The gravitational faraday rotation for cylindrical gravitational solitons

General Relativity and Gravitation, Vol. 21, No. 6, 1989

The Gravitational Faraday Rotation for Cylindrical Gravitational Solitons

Akira T o m i m a t s u I

Received May 3, 1988

By using cylindrical soliton solutions, the nonlinear behavior of gravitational waves in vacuum is studied in terms of a rotation of the polarization vector between two independent modes. This effect was emphasized by Piran, Sailer, and Stark as a gravitational analogue of the Faraday rotation. The polarization of the soliton wave is calculated, and it is found that the + mode which was dominant near the axis of symmetry is fully converted to the x mode at some interaction region and then the disturbance propagating along a light cone like a gravitational wave pulse contains both polarizations.

1. INTRODUCTION

Gravitational solitons in general relativity are found in the framework of the inverse-scattering problem technique [ 1 ]. It is of some physical interest to study their behavior which is due to the nonlinear nature of gravitational interaction. For example, the plane-symmetric soliton wave solution attracted much attention in the context of inhomogeneous anistropic cosmology 1-23, because it represents a nonlinear process of generation of gravitational waves from initial inhomogeneities.

This paper is motivated by a nonlinear interaction between two independent polarizations of cylindrical waves. By using an analytic approximation and a numerical calculation, Piran, Sailer, and Stark [3] showed the occurrence of a conversion of the + mode to the x mode, and vice versa. If an outgoing (or ingoing) cylindrical wave is linearly polarized, its polarization vector rotates as it propagates. This is a gravitational analogue of a phenomenon known as the Faraday rotation,

1 Department of Physics, Nagoya University, Nagoya 464-01, Japan.

613

0001-7701/89/0600-0613506.00/0 �9 1989 Plenum Publishing Corporation

614 Tomimatsu

but ingoing (or outgoing) x waves play the role of both plasma and magnetic field.

For the general cylindrical Kompaneets-Jordan-Ehlers line element the wave amplitudes corresponding to the + and x waves emitted from the axis of symmetry and those incident on it are defined in Ref. 3. Following such definitions, we calculate the respective wave amplitudes of the cylindrical soliton field. The polarization angles of outgoing and ingoing waves are introduced in terms of the ratio of the x wave amplitudes to the + ones. The soliton solution describes the collision of outgoing and ingoing waves in some finite region. Our purpose is to observe how their polarization angles change in the interaction region.

The single-soliton solution is studied in Section 2. The soliton disturbance exists only in the interior of a future light cone which can be seen as a shock front [1, 4]. The rotation between two polarizations proceeds as follows: although near the axis the + mode is dominant, at some intermediate region the x mode becomes dominant, and finally, at the wave front the outgoing wave contains both polarizations. In order to avoid the shock-like structure at the wave front, in Section 3 we consider the two- soliton solution. As the expressions of the wave amplitudes become very complicated, it is useful to make a special choice of the parameters. Then the solution looks like a single-soliton field. It is verified that the soliton field is a typical example of the gravitational field of which the nonlinear interaction causes a full rotation of the polarization vector of the wave. In Section 4 some discussion is added to the two-soliton solution.

2. POLARIZATION OF THE SINGLE SOLITON

Let us apply the so-called soliton transformation of Belinskii and Zakharov [1 ] to the cylindrical Minkowski metric. The cylindrical soliton solutions derived by this procedure can be written in the Kompaneets- Jordan-Ehlers form

d s 2 = e 2 ~ - q ' ) ( - d t 2 + d r 2 ) + e Z q ' ( d z +codq~)2 +r2e-2OdqJ 2 (1)

where three functions ~, co, and ~ depend on r and t only. The metric coefficients of the single-soliton solution are

e2~ �9 = w ( a 2 + 1)/(a z + w 2)

r = ra(1 -- w Z ) / w ( a 2 + 1)

e2~ = brl /2/( t 2 _ r2)1/2

(2)

(3)

(4)

Gravitational Faraday Rotation for Gravitational Solitons 615

where a and b are arbitrary constants and

w = { t - (t 2 - r2)'/2}/r (5)

These expressions of the metric coefficients are valid only at the interior of the future light cone t = r. (Here we are not concerned with the behavior in the past region t < 0.) In the exterior Of the light cone no disturbance of the gravitational field is present, i.e., ~p = c o = 7 =0. Thus the continuous functions ~ and co suffer discontinuities of their first derivatives on the light cone like gravitational shock waves.

