Current discontinuities on superconducting cosmic strings

ISSN 1063�7761, Journal of Experimental and Theoretical Physics, 2011, Vol. 113, No. 1, pp. 55–67. © Pleiades Publishing, Inc., 2011.Original Russian Text © E.Troyan, Yu.V. Vlasov, 2011, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 140, No. 1, pp. 66–79.

55

1. INTRODUCTION

Cosmic strings attract keen interest of researchersas the most probable objects responsible for the originof the large�scale structure of the Universe [1, 2]. Theemergence of strings dates back to early times after theBig Bang, when stable topological defects in the formof vortex�like solutions were formed in the course ofcosmological phase transitions associated with grandunified models. A string is a one�dimensional objectwith a mass per unit length m. At a curvature radius ofthe string much larger than its thickness, the stringtheory is built on a variation of the Goto–Nambuaction

(1)

with the Lagrangian

(2)

where

(3)

is the induced metric on the string world sheet withlocal coordinates σa (indices a and b can assume val�ues of 1 or 2) and gμν is the metric tensor of the globalspace (with coordinates xμ, where indices μ and νassume values 1, 2, 3, 4). The possibility of the exist�ence of superconducting strings [3] leads to a numberof new physical effects. For example, strings can be thesources of gamma�ray bursts and ultrahigh�energycosmic rays [4–6]. A superconducting string carryingpersistent currents like a superconductor possessesadditional internal degrees of freedom and itsLagrangian

(4)

S Λ dethab σ1 σ2dd∫=

Λ m2,–=

hab gμν∂xμ

∂σa�� ∂xν

∂σb�� gμνx a,

μ x b,ν= =

Λ Λ m χ,( )=

depends on the current amplitude

(5)

which is determined by the scalar field φ on the stringworld sheet and the external electromagnetic field Aμ.For a free string considered here, Aμ = 0.

The expression for Λ(m, χ) is called the equation ofstate for a superconducting string. The “linear” model[3, 7, 8] is used very often:

(6)

The dependence of the Lagrangian on the “current”term expands significantly the fundamental view of thestring theory. In particular, two classes of solutionswere found when infinitesimal perturbations wereintroduced into the equations of motion [9]: the per�turbations of the string world sheet or transverse per�turbations (called “wiggles”) and the perturbationspropagating within the string world sheet or longitudi�nal perturbations similar to acoustic waves (“jiggles”).Several studies are devoted to the stability of small per�turbations [10, 11].

In this paper, we will be interested in finite�ampli�tude perturbations. In this case, it is necessary to con�sider the appearance of discontinuous solutions whenthe current behind the discontinuity χ+ does not coin�cide with the current ahead of the discontinuity χ–.A finite current difference Δχ = χ+ – χ– can generatea shock wave traveling over the string world sheet. Thephysical possibility of the appearance of shock waveson strings is connected with the dependence of thestring Lagrangian (4) on current χ. From a mathemat�ical viewpoint, shock waves are discontinuous solu�tions of the equations of motion. In the original anal�ysis [12], the possibility of the existence of stable shockwaves in the case of space�like currents χ > 0 (in the

χ jaja– hab φ a, Aμx a,μ+( ) φ b, Aμx b,

μ+( ),–= =

Λ m2– χ2��.–=

Current Discontinuities on Superconducting Cosmic StringsE. Troyan* and Yu. V. Vlasov

Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700 Russia*e�mail: [email protected]

Received September 13, 2009; in final form, December 30, 2010

Abstract—The propagation of current perturbations on superconducting cosmic strings is considered. Theconditions for the existence of discontinuities similar to shock waves have been found. The formulas relatingthe string parameters and the discontinuity propagation speed are derived. The current growth law in a shockwave is deduced. The propagation speeds of shock waves with arbitrary amplitudes are calculated. The reasonwhy there are no shock waves in the case of time�like currents (in the “electric” regime) is explained; this isattributable to the shock wave instability with respect to perturbations of the string world sheet.

DOI: 10.1134/S1063776111060215

NUCLEI, PARTICLES, FIELDS,GRAVITATION, AND ASTROPHYSICS

56

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 113 No. 1 2011

TROYAN, VLASOV

so�called “magnetic” regime) was pointed out and theabsence of shock waves in the case of time�like cur�rents χ < 0 (in the “electric” regime) was postulated.This was confirmed through numerical simulations ofthe equations of motion [13, 14]: the continuous sinu�soidal profile of a small perturbation in the magneticregime degenerated into a discontinuity, while only asmooth current jump accompanied by a change in thestring geometry remained in the electric regime. Tofind the reason why no shock waves are possible in theelectric regime, it is necessary to investigate the condi�tions for the existence of discontinuous solutions. Wewill need to check whether the evolutionary conditionand the condition for the emergence of corrugationinstability of the shock front are met as well as to ana�lyze the relations at discontinuity and the analyticexpressions for low�intensity discontinuities asapplied to specific string models. The main result ofour paper is the derivation of the current growth lawΔχ > 0 and analysis of the shock wave stability withrespect to oscillations of the string world sheet. Thepatterns found provide an insight into the properties ofcurrent discontinuities on superconducting strings—apoorly studied phenomenon that can be of greatimportance in astrophysical applications.

Here, we will use a relativistic system of units inwhich the speed of light, the gravitational constant,and the Planck and Boltzmann constants are set equalto one, cL = G = � = kB = 1 and the signature of themetric tensor gμν is diag{–1, 1, 1, 1}.

2. THE EQUATIONS OF MOTIONAND SMALL PERTURBATIONS

For superconducting strings, the field theory can beformulated by analogy with continuum mechanics [9,15]. For example, the energy–momentum tensor isconstructed based on the string Lagrangian (4),

(7)

where the parameters U and T, which have an analogywith the internal energy and pressure of the fluid, areuniquely defined via Λ; uμ and vμ are the mutuallyorthogonal time� and space�like eigenvectors

(8)

related to the projective tensor

(9)

The string energy–momentum tensor (7) obeys theconservation law [15]

(10)

Tμν Uuμuν Tvμvν,–=

vμvμ uμuμ– 1, uμ

vμ 0,= = =

ημν habx a,μ x b,

νvμvν uμuν

.–= =

ην

ρ∇ρTμν 0.=

Transverse projection of Eq. (10) onto the stringworld sheet gives the equations

(11)

where = – .

