1/2

.

~123!

This expression forvm is Apm/4 times the speed of a tesmass, with the same charge-to-mass ratioQ/M as the object,that falls in to the surface of the object from a much lowvelocity far away, because the static force given by Eq.~15!between a compact object and a string ispm/4 times thatbetween the object and a copy of itself at the same separawithout the string.

One can now invert Eq.~122! and replacer m by R to getthe initial velocityv0 needed so that the critical capture ocurs with the minimum separation being given by the radR of the object:

v0'S 9p5

2048D1/4S 12

Q2

M2D 3/4

m1/4S M

R D 5/4

. ~124!

The corresponding critical impact parameter is

bcrit'M S 512

9p3D 1/4S 12Q2

M2D 21/4

m1/4S M

R D 27/4

. ~125!

For the initial velocity to obey the criterionv0!(12Q2/M2)1/3m2/3 for the stiff-string approximation to bevalid during the critical capture, one needs

S 12Q2

M2D S M

R D 3

!m!1. ~126!

This is an extremely weak restriction for solar system ojects, sinceM /R is approximately 2.1231026 for the Sun,6.96310210 for the Earth, and 3.14310211 for the Moon.

For the Sun, with massM'1.47662504 km, chargeQ'0, and radiusR'6.9603105 km, critical capture with theminimum separation beingR gives the initial velocity andimpact parameter as v0'0.827(m/1026)1/4 m/s'2.98(m/1026)1/4 km /hr and bcrit'4.6131011(m/1026)1/4 m'3.08(m/1026)1/4 AU ~astronomical units!. Therelative velocity of the Sun and string when the string jugrazes the sun is vm'547(m/1026)1/2 m/s'1970(m/1026)1/2 km /hr. When the string is grazing thSun, it gives the sun an acceleration ofa5pmM /(4r 2)'2.1531024(m/1026) m/s2, which is less than the averagacceleration of the Earth toward the Sun, 5.9331023 m/s2,unlessm>2.7631025. Long slow strings withm;1026 orless apparently do not give much danger of pulling the Saway from the Earth, but they could change the orbital chacteristics of the earth by a few percent, with possibly distrous climatic effects.

02351

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on

s

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t

nr-s-

For the Earth, with M'0.444 cm, Q'0, andR'6371 km, one gets v0'3.6531025(m/1026)1/4 m/s'13.1(m/1026)1/4 cm/hr, bcrit'1.7331012(m/1026)1/4 m'11.6(m/1026)1/4 AU, and vm'9.91(m/1026)1/2 m/s'35.7(m/1026)1/2 km /hr, if the Earth were isolated so thawe could neglect the larger effect of the Sun on the Eaduring the interaction of the string with the Earth. When tstring is grazing the Earth, it gives the Earth an acceleraof a'7.7131026(m/1026) m/s2, which is less than the average acceleration of the Earth toward the Sun ifm,7.6831024. Thus a string with roughly this small a tensionsmaller could not kidnap the Earth from the Sun. The avage acceleration of the Earth toward the Moon is ab3.3231025 m/s2, which is greater than the accelerationthe Earth toward a string grazing it ifm,4.3031026, so acosmic string withm;1026 would be unable to pull theEarth away from either the Sun or the Moon~barring con-trived resonance effects!, though there is not a large margiof safety for pulling the Earth away from the Moon.

For the Moon, with M'0.0546 mm, Q'0, and R'1738 km, one gets v0'7.5831027(m/1026)1/4 m/s'2.73(m/1026)1/4 mm/hr, bcrit'5.3431012(m/1026)1/4 m'35.7(m/1026)1/4 AU, and vm'2.10(m/1026)1/2 m/s'7.58(m/1026)1/2 km /hr, assuming the Moon were isolated. At the Moon’s surface, a string would give it an acceration ofa'1.2731026(m/1026) m/s2, which is less thanthe average acceleration of the Moon toward the Sun im,4.6631023, and is less than the average accelerationthe Moon toward the Earth, 2.7031023 m/s2, if m,2.1231023. Thus the Moon is in even less danger of beipulled away from the Sun than the Earth is, and even ifEarth-Moon system were isolated from the Sun, a string wreasonably small tension could not remove the Moon frthe Earth.

Another effect one can estimate is that of a string ontides. If the Earth had a string grazing it or running throuit, presumably the curvature of spacetime in and nearEarth would be altered by a fraction of orderm. The Rie-mann curvature tensor at the surface of the Earth hasorthonormal component Rr0r

0 52M /R3'3.0831026 s22

'40.0 h22, which is about 1.793107 times the corresponding curvature contribution from the Moon, and about 3.3107 times the corresponding curvature contribution frothe Sun. Therefore, if a cosmic string ran through the Eathe perturbation of the Earth’s tidal force would be of torder of 18(m/1026) times the tidal force of the Moon. If theEarth’s axis were not aligned with the direction of the strinas the string cuts a swath through the rotating Earth,resulting perturbation in the Earth’s tidal~curvature! gravita-tional field might produce Bay-of-Fundy size tides at macoastal cities.

ACKNOWLEDGMENTS

I thank Jean-Pierre De Villiers and Valeri Frolov for geting me interested in this subject, and I thank them aPatrick Brady, Gary Horowitz, and Brandon Carter for futher discussions. This work was supported in part byNatural Sciences and Engineering Research CouncilCanada.

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recala-

c.d,

-


@1# J.-P. De Villiers and V. Frolov, Int. J. Mod. Phys. D7, 957~1998!.

@2# J.-P. De Villiers and V. Frolov, Phys. Rev. D58, 105018~1998!.

@3# D.N. Page, Phys. Rev. D58, 105026~1998!.@4# J.-P. De Villiers and V. Frolov, Class. Quantum Grav.~to be

published!, gr-qc/9812016.@5# Y. Nambu, in Proceedings of the International Conference

Symmetries and Quark Models, Detroit, Michigan, 1969; Letures at the Copenhagen Summer Symposium~1970!.

@6# T. Goto, Prog. Theor. Phys.46, 1560~1971!.@7# B. Carter, Phys. Lett. B224, 61 ~1989!; J. Geom. Phys.8, 53

~1992!; Class. Quantum Grav.9, 19 ~1992!; Phys. Rev. D10,4835 ~1993!; in Formation and Interactions of TopologicaDefects, edited by R. Brandenberger and A.-C. Davis~Plenum,

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New York, 1995!, p. 304, hep-th/9611054; Tlaxcala lectunotes, 2nd Mexican School on Gravitation and MathematiPhysics, 1996, edited by A. Garcia, C. Lammerzahl, A. Mcias, and D. Nunez, hep-th/9705172.

@8# A. Vilenkin, Phys. Rev. D23, 852 ~1981!.@9# A. G. Smith, in The Formation and Evolution of Cosmi

Strings, edited by G. W. Gibbons, S. W. Hawking, and TVachaspati~Cambridge University Press, Cambridge, Englan1990!.

@10# V.P. Frolov, V.D. Skarzhinsky, A.I. Zelnikov, and O. Heinrich, Phys. Lett. B224, 255~1989!; B. Carter and V.P. Frolov,Class. Quantum Grav.6, 569 ~1989!.

@11# J. P. De Villiers~private communication!.@12# B. Carter~private communication!.

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Low velocity gravitational capture by long cosmic strings