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Dirac equation in bianchi I metrics
Volume 121, number 1
PHYSICS LETTERS A
30 March 1987
DIRAC EQUATION IN BLANCHI I METRICS L.P. CHIMENTO ‘and M.S. MOLLERACH Departamento de FIsjca, F.C.E. yN., Universidad de BuenosAires, Pab. 1, 1428 BuenosAires, Argentina Received 28 August 1986; revised manuscript received 30 December 1986; accepted for publication 9 January 1987
We present the Dirac equation generalization to spatially flat anisotropic Bianchi I metrics and we study their classical solutions. We find only two independentspinors, and we show that it is not possible to obtain the solutions from those of the Robertson—Walkercase by perturbing the metric.
The quantum field theory in curved spacetime has been a matter of great interest in recent years because of its applications to cosmology and astrophysics. The evidence of the existence of strong gravitational fields in our universe led to the study of the quantum effects of a classical gravitational field on other fields, such as creation ofparticles and antiparticles and vacuum polarization. After the appearance of Parker’s papers on scalar fields [1] and spin-i fields [2], several authors have studied this subject (see refs. [3—5]). In a recent work [61, we studied the particle creation effect for spin-i fields in Robertson—Walker metrics. Our purpose is to extend this study to the case of Bianchi I anisotropicspatially flat metrics. This is a very interesting point, as it has been shown that the creation of scalar particles by an anisotropic gravitational field leads to dissipation of the anisotropy when the universe expands [7,8]. To this effect, we shall startby finding the classical solution ofthe Dirac equation in this metric, which is the subject of this work. The generalization of the Dirac equation in curved spacetime is [9] (F~V~—m)’P(x,t)=0,
(1)
where rP=va~ya, Ti —~ 0~U’ ~/4”/4’
L
_1r ~—5LY
(2) a
,Y$117 J a
117$vj~,
are the Dirac matrices in Minkowski space and Va ~ is a set of four-vector fields called vierbein, whose components are related to the metrics by the equation yP
g~=V~av,s~,
(4)
where ,j,.~is the Minkowski metric. The generalized Dirac matrices satisfy the anticommutation relations {y~,y1}=—2g~’.
(5)
The metric of a spatially flat anisotropic Bianchi I universe is ds2 =dt2 —a~(t)dx~—a~(t)dx~—a~(t)dx~ .
Fellow ofthe Consejo Nacional de InvestigacionesCientlficas y Técnicas.
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(6)
Volume 121, number 1
PHYSICS LETTERS A
30 March 1987
Using (2) and (3) for this metric we obtain the matrices 1” and the spinorial connections
~
r°=y°,T’=a~’(t)y’
(7)
cr°0, ci=~(â1/a1)p°p’
(8)
,
(in eqs. (7) and (8) there is no summation over the indices). The Dirac equation (1) can be solved using separation of variables. We propose, in analogy with ref. [10], where the solution is studied in a Robertson—Walker metric, the following form for the solutions, dt’) exp(if~OaKdt’) exp(if~OAKdt’) exp (if~ 0fl~ dt’) exp(i.ftQx
312(a,a 2 ~P(x t)— (2m) 1 2a3)” k ‘
—
(
.
9
Inserting (9) into (1) we obtain the following matrix equation, Q~+m 0 -K3 0 A~+m —K~
-K K3
—K3 —K~
aK—rn
0
0
fl~—m
—K K3
f~(:)=0,
(10)
wherefx(t) is the column matrix definedin (9) and (11)
1C1=/ç•/a1, K~=K1±iK2.
The homogeneous equation system will have a non-trivial solution when 2 —(AK+rn)(aK—rn)][K2
—
(QK+m)(PK—m)]
+K~(QK—AK)(flK—aK)=0.
(12)
[K
Solving the system, we get the following expression forfK( t), —K K2—(AK+m)(aK—m) K 2—(QK+m)(PK—m) 3 K JK(t)=
—K(AK—QK)
1
K2—(QK+m)(aK—rn)
(
exP(iJAKdt”). “ to
(13)
/
(AK—~K)K~K
K~\~.K2~(QK+rn)(aK_m) ~~~+m)
Putting (13) into (9) it will solve eq. (1) ifthe functions QK, A~,aK and JJK satisfy the following differential equations, resulting from the application of the operator (y°O~ + iK~,y~ + m) to eq. (10) and the substitution of expression (13) into it, ~K_iQK(0~C+i(H 3)
A~_iA 1_w~+i(H3 ~ 8
2—(A~+rn)(a~—m) K K~(AK—QK) _~(HKY (QK+m)=0,
K
K
)
2 —(QK+rn)(aK—m) —jH K
3(A~+m)=0,
Volume 121, number I
PHYSICS LETTERS A
2 2 K2—(A1+rn)(a1—m) a~—ia~w~+iH3 —1 (HK) K .
