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Lens effect in a Kerr field
Vol. 29 (1991)
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.
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.