Projecting Eq. (10) along the string world sheetyields two other equations:

(12)

where the parameters

(13)

are defined as [16]

(14)

in the case of space�like currents χ > 0 (in the mag�netic regime) or as

(15)

in the case of time�like currents χ < 0 (in the electricregime). The parameter

(16)

is calculated based on a specific model of theLagrangian. For the Goto–Nambu string (2), U =⎯T = –m2, the quantities (13) become zero and theconservation laws (12) are obeyed identically.

For superconducting strings with a current, sepa�rating the equations of motion into the “extrinsic”(11) and “intrinsic” (12) ones allows us not only tosimplify considerably the problem but also to revealthe physical meaning of the string dynamics. Equa�tions (11) and (12) admit nontrivial solutions in theform of infinitesimal perturbations [9]. Equations (11)specify the perturbations of the string world sheet (theso�called wiggles) similar to transverse oscillationswhose propagation speed is

(17)

In contrast, Eqs. (12) specify the perturbations propa�gating within the string world sheet (jiggles). Theseperturbations are similar to acoustic waves propagat�ing with the speed

(18)

The stability region for a current�carrying string isdefined by the conditions [13]

(19)

⊥σ

μ Uuν∇νuσ Tvν∇νvσ–( ) 0,=

⊥σ

μ uν∇νvσ

vν∇νuσ–( ) 0,=

⊥σ

μ gσμ ησ

μ

ημ

ν∇ν Mvμ( ) 0, ημ

ν∇ν nuμ( ) 0,= =

M dUdn��, n dT

dM��–= =

M2 χ, n2 K2χ= =

M2 K2χ, n2– χ–= =

1K�� 2dΛ

dχ��–=

c⊥( )2 T

U�� .=

c2 dTdU��– n

M��dM

dn��.= =

1 c2 0, 1 c⊥( )2

0.>≥>≥


CURRENT DISCONTINUITIES ON SUPERCONDUCTING COSMIC STRINGS 57

The wiggles are purely geometric in nature andeach individual string configuration requires a numer�ical calculation of the equations of motion [10, 11, 13,14]. The jiggles are possible for any geometry of a cur�rent�carrying superconducting string. That is why thequestion about the existence of stable current discon�tinuities similar to shock waves arises.

3. THE EQUATION OF STATEFOR A SUPERCONDUCTING STRING

The expressions for the speed of the perturbationswithin the string world sheet can be derived fromEqs. (14)–(16) and (18) [17]:

(20)

(21)

where

(22)

For a string with the “linear” equation of state (6), thepropagation speed of the longitudinal perturbations isconstant and equal to the speed of light

c = 1. (23)

The physical model (6) describes a string in the limit ofa small current χ 0 [16]. The situation is com�pletely different when the Lagrangian of a supercon�ducting string has a more complex dependence. Thewell�known generalized models include the “tran�sonic” model [18]

(24)

the “polynomial” model [17, 19]

(25)

the “inverse�ratio” model [16]

(26)

and the “logarithmic” model [20]

(27)

Here, the parameter m∗ <2m and all four nonlinear

models (24)–(27) are reduced to model (6) in the caseof weak currents.

c2 KK 2K 'χ+��, χ 0,>=

c2 1 2K 'χK

��, χ+ 0,<=

K ' dKdχ��.=

Λ m m2 χ+ ,–=

Λ m2– χ2�� 1 χ

m*2

��–⎝ ⎠⎜ ⎟⎛ ⎞

,–=

Λ m2– χ2�� 1 χ

m*2

��+⎝ ⎠⎜ ⎟⎛ ⎞ 1–

–=

Λ m2– m*2 1 χ

m*2

��+⎝ ⎠⎜ ⎟⎛ ⎞

.ln–=

According to (20)–(22) and (16), the longitudinalperturbations calculated in terms of the transonicmodel (24) will propagate with the speed

(28)

in the case of space�like currents and

(29)

in the case of time�like currents. For the polynomialmodel (25), the speed of sound is defined as

(30)

(31)

For the inverse�ratio model (26), we have

(32)

(33)

In contrast, the speed of sound determined in terms ofthe logarithmic model (27) is

(34)

(35)

In expressions (29)–(35), the constraints on the currentamplitude χ are imposed by the string stability condi�tion (19). Thus, when the generalized models withLagrangians (24)–(27) are taken to describe supercon�ducting strings, the expression for the propagationspeed of the longitudinal perturbations within the stringworld sheet (18) turns out to be dependent on the cur�rent term χ. Note that the equations of motion werenumerically simulated [13, 14] only based on the twononlinear models (24) and (26). Therefore, it is impor�tant to find the common patterns inherent in all stringmodels (24)–(27).

4. THE EMERGENCE OF DISCONTINUITIES

The appearance of discontinuities is attributable tothe instability of small perturbations when a continu�ous solution of the equations of motion becomesimpossible, which gives rise to discontinuous solutions

c2 m2 χ+

m2 2χ+��, χ 0>=

c2 m2 2χ+

m2 χ+��, m2

2��– χ 0< <=

c2 m*2 2χ–

m*2 2χ+

��, m*2

2�� χ 0,> >=

c2 m*2 2χ+

m*2

2χ–��, m*

2

2��– χ 0.< <=

c2 m*2 χ+

m*2 5χ+

��, χ 0,>=

c2 m*2 5χ+

m*2 χ+

��, m*2

5��– χ 0.< <=

c2 m*2 χ+

m*2 3χ+

��, χ 0,>=

c2 m*2 3χ+

m*2 χ+

��, m*2

3��– χ 0.< <=

58


TROYAN, VLASOV

[21, 22]. The physical string parameters (6) and (24)–(27) depend on the constants m and m∗ and the vari�able χ(xμ). Within the string world sheet, any smallperturbation propagates with speeds (28)–(35). Con�sider a finite current perturbation Δχ = χ – χ0. At

