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Massless particle radiation of Vilenkin's planar domain wall
Volume 264, number 3,4
PHYSICS LETTERS B
1 August 1991
Massless particle radiation of Vilenkin's planar domain wall Anzhong Wang 1 Division of Theoretical Physics, Department of Physics, University of loannina, P.O. Box 1186, GR-451 I0 loannina, Greece Received 17 April 1991
A solution of the Einstein equations is found, which represents the massless particle radiation of Vilenkin's planar domain wall. The wall comes from negative infinity, and at the point (u, v) = (0, 0 ) it degenerates into two impulsive shells of null dust, moving in opposite directions at the velocity of light.
In 1975, Zel'dovich, Kobzarev and Okun' [ 1 ] discovered that spontaneous violation of CP-invariance would lead to the formation of a domain structure o f the vacuum. If there is no mechanism which causes such a structure to disappear at a sufficiently early time in the evolution of the universe, the inhomogeneities of the domains will induce unacceptable anisotropies on the microwave background radiation. Thus, either models with discrete symmetry breaking are ruled out by cosmology or there must exist some mechanisms to make the walls disappear. Zel'dovich et al. [ 1 ] (see also Kibble [2], Vilenkin [3,4] and Sikivie [ 5 ] ) suggested that if the discrete symmetry was not exact, the energy difference between the two vacua would cause the false v a c u u m to disappear and the real v a c u u m would eventually come to occupy all the space. Gelmini, Gleiser and Kolb [ 6 ] have shown that the introduction of a bias indeed destabilizes infinite domain walls, and that the resulting v a c u u m pressure causes the walls to intersect and break up into finite bubbles, which may supply seeds for the formation of the large-scale structure o f the universe. Another possibility is the interaction of the walls with particles and gravitons [ 1,2,4,7 ]. Massarotti [ 8 ] has studied the evolution of light domain walls [ 9 ] interacting with a major gaseous component o f the dark matter, and found that, unlike vacuum domain walls [ 10], interacting domain walls with the dark matter have negligible distortions o f the microwave Permanent address: Physics Department, Northeast Normal University, Changchun, Jilin, China. 274
background, but may provide a mechanism o f forming the large-scale structure of the universe. On the other hand, we have shown recently [ 11 ] that a plane thin domain wall with reflection symmetry interacts only with the surrounding matter fields and does not interact with any gravitational waves. This surprising result may be due to the high symmetry of the wall. For more realistic cases, one might expect it does. However, as shown by Massarotti [ 8 ], this interaction is secondary comparing with the one o f domain walls with matter fields. In this letter, we study the interaction o f a planar domain wall with impulsive (infinitely thin) shells o f null dust (dust consisting o f unidentified massless panicles [12 ] ). The solution of the Einstein equations representing such an event is given by ds2=2 exp[-ka(u,
v) ] u dv
- e x p [ k f l ( u , v) ] (dx2 + dy 2) ,
( 1)
where
o~(u, v) = - z [ H ( z ) - H ( - z ) z-uH(-u)
-vii(-v)
fl(u, v ) ~ u H ( - - u ) +vn(-v)
], ,
[l-H(z)+H(--z)]
[ l + H(z)-H(-z)
],
(2)
k is a positive constant, and H ( x ) denotes the Heaviside function, which is unity for non-negative arguments and otherwise zero. In the following, the coordinates will be numbered {x u} - (u, v, x, y}. For the sake of convenience, we divide the space-
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 264, number 3,4
PHYSICS LETTERS B
1 August 1991
Region lib ( u > 0 , v < 0 ) .
time into regions I - I I I (see fig. 1 ). In each region the metric and the non-vanishing first derivatives of the metric coefficients are given as follows. Region Ia (u, v < 0 , u < v ) .