From the position of the extremum of (2) one may interpret the soliton field as a disturbance propagating along the timelike world line W 2 = a2, i.e.,

r = 2 [at t / (a 2 + 1) (6)

(Because the range of w is 0 < w < 1 at the region t > r, the extremum exists only when a2< 1.) In any cylindrical system the gravitational energy can be defined in terms of a C-energy flux vector pu which satisfies the conser- vation law V~P~=0 ['5]. For the line element (1) the derivative Y,r plays the role of the C-energy density ~ P'. The single-soliton solution gives

~,r = ( t2 + ra ) /4r ( t 2 - r 2) (7)

The C-energy density is not concentrated on the world line (6) of the extremum. As shown in the following, it represents the propagation of a gravitational disturbance with the pure x mode.

Let us introduce the ingoing and outgoing coordinates defined by

u = ( t - r ) /2 (8)

v = (t + r ) /2 (9)

The metric coefficient y satisfies the equations r

(10)

where

r 2 + A 2 _ B 2) y,t = ~ (A2+ - B + (11)

A+ =2~'v (12)

B+ =2O.u (13)

A • = e2C'oo,v/r (14)

B• = e2Oco../r (15)

616 Tomimatsu

Therefore the quantities

A -- (A2+ + A 2 ),/2, B - (B2+ + B 2 )1/2 (16)

can be interpreted as the ingoing and outgoing wave amplitudes. The indices + and • denote the respective polarizations, and the respective polarization angles 0A and 0~ are defined by

tan 2O a = A • / A + , tan 2O n = B • / B + (17)

By substituting (2) and (3) into (12)-(15), the single-soliton solution gives

1 { u ' r a2--W 2 A+ = r \ v J a ~ - 5 (18)

1 ( v ) 1/2 a 2 - w 2 B + = - - r \ u } a-5-~w 2 (19)

l ( u ] '/2 2aw (20) A• =-~ \-~1 a 2 + w 2

1(/)~ 1/2 2aw (21) B• =rku/ a 2+w 2

The wave amplitudes

A = = - (22) r r

do not depend on the parameter a, which is related only to the polarization angles as

tan 0~ = - t a n 0B = w/a (23)

The C-energy density is shared between the ingoing and the outgoing waves at the ratio of u2/v 2, while it is shared between the + and the x waves at the ratio of (a 2 - w2)2/4a2w 2.

Let us see how 0B ( = -0A) changes along a null ray u = Uo > 0. At the axis r = 0 (i.e., w = 0) no x mode is present, i.e., tan 0B=0. The pure + wave emitted from the axis interacts with the ingoing x wave A• to generate B• When the null ray crosses the timelike world line (6), B+ vanishes completely, i.e., Itan 0B[ = 1. After that (w > [al) a part of B • is converted to B+ in order to amplify A• of ingoing waves propagating


toward the world line (6). The wave amplitudes become smaller as v becomes larger;

A v ~ " ul/2/Id3/2' B ~ o o > 1/(UoV) 1/2 (24)

The nonlinear interaction must stop in the limit v --* oo (i.e., w ~ 1), when 0" approaches the value

tan 0 , = - 1 / a (25)

The same behavior is also seen for the ingoing waves. As A vanishes on the light cone u = 0, no ingoing wave crosses it. Then

the polarization vector of the outgoing wave along u = 0 does not rotate at all and keeps the phase angle identical with (25). If lal ~ 1, both ingoing and outgoing waves contain little amount of the x mode near the boun- daries r = 0 (w = 0) and u = 0 (w = 1) of the interaction region. However, unless a = 0 , a complete conversion of A+ and B+ to A• and B• can occur through their propagation into the interaction region. This is a significant feature of the single-soliton field.

