Exact Faraday Rotation in the Cylindrical EM Waves

(1)

PHYSICAL REVIEW D VOLUME 42, NUMBER 2 15JULY 1990

Exact

Faraday rotation

in

the

cylindrical

Einstein-Maxwell

waves

M.

R.

Arafah,

S.

Fakioglu, and

M.

Halilsoy'

Department

of

Physics, King Abduiaziz University, P.O.Box9028, Jeddah 2I413,Saudi Arabia (Received 25 January 1990)

We obtain the exact behavior ofthe cross-polarized cylindrical Einstein-Maxwell waves that

gen-eralizes the well-known Einstein-Rosen waves. In the presence ofthe second mode ofpolarization the outgoing waves interact with the incoming ones toexhibit an analogous effect ofthe Faraday ro-tation.

I.

INTRODUCTION

Cylindrical gravitational waves were introduced first in a historical paper' by Einstein and Rosen

(ER).

In 1957 Weber and Wheeler analyzed their properties and drew the conclusion that

ER

waves satisfy the physical reality requirements. Since then, the cylindrical gravitational waves

of

ER

have continued to remain an important to-pic in the physics literature. By this token, in a recent study ' _{a connection} _{has been established} _between _the

Faraday rotation

of

the electromagnetic polarization vec-tor in a plasma and an analogous effect in the cross-polarized waves

of

ER.

The authors based their analysis on numerical integration

of

the field equations, leaving the exact correspondence between the two phenomena open. On this account, in this paper, we provide an exact solution tothe Einstein-Maxwell (EM)equations that ex-hibits Faraday rotation and which reduce to the

ER

waves in a particular limit. Our line element that de-scribes cross-polarized

EM

waves,

ds

=e

r &'(dt dr )

e—

&(dz+—

wdP)

re—

involves the functions _g,

_y,

and w to be determined as functions

of

t and r, through the EM equations. The EM equations are well known to reduce to the pair

of

Ernst equations

(g+ris)

—

1)V

$=2V(

(P'g+s)V'q)

~

(g'+ris)

—

1}V

ri=2Vri (gVg+s)Vs)),

(2)

where _{g and} _g represent the gravitational and elec-tromagnetic complex potentials, respectively. The differential operators here act on the cylindrically symmetrical Oat space-time suitable to the description

of

the cylindrical-wave problem.

II.

THESOLUTION

Our solution makes use

of

a functional dependence be-tween g and q,namely,

g=ago

n=(1

—

a'}'"go

(3)

where a is a real constant satisfying

0~a

~

1, and go is

the Ernst potential for the vacuum fields.

By this choice, for the sake

of

an exact solution, we electrify gravity in such amanner that the total field en-ergy

(C

energy) represented by the metric function _y remains the same with the y

of

the

ER

waves. We adopt now the previously known vacuum solution to the prob-lem in the construction

of

EM fields.

For

the reason, however, connected

to

the evolution

of

the waves, we shall express the solution in a different no-tation that is more appropriate

to

the initial-value prob-lem for the cylindrical waves. Following Piran and Safier and Piran, Safier, and Stark we introduce the am-plitudes

I+

=2(of+0.

»

0+

=2(et

e2ttt e2$ (w,

+w„),

0&&

=

(w,

—

w„),

T (4)

which satisfy the following set

of

first-order equations in the ingoing u

=

—,

'(t

r)

and outgoin—_g v

=

—,

'(t +r)

coordi-nates:

I+ —0+

+,Q

Ix+ox

I

X)M

I+

—

0+

+IxOx

0+

„=

+IxOx

2f

(&)

Ix+Ox

—

_I+Ox

~

Ox,

U

=

—

Ix

0+

2T

where we have used the abbreviations

A

=J,

(r)cost+Jo(r)sint,

B

=J,

(r)cost

—

Jo(r)sint,

C

=2a

cosh[Jo(r)cost]

—(1+a

2)(1+sin2a)'

_{sinh[Jo(r)cost]}

D

=(1+a

)(1+sin a)'~ cosh[Jo(r)cost],

—

_2a_si _h[

_J

_n(R

_{)ocost]+ I}

—

_a

(7) Here,

_I+,

_x]

represents the incoming amplitude with linear (cross) polarization which reflects at

r=0

and turns into the outgoing amplitude

_0+[

_x,

.

Our exact solution is

AC

_0+—

BC

(6)

I„=

—

A

_(1+a

₂_)sina,.

_0„=

—

B

_(1+a

_)sina,

(2)

438 M. R.

.