gn~,,=kexp(kv)
dSlZa= 2 e x p [ k ( u - v ) ] du dv
gnb 2 2 , v = gI~3b.~ , = _ 2k exp (2kv)
dS2ib = 2 exp(kv) du d v - e x p ( 2 k v ) ( d x 2 + d y
- e x p ( 2 k u ) ( d x 2 + dy 2) , la la g .... = -gu~,~ = k exp [ k ( u - v )
l, ~ ; ,la3 3 , u g22,u la
~ _
--
(3)
la
] du dv
- e x p ( 2 k v ) (dx2 + d y 2) , b Ib g I.... = - guv, v=-kexp[k(v-u)]
--- 2k exp (2kv) --
(4)
Region IIa ( u < 0 , v > 0 ) . dS~la = 2 e x p ( k u ) du d v - e x p ( 2 k u )
( d x 2 + d y 2) ,
lla g~.,u = k exp (ku) , gila alia 22,u ~/5 33,u ~
(6)
ds~n = 2 du d r - d x 2 - d y 2 ,
],
where gu~,~ = Og~,/Ov, etc. Region Ib (u, v < 0 , u > v ) .
glb --.lb 22,v --~33,v
,
Region III (u, v > 0 ) .
2kexp(2ku)
ds~b = 2 e x p [ k ( v - u )
2) ,
(5)
-2kexp(2ku)
gill
,~,,~ -- -, , .
(7)
It is easy to show that, inside each of the above five regions, the space-time is fiat, and that all the Weyl scalars, or equivalently the Weyl tensor, vanish. Therefore, in the present case no free gravitational field is present. Across the boundaries separating these regions the metric coefficients are continuous, but their first derivatives are not. Thus, on these boundaries we expect that matter will appear. Suppose each boundary is described by ~o(x u) = 0, then the normal vector to the boundary can be defined as k~, - a~°(x~) ~x~)=o" ~ 0x
(8)
Hence, the stress-energy tensor is given by [ 13 ] t
Tu, = zu~6(~o ) ,
V
U
(9)
where zu~ is the surface stress-energy tensor, and given by z~,,, = ½[y~( kakag,,,, - k~,k~ ) + ( k . y .a + k . ~ ) k a
'
lib
- kak~yu. - g u ~ k ~ k a y ~a] ,
(10a)
_= lira gu~.a(xa,~o)- lim gu~.~(xa.~o).
(10b)
~o~O +
1
/
n~
¢~o-
In the following, we consider each boundary separately. Let us first consider the one separating region Ia from region Ib. In this case, the normal vector is given by 2 ,a ~---'~/2 __ ]¢-Ia~lb={1, -- 1, 0, 0}
Fig. 1. The (u, v)-plane of the space-time model described in the text. An infinitely thin planar domain wall moves in from negative infinity, and at the point (u, v) = (0, 0 ) degenerates into two impulsive shells of null dust moving in opposite directions at the velocity of light.
(11)
Inserting eqs. (3), (4), (10b) and (11 ) into eq. (10a), we find z l~a - l b = 4 k [ g t , , - 2 u 2 ~ ( 2 ~ 2
~
) -- l ] = 4 k h u . ,
(12) 275
Volume 264, number 3,4
PHYSICS LETTERS B
where hu, is the metric projected into the hypersurface u = v . The a b o v e surface stress-energy tensor corresponds to a d o m a i n wall [ 3,14 ]. Thus, in region I a w l b , there is a thin p l a n a r d o m a i n wall with support on the surface u = v. The energy density is ~ = 4k, which is constant. If we define the new coordinates t - u + v, a n d z - u - v, then we find that the metric in region Ia u Ib can be written as d s 2 = ½ e x p ( - k [ z[ ) ( d t 2 - d z 2) -exp[k(t-
[zl ) ] (dx2 + dy 2) •
(13)
But, this is Vilenkin's p l a n a r d o m a i n wall solution [14]. On the b o u n d a r y I b - I I b , on the other hand, we have
~/2l__=--/2](Ib-~llb-~ { 1, 0, 0, 0),
~"r / 2Ib-Ilb u
=0
2knun, '
._
_,.llb~lll
{0, 1 0,0}
(15)
which corresponds to an impulsive shell o f null dust with support on the hypersurface v= 0 and u > 0 [ 15 ]. Note that the energy density o f the shell is constant, too. F r o m the s y m m e t r y shared by the metric coefficients g ~ , g~u~ a n d gu~na,~/2~-nb[see eqs. ( 3 ) - ( 6 ) ], it is easy to show that lla-lll -9kl I t ~1v ~ . . . . .
