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Fields of massless rigidly rotating charged dust (*)

Y. A. ABD-ELTWAB

Mathematics Department, Faculty of Science, South Valley University - Sohag, Egypt (ricevuto il 30 Luglio 1996; approvato il 4 Marzo 1997)

Summary. — A well-behaved class of solutions of the Einstein-Maxwell equations is

obtained when the source of the field is a massless rigidly rotating charged dust. The class of solutions possesses vanishing Riemann scalar, R 40 and nonvanishing Ricci tensor. A particular Som and Raychaudhuri solution is considered as an example, where some conservative quantities are discussed via a Ricci collineation vector field. Finally, it is shown that the energy density is conserved on some cylindrical surfaces in the local coordinate system.

PACS 04.20.Cv – Fundamental problems and general formalism. PACS 04.20 – Classical general relativity.

1. – Introduction

In 1980, Bonnor could obtain the general solution of the Einstein-Maxwell equations in the presence of axially symmetric distribution of charged dust in rigid rotating motion when the Lorentz force vanishes [1]. Bonnor began with the most general form of the axially symmetric metric [2]:

ds2 4 2em

(

( dx1₎2 1 ( dx2)2

₎

2 l( dx3)2 2 2 m dx3dx4 1 R (1.1)

where m, l, m and f are functions of x1and x2only. Bonnor assumed steadily rotating dust which means that

Ui

4 ( 0 , 0 , U3, U4_{) , A}

i4 ( 0 , 0 , c , f) , Ji4 ( 0 , 0 , J3, J4) ,

(1.2)

where Ui_{, A}

i and Ji are the four-velocity, the four-potential and the four-current,

respectively. All the components in (1.2) being functions of x1and x2only. Besides, the assumption (1.2) means that motion and currents parallel to the x1_{-axis are excluded.}

The Einstein-Maxwell equations for charged dust are (see ref. [2])

Ri k2 1 2d i kR 48pruiuk1 2 FiaFka2 1 2d i kFabFab, (1.3)

(*) The author of this paper has agreed to not receive the proofs for correction.

where Fabis the electromagnetic field tensor, defined by

Fab4 Ai , k2 Ak , i, F; bab4 4 pJa,

(1.4)

Rab denotes the Ricci tensor of the space-time and r is the matter density. A comma

denotes partial differentation and semicolon denotes covariant differentiation. If we let

lf 1m2

4 (x2)2_,

(1.5)

then the field equations (1.3) with (1.2) imply

R331 R444 0 , (1.6) R111 R224 28 pemr , (1.7) R114 1 (x2₎2

m

f

kg

¯c ¯x2

h

2 2

g

¯c ¯x1

h

2

l

1 l

kg

¯f ¯x1

h

2 2

g

¯f ¯x2

h

2

l

1 (1.8) 12 m

k

¯f ¯x2 ¯c ¯x2 2 ¯f ¯x1 ¯c ¯x1

l

}

2 4 pe m_{r ,} R224 2R112 8 pemr , (1.9) R124 2 (x2₎2

m

l ¯f ¯x1 ¯f ¯x2 2 f ¯c ¯x1 ¯c ¯x2 2 m

k

¯f ¯x1 ¯c ¯x2 1 ¯f ¯x2 ¯c ¯x1

ln

, (1.10) R442 R332 2 mF21R434 (1.11) 4 22 (x2₎2e m

m

_f21

k

₍

_m2 1 (x2)2

₎

g

¯f ¯x1

h

2 1

g

¯f ¯x2

h

2

l

1 2 m

k

¯f ¯x1 ¯c ¯x1 1 ¯f ¯x2 ¯c ¯x2

l

1 1f

kg

¯f ¯x1

h

2 1

g

¯c ¯x2

h

2

l

}

1 8 pr(u3u32 u 4 u31 2 mf21u 3 u4) , R434 2 (x2₎2e m

m

m

kg

¯f ¯x1

h

2 1

g

¯f ¯x2

h

2

l

1 f

k

¯f ¯x1 ¯c ¯x1 1 ¯f ¯x2 ¯c ¯x2

ln

2 8 pru 3 u4. (1.12)