(36)

the propagation speed of this finite perturbation willdepend on its amplitude:

(37)

This implies that the neighboring parts of the pertur�bation with a given profile

(38)

will move with different speeds and the finite perturba�tion (38) will change. In the long run, the initialsmooth perturbation profile is distorted to such anextent that a unique solution becomes impossible anda discontinuity then emerges in the solution. Numeri�cal simulations of the equations of motion for a super�conducting cosmic string [13, 14] show that an initialcurrent profile in the shape of a sine wave turns into asawtooth profile—in close analogy with the emer�gence of shock waves in continuous media [21, 22].Thus, the dispersion of the propagation speed forsmall perturbations (36) can cause the finite currentperturbations to degenerate into discontinuities orshock waves.

When a superconducting string is considered interms of the linear model (6), the propagation speed ofthe longitudinal perturbations is always constant andequal to the speed of light. Therefore,

(39)

and the small perturbations in the equations of motionwill always remain stationary, without giving rise to

dcdχ�� 0≠

c χ( ) c χ0( ) ∂c∂χ��Δχ.+=

Δχ xμ( ) f xμ( )=

dcdχ�� 0=

instabilities. The nonlinearity of Lagrangians (24)–(27) with respect to the current χ plays a crucial role inthe appearance of shock waves. In particular, the smallperturbations propagating with speed (28), (29) in thetransonic model (24) can develop into discontinuities,because

(40)

in the case of space�like currents dχ > 0, while

(41)

in the case of time�like currents dχ < 0. The same pos�sibility is not ruled out for the strings considered interms of the more complex polynomial model (25),the inverse�ratio model (26), and the logarithmicmodel (27), for which the speed of sound obeys thesame inequalities (40) and (41). In contrast to contin�uum mechanics, there are no fundamental restrictionson the fulfillment of both inequalities (40) and (41) forcosmic strings. Nevertheless, the possibility of thedegeneration of finite current perturbations into ashock wave by no means guarantees the appearance ofthe latter. Unstable discontinuities can break up into aseries of compression (rarefaction) waves in the formof smooth current jumps. We need to ascertain whattypes of discontinuities will be stable.

5. THE EVOLUTIONARY CONDITION

Due to the dispersion of the speed of sound (40),(41), a finite current perturbation Δχ can give rise to ashock wave. For a shock wave to exist without decay�ing, the evolutionary condition must be met [23–25].We emphasize that this condition is necessary but notsufficient. A sufficient condition is the stability withrespect to perturbations at the shock front (“corruga�tion”) [26, 27]. It will be discussed at the end of thepaper, but now we will investigate the evolutionarycondition for shock waves on superconducting cosmicstrings.

Consider the small acoustic perturbations emerg�ing at the discontinuity surface and propagating paral�lel to the shock wave motion but in two opposite direc�tions from the discontinuity surface with the speed ofsound. Let c– and c+ be the speeds of sound ahead ofand behind the discontinuity, respectively. These smallperturbations will be transported by the shock wavewhose speed will be denoted by w–. In a frame of ref�erence comoving with the discontinuity surface, thediscontinuity itself is stationary and the speed of mat�ter ahead of the discontinuity w– is equal to the shockwave speed. Denote the speed of matter behind thediscontinuity (in the same comoving frame of refer�ence) by w+. The small perturbations ahead of thefront can move relative to the discontinuity surfacewith the speed

dc2

dχ�� 0<

dc2

dχ�� 0>

w−

c−

c−

w+

c+

c+

Fig. 1. Direction of the speed of matter w and the speed ofthe acoustic perturbations c ahead of (the index –) andbehind (the undex +) the shock front.



(42)

or

(43)

Figure 1 explains the situation. The small perturba�tions behind the front will propagate with the speed

(44)

or

(45)

If the perturbation propagates from the discontinuitysurface to infinity (in the frame of reference associatedwith the shock front, see Fig. 1), this implies that

(46)

For

(47)

there are two such perturbations propagating withspeeds (43) and (44). If

(48)

then only (44) is such a perturbation. For

(49)

we have three types of perturbations propagating withspeeds (43)–(45). In contrast, for

(50)

there are again two perturbations, (44) and (45).Lest the discontinuity break up, it must be stable

with respect to the small perturbations emerging at itsfront. The small perturbations specified by the bound�ary conditions at the discontinuity surface will remainsmall as long as the number of equations of motionexactly coincides with the number of unknownparameters contained in it. If the number of equationsis greater than the number of independent parameters,then the problem on the evolution of a small perturba�tion has no solutions at all; if, alternatively, the numberof equations is less than the number of parameters,then there will be an infinite set of solutions. For theperturbations within the string world sheet under con�sideration, we have the two conservation laws (12).Exactly two independent parameters must correspondto them. One independent parameter (the speed w–)determines the shift of the shock wave itself at the sur�face of which the perturbation emerged. The evolutionof each small perturbation specified by the boundaryconditions at the discontinuity surface is also deter�mined by one independent parameter (the infinitesi�

C–

w– c–+1 w–c–+��=

C–w– c––

1 – w–c–

�� .=

C+

w+ c++1 w+c++��=

C+w+ c+–

1 – w+c+

�� .=

C– 0, C– 0, C+ 0, C+ 0.> >< <

w– c–, w+ c+< <

w– c–, w+ c+,<>

w– c–, w+ c+,><

w– c–, w+ c+,> >

mal increment in current χ). Only condition (48) cor�responding to one small perturbation (44) will corre�spond to the shock wave being evolutionary. Incontrast, under conditions (47), (49), or (50), thesmall perturbations emerging at the discontinuity sur�face will be characterized by the number of indepen�dent parameters greater than the number of equationsof motion (12).

