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Gravitational interaction of massless particles
Volume 141, number 5,6
PHYSICS LETFERS A
6 November 1989
GRAVITATIONAL INTERACTION OF MASSLESS PARTICLES Valeria FERRARI International Centerfor Relativistic Astrophysics — ICRA, Dipartimento di Fisica “G. Marconi’~Università di Roma, Rome, Italy
and Jesus IBAREZ’ Departamento de Fisica, Universidad de Las Islas Baleares, Spain Received 21 June 1989; revised manuscript received 31 July 1989; accepted for publication 6 September 1989 Communicated by J.P. Vigier
An exact solution of non-vacuum Einstein equations describing the collision of two clouds ofnull particles and their consequent gravitational interaction is presented. As a result, a source is produced which satisfies the Klein—Gordon equation for massless scalar fields, and a physical singularity finally appears.
The gravitational interaction of colliding shells of null dust has been recently investigated by Chandrasekhar and Xanthopoulos [1,2], and by Dray and ‘t Hooft [3]. In refs. [1] and [2], the null dust flies in flat spacetime following the leading edges of impulsive gravitational waves, and the collision produces a region of interaction where either a mixture of null dust moving in opposite directions is present, or a perfect fluid in which the energy density ~t is equal to the pressure p. The presence of different fluids in the region of interaction, is due, as clarified by Taub [4,51,to different assumptions on the energy—momentum tensor, which specify the nature of the interaction. In ref. [3] the collision of two impulsive planar shells of null dust has been considered, with a result similar to that described in ref. [1]. One of the most remarkable consequences of the interaction of null dust, is the possibility, shown in ref. [21, of a gravitationally induced transformation of massless particles, describing null trajectories, into a perfect fluid, although with a peculiar equation of state. We shall show in this paper that a similar transition can occur also when no gravitational waves are Guest ofthe Department ofPhysics ofthe University ofRome.
coupled to the dust, and that a source can be produced which behaves, at a macroscopical level, as a fluid with an equation of state different from p = and with an anisotropic distribution of pressure, being the components orthogonal to the direction of propagation different from the parallel component. The energy—momentum tensor satisfies the strong energy conditions. Unlike the solutions quoted before, the metric is continuous with its first derivatives, therefore no impulsive waves are present on the null hypersurfaces separating different regions. A physical singularity develops in the region of interaction. One might ask how is it possible that the gravitational interaction ofnull dust can produce a so large variety of sources, and whether these solutions really represent the interaction of some fields, and we shall try to answer this question. The null dust is defined by the following energy— momentum tensor: T~V ~ )k~k~ (1) —
where E is a positive function of a null coordinate u representing the energy of the dust, and k’~is a null vector. This definition presents an intrinsic ambiguity, since the energy—momentum tensor (1) can
0375-960l/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
233
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
represent any kind of massless field propagating as a plane wave along a definite direction. In this respect, it is not surprising that many solutions exist
equations in five dimensions, we can derive a solution of the non-vacuum Einstein equations in four dimensions, provided g44 satisfies the constraint
in which two waves of null dust collide and, due to the gravitational interaction, produce different sources, as shown in the references quoted above. However, when the specific equations satisfied by the null particles are identified, the problem is cornpletely defined, the solution is unique, and the ambiguity disappears. Thus, in our case, we have checked that the Klein—Gordon equation for a massless scalar field is satisfied both in the region of interaction andsatisfies in the precollision and that the scalar field the Junctionregion, conditions across the null boundaries separating different regiovs. Therefore we are able to identify the nature of the
(4b), and the energy—momentum tensor associated to the function 0 satisfies the energy conditions. From eqs. (3), the equation for the function q is
null dust and we can conclude that our solution represents the gravitational interaction of massless sealar waves. It is interesting to note that the source in the region of interaction mimics aofnon-perfect with an anisotropic distribution pressure, asfluid is shown by the analysis of the corresponding energy—mo•
.
—
~=0,
(6)
where the dot and the prime indicate, respectively, differentiation with respect to t and z; the function f can be derived by quadrature from the system 2+q2)—2/3t f=3tqq’ (7) f=4t(q’ We select the solution .
q=c 1log~1+c2logj~2+klogt
.
