The integrals in formula (3) do not depend on the number of periods N, consequently, the radiation spectrum of the undulator for N >> i coincides with the spectrum of an infinite indulator everywhere, with the exception of a neighborhood of the frequencies ~in and m2n. In the ultrarelativistic case for $11 + i the frequency of the upper boundary ~2n grows proportionally to the square of the energy, while the lower boundary is fixed ~in % n~0/2. The ground energy is radiated at frequencies m >> Wln, therefore for ultrarelativistic particles in (3) one can let U(xln) = i.

i.

2.

.

4.

.

LITERATURE CITED

N. A. Korkhmazyan and S. S. ~ibanyan, Dokl. Akad. Nauk SSSR, 203, 791-793 (1972). A. F. Medvedev, V. Ya. Epp, M. L. Shinkeev, and V. F. Zalmezh, Nucl. Instrm. Methods, A308, 124-127 (1991). L. D. Landau and E. M. Lifshits, Field Theory [in Russian], Nauka5 Moscow (1973). M. M. Nikitin and V. Ya. Epp, Undulatory Radiation [in Russian], Energoatomizdat, Moscow (1988). M. A. Abramovits and I. Stigan, Handbook of Special Functions [in Russian], Nauka, Moscow (1979).

COSMIC STRINGS IN A SPACE WITH TOPOLOGY M2 • V2

Yu. V. Grats UDC 523.11

For the case of a space-time with topology M 2 • V 2, we show that the equation of motion of an infinitely thin string has solutions that can naturally be called the laws of motion of a straight string. For the case of a locally plane conical space, the solutions are written explicitly. We show that such motions of a test string are not followed by gravitational radiation.

Recent increased interest in cosmic strings is due to a number of reasons. One of the main reasons is the role which cosmic strings may have played in the formation of the struc- ture of the universe [1-4]. The possibility for strings, created immediately after the Big Bang, to survive up to the present stimulated a search for their observable manifestations, and, as a consequence, the study of the behavioral characteristics of classical and quantum matter in conical spaces.

In quantum theory, the conical characteristics of space lead to vacuum polarization of quantum fields [5-8]. As is well known, the vacuum polarization effect is a consequence of a deformation of the standard solutions of the corresponding classical wave equation, which appears during changes in the space-time topology. Therefore, considering pure classical processes in which particles creating their own fields participate, it is natural to expect that the influence of the global structure of space-time can be significant. Actually, on the one hand, since the space~ime of a straight cosmic string locally coincides with Minkowski space, the Newtonian potential for the string equals zero, and a particle does not feel a force due to the string. On the other hand, for the case when a particle has its own field, the process becomes nonlocal since the particle field is determined by the structure of the manifold as a whole, and this should leave its mark on the dynamics of particles and on classical radiation processes. An example of such an effect is the phenomenon of repulsion of a test charge by a cosmic string found by Linet [9]. It was shown in [i0] that, under geodesic motion in a conical space, a retarded Green function acquires a nonvanishing radia- tion part and the particle begins to radiate scalar, electromagnetic, and gravitational waves.

If we consider a collision of two parallel strings, one of which is a test string, then every element of the latter will move due to the same law that governs a test particle.

M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. ii, pp. 75-79, November, 1991.

IO10 0038-5697/91/3411-1010512.50 �9 1992 Plenum Publishing Corporation

Moreover, although the test particle radiates gravitational waves, the radiation of the strings is absent [i0]. This result has a simple explanation. Indeed, in the case of parallel strings, the situation is invariant with respect to translations along their common direction. Therefore, radiation should take place in a direction perpendicular to the strings. Then we come to the problem concerning gravitational radiation associated with the flight of a point-like mass in a (2 + l)-dimensional conical space-time. This reduction of a special dimension explains the absence of radiation, since in the (2 + l)-dimensional case gravitational waves simply cannot exist.

The proof given for the case of parallel strings is strongly based on the particular symmetry of the problem. The question then arises, whether the statement concerning the ab- sence of radiation will remain valid if the strings are not parallel, and also in what sense we can speak about a straight string when the space is curved.

Below, for a space-time with topology M 2 • V2, we will show that there are solutions to the equation of the string motion that generalizes the concept of a straight, moving string, and give some reasons why such motions are not accompanied by gravitational radiation.

Throughout the paper, we assume the speed of light equals unity and use the metric of Minkowski space with the signature (+ ).