It should be also noted that the outgoing wave amplitude B becomes infinitely large at the wave front u = 0. For the cosmological soliton waves [2] the largest portion of the disturbance was shown to propagate asymptotically at the velocity of light. For the single soliton discussed here

i t s infinite C-energy density which should correspond to the largest disturbance lies on the light cone (implying speed-of-light propagation). This portion of the outgoing radiation behaves like a gravitational wave pulse propagating freely (with a fixed polarization vector), and some interaction region remains behind the wave front. The behavior shown here of the + and x waves in the interaction region may be due to a peculiar effect produced by the infinite wave amplitude at u = 0. To discuss this point, in the next section we consider the two-soliton solution free from the singular wave front.

3. ANALYSIS OF THE T W O - S O L I T O N S O L U T I O N

The cylindrical two-soliton solution is generated from the Minkowski metric via the soliton transformation with two pole trajectories. To make the metric coefficients real and to avoid their discontinuities at light cones, we choose the pole trajectories as

#1 = fi2 = iq -- t + [ ( iq -- t) 2 -- r 2 ] 1/2 (26)

618 Tomimatsu

where q is a real parameter and an overbar denotes complex conjugation. Then by means of purely algebraic operations in which a complex parameter a and a complex function

w = - # l / r (27)

are introduced, we obtain the solution

e 2q' = Iw] 2 F IG (28)

oo = - r H / F (29) where

a 2 + 1 2 ( [al2 q- 11)2 (30) F= ~ -\lwL2_

a2+w22 (la[~2 + ]--w12~ 2 (31) G= ~ - \ Lwl2_l ]

= 2 R e )'~ ( a 2 + l [a[2+ 1 ' ~ (32) H [W \ w 2 - 1 ~w-~---]/J

Because of the presence of q in w the metric coefficients are written in the above-mentioned form even at the region r > t. Both ~ and co approach zero at spacelike infinity t ,~ r ~ ~ , where [wl --* 1.

It is now straightforward to calculate the respective wave components A +, B+, A • B • but in general they are contaminated by the terms due to some interaction between two separated solitons and their expressions are rather complicated. Fortunately a suitable choice of the complex parameter a can make the calculation simpler. Let us take a = ~. Then the solution is regarded as a complete overlapping of two solitons and behaves like a single-soliton field which suffers some modification due to the elimination of the shocklike structure. We obtain the wave amplitudes A, B and the polarization angles 0A, On in forms similar to (22) and (23):

2 ( 1 - [ w l z) 2 (1-1wl 2) A = B = (33)

r l1 +w] 2 ' r l1 - w ] 2

tan 0A = - t a n 0 8 = ]wlZ/a (34)

The function Iwl 2 runs from zero to unity along an outgoing null ray u = Uo > 0. Therefore, if la[ < 1, the + mode is completely converted to the x mode on the timelike world line Iwl 2 = a , i.e.,

2 la[ ,/2 { (lal + 1-]2 q2"~'/2 r = l a l + l t2 + (35)

\ l a l - - l J J


This trajectory resembles (6) at late times, when the pure • wave propagates at a constant velocity. Any results obtained in the previous section are not essentially altered so far as the behavior of the polarization is concerned.

Let us discuss the ingoing and outgoing wave amplitudes A and B. At the initial time t = 0 we have

w = - - q + + 1 i r \ r

(36)

because Iw] must approach zero near the axis. (Without loss of generality we can assume q > 0.) Then the initial amplitudes are

A = B = 2q/r(r 2 + q2)1/2 (37)

The disturbance is localized in the region r ~< q. At late times t~>q we consider three asymptotic regions. ( i )A t

timelike infinity t ~> r we have

t - -*- 1 - - + 0 ( t , B--* 2 r l r

which approach twice the respective wave amplitudes (22) of the single soliton. (ii) At spacelike infinity t ~ r we have

A ~- B ~- 2q/r 2 (39)

The initial distribution (37) (r ~> q) still remains in this region. (iii) At null infinity v ~ ~ we have

A - {2u + (4u 2 + q2)1/2} 1/2/r3/2 (40)

B "~ 2{2u + (4u 2 + q2)1/2} m/(4u2 + q2)1/2 rl/2 (41)

Instead of the singular behavior of (22) at u = 0, we find that at u = q/2 x f 3 the C-energy density (7,r ~ rB2) has a local maximum which should correspond to a gravitational wave pulse propagat ing at the velocity of light. Its polarization angle 08 is now fixed to the value tan 08 = - l / a , although the + mode was dominant when it was emitted from the axis. The two-soliton solution can illustrate clearly a process of generation of the gravitational wave pulse from the initial disturbance localized near the axis as well as a full rotation of its polarization vector in the interaction region.