Besides, eqs. (1.4) give

f

y

¯ 2_c ¯(x1)2 1 ¯2c ¯(x2)2 2 1 x2 ¯c ¯x2

z

1 m

y

¯2f ¯(x1)2 1 ¯2f ¯(x2)2 2 1 x2 ¯f ¯x2

z

1 (1.13) 1 ¯f ¯x1 ¯c ¯x1 1 ¯f ¯x2 ¯c ¯x2 1 ¯m ¯x1 ¯f ¯x1 1 ¯m ¯x2 ¯f ¯x2 4 4 p(x 2 )2em_J3 , m

y

¯ 2_c ¯(x1)2 1 ¯2c ¯(x2)2 2 1 x2 ¯c ¯x2

z

2 l

y

¯2f ¯(x1)2 1 ¯2f ¯(x2)2 2 1 x2 ¯f ¯x2

z

1 (1.14) 1 ¯m ¯x1 ¯c ¯x1 1 ¯m ¯x2 ¯c ¯x2 2 ¯l ¯x1 ¯f ¯x1 2 ¯l ¯x2 ¯f ¯x2 4 4 p(x 2₎2_em_J4_.

Equations (1.6)-(1.14) yield a well-behaved class of solutions with suitable compatibility conditions (see ref. [1]).

2. – Rigidly rotating massless charged dust

In this section I developed a solution with vanishing Riemann tensor and nonvanishing Ricci tensor via the assumption that the rigidly rotating charged dust is massless. First, let us write the Einstein-Maxwell equations for charged dust in the form [3] Ri k2 1 2d i kR 428p

k

ruiUk2 1 4 p

g

1 2d i k2 uiuk

h

(E21 B2) 2 (2.1) 2 1 4 p(E i_E k1 BiBk) 2 1 4 p(U i_S k1 UiSk)

z

,

where Sk_{, E}k _{and B}k _{are the Poynting vector, the electric field, and the}

magnetic-induction vector, respectively.

If we assume the current flow such that

Ji_{4 sU}i, (2.2)

where s is the charge density, taking the divergence of (2.1) yields

rUi_{4 sE}i. (2.3)

For massless charged dust (r 40) we get

Ei

4 0 , i 41, 2, 3 .

(2.4)

That is, the field of massless dust is just pure magnetic field. Second, in case when r 40 the field equations (1.7)-(1.12) become

R111 R224 0 , (2.5) R112 R224 2(x2)22

kg

¯c ¯x2

h

2 1

g

¯c ¯x1

h

2

l

, (2.6) R124 22(x2)22f ¯c ¯x1 ¯c ¯x2 , (2.7) R4 42 R332 2 mf21R434 2(x2)22e2mf

kg

¯c ¯x2

h

2 1

g

¯c ¯x1

h

2

l

, (2.8) R434 0 . (2.9)

The contravariant metric tensor associated with (1.1) is gmn 4

.

`

`

`

´

2e2m 0 0 0 0 2e2m 0 0 0 0 2f lf 1m2 2m lf 1m2 0 0 2m lf 1m2 l lf 1m2

ˆ

`

`

`

˜

. (2.10)

Equations (2.5), (2.6) and (2.10) yield

R1 14 g11R114 2(x2)22e2m

kg

¯c ¯x2

h

2 1

g

¯c ¯x1

h

2

l

, (2.11) R2 24 g22R224 e2m

g

2(x2)22

kg

¯c ¯x2

h

2 1

g

¯c ¯x1

h

2

lh

. (2.12)

Equations (2.11) and (2.12) imply

R111 R224 0 .

(2.13)

From (1.6) and (2.13) we get

R 4R111 R221 R331 R444 0 .

(2.4)

Hence I obtained the following theorem:

Theorem (2.1): An axially symmetric metric of a rigidly rotating charged massless

dust possesses zero Riemannian scalar. The associated field of the matter content is just pure electric field. Besides, if the origin of the comoving coordinate system and the boundary of the manifold are excluded, then the Ricci tensor does not vanish.