MOTION OF A TEST STRING IN A SPACE WITH TOPOLOGY M 2 • V 2

Let us consider an infinitely thin test string moving in a space-time with metric g~v. During motion, the string sweeps over a two-surface (a world-sheet), the points on which are specified by introducing two parameters: the time parameter �9 and the space parameter o. The only scalar characterizing the world-sheet of the string is its area. Therefore, for the action of the string, we choose the expression (the Nambu action) [i0]

S = - - iz [ ( - - 7)I/2 dzdz, ( 1 )

where u is the determinant of the internal metric on the world-sheet. The value ~ has dimensionality mass/length and is identified as the linear mass density of the cosmic string.

Sufficiently large freedom in choosing a parametrization for the world-sheet allows us to apply the orthonormal conditions

g ~ X~ X ~ = O,

g ~ (X~X~ + X'~ r = O,

where the dot and prime mean derivatives with respect to T and o, respectively. tion of motion is then given in its simplest form

( 2 )

The e q u a -

x~ - x"~ + r ~ ( ~ ~ - x " x 'D = O. ( 3 )

Taking into account further applications, we restrict our discussion to spaces with topology M 2 x V2, where M 2 is the two-dimensional pseudo-Euclidean space (coordinates t and z), and V 2 is the two-dimensional Riemann space (coordinates x I and x2), i.e., we assume that the metric has the form

ds ~ dt 2 _ dz ~ ~(2~ = -- gab (X c) dxadxb. (4)

Here, g(b)(X c) is a metric on V2; a, b, c = i, 2.

In the case under consideration, the only nonvanishing connections are F~c (2)F~c,

where (2)Fbca denotes the two-dimensional Kronecker symbols constructed from the two-space

metric g(2~. Therefore, the law governing the string motion in the space-time (4) is deter-

mined from the solution to the system of equations

t = t"; z = z'~; x '~ - x " + c2~rL (x~ .,~ - x '~ x ' 9 = o . ( 5 )

The orthonormal conditions (2) then take the form

g ( ~ /c ~ x '~ = i t ' - z z ' ,

g~2, (x ~ x~ + x,~ x ' D (~)~ + (t,)~ (~)2 (z,)2. (6 )

i01!

The equations of motion (5) and the connection equations (6), will be satisfied if for the parameter T we choose time t, and assume that the spatial coordinates depend only on the spatial variable o, and, being functions of o, are solutions of the geodesic equation for the cross section t = const of the space-time (4):

-.- ~- (2)Fa X'O X'C t ~; z ~ c o s = + Z o ; x ~ O; x " ~ + o~ = 0 ;

gO) ..,a ~,0 ab .~ .4. ~- S| I12a. (7)

We will call the parameter ~, which defines the solution of (7), the slope angle of the test string with respect to the z axis.

Thus, in the case under consideration, we see that the equation of string motion (3) has a family of solutions, each of which is characterized by the fact that at every instant of time t the string is along some quite definite (but completely arbitrary) geodesic of the corresponding 3-space. Clearly, such solutions give us a natural generalization of the con- cept of a straight stationary string for the case of a space-time with the metric (4).

We will say that a moving string is straight (minimally stretched) if its position in space changes with time in such a way that at any instant of time t the string is located on some geodesic of a constant time cross section. The fact that such solutions certainly exist follows from the invariance of the metric coefficients (4) and the Christoffel symbols under Lorentz rotations in the (t, z)-plane. This invariance results in the fact that any solution of the system (5), satisfying the additional conditions (6), is transformed by this rotation into some new solution. If the solution (7), corresponding to a stationary string, under- goes the transformation, then, as one can easily check, a new law of motion can be obtained from (7) by using the replacements

, !

z o ~ Z o + V t , ~ , ~--+~', (8)

w h e r e V i s a s p e e d a l o n g t h e z a x i s ( a p a r a m e t e r t h a t d e t e r m i n e s L o r e n t z r o t a t i o n ) , and O! ! , a , and z~ a r e r e l a t e d , r e s p e c t i v e l y t o o, a , and z 0 t h r o u g h t h e u s u a l e q u a t i o n s f o r c h a n g - i n g l e n g t h s and a n g l e s u n d e r t h e L o r e n t z t r a n s f o r m a t i o n s :

zo = (1 - V=)~12Zo; ~ ' = ( I - - V ~ c o s 2 ~ ) ~ a ~ ;

c o s = ' = (1 - - V D ~'2 c o s = / (1 - V = c o s a =)~f2. ( 9 )

We should separately discuss the case when the string is parallel to the z axis, since at a = 0 the Lorentz transformation maps the solution (7) into itself while only the para- metrization of the world-sheet changes.