T/2v
,/.la-lla~ vO .
-Izu
(16)
It follows that the b o u n d a r y v - - 0 a n d u < 0 is free o f matter, while the b o u n d a r y u = 0 a n d v > 0 is occupied by an impulsive shell o f null dust with the same energy density as the one with support on the surface v = 0 and u > 0. S u m m a r i z i n g the above results, we find that the solution given by eqs. ( 1 ) a n d ( 2 ) represents Vilenkin's planar d o m a i n wall coming from t = - oo with a constant energy density a ( = 4 k ) , and that at the
276
p o i n t (u, v ) = (0, 0) the wall degenerates into two constant-profile impulsive shells o f null dust m o v i n g in opposite directions at the velocity o f light. The significance o f the above solution is that it shows that it m a y be possible for d o m a i n walls to disappear through particle radiation. It also suggests that, like cosmic strings [ 16 ], d o m a i n walls might not be gravitationally topological stable defects at all, and that the " u n d e s i r a b l e " d o m a i n walls f o r m e d from G U T phase transitions might have partially ( i f not totally) degenerated into another form o f energy before they d o m i n a t e d the universe.
References
(14)
Thus, the u = 0 and v < 0 b o u n d a r y is free o f matter. Similarly, one can easily show that, across the b o u n d a r y v = 0 and u > 0 , the surface stress-energy tensor is given by rllb-Ill
1 August 1991
[ 1] Ya.B. Zel'dovich, I. Yu. Kobzarev and L.B. Okun', Sov. Phys. JETP 40 ( 1975 ) 1. [ 2 ] T.W.B. Kibble, J. Phys. A 9 ( 1976 ) 1387, [3] A. Vilenkin, Phys. Rev. D 23 ( 1981 ) 825. [4] A. Vilenkin, Phys. Rep. 121 (1985) 264. [ 5 ] P. Sikivie, Phys. Rev. Lett. 48 (1982) 1156. [6] G.B. Gelmini, M. Gleiser and E.W. Kolb, Phys. Rev. D 39 (1989) 1558. [7] A.E. Everett, Phys. Rev. D 10 (1974) 3161. [8] A. Massarotti, Phys. Rev. D 43 ( 1991 ) 346. [9] C. Hill, D. Schramm and J. Fry, Comm. Nucl. Part. Phys. 19 (1989) 25. [ 10] W.H. Press, B.S. Ryden and D.N. Spergel, Astrophys. J. 347 (1989) 590; L. Kawano, Phys. Rev. D 41 ( 1990 ) 1013. [ 11 ] A.Z. Wang, Plane walls interacting with gravitational waves and matter fields, preprint ( 1991 ). [ 12 ] D. Kramer, H. Stephani, H. Herlt, M. MacCallum and E. Schmutzer, Exact solutions of Einstein's field equations (Cambridge U.P., Cambridge, 1980). [ 13] A.H. Taub, J. Math. Phys. 21 (1980) 1423. [ 14] A. Vilenkin, Phys. Lett. B 133 (1983) 177; J. lpser and P. Sikivie, Phys. Rev. D 30 (1984) 712. [ 15] T. Dray and G. 't Hooft, Class. Quantum Grav. 3 (1986) 825; A.H. Taub, J. Math. Phys. 29 (1988) 690; D. Tsoubelis and A.Z. Wang, Gen. Rel. Grav. 22 (1990) 1091. [ 16] R. Gleiser and J. Pullin, Class. Quantum Grav. 6 (1989) L141.