ARAFAH,AH S.FAKIOGLU, AND M. HALILSOY 42

w ich

J

and J& are Bessel functions

of

orders

0

and 1, and

a

is the cross-polarization angle. No e

litude and the separation constants that a ear in the original

ER

solution both to euni y.

d to the pure electromagnetic

a=

=0

=

_a_{our solution re uces} _o

il t at wave solution o

f

ER

type.

It

can be checked easi _y a

~ ~

in the above solution (6) the boundary conditions at the symmetry axis

(r=0),

I+=0+, Ix=

Ox

~

are satis e' fi d

.

Furtheru our solution satisfies, at

t=0,

t

p,8 O.6 04

0-o.p 0.8 o4

I

₊

= —0+,

Ix

=Ox

and it has the property that for

(t

——

=

r

=0),

as in the case

of

a nodal point, all amplitudes reduce to zero.

III.

ROTATION OF THEPOLARIZATION VECTOR Figure 1 showss the linearly polarized

ER

amplitudes

(I+,

,

0

+)wh' h correspondsic to the values

a=,

o—

(6). In

Fig.

2 (for

a=10',

a=0.

5) and

Fi.

'g. 3

=60

a=0.

5) we display all the amp i

(for

a=,

a=

I,

O+,

Ix,

Ox

) for the indicated ranges

of

t and

r.

1' _d '

ll s mmetrical soliton solutions we define the relative polarization angle 8 in accor ance with p6.g 04. Q, Z p,0 -0~ -O,4 -o.6'

Ix

tanL9=

I+

0+

Ox

_=(1+a

_p sina C (10) ().8 o.~ 0,0 04. o.~ 0,$8])) p.oS p.p& poo pP4 -O.o O)8

j.

Z)))), 0,8 04. 0.0 04. g.8 0,(8ii p08 pp4 ppo -0.04 008 0 f8',

FIG.

1. The linearly independent polarized amplitudes the domain

(0(r

~10,0

{I+,

0+

) of Einstein and Rosen in t e

(t

&10).

I

0

)for the particular

FIG.

2. The amplitudes

(I+,

0+,

I

~,Ox

=

10 nd

a=0.

5 in the domain (0+r

(

10,0 parameters o.

=

10 an a

=

(3)

42 EXACTFARRADAY ROTATION ININ THE CYLINDRICAL . . 439 p6& 0$ 0,3 O.0 {3,Z ppi $.Z ~,0 o.s 0.6 O4 a.~ p6i( 0$ D,g p.0 0,Z 0$, ),Gi f.8 [,0i 0,8 06 p$ 0,8 p6{)) p40 0g0 000 0,Z0 040 {j60 p40 0@0 000

0)0

040

FIG.4. Th e absolute value

a=10',

a=0.

5 ue of0 for the sp p 10 0) respectively. The

x)

energy densit o these amplitudes by i _y

_y,

is expressed in terms

(z.

2

+z2

_)li2 X

By studying the ratio

~X

₌

~tanH, (13)

wherere, the function

C

is given in

ll m 1g.(4). d i 11o '

F'

nc ion

inally, in orderer toto see the ratio o

of

e e ne the total lin

r,

=(r',

2

+0' )'"

y (y2

+02

1/2

we conclude that th d

1

e ominant

c

ion a ternates betw

e resul-cross modes.

e ween the linear and the

IV. CONCLUSION

FIG.

3. Thehe

he,

amplitudes

(I

0

,

I,

O„)

fo _p

(4)

singulari-M.

R.

ARAFAH, S.FAKIOGLU, AND M. HALILSOY 42

ty, unbounded amplitudes do not develop in the cylindrical-wave problem. Instead, the amplitude pat-terns extend from the symmetry axis to infinity smoothly and with decreasing magnitudes in asymptotic regions. Although the labeling may imply the opposite, it can be observed from the general solution (6) that the linearly

polarized amplitudes

(I+,0+

)are not independent from

the cross-polarization angle

e.

For

this reason the graphs

of

(I+,

I„,

O+,

0„)

are not exactly the same as the numerical graphs

of

Piran and Safier. We recall that they had chosen particular input pulses in their analysis.

*Present address: Department of Physics, Eastern

Mediterra-nean University,

G.

Magosa Mersin 10, Turkey.

'A. Einstein and N.Rosen,

J.

Franklin Inst. 223, 43(1937).

J.

Weber and

J.

A.Wheeler, Rev. Mod.Phys. 29,509(1957). 3T. Piran and P.N.Safier, Nature (London) 318,271(1985). 4T. Piran, P.N. Safier, and

R.

F.

Stark, Phys. Rev. D 32, 3101

(1985).

5F.

J.

Ernst, Phys. Rev. 168,1415(1968). M.Halilsoy, Phys. Rev.D 39, 2172 (1989). 7F.

J.

Ernst, Phys. Rev. 167, 1175 (1967).