620 Tomimatsu

4. D I S C U S S I O N

Let us extend our investigation to the general case a v~ 4 for which the two-soliton field should cause the rotation between two polarizations in a more complicated manner. For example, the trajectory of A + = 0 does not coincide with that of B§ =0 . However, for the case la[ ~ 1 the difference between the two trajectories is rather negligible. To understand this, it is convenient to consider the region such that

[w[ 2 = O(la[) ~ 1 (42)

because both A+ and B+ are expected to vanish there. [ I f a = & we obtain the trajectory I w[2= l al .] By virtue of (42) we can derive the approximate forms of A+ and B§ which are found to be simultaneously zero at the world line

{Im(ar~)} 2 = [w[ 4 (Im w) 2 (43)

At the initial time w becomes pure imaginary [see (36)], and then (43) reduces to

Iwl 2 = IRe al (44)

which is identical with the case a = 4 I-(35) at t = 0]. However, unless a is real, at sufficient late times such that Jim wl ~ IwL the solution of (43) becomes inconsistent with the condition (42). This suggests that the world line on which A§ = 0 or B§ = 0 must stop at a late time. All of the + mode of one of the two solitons may be converted to the x mode at some region, but the wave there is contaminated by the + mode of the other. The disappearance of the pure x wave will happen when the two solitons are well separated. To arrive at a more definite conclusion on the behavior of the polarization for the general case a ~ 4, it is necessary to analyze the two-soliton field in detail.

There exists some difference between the two cases a = 4 and a ~ 4 on the space-time structure at the axis too. The metric coefficient 7 can be written as

e2r = -- (Iwl 2 -- 1) 2 F/(lal 2 + 1) 2 (45)

in which ? approaches zero at spacelike infinity, but near the axis r ~ 0 it goes like

e2~ ~ q2r2/(t2 + q2)2 (46)


for a = 8, and

e 2~ --* 4(Im a)2/(]al 2 + 1) 2 (47)

for a r Only the solution corresponding to the case a C a becomes regular at the axis. However, for both the cases the disturbance near the axis does not decay even in the limit t-~ ~ . (Except the case a2= - 1 , for which the space-time becomes flat everywhere, 7 does not approach zero at the axis.) This implies that some (nonstationary) matter field must be present there as a possible source of the two-soliton field.

If one considers a four-soliton solution, it is possible to impose the condition that the disturbance near the axis decays with the lapse of time, leaving gravitational waves propagating along light cones. In this soliton field also the x mode will be dominant at an intermediate region. In the context of plane-symmetric cosmological waves the collision of a four- soliton wave and another gravitational wave pulse was studied [6]. We can treat the same process in terms of cylindrical waves. An interesting problem is whether a large part of the + mode of the nonsoliton wave can be converted to the x mode when it crosses the soliton wave. This remains for future investigation.

ACKNOWLEDGMENT

This work was partially supported by Grant-in-Aid for Scientific Research 62540279 from the Ministry of Education, Science and Culture.

REFERENCES

1. Belinskii, V. A., and Zakharov, E. E. (1979). Soy. Phys. JETP, 49, 985. 2. Belinskii, V. A., and Fargion, D. (1980). Nuovo Cim., 59B, 143; Carr, B.J., and

Verdaguer, E. (1983). Phys. Rev., D28, 2995. 3. Piran, T., and Sailer, P. N. (1985). Nature, 318, 27l; Piran, T., Sailer, P.N., and Stark,

R. F. (1985). Phys. Rev., D32, 3101. 4. Belinskii, V. A., and Francaviglia, M. (1982). Gen. ReL Gray., 14, 213. 5. Thorne, K. S. (1965). Phys. Rev., 138, B251. 6. Cespedes, J., and Verdaguer, E. (1987). Phys. Rev., D36, 2259.

842/2~/6-6

Documents

The gravitational faraday rotation for cylindrical gravitational solitons