Using the definition of Ricci tensor and eqs. (1.6)-(1.12) we get

R114 f (x2)2

kg

¯c ¯x2

h

2 2

g

¯c ¯x1

h

2

l

4 2R22, (2.15) R334 gi3R3i4 g33R331 g43R344 lfe2m (x2₎2

kg

¯c ¯x2

h

2

g

¯c ¯x1

hl

, (2.16) R444 gi4R4i4 g34R431 g44R444 f2_e2m (x2)2

kg

¯c ¯x2

h

2 2

g

¯c ¯x1

h

2

l

, (2.17) R124 gi1R2i4 g11R214 22 f (x2₎2

g

¯c ¯x1

hg

¯c ¯x2

h

, (2.18) R134 gi1R3i4 0 , (2.19) R144 2 gi1F4 agisFstgta4 0 , (2.20)

R234 2 gi2F3 agisFstgta4 2 l (lf 1m2_)(x2₎2

g

¯c ¯x2

hg

c x2 1 ¯c ¯x2

h

, (2.21) R244 2 gi2F4 agisFstgta4 2 m (lf 1m2_)(x2₎2

g

¯c ¯x2

hg

c x2 1 ¯c ¯x2

h

, (2.22) R344 2 gi3F4 agisFstgta4 2 g33F4 ag3 sFstgta1 2 g43F4 ag4 sFstgta4 (2.23) 4 2 g33F4 ag33F3 tgta1 2 g33F4 ag34F4 tgta1 2 g43F4 ag43F3 tgta1 2 g43F4 ag44F4 tgta4 4 2 g33F4 ag33F34g4 a1 2 g33F4 ag34(F42g2 a1 F43g3 a) 1 12 g43F4 ag43F34g4 a1 2 g43F4 ag44(F42g2 a1 F43g3 a) 4 2 m (lf 1m2₎

g

¯c ¯x2

hg

c x2 1 ¯c ¯x2

h

.

3. – A Ricci collineation vector field for massless rigidly rotating charged dust

If a Riemann space admits a vector zi _{such that}

LzRih4 Rmn , lzl1 Rlnzl, m1 Rmlzl, n4 0

(3.1)

holds, we say that the Riemann space admits a “Ricci collineation” (RC), where Lz

denotes the Lie derivative with respect to the vector zi_[4].

Using the Ricci tensor components (2.15)-(2.23) and the definition (3.1) we get

z1, 14 0 , (3.2a) R22z2, 21 R32z3, 21 R42z4, 24 0 , (3.2b) R33 , 2z21 2(R23z2, 3R33z2, 31 R43z4, 3) 40 , (3.2c) R44 , 2z21 2(R24z2, 4R34z2, 41 R44z4, 4) 40 , (3.2d) R22z2, 11 R32z3, 11 R42z4, 11 R11z1, 24 0 , (3.2e) R23z2, 11 R33zl, 11 R43z4, 11 R11z1, 34 0 , (3.2f ) R24z2, 11 R34zl, 11 R44z4, 11 R11z1, 44 0 , (3.2g) R23 , 2z21 R23z, 22 1 R33z3, 21 R43z4, 21 R22z2, 31 R23z3, 31 R24z4, 34 0 , (3.2h) R24 , 2z21 R24z, 22 1 R22z2, 41 R23z3, 41 R24z4, 41 R34z31 R44zl, 24 0 , (3.2i) R34 , 2z21 R24z, 32 1 R34z3, 31 R44z4, 31 R32z2, 41 R33z3, 41 R34z4, 44 0 . (3.2j)

Substituting the Ricci tensor components of any of Bonnor’s solutions with zero mass into eq. (3.2), one would solve these equations to obtain a well-behaved RC vector. As an example, I consider the Som and Raychaudhuri solution in cylindrical symmetry coordinate system and only longitudinal magnetic field as the source of the

metric. It is an example of Bonnor’s work (see ref. [5]). For this solution we have v 4k1(x2)2, (3.3) c 4k2(x2)2, (3.4) m 4 (4k222 k12)(x2)2, (3.5) r 4 e 2m p

g

1 2k 2 12 k22

h

, (3.6) s 4 e 2m p k1k2, (3.7) f 4const41 (say ). (3.8)