PHYSICS LETTERS B
1 August 1991
Massless particle radiation of Vilenkin's planar domain wall Anzhong Wang 1 Division of Theoretical Physics, Department of Physics, University of loannina, P.O. Box 1186, GR-451 I0 loannina, Greece Received 17 April 1991
A solution of the Einstein equations is found, which represents the massless particle radiation of Vilenkin's planar domain wall. The wall comes from negative infinity, and at the point (u, v) = (0, 0 ) it degenerates into two impulsive shells of null dust, moving in opposite directions at the velocity of light.
In 1975, Zel'dovich, Kobzarev and Okun' [ 1 ] discovered that spontaneous violation of CP-invariance would lead to the formation of a domain structure o f the vacuum. If there is no mechanism which causes such a structure to disappear at a sufficiently early time in the evolution of the universe, the inhomogeneities of the domains will induce unacceptable anisotropies on the microwave background radiation. Thus, either models with discrete symmetry breaking are ruled out by cosmology or there must exist some mechanisms to make the walls disappear. Zel'dovich et al. [ 1 ] (see also Kibble [2], Vilenkin [3,4] and Sikivie [ 5 ] ) suggested that if the discrete symmetry was not exact, the energy difference between the two vacua would cause the false v a c u u m to disappear and the real v a c u u m would eventually come to occupy all the space. Gelmini, Gleiser and Kolb [ 6 ] have shown that the introduction of a bias indeed destabilizes infinite domain walls, and that the resulting v a c u u m pressure causes the walls to intersect and break up into finite bubbles, which may supply seeds for the formation of the large-scale structure o f the universe. Another possibility is the interaction of the walls with particles and gravitons [ 1,2,4,7 ]. Massarotti [ 8 ] has studied the evolution of light domain walls [ 9 ] interacting with a major gaseous component o f the dark matter, and found that, unlike vacuum domain walls [ 10], interacting domain walls with the dark matter have negligible distortions o f the microwave Permanent address: Physics Department, Northeast Normal University, Changchun, Jilin, China. 274
background, but may provide a mechanism o f forming the large-scale structure of the universe. On the other hand, we have shown recently [ 11 ] that a plane thin domain wall with reflection symmetry interacts only with the surrounding matter fields and does not interact with any gravitational waves. This surprising result may be due to the high symmetry of the wall. For more realistic cases, one might expect it does. However, as shown by Massarotti [ 8 ], this interaction is secondary comparing with the one o f domain walls with matter fields. In this letter, we study the interaction o f a planar domain wall with impulsive (infinitely thin) shells o f null dust (dust consisting o f unidentified massless panicles [12 ] ). The solution of the Einstein equations representing such an event is given by ds2=2 exp[-ka(u,
v) ] u dv
- e x p [ k f l ( u , v) ] (dx2 + dy 2) ,
( 1)
where
o~(u, v) = - z [ H ( z ) - H ( - z ) z-uH(-u)
-vii(-v)
fl(u, v ) ~ u H ( - - u ) +vn(-v)
], ,
[l-H(z)+H(--z)]
[ l + H(z)-H(-z)
],
(2)
k is a positive constant, and H ( x ) denotes the Heaviside function, which is unity for non-negative arguments and otherwise zero. In the following, the coordinates will be numbered {x u} - (u, v, x, y}. For the sake of convenience, we divide the space-
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 264, number 3,4
PHYSICS LETTERS B
1 August 1991
Region lib ( u > 0 , v < 0 ) .
time into regions I - I I I (see fig. 1 ). In each region the metric and the non-vanishing first derivatives of the metric coefficients are given as follows. Region Ia (u, v < 0 , u < v ) .
