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Motion of test particles and photons in the gravitational field of a cosmic string
Physics Letters A 160 ( 1991 ) 119-122 North-Holland
P IdYSICS LETTERS A
Motion of test particles and photons in the gravitational field of a cosmic string A.
Banerjee and N. Banerjee
1
Relativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Calcutta 700032, India Received 29 May 1991; revised manuscript received 30 August 1991;accepted for publication 12 September 1991 Communicated by J.P. Vigier
The motion of light rays and test particles are studied in the gravitational field of a cosmic string. Using Hiscock's semiclassical cosmic string metric, it is found that some of the outgoing particles will traverse an infinite distance but some particles may be trapped by the string at a finite distance from the axis. For incoming particles also, some will reach the boundary of the string but some may face a potential barrier and cannot penetrate beyond a certain distance from the string.
1. Introduction Phase transitions o f q u a n t u m fields in the early universe might have produced some stable topological defects [ 1 ]. One of these defects, cosmic strings, have attracted a lot o f interest in recent years, mainly because o f its possible candidature as seeds for the formation of galaxies. The gravitational effect o f an infinite straight cosmic string is the removal from spacetime of a wedge of angular size 8~/z where/z is the string field energy per unit length of the string. The possible observable effect o f this way be the formation o f double images o f quasars [2]. The formation, physical properties and cosmological evolution of strings and other topological defects such as domain walls and monopoles have been reviewed by Vilenkin [ 3 ]. In this paper, we use Hiscock's semiclassical cosmic string metric [4] and investigate the motion of photons and timelike test particles in the gravitational field o f such a string and show how the test particles and light rays may be trapped in the field o f this string. With this semiclassical cosmic string metric, where the vacuum expectation value o f the stressenergy tensor o f an arbitrary collection o f massless free q u a n t u m fields is nonzero, the exterior spacePermanent address: Physics Department, Sripat Singh College, Jiaganj, Murshidabad 742123, India.
time is no longer flat. In this case we find that not all the particles can escape to infinity. Some particles, depending on their energy to angular momentum ratio, will escape and some others will be trapped by the string at a certain distance from the core. For incoming particles also, some of the particles will not be able to penetrate within a certain region. This feature o f a "forbidden zone" is not observed in the case o f a flat conical exterior metric o f a gauge string. In the next section we discuss the geodesic equations and study the motion of outgoing or incoming particles in the field o f a string.
2. Motion of particles in the field of a cosmic string The well known metric outside an indefinitely long straight infinitesimally thin gauge string is d S 2 = - d t E + d r E + d z 2 + r 2 ( 1 - 8/~) dO 2 ,
( 1)
which represents a cylindrically symmetric flat Minkowski space with a wedge of angular width 8n/z removed from the spacetime. Here # is the energy per unit length o f the string. Vilenkin [2 ] derived this metric within the linearized theory of general relativity. Gott [ 5 ], Hiscock [ 6 ] and L i n e t [ 7 ] accomplished the derivation o f this metric as an exact solution o f the Einstein equations. For a string represented by the metric (1), it can
0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
1 19
Volume 160, number 2
PHYSICS LETTERS A
be shown that photons or timelike particles, once emitted from the string, can escape to infinite distance. On the other hand for incoming particles, there is a value for r, when the particles bounce back, i.e., face a potential barrier, indicating a repulsive field. A detailed analysis of the geodesics of such a string is given by Gott [5]. In a recent communication, Demiansky [8] considered an infinite straight massive conducting cosmic string which reduces to a gauge string given by eq. ( 1 ) if the current flowing on the string and the mass of the string are put equal to zero. Demiansky's metric reduces to that due to an infinite cylindrically symmetric mass distribution if the string field energy (//) and the current are taken to be zero. He found that the escape of particles depends on the mass per unit length of the string while the string field energy has no significant role. The value of the linear mass density, for which photons, emitted in any direction, can traverse an infinite distance, is similar to that noticed by Banerjee [9 ]. As had already been mentioned, the spacetime geometry of the exterior of a cosmic string given by eq. ( 1 ) is flat even in the full nonlinear theory of general relativity. The stress-energy tensor of the exterior of such a string is zero. Hiscock [4] argued that the conical spacetime geometry of a string may not be globally identical to Minkowski space and the stressenergy tensors of the q u a n t u m fields will have nonzero vacuum expectation values. He calculated the vacuum stress-energy tensor for an arbitrary collection of conformal massless free fields for the conical string metric to be
( T ~ ) =Ahr -4 diag(1, 1, - 3 , 1 ) ,
(2b)
d S 2 = ( 1 --air 2) ( -- d r 2 + d z 2) + d r 2
120
t:+ ~33~2_ ( 1 - 4 / / ) 2 r ~ 2 = 0 ,
2a
(4)
?i=O ,
(5)
(1 +4a/r2)~'+ 2t:~ = 0 , r
(6)
i ' + r3 ( 1 - a / r 2)
with
(1-a/r2)i2-k 2 --r2( 1 --4//)2( 1 +4a/r2)~2=O
(7)
for photons, and
(1--a/r2)t2--~ 2 --r2( 1 --4//)2( 1 +4a/r2)O2>O
(8)
for timelike particles. Here a dot represents differentiation with respect to some affine parameter 2. If the geodesics are not null, one can choose it to be equal to s and hence relation (8) has the exact form
( l--a/r2)t2--~ 2 --r2( 1 --4//)2( 1 +4a/r2)~2= 1 .