Condition (48) is the evolutionary condition for ashock wave on a cosmic string. The discontinuityemerging when the perturbations propagate on a cos�mic string will not break up precisely when condition(48) is met.

Note that three conservation laws correspond tothree perturbation parameters (shock wave entropy,pressure, and speed) in continuum mechanics and theevolutionary condition for ordinary shock waves [23,24] coincides with our evolutionary condition (48) fordiscontinuities on cosmic strings. This is nothing morethan a coincidence, because the necessity of inequality(48) for continuous media is also determined by ther�modynamic principles, while there are no fundamen�tal restrictions on all four possibilities (47)–(50) in thetheory of cosmic strings.

6. THE RELATIONS AT DISCONTINUITY

Let us derive the relations at discontinuity. We needto find the jumps in quantities in the equations ofmotion. We will use a standard mathematical tech�nique for studying shock waves in relativistic hydrody�namics [28, 29]. We consider a finite�amplitude per�turbation and introduce a unit normal vector to theperturbation propagation front in the form

(51)

When perturbation propagate in the medium, theincrement in the gradient of an arbitrary quantityΔ(∇μQ) λμΔQ occurs along the characteristic vec�tor λμ and is proportional to the increment in thisquantity itself ΔQ. Here, w is the shock wave speed. Inthe acoustic limit, w turns into the speed of sounddefined by (18) and the vector (51) describes an infin�itesimal perturbation similar to an acoustic wave [9].

Let us again pass to the frame of reference in whichthe shock front is stationary. The indices – and + willdenote the states ahead of and behind the front,respectively. After the substitution of (51) into theequations of motion (12), we find the relations for thejumps in quantities at the discontinuity front:

(52)

(53)

λν

wuν vν+

1 w2+�� .=

η+μ

ν λ+νM+v+μ η μ–

ν λ ν– M–v–μ

,=

η+μ

ν λ+νn+u+μ η μ–

ν λ ν– n–u–μ

.=

60


TROYAN, VLASOV

Substituting expressions (9) for the tensor and (51) forthe normal vector into (52) and (53) yields two equa�tions

(54)

(55)

From these equations, we find the speed ahead of thediscontinuity front

(56)

and the speed behind the front

(57)

Substituting in the parameters μ and n defined by (14)and (15) in the magnetic regime for χ > 0, we have

(58)

(59)

In contrast, in the electric regime for χ < 0, thespeed ahead of the shock front is defined by the for�mula

(60)

while the speed behind the front is

(61)

7. LOW�AMPLITUDE PERTURBATIONS

In the acoustic limit, when the current differenceΔχ = χ+ – χ– is small compared to the value of itself, Eqs. (60) and (61) are expanded into a series insmall quantity Δχ. Given definition (22), we find anapproximate expression of speed (58) in the magneticregime:

(62)

w+2 n+

2

w+2 1–

��w–

2 n–2

w–2 1–

��,=

M+2

w+2 1–

��M–

2

w–2 1–

�� .=

w–2 n+

2

M+2

��M+

2 M–2–

n+2 n–

2–��=

w+2 n–

2

M–2

��M+

2 M–2–

n+2 n–

2–�� .=

w–2 K+

2 χ+ χ––

K+2 χ+ K–

2 χ––��,=

w+2 K–

2 χ+ χ––

K+2 χ+ K–

2 χ––��.=

w–2 1

K+2

��K+

2 χ+ K–2 χ––

χ+ χ––��,=

w+2 1

K–2

��K+

2 χ+ K–2 χ––

χ+ χ––��.=

χ

w–2

c–2

�� 1 c–2 3 K–'( )

2K–K–''–

K–2

��χ–Δχ,+=

where

(63)

An exactly analogous expression follows from Eq. (59):

(64)

Relations (62) and (64) correspond to the evolutionarycondition for a shock wave (48) when

(65)

and

(66)

In the electric regime, for negative currents χ < 0,Eqs. (60) and (61) after the substitution of the smallincrement Δχ = χ+ – χ– give

(67)

and

(68)

Hence it follows that the evolutionary condition (48)for the shock wave propagating along the string in theelectric regime (χ– < 0 and χ+ < 0) again requires thefulfillment of inequalities (65) and (66). In principle,the current χ can change its sign when passing throughthe shock front, but conditions (65) and (66) remainvalid. It follows from (65) and (66) that the sign of thecurrent increment in the shock wave Δχ depends onthe sign of the quantity

(69)

If κ is positive on both sides of the front, then Δ is alsopositive. If, alternatively, κ < 0, then the current incre�ment will be negative, Δχ < 0. Suppose that the quan�tity (69) changes its sign when passing through theshock front. Conditions (65) and (66) then will not besatisfied simultaneously, implying the impossibility ofshock waves.

8. THE CURRENT GROWTH LAW

The transonic model (24) is applicable in the rangeof currents (28), (29). Calculating (16), (22), and (63)for this model and substituting them into (69), we findthat

(70)

K '' d2K

dχ2��.=

w+2

c+2

�� 1 c+2 3 K+'( )

2K+K+''–

K+2

��χ+Δχ.–=

3 K–'( )2

K–K–''–[ ]Δχ 0>

3 K+'( )2

K+K+''–[ ]Δχ 0.>

w–2

c–2

�� 1 1

c–2

��K–K–'' 3 K–'( )

2–

K–2

��χ–Δχ+=

w+2

c+2

�� 1 1

c+2

��K+K+'' 3 K+'( )

2–

K+2

��χ+Δχ.–=

κ 3 K '( )2 KK ''.–=

κ 1

m2 χ+( )m2��, χ m2

2��.–>=



Similarly, for the polynomial model (25), given(30) and (31), we have

(71)

For the inverse�ratio model (26), given (32) and(33), we obtain

(72)

Finally, for the most complex logarithmic model(27), our calculation gives

(73)

κ 4m*4

m*2 2χ–( )

4��, m*

2

2�� χ m*

2

2�� .> >=

κ 10 m*2 χ+( )

2

m*8

��, χ m*2

5��.–>=

κ 3

m*4

��.=

It follows from (70)–(73) that κ is positive in theentire range of admissible currents χ. Given (65), (66),and (69), this implies that only the shock waves inwhich

Δχ > 0 (74)

irrespective of the sign of the current χ are evolution�ary. Thus, the current jump in the shock wave Δχ =χ+ – χ– is always positive. Condition (74) may becalled the current growth law. Note that the derivationand rigorous justification of condition (74) are givenhere for the first time, although the current growth canbe detected in the shock waves computed by numeri�cally simulating the equations of motion [13, 14]. In

the latter case, a growth in parameter n = K wasobserved in the magnetic regime. This is equivalentprecisely to the growth in χ, because K(χ) is a strictlyincreasing function for all string models (24)–(27).

