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REPORTS

ON

MATHEMATICAL

LENS EFFECT

No. 3

PHYSICS

IN A KERR FIELD

JERZY PLUTA Institute

of Fundamental

Technological (Received

Research, December

Polish Academy

of Sciences, Warsaw

7, 1989)

A ray of light passing through a gravitational field of a rotating black a deflection. The equations describing an asymptotic shape of its trajectory for a ray passing in the plane 0 = 7c/2, next for the general case. An image seen by the distant observer (the gravitational lens effect) is described

hole experiences are derived, first of a light source in both cases.

1. Trajectories of light rays passing in the plane 8 = 742 The path

of a light ray is determined

by the eikonal

equation:

IC/.&qg@ = 0. In Boyer-Lindquist ds2 = g,gdx”dxB

coordinates =

(

1 -y

(1)

the metric around a Kerr black hole is (with c = 1):

>

dt2-;dr2-zd@22r ra .

sin2 8 sin2 tldcp2+ ---Bs1n20dcpdt, c

(2)

where A = r2-rTgr+a 2, C = r2 + a2 cos20, rg = 2Gm. Because an asymptotic shape of trajectories is considered, in all further calculations I will keep the terms containing rJr and a/r in at most first power. Due to the general procedure for solving the eikonal equation, we look for $ in the form: $ = --t+Mq+$,(r). Combining

equations

(lH3)

we calculate

(3)

$,. Its asymptotic

t,b,(r) = +5$“(r)+Wrgarcoshi-_

shape (for r + rJ is:

l-f, F

where p = M/o. The first term in (4) corresponds to a straight line r = /?/cos cp, passing a distance p from the origin, the second one- to a relativistic correction

C2891

at in

290

J.

PLUTA

a centrally symmetric gravitational field, while the third term is related rotation of a black hole. The trajectory of the ray is described by the equation: ati aM = const. So finally, for the ray of light passing

from infinity

to the

(5) to infinity

we obtain:

(6) On Fig. 1 the broken line-

line corresponds

to a = 0 (Schwarzschild), 2ar to a # 0 (Kerr). The angle a equals --A P’ *

big.

while the solid

1

2. The image of a light source seen by the observer (for the light rays passing in the plane 8 = 42 only) We look for the images 0, and 0, seen by the observer values 6, and 6, (Fig. 2). Because

in R, defined by the

RB -OBSERVER

Fig. 2

LENS

EFFECT

IN A KERR

FIELD

291

and

(see

Fig. 2), then from (6):

(7)

61.2 =

To find PI and p2 let us notice

(Fig. 2), that A&

(8)

= R,

where A#) is a change of angle cp from R, to R,. On the other hand, from (4), (5) and (6) we obtain: (9) where --

aAlp dM

(see Fig. 2). Combining

=

n-PI.2

f+f ( .A

B>

(8) and (9) we obtain:

The solutions to the equations are given by:

(lo), consistent

with our asymptotic

2r,R*RB+~ Pl.2 = J

RA+RB -2’

approximation,

(11)

Then from (7) and’(11) it follows that the images of light source tire given in R, by: (12) Knowing the values S,, 6,, R, and R,, from equations (12) we can calculate the mass and the angular momentum a of the black hole.

m

3. Trajectories of light rays in the general case As in the first paragraph, to determine the trajectory equation (1) and we look for $ in the form:

of the ray we use the eikonal

J. PLUTA

292 ti = -

ot + Mv + h(r) + tide)

(13)

(as it was shown by Carter Cl], this method is successful). The asymptotic forms of $Jr) and II/Jo) that we obtain from calculations

are: (14)

(15) where x =

s+ 2@? and K is a new constant of motion specified in [l], [2]. J Comparing (14) with (4) we can say that 3t corresponds to a distance between the ray and the origin. The trajectory of the ray can be found by solving the equation

alC/ _ !?!!!I+!!!!! = const

aK_ aK aK

(16)

’

where

wr aK -

1

r9

-arccosX-2cX.t

+

r

aDr, fm4

1-g

2cux=

\i-

I-$,

(17)

$_

ah =-20x

_-

1

(

cOse

arccos

--

82

7[: 2

.

(18)

> 1-2 $_

It is complicated to find the analytic form of 0(r) in the general case, but it is easy to calculate it for the case /I = 0, i.e. the ray of light pointed at the symmetry axis, and to generalize it afterwards to all other cases. For the ray passing from infinity to infinity we obtain: (a) from (16H18) delBXO =

do,+%,

(19)

where de, corresponds to a straight line. So in the plane 8, the ray passing at a distance x from the centre experiences a deflection like in a centrally symmetric gravitational field. (b) from (5), (17) ‘and (18)

&l/J=0 = 2ar,sin-le x2

7

(20)

LENS

EFFECT

IN A KERR

FIELD

293

where 8’ is the final value of 6 for this ray. So in the plane cpthe same ray experiences a deflection related to the rotation of the black hole. Comparing the results obtained in the expressions (6), (19) and (20) we can generalize them to any other case. Looking at Fig. 3 we have 0 -light-- source, C-centre of the field, P, -plane 0 = n/2, P, -plane enclosing axis OC, OC I AB, P, -initial plane of motion of the ray, CI- angle between P, and P,, Z-normal vector to P,, Z1-its component orthogonal to both vector d and AB; le',l = ~0~acos0, e', -its component orthogonal to 2,; (&,I= (sin20-t +cos20 sin2a)‘j2.

Fig. 3

The deflection of the ray related to the rotation of the black hole can be devided into two parts. The part which lies in the plane 8 is proportional to le’,l, while the part which lies in the plane cp- to lZ21.So the final asymptotic values of 9’ and Acpfor the ray with the initial values 8 and c1(Fig. 3) passing at a distance x from the origin, are (a) the classical ray 8’=

n-8,

(21)

294

J. PLUTA

Aq=O, (b) the Schwarzschild

(22)

metric 8’ = n-O+?sinar,

(23)

Aq = scosccsin-lO’, x

(24)

(c) the Kerr metric 8’=

A&5 x

x-

0+3 x

(

sincc+acoscrcos8 x

(cosa+~ x

, >

sin20+cos20sin2a

sin-l8’. >

(25)

(26)

4. The image of a light source seen by the observer in the general case The angle between the axis light source (R,J- observer (I&) and the symmetry axis is 8, (0, > z/2). Modifying expression (12) we obtain (Fig. 4) (M, N-images of the light source-see Fig. 2)

Fig. 4

6

J

25

.R, 4sinfAd

lv2= R,+R, R,+

2R, '

(27)

a 1~0s e,l

11,2=

R,.

6,,2

’

128)

LENS

From

EFFECT

IN A KERR

FIELD

295

(27) and (28) it follows that

Knowing calculate

the values 6 1, 6,, R,, R, and y, from the equations (27) and (29) we can the mass m and the angular momentum a of the Kerr black hole.

I am greatly indebted

to Prof. Marek Demiariski for his help and encouragement.

REFERENCES [l]

Carter, B.: Phys. Rev. 174 (1968), 1559.

[2]

Demiahski,

M.: Relativistic

Astrophysics, PWN,

Warszawa

1985.

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