A C L A S S O F A X I A L L Y - S Y M M E T R I C E L E C T R O M A G N E T I C

M A S S M O D E L S

SAIBAL RAY, D I P A N K A R RAY

Department of Mathematics, Jadavpur University, Caleutta, India

and

R. N. TIWARI

Department of Mathematics, Indian Institute of Technology, Kharagpur, India

(Received 9 April, 1992)

Abstract. This paper presents a set of solutions of coupled Einstein-Maxwell equations with matter for the Levi-Civita's metric which can be interpreted as electromagnetic mass models which are extensions of the electromagnetic mass models obtained previously. It may be pointed out that electromagnetic mass models are solutions of coupled Einstein-Maxwell equations with matter where all the characteristics of matter vanish when the charge vanishes. Existence of such solution tends to confirm Lorentz's conjecture that the mass of an electron may be of purely electromagnetic origin.

1. Introduction

Exact solutions of Einstein's equations in the general relativity that include marter as

well as electromagnetic field are of some special interest in their own right. However,

a particular interest in such solutions arises from the fact that some of these solutions

have been interpreted as electromagnetic mass models, in the sense that for such

solutions all the physical characteristics of matter are dependent only on the charge

density of the electromagnetic field and vanish when the charge density vanishes. Such

solutions rend to confirm Lorentz 's conjecture that the mass of an electron may be

originating from its electromagnetic field itself. In the present work, which is a con-

tinuation of our earlier work (Tiwari et al., 1991; Tiwari and Ray, 1991a, b) we obtain

a set of solutions o f coupled Einstein-Maxwell equations containing matter which is

of electromagnetic origin. Incidentally, it may be added that the set of solutions belong

to a particular class known as W e y l - M a j u m d a r - P a p a p e t r o u class.

The work has been arranged as follows: in Section 2 the field equations are provided;

Section 3 is devoted for obtaining a class of solution; and the concluding remarks are

made in Section 4.

2. Field Equations

The Einstein-Maxwell field equations for the case of charged dust distribution are given

by

Gj: = R } - ig)R1 i = - 8~[T~ (m). + T} («m»] , (1)

( x / ~ g F'J), y = 4 ~ x ~ - g J~ (2)

Astrophysics and Space Science 199: 333-337, 1993. © 1993 Kluwer Academic Publishers. Printed in Belgium.

334

and

where

S. RAY ET AL.

F[ij, k] = 0 ,

Tj. (m) and Tj- (e'° correspond to the marter and the

(3)

electromagnetic energy-

where

and

and

V2U = 4rcae2~ 3fl (7e)

U 1 = - e - 2 f l ~ 1 (8a)

and

where

U 2 = - e - 2 / ~ ~ 2, (8b)

where the suffices 1 and 2 after the variables denote partial derivatives with respect to

the coordinates r and z, respectively.

The equations of continuity, viz., Tj; i = 0 yield

Pfi l = e a ( 2 B - «) U1 V2 U/4= (9a)

Pfl2 = e2(2/~- «) U2 V2 U / 4 ~ ,

= - + - - ( 1 0 )

r (~z 2

is the Laplacian operator in the two-dimensional cylindrical coordinates.

(9b)

m o m e n t u m tensors, respectively, and are given by

Tj (m) = p # u j (4)

and

= - - 1 i ~ r, k l l Tj(«ù, ) 1 [ _ FjkFik + ~ g ) r k / r J. (5) 4re

In view of the above, the Eins te in-Maxwel l field equations for static axially-symmetric

Levi-Civitä 's metric ds 2 = e2/~dt2 - e 21~[e2«(dr2 + dz 2) + r2dq52], (6)

where c~ and fl are functions of the coordinates r and z only, can be expressed in the form (Tiwari et al., 1991) as

V2fi = 4 n p e 2(«-~) + e2~(U~ + U~) , (7a)

0{11 q- ~22 q- f i? + f12 = f f 2 f l ( U 2 q_ U 2 2 ) , (7b)

fl~ - flä - Œ1/r = e2~(U12 - U~) , (7c)

2filf12 - ~2/r = 2e2ôU1 U2 (7d)

AXIALLY-SYMMETRIC ELECTROMAGNETIC MASS MODELS 335

3. Static Charged Dust Sources of Purely Electromagnetic Origin

Equations (8a, b) and (9a, b) immediately yield

UI _ 01 ~ 1 - (11)