χ

5

0.2

0.6

30 1Δχ

42

(c)

0.4

c+2

w+2

0.8

1.0

5

0.2

0.6

30 1Δχ

42

(d)

0.4

c+2

w+2

0.8

1.0

50.2

0.6

30 1Δχ

42

(a)

0.4

c+2

w+2

0.8

1.0

0.10

0.2

0.6

0.060 0.02Δχ

0.080.04

(b)

0.4

w−2

c−2

c+2

w+2

0.8

w−2

c−2

w−2

c−2

c−2

w−2

Fig. 2. Speed w and speed of sound c behind (index +) and ahead of (index –) the shock front versus current increment Δχ in themagnetic regime: (a) for the transonic model (24), (b) for the polynomial model (25), (c) for the inverse�ratio model (26), and(d) for the logarithmic model (27).

62


TROYAN, VLASOV

Just as the change in the density of matter in a shockwave propagating through a continuous medium, thecurrent increment Δχ, in principle, can be negative,but it turned out that κ > 0 for all string models (24)–(27) and, hence, always Δχ > 0.

As we have already seen, no discontinuities canexist for the linear model of a superconducting stringwith Lagrangian (6), because there is no dispersion ofthe speed of sound (23) that is always constant andequal to c = 1.

9. NUMERICAL ANALYSIS

It is easy to verify that both K and its derivative (22)are always positive for all models of superconductingstrings (24)–(27). Therefore, in view of the currentgrowth law (74), the following condition is always met:

(75)K+ K–.>

If the string ahead of the shock front is in the mag�netic regime (χ– > 0), then, in view of the currentgrowth law (74), χ+ > 0 always behind the front and thestring will remain in the magnetic regime. In this case,there are no restrictions on the shock wave amplitude;only the polynomial model (30) applicable at χ <

/2 constitutes an exception.

From (58) and (59), we find that in the magneticregime

(76)

Given condition (75), this implies that

(77)

When χ– 0, the speeds of sound (28), (30), (32),and (34) will approach the limit c– 1, while,according to (58) and (59), w– 1. The results of

m*2

w–2

w+2

��K+

2

K–2

��.=

w– w+.>

0.05

1.2

0.6

0.030 0.01Δχ

0.040.02

(c)

0.4

c+2 w+

2

1.0

0.10

0.2

0.6

0.060 0.02Δχ

0.080.04

(d)

0.4

c+2

w+2

0.8

0.2

0.8

1.0

1.2

0.5

0.6

0.30 0.1Δχ

0.40.2

(a)

0.4

c+2

w+2

1.0

0.10

0.2

0.6

0.060 0.02Δχ

0.080.04

(b)

0.4

c+2 w+

2

0.80.8

1.0

1.2

c−2

w−2

w−2

c−2

w−2

c−2 c−

2w−

2

Fig. 3. The same dependences as those in Fig. 2 calculated in the electric regime at χ– = –0.4 (a), ⎯0.09 (b), ⎯0.04 (c), and⎯0.08 (d).



our numerical calculation based on Eqs. (58) and (59)for shock waves with an arbitrary intensity are shownin Fig. 2. The speed w and the speed of sound c werecalculated at m = 1 and m∗ = 0.5; the initial current

was chosen to be χ– = 0.4 (except for χ– = 0.025 whencomputing model (25), which is applicable when χ <0.125).

In the acoustic limit, when Δχ � χ–, the solutioncoincides with the approximate expressions (62) and(64).

If, alternatively, the string ahead of the shock frontis in the electric regime χ– < 0, then, in view of the cur�rent growth law (74), the current behind the shockfront may turn out to be χ > 0, i.e., a changeover fromthe electric regime to the magnetic one will occur. Inthis case, the current jump in the shock wave Δχ is nolonger small compared to the value of itself andthe analytic expansion of Eqs. (58)–(61) into a seriesin small parameter is inadmissible. The numericalsolution of Eqs. (58) and (59) is shown in Fig. 3. Thespeed w and the speed of sound c were calculated atm = 1 and m∗ = 0.5; the initial current was taken to be

close to the maximum admissible values for eachmodel (29), (31), (33), and (35). A stable shock waveexists only at Δχ < . If the string behind the shockfront were in the magnetic regime with χ+ > 0, then thecondition w+ > c+ would necessarily be met, which isin conflict with the evolutionary condition (48). Inaddition, the speed of sound c+ would be higher thanthe speed of light. Thus, no changeover from the elec�tric regime to the magnetic one is possible—the shockwave immediately loses its stability.