Massless charged dust requires 4 k2 24 2 k124 2 k2. (3.9) Since f 41 then, m 4k(x2)2, (3.10) m 4v4k(x2₎2_, (3.11) c 6 1 k2k(x 2₎2_, (3.12) l 4 (x2)2 2 k2(x2)44 (x2)2

(

1 2k2(x2)2

)

. (3.13)

Substituting from (3.10)-(3.13) into eq. (3.2) we get

R114 2R224 2 k2, (3.14) R334 2 k2(x2)2

(

1 2k2(x2)2

)

e2k 2_(x2₎2 , (3.15) R444 22 k2e2k 2_(x2₎2 , (3.16) R124 R134 R144 0 , (3.17) R234 6 k2x2

(

1 2k2(x2)2

)

, (3.18) R244 6 k3x2, (3.19) R344 9 k3(x2)2. (3.20)

Using (3.14)-(3.20) into eqs. (3.2) we get z1, 14 0 , (3.21) 22 k2z2 , 21 6

(

1 2k2(x2)2

)

k2(x2)2z3, 21 6 k3x2z4, 24 0 , (3.22)

(

2 k6(x2)5_{2 6 k}4(x2)3_{1 2 k}2x2

)

e2k2(x2₎2 z2_{1 6 k}2x2

(

1 2k2(x2)2

)

z2, 31 (3.23) 12 k2(x2₎2

₍

1 2k2_(x2₎2

₎

_e2k2(x2)2_z3 , 31 9 k3(x2)2z4, 34 0 , 2 k4_x2_e2k2(x2)2_z2 1 6 k3x2_z2 , 41 9 k3(x2)2z3, 42 2 k2e2k 2_(x2₎2 z4 , 44 0 , (3.24) 22 k2z2 , 11 6

(

1 2k2(x2)2

)

k2(x2) z3, 11 6 k3x2z4, 11 2 k2z1, 24 0 , (3.25) 6 k2_x2

₍

1 2k2_(x2₎2

₎

_z2 , 11 2 k2(x2)2

(

1 2k2(x2)2

)

e2k 2_(x2₎2 z3 , 11 (3.26) 19 k3(x2)2z4, 11 2 k2z1, 34 0 , 6 k3_x2_z2 , 11 9 k3(x2)2z3, 12 k2e2k 2_(x2₎2 z4 , 11 2 k2z1, 44 0 , (3.27) 6 k2

₍

1 23k2_(x2₎2

₎

_z2 1 6 k2x2

₍

1 2k2_(x2₎2

₎

_z2 , 21 6 k2x2

(

1 2k2(x2)2

)

z3, 32 (3.28) 22 k2z2 , 31 2 k2(x2)2

(

1 2k2(x2)2

)

e2k 2_(x2₎2 z3 , 21 9 k3(x2)2z, 24 1 6 k3x2z4, 34 0 , 6 k3z2_{1 6 k}3x2z2, 21 9 k3(x2)2z3, 22 2 k2e2k 2_(x2₎2 z4, 22 (3.29) 22 k2z2 , 41 6 k2x2

(

1 2k2(x2)2

)

z3, 41 6 k3x2z4, 44 0 , 18 k3_x2_z2 1 6 k3x2_z2 , 31 9 k3(x2)2z3, 32 2 k2e2k 2_(x2₎2 z4 , 31 (3.30) 16 k2x2

(

1 2k2(x2)2

)

z2, 41 2 k2(x2)2

(

1 2k2(x2)2

)

e2k 2_(x2₎2 z3, 41 9 k 3 (x2)2z4, 44 0 .