.
•
,
•
•
reduction of a solution of Einstein s equations in fivedimensional vacuum. Let us consider the following form of the metric: 2=e~[—(dt)2+(dz)2] th 2/s[e~ (dxi)2+e~(dx2)2+e2~(dx4)21 +t ‘2’
where 2--t2 ~i2= 1 —z+~,/( 1 —z) k + 9 c 2)~ and c1 and c2 are constants. We shall consider this solution in the region —
— — ~
0z~t~l t—l.~z~l—t.
‘•
f=3((kcl+c~)lOg/1l+(kC 2+c4)log~L2 2c’1c2
(A,
B=0, 4),
(3)
are equivalent to the following system of equations in four dimensions: R~~=O~q~,~p,
(4a)
Ø’~’.,,~=0(~,v=0, 3)
(4b)
where ØJ~(I/3e_~
Cui+u 2) 2
—
4) log t~.
+ (4k
(11)
I
To investigate the physical meaning of the solution, we shall express the components of the energy— momentum tensor TppØ~Ø;p;p, (12) with respect to an orthonormal basis. The resulting canonical form of TM~is defined by the following eigenvalues:
(5)
Therefore, from a given vacuum solution of Einstein’s 234
+1
~
The metric functions f~ q, are assumed to depend on and z only. It is known [6,71 that Einstein equations written in five dimensions for the the vacuum,
(10)
Consequently, f will be
.
RAB—O
(8)
•
mentum tensor in a suitably chosen orthonormal basis, The solution has been obtained by a dimensional •
q” ~q—
Volume 141, number 5,6
=
e-1 {
—
O;0;o +
PHYSICS LETTERS A
6 November 1989
I
12
033
E(0;o;o+0;3;3)2_40,2o;31h/’2}
,
i
(13a)
t 0-1
e —f
:
23{0;0;O+0;3;3
6
+[(0;o;o+0;3;3)240~o;31h/2}
,
(13b) (13c)
To ensure the reality of the eigenvalues, and to make them finite on the null boundaries z=t— 1,
z= 1
—t
,
0
(14)
—0.5
0
z
05
we must assume c
1 = c2 = 1 /,.j~. 21, 2 2~ 23 correspond, respectively, to a timelike vector e0 and to three spacelike vectors e1, e2, e3 provided O;o;o + 0; 3.3>0. This inequality is always satisfied in the region defined by eqs. (10). Hence we can identify the energy density, as measured by —
e0, and the pressure in the three spacelike directions ea as 21,
~u=—A0,
P1P2
p
3=13.
_________________________________________ i0.(~
-
t 01
7.5
(15)
With this identification, the strong energy conditions an observer whose world-line has unit tangent vector ~t+p1>O, u+~p1>O, 1=1,3, are satisfied. Plots of —2~and
Fig. 1. The logarithm of the energy density p and of the p~component of the pressure are plotted for selected values of time, in the corresponding interval of z, in the region of interaction. The superscribed + refers to i’~ the continuous line to ~
23,
and A~=22 are
corresponding given It should in figs. be 1noted z-interval. and 2, that forthe different velocity values field associated of tin the to the eigenvalue A~is timelike, and that the source in the region of interaction therefore behaves as a fluid whose streamlines are tirnelike trajectories. The fluid has an anisotropic distribution of pressure, as shown by eqs. (15), and the component along the direction of propagation P3 is always greater than the components parallel to the wavefront Pi =P2. The hydrodynamical equations, ~ ~.= 0, are satisfied as a consequence of eqs. (4) and the Ricci identities. When t—~0there a physical singularity: ergy density and theispressure diverge like the en~,p 313. (16) In1,p3~t’ order to extend the metric across the null boundaries (14), by using the standard Khan—
~°
2.5
-
‘-0.5
0
0.5
Z
Fig. 2. The logarithm of the Pi =P2 components of the pressure is plotted forthe same values oft and z used in fig. 1.