Let us analyze the transition from the electricregime to the magnetic one. For this purpose, we setχ– = χ1 < 0 and χ+ = 0 in formula (60) and χ– = 0 andχ+ = χ2 > 0 in formula (59). This gives

(78)

From (60) and (61), we find that in the electricregime

(79)

whence, given condition (75), we have

(80)

In the frame of reference in which the shock wavepropagates with the speed w– relative to the stationarystring, behind the shock front there will be a motionwith the relative speed

(81)

χ–

χ–

w–2

w+2

�� K–2 χ1( )K+

2χ2( ).=

w–2

w+2

��K–

2

K+2

��,=

w– w+.<

w0w– w+–

1 w–w+–��.=

According to inequality (80), the speed w0 is negative,implying the motion in a direction opposite to thedirection of shock wave propagation. In continuummechanics [30], as in the magnetic regime of super�conducting strings, the reverse inequality (77) isalways valid, implying that the medium behind thefront will be dragged behind the shock wave motion.However, this by no means rules out the existence ofshock waves in the electric regime unless the currentjump Δχ in the shock wave exceeds the value of itself, which corresponds to the maximum possiblecurrent behind the front χ+ 0. In the latter case,the speed of sound tends to the limit c+ 1, whilethe speed w+ 1, according to (60) and (61). Allfour models of superconducting strings (29), (31),(33), and (35) in the electric regime have a limitedapplication at a current χ that does not exceed in mag�

nitude a certain value (m2/2, /2, /5, and /3,respectively). Thus, the evolutionary condition in theelectric regime does not forbid finite�amplitude shockwaves.

It is important to note that the constraint on theamplitude of shock waves in the electric regime isdetermined by requirement (19). The latter defines therange of currents at which the stability of the string perse rather than the stability of discontinuities isretained. Figures 2 and 3 show a qualitative similaritybetween the shock waves considered in terms of allfour models (24)–(27). In this case, the following ine�qualities are valid:

(82)

in the magnetic regime and

(83)

in the electric regime.In the original analysis [12], it was postulated that

the current discontinuities in the electric regime willbe unstable in view of inequality (83) and the negativesign of speed (81), which is atypical of the theory ofshock waves [23]. In fact, Eqs. (81) and (83) have noinfluence on the evolutionary condition (48) and,hence, cannot forbid the shock waves in the electricregime.

Nevertheless, the numerical solution of the equa�tions of motion [13, 14] revealed no shock waves in theelectric regime—smooth current jumps without anyemergence of discontinuities were present instead ofshock waves. This cannot be explained based exclu�sively on the equations of motion within the stringworld sheet (12) without allowance for the equationsof motion for the world sheet itself (11). The evolu�tionary condition necessary for the existence of ashock wave may turn out to be insufficient if the shockwave is sensitive to the so�called corrugation instability[26, 27, 31]. In the case of cosmic strings, the latter isgenerated precisely by the wiggles of the world sheet.

χ–

m*2 m*

2 m*2

w– c– c+ w+> > >

c+ w+ w– c–> > >

64


TROYAN, VLASOV

10. CORRUGATION INSTABILITY

When a shock wave propagates along a cosmicstring, the states behind and ahead of the shock frontare uniquely defined by the equations, while the cur�rent undergoes a jump from χ– to χ+. This implies thatthe propagation speed of the wiggles (17) will also

undergo an abrupt change from to . The equa�tions of motion for the string world sheet itself (11) aresignificantly different from the “intrinsic” equationsof state (12) and the result of a change in the speed ofthe transverse world sheet oscillations will be not theemergence of a “transverse” discontinuity but achange in the radius of curvature [13, 14]. Since thestability of a cosmic string depends on its geometricconfiguration [10, 11], the reasonable question arisesas to how the geometric string characteristics affect thestability of current discontinuities. Formulas (58),(59) and (60), (61) uniquely define a discontinuoussolution and the current growth law (74) is a necessarycondition for its stability. We need to find a sufficientcondition. For this purpose, we will have to investigatethe instability of the discontinuous solution withrespect to oscillations of the string world sheet.

It is well known from the physics of shock wavesthat, although the evolutionary condition (48) is met,the shock front can be subjected to corrugation insta�bility [26, 27, 31]. The corrugation instability is theinstability of discontinuities with respect to the pertur�bations propagating in a direction transverse to theshock wave motion. This instability manifests itself inoscillations of the shock front geometry and the per�turbation produced at the front continues to emitoscillations, carrying away energy from the shockwave. This leads to a nonstationary, expanding withtime, transition region from the – state to the + state,i.e., to smearing of the shock front, which turns into asmooth jump. In the three�dimensional or two�dimensional formulation of the problem (e.g., as in thecase of membranes), the instability will manifest itselfin the emergence of ripples or corrugation at the shock

c–⊥ c+

⊥

wave surface. For cosmic strings, the perturbations ofthe string world sheet (wiggles) similar to the trans�verse oscillations (17) can serve as an analog of thecorrugation instability of a shock wave. Obviously, toretain the instability, these perturbations should notbecame independent and propagate from the shockfront to infinity. By analogy with the corrugation insta�bility of shock waves in continuous media [26, 27, 31],we will perform a qualitative analysis for shock waveson cosmic strings.

Let us again pass to the frame of reference associ�ated with the shock front. In this frame of reference,the front is stationary, while the speeds ahead of andbehind the front are w– and w+, respectively. Considerthe perturbations of the string world sheet produced bythe shock wave itself and propagating from the front intwo opposite directions, as shown in Fig. 4. The wig�gles of the string world sheet traveling with the speed

(84)

or

(85)

emerge at the leading edge of the shock front. The wig�gles of the string world sheet traveling with the speed

(86)

or

(87)

emerge at the trailing edge of the shock front.

Note that the relativistic transformations (84)–(87) are similar to Eqs. (42)–(45) for longitudinalacoustic perturbations (also in the frame of referenceassociated with the shock front). While there is apurely formal similarity, the formulas have a com�pletely different physical meaning. The acoustic per�turbations (42)–(45) determine the number of inde�pendent parameters in the equation of motion withinthe string world sheet (12) and the evolutionary condi�tion (48) is a necessary condition for the existence of aunique solution. Small oscillations of the string worldsheet do not determine a new evolutionary condition,because the states behind and ahead of the shock fronthave already been uniquely defined by Eqs. (56) and(57). Therefore, there are no constraints on the signsof speeds (84)–(87). Nevertheless, the relations