gn~,,=kexp(kv)
dSlZa= 2 e x p [ k ( u - v ) ] du dv
gnb 2 2 , v = gI~3b.~ , = _ 2k exp (2kv)
dS2ib = 2 exp(kv) du d v - e x p ( 2 k v ) ( d x 2 + d y
- e x p ( 2 k u ) ( d x 2 + dy 2) , la la g .... = -gu~,~ = k exp [ k ( u - v )
l, ~ ; ,la3 3 , u g22,u la
~ _
--
(3)
la
] du dv
- e x p ( 2 k v ) (dx2 + d y 2) , b Ib g I.... = - guv, v=-kexp[k(v-u)]
--- 2k exp (2kv) --
(4)
Region IIa ( u < 0 , v > 0 ) . dS~la = 2 e x p ( k u ) du d v - e x p ( 2 k u )
( d x 2 + d y 2) ,
lla g~.,u = k exp (ku) , gila alia 22,u ~/5 33,u ~
(6)
ds~n = 2 du d r - d x 2 - d y 2 ,
],
where gu~,~ = Og~,/Ov, etc. Region Ib (u, v < 0 , u > v ) .
glb --.lb 22,v --~33,v
,
Region III (u, v > 0 ) .
2kexp(2ku)
ds~b = 2 e x p [ k ( v - u )
2) ,
(5)
-2kexp(2ku)
gill
,~,,~ -- -, , .
(7)
It is easy to show that, inside each of the above five regions, the space-time is fiat, and that all the Weyl scalars, or equivalently the Weyl tensor, vanish. Therefore, in the present case no free gravitational field is present. Across the boundaries separating these regions the metric coefficients are continuous, but their first derivatives are not. Thus, on these boundaries we expect that matter will appear. Suppose each boundary is described by ~o(x u) = 0, then the normal vector to the boundary can be defined as k~, - a~°(x~) ~x~)=o" ~ 0x
(8)
Hence, the stress-energy tensor is given by [ 13 ] t
Tu, = zu~6(~o ) ,
V
U
(9)
where zu~ is the surface stress-energy tensor, and given by z~,,, = ½[y~( kakag,,,, - k~,k~ ) + ( k . y .a + k . ~ ) k a
'
lib
- kak~yu. - g u ~ k ~ k a y ~a] ,
(10a)
_= lira gu~.a(xa,~o)- lim gu~.~(xa.~o).
(10b)
~o~O +
1
/
n~
¢~o-
In the following, we consider each boundary separately. Let us first consider the one separating region Ia from region Ib. In this case, the normal vector is given by 2 ,a ~---'~/2 __ ]¢-Ia~lb={1, -- 1, 0, 0}
Fig. 1. The (u, v)-plane of the space-time model described in the text. An infinitely thin planar domain wall moves in from negative infinity, and at the point (u, v) = (0, 0 ) degenerates into two impulsive shells of null dust moving in opposite directions at the velocity of light.
(11)
Inserting eqs. (3), (4), (10b) and (11 ) into eq. (10a), we find z l~a - l b = 4 k [ g t , , - 2 u 2 ~ ( 2 ~ 2
~
) -- l ] = 4 k h u . ,
(12) 275
Volume 264, number 3,4
PHYSICS LETTERS B
where hu, is the metric projected into the hypersurface u = v . The a b o v e surface stress-energy tensor corresponds to a d o m a i n wall [ 3,14 ]. Thus, in region I a w l b , there is a thin p l a n a r d o m a i n wall with support on the surface u = v. The energy density is ~ = 4k, which is constant. If we define the new coordinates t - u + v, a n d z - u - v, then we find that the metric in region Ia u Ib can be written as d s 2 = ½ e x p ( - k [ z[ ) ( d t 2 - d z 2) -exp[k(t-
[zl ) ] (dx2 + dy 2) •
(13)
But, this is Vilenkin's p l a n a r d o m a i n wall solution [14]. On the b o u n d a r y I b - I I b , on the other hand, we have
~/2l__=--/2](Ib-~llb-~ { 1, 0, 0, 0),
~"r / 2Ib-Ilb u
=0
2knun, '
._
_,.llb~lll
{0, 1 0,0}
(15)
which corresponds to an impulsive shell o f null dust with support on the hypersurface v= 0 and u > 0 [ 15 ]. Note that the energy density o f the shell is constant, too. F r o m the s y m m e t r y shared by the metric coefficients g ~ , g~u~ a n d gu~na,~/2~-nb[see eqs. ( 3 ) - ( 6 ) ], it is easy to show that lla-lll -9kl I t ~1v ~ . . . . .