(9)
After integration, eqs. (5) and (6) give i = l _ aa/ r 2 ,
(10)
and
~= ( l + 4 a / r 2 ) r 2 ,
This stress-energy tensor is proportional to that calculated by Helliwell and Konkowski [ 10]. Using this expression for T~ as the source in the linearized semiclassical Einstein equations, Hiscock obtained a solution to the string metric. In the coordinate system where the radial coordinate is a measure of the proper distance (i.e. grr= 1 ), the metric is given by
+ ( 1 - 4//)2r2( 1 +4a/r 2) dq~2 ,
where a = 4nAh. The geodesic equations for this metric for motions along a z = c o n s t plane are
(2a)
where A = ( 1440n 2) - l [ ( 1 - 4 / / ) - 4 _ 1 ] .
11 November 1991
(3)
(11)
respectively, where oe and fl are constants of integration. As the Lagrangian
5fl= -- (1--a/ r2)i2 + i"2 + r 2 ( 1 --4//)2( 1 +4a/r2)~ 2 is cyclic in t and 0, and c~ and fl are nothing but 0~¢/0t and 0~'/00, they are identified to be the total energy and angular m o m e n t u m , respectively, which are constants of motion. Using these expressions for i and 0 in eq. (7) one
Volume 160, number 2
PHYSICS LETTERS A
obtains after some long but straightforward calculations the equation
dO/ (
11 November 1991
For light rays, however, the first term on the righthand side does not appear. This equation, along with ( 11 ), yields
(dr
6 /2(l+4a/rZ)r 2
dO,/
-- 1)
02
= r2( 1 --4,//)2( 1 +4a/r 2)
Xr2( 1 -- 4/t)2 ( 1 +4a/r 2)
(12)
for photons. Here l=ol/fl. In view of (12) one can then conclude 12> ( 1 - 4 l t ) Z ( l - a / r 2 )
r2( 1 +4a/r 2)
(13)
The right-hand side of this inequality is not a monotonically increasing or decreasing function of r. We now denote the function
(1--a/r2)(1--4ll) 2 rE( 1 +4a/r 2)
( 1 +V/-5)2a
and this occurs at r = x / ~ (1 + x / ~ ) . Hiscock's semiclassical cosmic string has a finite physical radius Rs, which is, however, much greater than 10a t/2 as shown in ref. [ 4 ] and the metric (3) is valid in the exterior region r > Rs. As the function Q(r) has its m a x i m u m in the region r < 10a 1/2 and decreases to zero as r - - , ~ one can arrive at the following conclusions. The photon once emitted from the string in the outward direction independent of the angle of emission will traverse an infinite distance, because the inequality (13) once satisfied at a finite r will remain satisfied up to r--}~. It means such photons will escape. On the other hand for the incoming photons those with 12> Q(RD will reach the string. But those having 12< Q(Rs) will turn back somewhere in the exterior region and the radius at which they turn back depends on the value of 12. The analysis of the timelike trajectories is somewhat different. We use the expressions for i and g~ from eqs. (10) and ( 11 ) in eq. (9) to obtain Cg2
1)
(15)
As (dr~dO) z is positive, we obtain the condition
l 2 > P(r)
(16)
where
P(r)=Q(r)+
1 --air 2 fl--------~-
Ar4 + b2r2 + c2=O ,
(1-4/0 2
~ 2 = - - 1 + l _ a / r ~2
r2(l+4a/r2) fl2(1_4/~)2
One can now obtain the m a x i m u m or m i n i m u m of the function P(r) by taking its derivative with respect to the radial coordinate r and equating it to zero. Such operation yields a relation
as Q(r). The m a x i m u m of this function is given by Omax --
×(Q
(17)
where A, b, c are constants, the explicit expression for A being A=
1 flZ(1--4a)2
1
a"
When A is positive or zero, eq. (17) cannot be satisfied for real values of r and one does not get any extremum of the function P(r). It increases steadily from zero at r ~ x / ~ to lift 2 at r ~ o o . So, i f / z > lift 2, the outgoing test particles will escape to infinity, but if l 2 < 1/f12, the outgoing particles will be trapped at some distance from the string. This feature of trapping of particles is clearly different from the case of a classical string represented by the metric ( 1 ). For incoming particles, however, once the inequality (16) is satisfied, it remains so up to the radius of the string as P(r) decreases with the decrease of r. On the other hand, when A < 0, there is only one solution of eq. (17) for real and positive r, where there is a m a x i m u m . The picture is then analogous to that of photons and the conclusions for timelike particles are exactly identical.