Since the line element (1.1) is symmetric with respect to the x1_{-axis and is stationary,}

one may take

z1 4 0 , (3.31) za , 14 0 , (3.32) za , 44 0 . (3.33)

The system (3.21)-(3.30) is, thus, reduced to

(

1 2k2(x2)2

)

x2z3, 21 kz4, 24 0 , (3.34) 2

(

1 2k2_(x2₎2

₎

_e2k2(x2)2_z3 , 31 9 kz4, 34 0 , (3.35) 6

(

1 2k2_(x2₎2

₎

_z3 , 311 2(x2)2

(

1 2k2(x2)2

)

e2k 2_{) x}2₎2 z3 , 21 9 kx2z4, 21 6 kz4, 34 0 , (3.36) 9 k(x2)2z3, 32 2 e2k 2_(x2₎2 z4, 34 0 , (3.37)

which has a solution of the form z1_{4 z}2_{4 0 ,} (3.38) z3₄ C

(

1 1 (3/2) x 3

₎

k2x2

(

1 2k2(x2)2

)

( 2 e2k2(x2₎2 2 9 ) , (3.39) z4₄ 2Cx 3 ( 2 e2k2(x2₎2 2 9 ) 2 6 C

(

1 1 (3/2) x3

)

6 k3_x2_{( 2 e}2k2(x2)2 2 9 ) . (3.40)

4. – Conservative quantities in the zero-mass solution of Som and Raychaudhuri

First I mention the following known theorem (see ref. [4]):

Theorem: If a space-time V4with R 40 and Ri jc0 admits an RC, then there exists

a covariant conservative law generator of the form ( g1 /2Tm

l zl), m4 0 ,

(4.1)

where g 4NDet gi jN , zl is defined by LzRi j4 0 and Tlm4 Tnlgnm with Tnl being the

energy-momentum tensor. From (2.15) we get g1 /2₄ e 2k2(x2₎2 x2 . (4.2)

In general relativity, the energy-momentum tensor of a massless electromagnetic field is given by (see ref. [2])

Trs4 1 4 p

k

1 4grsfabf ab 2 fra fsa

l

, (4.3) where fra fsa4 1 2fabfmg( g am_db rdns1 gbndardms) , (4.4)

and frais the electromagnetic-field–strength tensor.

Substituting (3.38)-(3.40) and (4.2)-(4.4) into (4.1) we obtain the four conservative quantities which are corresponding to the values m 41, 2, 3, and 4.

Example (4.1): On the hypersurfaces x2_{(4 r) 4const, provided this constant is}

real and does not equal zero or 61/k, we obtain

z1_{4 z}2_{4 0 ,} (4.5) z3 4 C1x31 C2, (4.6) z4_{4 C}3x31 C4. (4.7)

The coefficients C1, C2, C3and C4are constants. For m 43, (4.1) yields T3 3 , 3z31 T33C11 T4 , 33 z41 T43C34 const , (4.8) From (5.3) we get T334 2 1 8 p(x 2₎2

₍

1 2k2_(x2₎2

₎

_H2_, (4.9) T344 2 k 8 p(x 2₎2_H2_, (4.10) T444 1 8 p(x 2₎2_H2_. (4.11) Also, we have T3 34 21 (x2₎2T332 kT34, (4.12) T3 44 21 (x2₎2T342 kT44. (4.13)

Substituting (4.9)-(4.11) into (4.12) and (4.13) we get

T334 H2 8 p , (4.14) T3 44 0 . (4.15)

Using (1.1) and (3.12) we obtain

C1H2 8 p 4 const or , H2 4 const . Since Ei

4 0 , the energy density is conserved on the coaxial cylindrical surfaces (in the local coordinates) which are not singular relative to the vector za given by (3.38)-(3.40).

R E F E R E N C E S

[1] BONNORW. B., J. Phys. A, 13 (1980) 3465.

[2] CARMELJM., Classical Field: General Relatvity and Gauge Theory (John Wiley & Sons, Inc.) 1982.

[3] RAYCHAUDHURIA. K., J. Phys. A, 15 (1982) 831.

[4] KATZING. H., LEVINEJ. and DAVISW. R., J. Math. Phys., 10 (1969) 4. [5] SOMM. and RAYCHAUDHURIA. K., Proc. R. Soc. London, Ser. A, 304 (1968) 8.