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INVERSE PROBLEM OF DYON AND MAGNETIC MONOPOLE DYNAMICS. V. K. Shchigolev. UDC 530.12:531.51. The problem of finding an electromagnetic ...
2, 3. 4.

Ya. B. Zel'dovich and I. D. Novikov, Structure and Evolution of the Universe [in Russian], Nauka, Moscow (1975). E. M. Lifshitz, Zh. Eksp. Teor. Fiz., 16, No. 7, 587 (1946). N. A. Shvetsova and V. A. Shvetsov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 3, 83 (1970).

INVERSE PROBLEM OF DYON AND MAGNETIC MONOPOLE DYNAMICS V. K. Shchigolev

UDC 530.12:531.51

The problem of finding an electromagnetic field allowing the movement of dualcharged particles on the background of an arbitrary static space-time to have a priori given properties, is considered. A solution of this problem is given, and the degree of arbitrariness of the obtained solution is found. Two particular examples of application of the general results are considered.

Classical inverse problems of dynamics [i] and their generalization for the case of relativistic movement [2-4] have been solved for netrual bodies or charged test particles. The opinion was expressed [3] that these methods can be of help in solving problems of control over the movement of particles in various devices and experimental setups. In such cases. inverse problems of dynamics of magnetically charged particles, i.e., dyons or magnetic monopoles, are especially interesting, because their solution can help constructing various focusing, accumulating and other devies in experiments searching for magnetic charge. In this paper we consider one inverse problem of dynamics of dyons (and, as a particular case, magnetic monopoles)moving in the electromagnetic field on the static spacetime background. i. Let us assume that we know (or have been given) ic dyon in a certain domain D { x ~ :

the velocity field of a relativist-

(1)

dx~/dt = V ~ (x~),

where V~(x ~) are given functions of the coordinates x~, differentiable in D. Roman indices run from 0 to 3, and Greek ones from 1 to 3. Suppose that the metric properties of space-time are described by the static interval of the type [5]:

ds 2 ~ EtjdxZdx } = h dt 2 where h(x a) and

?=~(x~) a r e given functions,

-

-

~ d x ~ d x ~,

(2)

and the speed of light c = I.

The equation describing the movement of a particle carrying the electric charge eo and magnetic charge go, has the form [6]:

mSuZ/ds ~ m (duZ/ds + F}~uJu ~) = eoFiiuj + g~Fqaj,

(3)

where m is the rest mass of the particle, u i = dxi/ds is its four velocity, Fij is the electromagnet ie field tensor, and *Fij = ( ~ - 7 ~ ) e i j k m Fkm is its dual. Assuming that the given functions V ~ from Eq. (i) are particular integrals of the equation of motion :(3) in the space (2), we reconstruct the components of the tensor F.. l], i.e ", we construct such a configuation of the electromagnetic field that allows the movement (I). 2. Let us find the general solution of the problem of Sec. i. First of all, let us introduce the vectors E~ of the electric and Ha of the magnetic filed strength [5]: Bashkir Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, NO. i, pp. 13-16, January, 1983. Original article submitted June 29, 1982.

0038-5697/83/2601-0011507.50

9 1983 Plenum Publishing Corporation

ii

and use the definition of the dual tensor *Fij, in order to write

(4)

Fo~ = -- F~o = E~; *Fo~ = -- *F~o = H~; F ~ = h -3;2 ]/Z,%~H"; * F ~ = - - h -~i~ ]/:f-=.,~eL

the three indices here and belowbeing moved by means of 7~ and ~" , ] = det(7~). 9 Besides, notice that the normalization condition for the four velocity utui = 1 and Eq. (i) imply that u~

=

(h

- -

(5)

' f ~ V ~ V ~ ) -as~ ,

and, consequently, u ~ + u~ ~. Substituting Eqs.

(4) into the equations of motion

goo gu~

(3), we have

(6)

- (eoE~ + golly) u~

~/r "f~ ds -

-

= u~ [e~ (E~ @ h -~/~ 1 / ~ - ~ V ~ t P ) -~ go (H~ -- h - ~;z p ~ i ~ ~ V~s

.

(7)

Therefore, the problem has been reduced to finding the vectors E~ and Ha, the components of the tensor Fij being found by means of Eqs' ~4). Contracting Eq. (7) with u ~ and using the condition uiu i = i, one can easily see that Eq. (6) follows from this condition and Eq. (7). Therfore, the vectors E~ and H~ can be found from the system of equations (7), and, evidently, the general solution will contain three arbitrary functions. It is convenient to solve system (7) as follows. Multiplying Eq. over a, we obtain the equation that can be satisfied by setting

(7) by V ~ and summing

(8)

eoE~ + goH~ --~- m ~'4 __ h_~; 2 l/-:~z~, ~ V~P~, It ~ ds

where pY is an arbitrary field differentiable in D. Using Eq. (8) to express, Ea, and substituting it into Eq. (7), we shall obtain the equation:

~

m ~tl~ V;~ [(e~ _.L, 6o,"-~'~l t f + tneoPv - - go uo ds

mg~

for example,

I/~,~o~V~P ~] = O,

which can be turned into an identity setting the expression in the square brackets equal to e,~qzFVv , where ? is an arbitrary regular function of x ~. Hence, we obtain the expression for HY. Substituting it into Eq. (8), we find, evidently, EY. Using the arbitrarines of p~ and introducing a new field R~== ~ V ~ @ P= , we can finally write:

E~ = m (e~ + g~)-~ (goR ~ + eo~ "~ + eoh-'/2 7o~ V - : { ~ . V ~ R O ,

(9)

H ~ = n: (e~, + g~)-i ( _ eoR~ + go~.~ + g~h-mT~'~ ]/Tia~V~R~),

where 6h

(io)

OX;s '

1

and {~} ----~- 5"~ (-- 0o7~~ 4- 0~5%~? 0~7~) , and Uo can be computed using Eq.

12

(5) 9

The general solution

(9) contains three arbitrary functions R ~. Notice that the obtained solutions of the inverse problem (9) can be evidently written in a vector form if we make use of the fact that the last term in (9) is proportional to the vector product

(v

1 / -T-::;,. ~w , R , =

• R)~.

Substituting the solutions (9) into the Maxwell equations (which is not a part of the problem set in Sec. i, but does not contradict it), one can obtain differential relations between the components of the field R ~, or between them and the sources of the electromagnetic field. Other theoretical or experimental considerations limiting the functional arbitrariness in the choice of R ~, can also be stated [3]. 3.

Consider two particular

examples of application of the general results of Sec. 2.

A. Let the velocity of a magnetic monopole {x ~} = {x, y, z} as follows:

(go) be represented

v~ -~ (x~ - 1p'~/x, v~ =