f12(1--4fl)2
r2(l+4a/r2 ) .
(14) 121
Volume 160, number 2
References [ 1 ] T.W.B. Kibble, J. Phys. A 9 (1976) 1387. [2] A. Vilenkin, Phys. Rev. D 23 ( 1981 ) 852. [ 3 ] A. Vilenkin, Phys. Rep. 121 ( 1985 ) 263. [4] W.A. Hiscock, Phys. Lett. B 188 (1987) 317. [5] J.R. Gott III, Astrophys. J. 288 (1985) 422.
122
PHYSICS LETTERS A
11 November 1991
[6] W.A. Hiscock, Phys. Rev. D 31 (1985) 3288. [7] B. Liner, Gen. Rel. Grav. 17 (1985) 1109. [ 8] M. Demiansky, Phys. Rev. D 38 (1988 ) 698. [9] A. Banerjee, Proc. Phys. Soc. 1 (1968) 495. [10]T.M. Helliwell and D.A. Konkowski, Phys. Rev. D 34 (1986) 1918.
P IdYSICS LETTERS A
Motion of test particles and photons in the gravitational field of a cosmic string A.
Banerjee and N. Banerjee
1
Relativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Calcutta 700032, India Received 29 May 1991; revised manuscript received 30 August 1991;accepted for publication 12 September 1991 Communicated by J.P. Vigier
The motion of light rays and test particles are studied in the gravitational field of a cosmic string. Using Hiscock's semiclassical cosmic string metric, it is found that some of the outgoing particles will traverse an infinite distance but some particles may be trapped by the string at a finite distance from the axis. For incoming particles also, some will reach the boundary of the string but some may face a potential barrier and cannot penetrate beyond a certain distance from the string.
1. Introduction Phase transitions o f q u a n t u m fields in the early universe might have produced some stable topological defects [ 1 ]. One of these defects, cosmic strings, have attracted a lot o f interest in recent years, mainly because o f its possible candidature as seeds for the formation of galaxies. The gravitational effect o f an infinite straight cosmic string is the removal from spacetime of a wedge of angular size 8~/z where/z is the string field energy per unit length of the string. The possible observable effect o f this way be the formation o f double images o f quasars [2]. The formation, physical properties and cosmological evolution of strings and other topological defects such as domain walls and monopoles have been reviewed by Vilenkin [ 3 ]. In this paper, we use Hiscock's semiclassical cosmic string metric [4] and investigate the motion of photons and timelike test particles in the gravitational field o f such a string and show how the test particles and light rays may be trapped in the field o f this string. With this semiclassical cosmic string metric, where the vacuum expectation value o f the stressenergy tensor o f an arbitrary collection o f massless free q u a n t u m fields is nonzero, the exterior spacePermanent address: Physics Department, Sripat Singh College, Jiaganj, Murshidabad 742123, India.
time is no longer flat. In this case we find that not all the particles can escape to infinity. Some particles, depending on their energy to angular momentum ratio, will escape and some others will be trapped by the string at a certain distance from the core. For incoming particles also, some of the particles will not be able to penetrate within a certain region. This feature o f a "forbidden zone" is not observed in the case o f a flat conical exterior metric o f a gauge string. In the next section we discuss the geodesic equations and study the motion of outgoing or incoming particles in the field o f a string.