Equation (11) implies that there taust exist some functional relationship between U and B (also between fi and T, and T and U). Hence, in general, we can assume

U = f ( f i ) , (12)

where f is an arbitrary function of fi. By use of relation (12), Equations (7c) and (7d) can be expressed as

and

r(1 - e : ~ f ~ ) (fi~ - fl~) = cq (13a)

2r(1 - e ; ~ f ~ ) f l l f i 2 = ca. (13b)

Furthermore, assuming that

7 = f (1 - e2'~f~) I/2 df l (14)

and if we use this relation in (13a) and (13b), we obtain

F(7 2 -- 7 2 ) = ~1 (1ga)

and

2r71 72 = ~2" (15b)

Now, (12), (15a), and (15b), when used in (7b), we have

71727 = 0. (16a)

Also, using the compatibility condition, viz., cq2 = g21 in (15a) and (15b), we get

72727 = 0. (16b)

The above two equations (16a) and (16b) together provide the following possibilities: either (i) 7~ = 72 = 0 (and, therefore, 727 = 0) or (il) 71 ¢ 22 ¢ 0, 727 = 0. We consider Case (i) only.

In view of the field equations (15a) and (15b), the first case immediately implies that

«-- constant (say, d) . (17a)

Equation (14) (alternatively (13a, b)) then provide

fB = _+e -~ . (17b)

336 s. RAY ET AL.

If we use (17b) in (8) and (12), we have

B : log(b + ~ )

where b is the constant of integration. Also, (9a) (alternatively (9b)) and (Te) with (12) and (17b), yield

0-= +--D,

which provides the equilibrium of a charged fluid in the Newtonian theory. Now, Equation (7e) is also expressible in the form

where

(17c)

(17d)

and

I 11 4~za° r n+2 + b (21c) W = _+ l (n + 2 ) 2 - "

U = ~ ( b ~ tl/) 1 = ~ e - / 3 ( 1 8 b )

If we assume U = 1 + V, Equation (18a) can be expressed in the form

V2V = +4~pe2a(1 + V) 3 , (18c)

which for small values of V reduces to Poisson equation (Majumdar, 1947) of the Newtonian theory.

Example

As an example we consider the case when «,/3, p, o, etc., are functions of the coordinate r only. Let us further assume that the charge density a satisfies the relation

cre2aU 3 = aor ' , (n > - 2 ) . (19)

Then, from Equation (18a), we get

Ull + U1/r = +4~Oo r" . (20)

This, using the condition of regularity at the origin r = 0, on integration yields,

I 4~°° 1 U = +_ l ( n + 2 ) 2 rn+2 ' (2la)

where l is an arbitrary constant. Hence, the values of/3 and ~ are obtained as

e 2 / ~ = - l o g [ l - 4 7 c a ° rn+21 (21b) (n + 2) 2

V2U = + 4~zo'e2dU 3 , (18a)

AXIALLY-SYMMETRIC ELECTROMAGNETIC MASS MODELS 337

The relativistic mass m of a cylinder of height h and radius a will be given by

[ 1-1 2rca°ha~+2 4~za° a n+z (22) m - ( n + 2 ) I ( n + 2 ) 2

which vanishes when the charge density a (i.e., %) is zero.

4. Conclusions

The solution set obtained is of purely electromagnetic origin. It is also observed that the solution of/~ in (17c) provides the Majumdar 's condition, viz.,

goo = (b + ~ ) 2 . (23)

Thus, the solution corresponds to the Weyl-Majumdar-Papapet rou- type (Weyt, 1917; Majumdar, 1947; Papapetrou, 1947) of static charged dust sources.

However, further work is needed to know whether the other case (i.e., Case (il)) also corresponds to the Weyl-Majumdar class only or not.

Refërences

Majumdar, S. D.: 1947, Phys. Rer. 72, 390. Papapetrou, A.: 1947, Proc. Roy. Irish Acad. A51, 191. Tiari, R. N. and Ray, Saibal: 1991a, Astrophys. Space Sci. 180, 143. Tiari, R. N. and Ray, Saibal: 1991b, Astrophys. Space Sci. 182, 105. Tiwari, R. N., Rao, J. R., and Ray, S.: 1991,Astrophys. Space Sci. 178, 119. Weyl, H.: 1917, Ann. der Physik 54, 117.