C–⊥ w– c–

⊥+

1 w–c–⊥+

��=

C–⊥ w– c–

⊥–

1 w–c–⊥–

��=

C+⊥ w+ c+

⊥+

1 w+c+⊥+

��=

C+⊥ w+ c+

⊥–

1 w+c+⊥–

��=

w−

c−⊥

w+

c+⊥c−

⊥

c+⊥

Fig. 4. Direction of the speed of matter w and the propaga�tion speed of the string world sheet wiggles c⊥ ahead of (theindex –) and behind (the index +) the shock front.



between the speeds define the conditions for the emer�gence of corrugation instability

What will happen at < < 0? The perturba�tion emerged at the leading edge of the shock front and

traveling with the speed will become independent,because the perturbation arriving from the “rear” with

the speed will not be able to overtake it and, hence,will not affect it. Thus, the shock front will actuallydivide the string into two independent regions. In con�trast to the corrugation instability of a shock wave in acontinuous medium [26, 27, 31], the energy of theshock wave under consideration will go not into theacoustic oscillations but into the oscillations of thestring world sheet, which, however, does not changethe result. The current discontinuity will become arbi�trarily small and, having expended its energy, theshock wave will cease to exist. Lest this happen, theinequality

> (88)

must hold. A similar situation of instability arises at

< , when the wiggles arrived from infinity wherethe shock wave (with the index –) has not yet passedcannot overtake the wiggles emerged at the trailingedge of the shock front (with the index +). Here, thestability criterion will be the condition

> . (89)

Thus, inequalities (88) and (89) are sufficient condi�tions for the stability of a shock wave on a supercon�ducting cosmic string.

C–⊥

C+⊥

C–⊥

C+⊥

C–⊥

C+⊥

C–⊥ C+

⊥

C–⊥ C+

⊥

The propagation speed of the wiggles (17) is calcu�lated from the formulas [17]

(90)

(91)

In the transonic model (24) the velocities of the wig�gles and jiggles are

c⊥ = c, (92)

while the wiggles for all the remaining nonlinear mod�els (25)–(27) are supersonic [16, 17, 20]: c⊥ > c. Theplot of c⊥ against χ is shown in Fig. 5 (the calculationwas performed at m = 1 and m∗ = 0.5). The depen�

dence c⊥(χ) has the same behavior in all models: amonotonic increase at χ < 0 and a decrease at χ > 0.

Given the current growth law in the shock wave (74),we have χ+ > χ–. Therefore, the following inequalitiesalways hold:

(93)

in the magnetic regime and

(94)

in the electric regime.It follows from (82) and (93) that in the magnetic

regime w– + > w+ + and condition (89) isalways valid. In contrast, condition (88) requires amore careful check. According to (92), condition (88)for the transonic model (24) is equivalent to the ine�

c⊥ Λ χ/K+Λ

��, χ 0,>=

c⊥ ΛΛ χ/K+��, χ 0.<=

c–⊥ c+

⊥>

c–⊥ c+

⊥<

c–⊥ c+

⊥

0.98

0−0.04χ

0.02−0.02

c⊥

0.96

1.00

0.04

Fig. 5. Speed c⊥ versus current χ. The solid, dashed, anddotted curves correspond to models (27), (26), and (25),respectively.

1.000004

0.4Δχ

0.60.2

C–⊥ /C+

⊥

1.0000000

1.000008

1.000012

Fig. 6. Ratio / versus current jump Δχ+ in the shock

wave in the magnetic regime for χ– 0–. The designa�tions of the curves are the same as those in Fig. 5. Model (25)shown by the dotted curve is applicable only for Δχ < 0.125.

C–⊥

C+⊥

66


TROYAN, VLASOV

quality w– – c– > w+ – c+. However, according to(82), in the magnetic regime w– > c–, while w+ < c+

and, hence, condition (88) is met. For the remainingthree string models (25)–(27), condition (88) in themagnetic regime is also met, which is verified by ourcalculation (see Fig. 6).

In the electric regime, as follows from (83) and(94), condition (89) is never satisfied. This implies thatno shock waves are possible in the electric regime. Thebreaking of inequality (89) in the electric regime, infact, expresses the breaking of the relationshipbetween the oscillations of the string world sheet ondifferent sides of the discontinuity. This explains whythe external geometry of the string observed in thenumerical solution of the equations of motion in theelectric regime changes [13, 14]. If, however, a currentdiscontinuity is artificially produced in the electricregime, then this discontinuity will be unstable, itsenergy will begin to dissipate by going into the wiggles,and the discontinuity will disappear.

11. CONCLUSIONS

Thus, the longitudinal perturbations within theworld sheet of a superconducting cosmic string can giverise to stable discontinuities similar to shock waves.Shock waves are inherent in all string models (24)–(27), which can have an essentially nonlinear depen�dence of the Lagrangian Λ on current χ and generalizethe linear model (6) originally proposed to describesuperconducting strings [3]. The linear model (6)turns out to be insufficient for the detection of shockwaves, because the speed of the longitudinal perturba�tions within the string world sheet is constant andequal to the speed of light.

Stable discontinuities satisfy the evolutionary con�dition (48). The latter requires that the speed of theshock wave be supersonic (w– > c–), while the flowspeed behind the shock front be subsonic (w+ < c+),just as in continuum mechanics. The propagationspeed of the shock wave w– is defined by Eqs. (58) and(59) in the magnetic regime (in the case of space�likecurrents χ > 0) or by Eqs. (60) and (61) in the electricregime (in the case of time�like currents χ < 0). Thespeeds of sound ahead of and behind the discontinuityfront (c– and c+), along with the parameter K(χ) cal�culated from Eq. (16), are uniquely defined via theLagrangian of a superconducting string (24)–(27).The acoustic limits for low�intensity shock waves aredefined by Eqs. (62), (64) and (67), (68), respectively.

For all of the existing superconducting string mod�els (24)–(27), the evolutionary condition (74) requiresa current growth in the shock wave Δχ > 0 in both mag�netic and electric regimes. It is important to empha�size that the current growth law does not forbid theexistence of shock waves in the electric regime,although condition (74) imposes significant con�straints on the currents ahead of and behind the shockwave.