Penrose algorithm, we introduce a pair of null coordinates u and v, related to t and z by the following equations: 4—v4, z=u4—v4. (17) t=1—u This choice ensures the functions f be finite on the boundaries (14), which correspond to the hypersurfaces u=0 and v=0. In terms of u and v, the metric becomes 235
Volume 141, number 5,6
PHYSICS LETTERS A
213e’~[(cjv1)2+ _eF du dv +t
~2
(dx2)2],
(18)
where e F= 64u 3v3 e-’i The only non-vanishing Weyl scalar in the region defined by eqs. (10) is ~P 2:
(
6v~e! =
—
1 6u
(1
—
u
—
v4)2 (19)
+l_~2_q~2)+~),
therefore no gravitational radiation is present in this region. ~P 4+v4= 1, which corwhen u responds to2ist =singular 0. When u—U0 ± (v—~0 ~), ~ and P2 tend to zero, ~ and p 3 tend to the same finite value, and q12 tends to zero. Because all the metric functions, and all the physical quantities are finite on the null boundaries, we can extend the metric by the substitution u—~uH( u), v—~vH(v)
,
where H is the Heaviside step function, which ensures the continuity of the metric. In addition, the first derivatives of the metric are continuous, and therefore, no impulsive waves are present, The extended metric in the region V~<0, 0 ~ U < 1 is 2— —8(1—u4) du dv ds + (1 —u4)213~‘~(1+u2) ~ 1/12
x
[(~
1)2 + (~2) 2]
.
(20)
The only non-zero component ofthe Ricci tensor is ~ and it is always positive in the region we are considering. Therefore the solution (20) allows an interpretation in terms of a distribution of null dust moving in the v-direction. The situation is symmetnc in the region (u ~ 0, 0 ~ v < 1). When u <0 and v< 0 the spacetime is flat. Atthis stage we can say that the solution presented in this Letter represents the collision and the following gravitational interaction of two planar sandwiches of massless particles propagating in flat spacetime. However, some further informations can be added. The source of the Einstein equations satisfies the Klein—Gordon equations for a massless scalar field 236
~
6 November 1989
,
(21)
both in the interaction region and in the precollision region. The function 0 is continuous everywhere with its first derivatives, and all the required junction conditions are tensor satisfied on by thethe nullRicci boundaries the stress—energy and tensor.byThis means, in particular, that ~ and ~ are continuous on (v=0, 0
~
U;
lution represents the collision, and the following interaction, of two massless scalar fields, which produce, in the region ofinteraction, a scalar field which behaves, at a macroscopical level, as a fluid. The interaction is not affected by any impulsive phenomenon. As in the collision of plane fronted gravitational waves, a singularity develops after the collision of the scalar waves. It should be noted that the stress—energy tensor we use (eq. (12)) is not in the form which is standard for a scalar field. However, it has been shown [81 that it can be reduced to that form by a conformal transformation of the metric, accompanied by a defmite transformation of the scalar potential. It is believed [91 that in the very early universe, at the time when the various interactions were unifled, the universe contained predominantly massless particles. At that time phase transitions areofsuggested to occur which would generate bubbles the new broken-symmetry fase. These bubbles, accelerated by the energy released in the transition, would possibly collide and generate pairs of scalar waves, whose following gravitational interaction is generally negiected in these models. The results presented in this paper suggest that, due to its possible relevance to the physics of the early universe, the gravitational interaction of null dust deserves a deeper investigation.
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
References
[5] A.H. Taub, Collision of impulsive gravitational waves followed by dust clouds, preprint.
[1] S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. A 403 (1986) 189. [2JS. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. A 402 (1985) 37. [3] T. Dray and G. ‘t Hooft, Class. Quantum Gray. 3 (1986) 825. [4] A.H. Taub, J. Math. Phys. 29 (3) (1988) 690.
[6] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. K1. 54 (1921)966. [7]0. Klein,Z. Phys. 37 (1926) 875. [8] V.A. Belinskii and I.M. Khalaktnikov, Sov. Phys. JETP 36 (1972) 591. [9] S.W. Hawking, I.G. Moss andJ.M. Stewart, Phys. Rev. D 26 (1982) 2681.