2. Motion of particles in the field of a cosmic string The well known metric outside an indefinitely long straight infinitesimally thin gauge string is d S 2 = - d t E + d r E + d z 2 + r 2 ( 1 - 8/~) dO 2 ,
( 1)
which represents a cylindrically symmetric flat Minkowski space with a wedge of angular width 8n/z removed from the spacetime. Here # is the energy per unit length o f the string. Vilenkin [2 ] derived this metric within the linearized theory of general relativity. Gott [ 5 ], Hiscock [ 6 ] and L i n e t [ 7 ] accomplished the derivation o f this metric as an exact solution o f the Einstein equations. For a string represented by the metric (1), it can
0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
1 19
Volume 160, number 2
PHYSICS LETTERS A
be shown that photons or timelike particles, once emitted from the string, can escape to infinite distance. On the other hand for incoming particles, there is a value for r, when the particles bounce back, i.e., face a potential barrier, indicating a repulsive field. A detailed analysis of the geodesics of such a string is given by Gott [5]. In a recent communication, Demiansky [8] considered an infinite straight massive conducting cosmic string which reduces to a gauge string given by eq. ( 1 ) if the current flowing on the string and the mass of the string are put equal to zero. Demiansky's metric reduces to that due to an infinite cylindrically symmetric mass distribution if the string field energy (//) and the current are taken to be zero. He found that the escape of particles depends on the mass per unit length of the string while the string field energy has no significant role. The value of the linear mass density, for which photons, emitted in any direction, can traverse an infinite distance, is similar to that noticed by Banerjee [9 ]. As had already been mentioned, the spacetime geometry of the exterior of a cosmic string given by eq. ( 1 ) is flat even in the full nonlinear theory of general relativity. The stress-energy tensor of the exterior of such a string is zero. Hiscock [4] argued that the conical spacetime geometry of a string may not be globally identical to Minkowski space and the stressenergy tensors of the q u a n t u m fields will have nonzero vacuum expectation values. He calculated the vacuum stress-energy tensor for an arbitrary collection of conformal massless free fields for the conical string metric to be
( T ~ ) =Ahr -4 diag(1, 1, - 3 , 1 ) ,
(2b)
d S 2 = ( 1 --air 2) ( -- d r 2 + d z 2) + d r 2
120
t:+ ~33~2_ ( 1 - 4 / / ) 2 r ~ 2 = 0 ,
2a
(4)
?i=O ,
(5)
(1 +4a/r2)~'+ 2t:~ = 0 , r
(6)
i ' + r3 ( 1 - a / r 2)
with
(1-a/r2)i2-k 2 --r2( 1 --4//)2( 1 +4a/r2)~2=O
(7)
for photons, and
(1--a/r2)t2--~ 2 --r2( 1 --4//)2( 1 +4a/r2)O2>O
(8)
for timelike particles. Here a dot represents differentiation with respect to some affine parameter 2. If the geodesics are not null, one can choose it to be equal to s and hence relation (8) has the exact form
( l--a/r2)t2--~ 2 --r2( 1 --4//)2( 1 +4a/r2)~2= 1 .
(9)
After integration, eqs. (5) and (6) give i = l _ aa/ r 2 ,
(10)
and
~= ( l + 4 a / r 2 ) r 2 ,
This stress-energy tensor is proportional to that calculated by Helliwell and Konkowski [ 10]. Using this expression for T~ as the source in the linearized semiclassical Einstein equations, Hiscock obtained a solution to the string metric. In the coordinate system where the radial coordinate is a measure of the proper distance (i.e. grr= 1 ), the metric is given by
+ ( 1 - 4//)2r2( 1 +4a/r 2) dq~2 ,
where a = 4nAh. The geodesic equations for this metric for motions along a z = c o n s t plane are
(2a)
where A = ( 1440n 2) - l [ ( 1 - 4 / / ) - 4 _ 1 ] .