T/2v
,/.la-lla~ vO .
-Izu
(16)
It follows that the b o u n d a r y v - - 0 a n d u < 0 is free o f matter, while the b o u n d a r y u = 0 a n d v > 0 is occupied by an impulsive shell o f null dust with the same energy density as the one with support on the surface v = 0 and u > 0. S u m m a r i z i n g the above results, we find that the solution given by eqs. ( 1 ) a n d ( 2 ) represents Vilenkin's planar d o m a i n wall coming from t = - oo with a constant energy density a ( = 4 k ) , and that at the
276
p o i n t (u, v ) = (0, 0) the wall degenerates into two constant-profile impulsive shells o f null dust m o v i n g in opposite directions at the velocity o f light. The significance o f the above solution is that it shows that it m a y be possible for d o m a i n walls to disappear through particle radiation. It also suggests that, like cosmic strings [ 16 ], d o m a i n walls might not be gravitationally topological stable defects at all, and that the " u n d e s i r a b l e " d o m a i n walls f o r m e d from G U T phase transitions might have partially ( i f not totally) degenerated into another form o f energy before they d o m i n a t e d the universe.
References
(14)
Thus, the u = 0 and v < 0 b o u n d a r y is free o f matter. Similarly, one can easily show that, across the b o u n d a r y v = 0 and u > 0 , the surface stress-energy tensor is given by rllb-Ill
1 August 1991
[ 1] Ya.B. Zel'dovich, I. Yu. Kobzarev and L.B. Okun', Sov. Phys. JETP 40 ( 1975 ) 1. [ 2 ] T.W.B. Kibble, J. Phys. A 9 ( 1976 ) 1387, [3] A. Vilenkin, Phys. Rev. D 23 ( 1981 ) 825. [4] A. Vilenkin, Phys. Rep. 121 (1985) 264. [ 5 ] P. Sikivie, Phys. Rev. Lett. 48 (1982) 1156. [6] G.B. Gelmini, M. Gleiser and E.W. Kolb, Phys. Rev. D 39 (1989) 1558. [7] A.E. Everett, Phys. Rev. D 10 (1974) 3161. [8] A. Massarotti, Phys. Rev. D 43 ( 1991 ) 346. [9] C. Hill, D. Schramm and J. Fry, Comm. Nucl. Part. Phys. 19 (1989) 25. [ 10] W.H. Press, B.S. Ryden and D.N. Spergel, Astrophys. J. 347 (1989) 590; L. Kawano, Phys. Rev. D 41 ( 1990 ) 1013. [ 11 ] A.Z. Wang, Plane walls interacting with gravitational waves and matter fields, preprint ( 1991 ). [ 12 ] D. Kramer, H. Stephani, H. Herlt, M. MacCallum and E. Schmutzer, Exact solutions of Einstein's field equations (Cambridge U.P., Cambridge, 1980). [ 13] A.H. Taub, J. Math. Phys. 21 (1980) 1423. [ 14] A. Vilenkin, Phys. Lett. B 133 (1983) 177; J. lpser and P. Sikivie, Phys. Rev. D 30 (1984) 712. [ 15] T. Dray and G. 't Hooft, Class. Quantum Grav. 3 (1986) 825; A.H. Taub, J. Math. Phys. 29 (1988) 690; D. Tsoubelis and A.Z. Wang, Gen. Rel. Grav. 22 (1990) 1091. [ 16] R. Gleiser and J. Pullin, Class. Quantum Grav. 6 (1989) L141.