237
PHYSICS LETFERS A
6 November 1989
GRAVITATIONAL INTERACTION OF MASSLESS PARTICLES Valeria FERRARI International Centerfor Relativistic Astrophysics — ICRA, Dipartimento di Fisica “G. Marconi’~Università di Roma, Rome, Italy
and Jesus IBAREZ’ Departamento de Fisica, Universidad de Las Islas Baleares, Spain Received 21 June 1989; revised manuscript received 31 July 1989; accepted for publication 6 September 1989 Communicated by J.P. Vigier
An exact solution of non-vacuum Einstein equations describing the collision of two clouds ofnull particles and their consequent gravitational interaction is presented. As a result, a source is produced which satisfies the Klein—Gordon equation for massless scalar fields, and a physical singularity finally appears.
The gravitational interaction of colliding shells of null dust has been recently investigated by Chandrasekhar and Xanthopoulos [1,2], and by Dray and ‘t Hooft [3]. In refs. [1] and [2], the null dust flies in flat spacetime following the leading edges of impulsive gravitational waves, and the collision produces a region of interaction where either a mixture of null dust moving in opposite directions is present, or a perfect fluid in which the energy density ~t is equal to the pressure p. The presence of different fluids in the region of interaction, is due, as clarified by Taub [4,51,to different assumptions on the energy—momentum tensor, which specify the nature of the interaction. In ref. [3] the collision of two impulsive planar shells of null dust has been considered, with a result similar to that described in ref. [1]. One of the most remarkable consequences of the interaction of null dust, is the possibility, shown in ref. [21, of a gravitationally induced transformation of massless particles, describing null trajectories, into a perfect fluid, although with a peculiar equation of state. We shall show in this paper that a similar transition can occur also when no gravitational waves are Guest ofthe Department ofPhysics ofthe University ofRome.
coupled to the dust, and that a source can be produced which behaves, at a macroscopical level, as a fluid with an equation of state different from p = and with an anisotropic distribution of pressure, being the components orthogonal to the direction of propagation different from the parallel component. The energy—momentum tensor satisfies the strong energy conditions. Unlike the solutions quoted before, the metric is continuous with its first derivatives, therefore no impulsive waves are present on the null hypersurfaces separating different regions. A physical singularity develops in the region of interaction. One might ask how is it possible that the gravitational interaction ofnull dust can produce a so large variety of sources, and whether these solutions really represent the interaction of some fields, and we shall try to answer this question. The null dust is defined by the following energy— momentum tensor: T~V ~ )k~k~ (1) —
where E is a positive function of a null coordinate u representing the energy of the dust, and k’~is a null vector. This definition presents an intrinsic ambiguity, since the energy—momentum tensor (1) can
0375-960l/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
233
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
represent any kind of massless field propagating as a plane wave along a definite direction. In this respect, it is not surprising that many solutions exist
equations in five dimensions, we can derive a solution of the non-vacuum Einstein equations in four dimensions, provided g44 satisfies the constraint
in which two waves of null dust collide and, due to the gravitational interaction, produce different sources, as shown in the references quoted above. However, when the specific equations satisfied by the null particles are identified, the problem is cornpletely defined, the solution is unique, and the ambiguity disappears. Thus, in our case, we have checked that the Klein—Gordon equation for a massless scalar field is satisfied both in the region of interaction andsatisfies in the precollision and that the scalar field the Junctionregion, conditions across the null boundaries separating different regiovs. Therefore we are able to identify the nature of the
(4b), and the energy—momentum tensor associated to the function 0 satisfies the energy conditions. From eqs. (3), the equation for the function q is
null dust and we can conclude that our solution represents the gravitational interaction of massless sealar waves. It is interesting to note that the source in the region of interaction mimics aofnon-perfect with an anisotropic distribution pressure, asfluid is shown by the analysis of the corresponding energy—mo•
.
—
~=0,
(6)
where the dot and the prime indicate, respectively, differentiation with respect to t and z; the function f can be derived by quadrature from the system 2+q2)—2/3t f=3tqq’ (7) f=4t(q’ We select the solution .
q=c 1log~1+c2logj~2+klogt
.