If the string remains parallel to the z axis at all instants of time, then the parameter o could be chosen in such a way that z = a and the two other spatial coordinates would not depend on o. The corresponding solution to the system of Eqs. (5) and (6) can be obtained from (7) if we exchange t with z, and also �9 with o, and then replace the trigonomic sine and cosine by the corresponding hyperbolic functions:

t = ~ c h a + t o ; z = a ; x ' a = O ;

~;,, + c~)r~ x~ x ~ = o; g ~ ,~" ,/~ = s h ~ ~ . (io)

Here, ~ is the velocity parameter and defines the speed of the test string as measured by a stationary inertial observer. Thus, if the string is parallel to the z-axis, then the seg- ment corresponding to fixed o moves at a constant speed along the geodesic V 2. Such a type of solution was discussed in [i0].

ABSENCE OF GRAVITATIONAL RADIATION DURING MOTION OF A TEST STRING IN A CONICAL SPACE

The metric for the space-time of an infinitely thin straight cosmic string in cylindri- cal coordinates has the form [2]

d s i = d t ~ _ _ d z i _ _ d r 2 _ _ r i d ~ 2 . (ii)

1012

This differs from the metric of Minkowski space only by the variational limits of the azimuthal angle ~:

O ~ 2 ~ b , b = l - - 4 G ~ , (12)

where D i s t h e l i n e a r mass d e n s i t y f o r t h e a c t i o n , a p p e a r i n g in Eq. ( 1 ) . Thus, t h e c r o s s section defined by t = const and z = const of the space-~time (ii) is a conical surface that locally coincides with the Euclidean plane. Therefore, the equation of a geodesic in the 3- space t = const is identical to the equation of a straight line in 3-dimensional Euclidean space, and so we have

z--~ (~cosct-l-z0; r2-~dg-}-o2sin~a;

tgcp= osinct/d. (13)

In this solution, the constant d represents the distance between the strings.

The law of motion governing a straight test string, which is passing by at impact dis- tance d from the z axis and at an angle a in a locally plane conical space, is obtained from (13) by substituting in (8) and (9). It has the form (the primes are omitted)

z-= ecosc, ,-kzoq- Vt; r2-- d2-}-o2sin2ct;

tgqc. --=- osin c~/d.

(14)

The solution obtained for the case of parallel strings, as discussed in [I0], exhausts all possible laws of motion in which the string is straight at all instants of time (there is no crossing on the z axis, on which the Riemann tensor of the background space-time is infinite). At a nonvanishing angle between the strings (the case ~ = 0 was considered above) and with the help of the Lorentz transformations and, possibly, world-sheet repara- metrization, we have shown that the problem reduces to the problem of a stationary straight string in the space-time (ii). This shows that such motions of a test string are not accompanied by gravitational radiation. Thus, we have succeeded in generalizing the results of i0] to the more general case of colliding straight strings. Moreover, the conclusion on the absence of gravitational radiation of a moving straight string, governed by the law (8) or (i0), is valid in all cases when the metric of the background space has the form (4). However, in the case of the space-time (Ii), these solutions are the only physically interesting solutions of the type considered, while in the general case this is not so. The question whether more solutions of Eq. (3) exist, which represent the law of motion of the straight string in space-time with the metric (4), will be discussed separately.

1 2 3 4 5 6 7. 8. 9.

i0. ii.

LITERATURE CITED

Ya. B. Zel'dovich, Mon. Not. R. Astron. Soc., 192, 663 (1980). A. Vilenkin, Phys. Rev. Lett., 46, 1169 (1981). N. Turok, Nucl. Phys., B242, 520 (1984). H. Sato, Progr. Theer. Phys., 75, 1342 (1986). T. M. Helliwell and D. A. Konkowski, Phys. Rev., D34, 1918 (1986). J. S. Dawker, Phys. Rev., D36, 3742 (1987). W. A. Hiscock, Phys. Lett., B18____88, 317 (1987). V. P. Frolov and E. M. Serbriany, Phys. Rev., D35, 3779 (1987). B. Linet, Phys. Rev., D33, 1833 (1986). A. N. Alley and D. V. Gal'tsov, Ann. Phys., 193, 142 (1989). H. B. Nielsen, P. Olesen, Nucl. Phys., 61, 45 (1973).

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