11 November 1991
(3)
(11)
respectively, where oe and fl are constants of integration. As the Lagrangian
5fl= -- (1--a/ r2)i2 + i"2 + r 2 ( 1 --4//)2( 1 +4a/r2)~ 2 is cyclic in t and 0, and c~ and fl are nothing but 0~¢/0t and 0~'/00, they are identified to be the total energy and angular m o m e n t u m , respectively, which are constants of motion. Using these expressions for i and 0 in eq. (7) one
Volume 160, number 2
PHYSICS LETTERS A
obtains after some long but straightforward calculations the equation
dO/ (
11 November 1991
For light rays, however, the first term on the righthand side does not appear. This equation, along with ( 11 ), yields
(dr
6 /2(l+4a/rZ)r 2
dO,/
-- 1)
02
= r2( 1 --4,//)2( 1 +4a/r 2)
Xr2( 1 -- 4/t)2 ( 1 +4a/r 2)
(12)
for photons. Here l=ol/fl. In view of (12) one can then conclude 12> ( 1 - 4 l t ) Z ( l - a / r 2 )
r2( 1 +4a/r 2)
(13)
The right-hand side of this inequality is not a monotonically increasing or decreasing function of r. We now denote the function
(1--a/r2)(1--4ll) 2 rE( 1 +4a/r 2)
( 1 +V/-5)2a
and this occurs at r = x / ~ (1 + x / ~ ) . Hiscock's semiclassical cosmic string has a finite physical radius Rs, which is, however, much greater than 10a t/2 as shown in ref. [ 4 ] and the metric (3) is valid in the exterior region r > Rs. As the function Q(r) has its m a x i m u m in the region r < 10a 1/2 and decreases to zero as r - - , ~ one can arrive at the following conclusions. The photon once emitted from the string in the outward direction independent of the angle of emission will traverse an infinite distance, because the inequality (13) once satisfied at a finite r will remain satisfied up to r--}~. It means such photons will escape. On the other hand for the incoming photons those with 12> Q(RD will reach the string. But those having 12< Q(Rs) will turn back somewhere in the exterior region and the radius at which they turn back depends on the value of 12. The analysis of the timelike trajectories is somewhat different. We use the expressions for i and g~ from eqs. (10) and ( 11 ) in eq. (9) to obtain Cg2
1)
(15)
As (dr~dO) z is positive, we obtain the condition
l 2 > P(r)
(16)
where
P(r)=Q(r)+
1 --air 2 fl--------~-
Ar4 + b2r2 + c2=O ,
(1-4/0 2
~ 2 = - - 1 + l _ a / r ~2
r2(l+4a/r2) fl2(1_4/~)2
One can now obtain the m a x i m u m or m i n i m u m of the function P(r) by taking its derivative with respect to the radial coordinate r and equating it to zero. Such operation yields a relation
as Q(r). The m a x i m u m of this function is given by Omax --
×(Q
(17)
where A, b, c are constants, the explicit expression for A being A=
1 flZ(1--4a)2
1
a"
When A is positive or zero, eq. (17) cannot be satisfied for real values of r and one does not get any extremum of the function P(r). It increases steadily from zero at r ~ x / ~ to lift 2 at r ~ o o . So, i f / z > lift 2, the outgoing test particles will escape to infinity, but if l 2 < 1/f12, the outgoing particles will be trapped at some distance from the string. This feature of trapping of particles is clearly different from the case of a classical string represented by the metric ( 1 ). For incoming particles, however, once the inequality (16) is satisfied, it remains so up to the radius of the string as P(r) decreases with the decrease of r. On the other hand, when A < 0, there is only one solution of eq. (17) for real and positive r, where there is a m a x i m u m . The picture is then analogous to that of photons and the conclusions for timelike particles are exactly identical.
f12(1--4fl)2
r2(l+4a/r2 ) .
(14) 121
Volume 160, number 2
References [ 1 ] T.W.B. Kibble, J. Phys. A 9 (1976) 1387. [2] A. Vilenkin, Phys. Rev. D 23 ( 1981 ) 852. [ 3 ] A. Vilenkin, Phys. Rep. 121 ( 1985 ) 263. [4] W.A. Hiscock, Phys. Lett. B 188 (1987) 317. [5] J.R. Gott III, Astrophys. J. 288 (1985) 422.
122
PHYSICS LETTERS A
11 November 1991
[6] W.A. Hiscock, Phys. Rev. D 31 (1985) 3288. [7] B. Liner, Gen. Rel. Grav. 17 (1985) 1109. [ 8] M. Demiansky, Phys. Rev. D 38 (1988 ) 698. [9] A. Banerjee, Proc. Phys. Soc. 1 (1968) 495. [10]T.M. Helliwell and D.A. Konkowski, Phys. Rev. D 34 (1986) 1918.