.
•
,
•
•
reduction of a solution of Einstein s equations in fivedimensional vacuum. Let us consider the following form of the metric: 2=e~[—(dt)2+(dz)2] th 2/s[e~ (dxi)2+e~(dx2)2+e2~(dx4)21 +t ‘2’
where 2--t2 ~i2= 1 —z+~,/( 1 —z) k + 9 c 2)~ and c1 and c2 are constants. We shall consider this solution in the region —
— — ~
0z~t~l t—l.~z~l—t.
‘•
f=3((kcl+c~)lOg/1l+(kC 2+c4)log~L2 2c’1c2
(A,
B=0, 4),
(3)
are equivalent to the following system of equations in four dimensions: R~~=O~q~,~p,
(4a)
Ø’~’.,,~=0(~,v=0, 3)
(4b)
where ØJ~(I/3e_~
Cui+u 2) 2
—
4) log t~.
+ (4k
(11)
I
To investigate the physical meaning of the solution, we shall express the components of the energy— momentum tensor TppØ~Ø;p;p, (12) with respect to an orthonormal basis. The resulting canonical form of TM~is defined by the following eigenvalues:
(5)
Therefore, from a given vacuum solution of Einstein’s 234
+1
~
The metric functions f~ q, are assumed to depend on and z only. It is known [6,71 that Einstein equations written in five dimensions for the the vacuum,
(10)
Consequently, f will be
.
RAB—O
(8)
•
mentum tensor in a suitably chosen orthonormal basis, The solution has been obtained by a dimensional •
q” ~q—
Volume 141, number 5,6
=
e-1 {
—
O;0;o +
PHYSICS LETTERS A
6 November 1989
I
12
033
E(0;o;o+0;3;3)2_40,2o;31h/’2}
,
i
(13a)
t 0-1
e —f
:
23{0;0;O+0;3;3
6
+[(0;o;o+0;3;3)240~o;31h/2}
,
(13b) (13c)
To ensure the reality of the eigenvalues, and to make them finite on the null boundaries z=t— 1,
z= 1
—t
,
0
(14)
—0.5
0
z
05
we must assume c
1 = c2 = 1 /,.j~. 21, 2 2~ 23 correspond, respectively, to a timelike vector e0 and to three spacelike vectors e1, e2, e3 provided O;o;o + 0; 3.3>0. This inequality is always satisfied in the region defined by eqs. (10). Hence we can identify the energy density, as measured by —
e0, and the pressure in the three spacelike directions ea as 21,
~u=—A0,
P1P2
p
3=13.
_________________________________________ i0.(~
-
t 01
7.5
(15)
With this identification, the strong energy conditions an observer whose world-line has unit tangent vector ~t+p1>O, u+~p1>O, 1=1,3, are satisfied. Plots of —2~and
Fig. 1. The logarithm of the energy density p and of the p~component of the pressure are plotted for selected values of time, in the corresponding interval of z, in the region of interaction. The superscribed + refers to i’~ the continuous line to ~
23,
and A~=22 are
corresponding given It should in figs. be 1noted z-interval. and 2, that forthe different velocity values field associated of tin the to the eigenvalue A~is timelike, and that the source in the region of interaction therefore behaves as a fluid whose streamlines are tirnelike trajectories. The fluid has an anisotropic distribution of pressure, as shown by eqs. (15), and the component along the direction of propagation P3 is always greater than the components parallel to the wavefront Pi =P2. The hydrodynamical equations, ~ ~.= 0, are satisfied as a consequence of eqs. (4) and the Ricci identities. When t—~0there a physical singularity: ergy density and theispressure diverge like the en~,p 313. (16) In1,p3~t’ order to extend the metric across the null boundaries (14), by using the standard Khan—
~°
2.5
-
‘-0.5
0
0.5
Z
Fig. 2. The logarithm of the Pi =P2 components of the pressure is plotted forthe same values oft and z used in fig. 1.