The reason why no shock waves are possible in theelectric regime is the instability with respect to pertur�bations of the string world sheet, which is similar to thecorrugation instability of shock waves in continuousmedia. This imposes the additional requirements (88)and (89) on the propagation speed of the transverseperturbations (90) and (91). In the magnetic regime,both conditions (88) and (89) are valid and the shockwaves can propagate without any restrictions. In theelectric regime, no shock waves satisfying inequality (89)can exist. If, however, a current discontinuity is artifi�cially produced in the electric regime of the string,then the energy will be emitted in the form of stringworld sheet oscillations, the discontinuity will beunstable and will turn into a continuous jump.

Although the equations for the speeds of shockwaves on strings (58) and (59) do not depend on geo�metric factors, the appearance of current discontinui�ties can lead to an energy loss in the form of transverseoscillations (wiggles) and can cause a change in thestring configuration. In this case, the string emission[32] will be a consequence important for observationsand the calculation is complicated significantly in thecase of collisions between strings with different cur�rents. The problem of a string under the action of anexternal electromagnetic field [33] and the emergenceof shock waves on such a string is of independent inter�est. However, these questions are beyond the scope ofthis paper and require a separate consideration. Fur�ther studies of shock waves and related phenomenawill provide a better knowledge of the dynamics ofsuperconducting cosmic strings. This is of greatimportance in astrophysical applications, includingthe search for the sources of gamma�ray bursts and theobservational data needed to test the hypotheses ofmodern cosmology.

ACKNOWLEDGMENTS

We are grateful to L.I. Menshikov and K.V. Stepan�yants for constructive discussions.

REFERENCES

1. T. W. B. Kibble, Phys. Rep. 67, 183 (1980). 2. A. Vilenkin, Phys. Rep. 121, 264 (1985). 3. E. Witten, Nucl. Phys. B 249, 557 (1985). 4. V. Berezinsky, B. Hnatyk, and A. Vilenkin, Phys. Rev.

D: Part. Fields 64, 043004 (2001). 5. F. Ferrer and T. Vachaspati, Int. J. Mod. Phys. D 16,

2399 (2006). 6. O. S. Sazhina, M. V. Sazhin, and V. N. Sementsov, Zh.

Eksp. Teor. Fiz. 133 (5), 1005 (2008) [JETP 106 (5),878 (2008)].

7. D. N. Spergel, T. Piran, and J. Goodman, Nucl. Phys.B 291, 847 (1987).

8. E. Copeland, M. Hindmarsh, and N. Turok, Phys. Rev.Lett. 58, 1910 (1987).

9. B. Carter, Phys. Lett. B 228, 466 (1989).



10. Y. Lemperiere and E. P. S. Shellard, Nucl. Phys. B 649,511 (2003).

11. M. Lilley, X. Martin, and P. Peter, Phys. Rev. D: Part.Fields 79, 103514 (2009).

12. G. V. Vlasov, arXiv:hep�th/9905040. 13. X. Martin and P. Peter, Phys. Rev. D: Part. Fields 61,

43510 (2000). 14. A. Cordero�Cid, X. Martin, and P. Peter, Phys. Rev. D:

Part. Fields 65, 083522 (2002). 15. B. Carter, Phys. Lett. B 224, 61 (1989). 16. B. Carter and P. Peter, Phys. Rev. D: Part. Fields 52,

1744 (1995). 17. B. Carter, Ann. Phys. (Weinheim) 9, 247 (2000). 18. N. K. Neilsen and P. Olsen, Nucl. Phys. B 291, 829

(1987). 19. R. Battye and P. Sutcliffe, Phys. Rev. D: Part. Fields 80,

085024 (2009). 20. B. Hartman and B. Carter, Phys. Rev. D: Part. Fields

77, 103516 (2008). 21. R. Courant and K. Friedrichs, Supersonic Flow and

Shock Waves (Interscience, New York, 1948; Inostran�naya Literatura, Moscow, 1950, p. 105).

22. L. D. Landau and E. M. Lifshitz, Course of TheoreticalPhysics, Vol. 6: Fluid Mechanics (Nauka, Moscow,1986, p. 529; Butterworth–Heinemann, Oxford, 1987).

23. L. D. Landau and E. M. Lifshitz, Course of TheoreticalPhysics, Vol. 6: Fluid Mechanics (Nauka, Moscow,1986, p. 466; Butterworth–Heinemann, Oxford, 1987).

24. K. S. Thorne, Astrophys. J. 179, 897 (1973). 25. V. I. Zhdanov and P. V. Titarenko, Zh. Eksp. Teor. Fiz.

114 (3), 881 (1998) [JETP 87 (3), 478 (1998)]. 26. S. P. D’yakov, Zh. Eksp. Teor. Fiz. 27, 288 (1954). 27. N. M. Kuznetsov, Usp. Fiz. Nauk 159 (3), 493 (1989)

[Sov. Phys.—Usp. 32 (11), 993 (1989)]. 28. A. H. Taub, Annu. Rev. Fluid Mech. 10, 301 (1978). 29. Yu. V. Vlasov, Zh. Eksp. Teor. Fiz. 111 (4), 1320 (1997)

[JETP 84 (4), 729 (1997)]. 30. L. D. Landau and E. M. Lifshitz, Course of Theoretical

Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow,1986, p. 463; Butterworth–Heinemann, Oxford, 1987).

31. K. A. Bugaev, M. I. Gorenshtein, and V. I. Zhdanov,Teor. Mat. Fiz. 80 (1), 138 (1989) [Theor. Math. Phys.80 (1), 767 (1989)].

32. D. V. Gal’tsov, Yu. V. Grats, and A. B. Lavrent’ev,Pis’ma Zh. Eksp. Teor. Fiz. 59 (6), 359 (1994) [JETPLett. 59 (6), 385 (1994)].

33. P. O. Kazinski, Zh. Eksp. Teor. Fiz. 128 (2), 312 (2005)[JETP 101 (2), 270 (2005)].

Translated by V. Astakhov

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Current discontinuities on superconducting cosmic strings