Penrose algorithm, we introduce a pair of null coordinates u and v, related to t and z by the following equations: 4—v4, z=u4—v4. (17) t=1—u This choice ensures the functions f be finite on the boundaries (14), which correspond to the hypersurfaces u=0 and v=0. In terms of u and v, the metric becomes 235
Volume 141, number 5,6
PHYSICS LETTERS A
213e’~[(cjv1)2+ _eF du dv +t
~2
(dx2)2],
(18)
where e F= 64u 3v3 e-’i The only non-vanishing Weyl scalar in the region defined by eqs. (10) is ~P 2:
(
6v~e! =
—
1 6u
(1
—
u
—
v4)2 (19)
+l_~2_q~2)+~),
therefore no gravitational radiation is present in this region. ~P 4+v4= 1, which corwhen u responds to2ist =singular 0. When u—U0 ± (v—~0 ~), ~ and P2 tend to zero, ~ and p 3 tend to the same finite value, and q12 tends to zero. Because all the metric functions, and all the physical quantities are finite on the null boundaries, we can extend the metric by the substitution u—~uH( u), v—~vH(v)
,
where H is the Heaviside step function, which ensures the continuity of the metric. In addition, the first derivatives of the metric are continuous, and therefore, no impulsive waves are present, The extended metric in the region V~<0, 0 ~ U < 1 is 2— —8(1—u4) du dv ds + (1 —u4)213~‘~(1+u2) ~ 1/12
x
[(~
1)2 + (~2) 2]
.
(20)
The only non-zero component ofthe Ricci tensor is ~ and it is always positive in the region we are considering. Therefore the solution (20) allows an interpretation in terms of a distribution of null dust moving in the v-direction. The situation is symmetnc in the region (u ~ 0, 0 ~ v < 1). When u <0 and v< 0 the spacetime is flat. Atthis stage we can say that the solution presented in this Letter represents the collision and the following gravitational interaction of two planar sandwiches of massless particles propagating in flat spacetime. However, some further informations can be added. The source of the Einstein equations satisfies the Klein—Gordon equations for a massless scalar field 236
~
6 November 1989
,
(21)
both in the interaction region and in the precollision region. The function 0 is continuous everywhere with its first derivatives, and all the required junction conditions are tensor satisfied on by thethe nullRicci boundaries the stress—energy and tensor.byThis means, in particular, that ~ and ~ are continuous on (v=0, 0
~
U;
lution represents the collision, and the following interaction, of two massless scalar fields, which produce, in the region ofinteraction, a scalar field which behaves, at a macroscopical level, as a fluid. The interaction is not affected by any impulsive phenomenon. As in the collision of plane fronted gravitational waves, a singularity develops after the collision of the scalar waves. It should be noted that the stress—energy tensor we use (eq. (12)) is not in the form which is standard for a scalar field. However, it has been shown [81 that it can be reduced to that form by a conformal transformation of the metric, accompanied by a defmite transformation of the scalar potential. It is believed [91 that in the very early universe, at the time when the various interactions were unifled, the universe contained predominantly massless particles. At that time phase transitions areofsuggested to occur which would generate bubbles the new broken-symmetry fase. These bubbles, accelerated by the energy released in the transition, would possibly collide and generate pairs of scalar waves, whose following gravitational interaction is generally negiected in these models. The results presented in this paper suggest that, due to its possible relevance to the physics of the early universe, the gravitational interaction of null dust deserves a deeper investigation.
Volume 141, number 5,6
PHYSICS LETTERS A
6 November 1989
References
[5] A.H. Taub, Collision of impulsive gravitational waves followed by dust clouds, preprint.
[1] S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. A 403 (1986) 189. [2JS. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. A 402 (1985) 37. [3] T. Dray and G. ‘t Hooft, Class. Quantum Gray. 3 (1986) 825. [4] A.H. Taub, J. Math. Phys. 29 (3) (1988) 690.
[6] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. K1. 54 (1921)966. [7]0. Klein,Z. Phys. 37 (1926) 875. [8] V.A. Belinskii and I.M. Khalaktnikov, Sov. Phys. JETP 36 (1972) 591. [9] S.W. Hawking, I.G. Moss andJ.M. Stewart, Phys. Rev. D 26 (1982) 2681.
237