2
2
flx—1fix—wx
~1 ~
-
30 March 1987
2—(Q K
~
1+m)(a1—rn) —0,
=
0
‘
14
2 + rn2. with ±=H,isK1 ± iH2K2 w~= k WeH1 see=â1/a1 that inand this(HK) case there a significant difference with respect to the isotropic case (see refs. [6,10]). In that case, the form ofthe Dirac equation admits a solution with Q~=A~ and a~=/i 1,and the condition that the homogeneous system equation determinant be equal to zero makes that two equations are linearly dependent on the other two, and the subspace of solutions has dimension two. In this way, two spinors of the basis are obtained and the other two result from applying the charge conjugation operation to the formerones. As distinguished from this, in the Bianchi I metric the condition of zero determinant (12) makes that only one of the equations (10) is linearly dependent, and then we obtain only one spinor of the basis and another one by applying the charge conjugation operation to the former, 2J~.(t). (15) f~(t)=y
On the other hand, in the isotropic case it can be seen that all the spinors can be obtained from one of them by appying to this one the charge conjugation operation and transforming the resulting two spinors with the change of coordinates x~~-~x The 2.matrix which transforms the spinors is 4 S=e”~’
{
~
9]
(16)
.
It can be seen that iff~(t)is a solution of eq. (10), then also isf~(t)= (Jj~{t))”’2.(f~) 12 indicates that the change K,i—~’K 2must be made inf1(t). When we try to obtain a four-spinor base in a Bianchi I metric using this method, we find that the transformation (16) applied to the spinors obtained from (13) and (15) do not lead to independent spinors, but leave them invariant, as we shall see below. The proof follows from multiplying (13) and (15)(ax)”2 by the matrix Sand changing K~~-~K2 in the expressions of 2, (A~)~2, and (fl~)”2 which appear in (J~)’2 and j~)12 f,(t) are theandf~(t). solutionsTof hethe functions system of (Q~)’~ differential equations (14) when the changes K,~-~’K and 2 H1.-’H2 are made. Using the condition of zero determinant (12), it can be seen that ~
~
\1—.2_ A
-“K,
IA
~1-2_r
“K)
I
\1’—.2_D
_~~x’ ~aJ()
PK,
ID
~pK
\1—’2_
—ax.
By means of the relations (17) and (12) it follows thatf~(1) andf~(t) transform into themselves apartfrom a phase shift. Therefore, in a Bianchi I metric the coordinate transformation x,l—~x2does not transform spinors characterizing particles or antiparticles of one helicity to the opposite as in the isotropic case. Note that in the Robertson—Walker metric this transformation not only leaves the Dirac equation invariant, but it is also a symmetry ofspacetime. In theanisotropic case the transformation leaves the Dirac equation invariant, but not the geometry. In this way, choosing the ansatz (9), we obtain only two independent spinor solutions ofthe Dirac equation (1), which are the ones obtained from substituting (13) and (15) into (9). This is another interesting feature ofthe spinor field in Bianchi type metrics [5]. On the other hand, we shall prove now that making an anisotropic perturbation to the Robertson—Walker 9
Volume 121, number 1
PHYSICS LETTERS A
30 March 1987
metric, it is not possible to obtain the Dirac equation solution spinors as a perturbation of the spinors of the isotropic case. Therefore, there is a manifest discontinuity in the solutions of the Dirac equation when we go from a Robertson—Walker metric to a Bianchi I metric, even if this is a small perturbation to the former. Ifineq. (6)wetake a=a+ö1, ö14~a,
(18)
and we consider in (9) +Qk, AX=QRW +A~, aK=aRw +a~, PK=aRW +flj~ (19) and axw are functions corresponding to isotropic case spinors (see ref. [6]) and Q~,A~,ag and where /J,~are £small perturbations, we can write the matrix equation (10) as I2K=QRW 2RW
MRwf~(t)= —M’fg,~w(t),
(20)
where MRW is the matrix analog to (10) in the Robertson—Walker metric and ~ the column matrix which solves MRwf,.,~w= 0. The matrix M’ is the first-order correction to the matrix (10). In addition, we have neglected the term M’fk because it is a second-order correction. Eq. (20) is a non-homogeneous system of four equations in fk. On the other hand, we know that the condition det MRW = 0 must be fulfilled, and also that this condition makes that two equations are linearly dependent on the other two. This fact imposes some restrictions to the coefficients of —M’f,~~ in order that the system be compatible. These conditions imply (21) However, it can be seen that if in the ansatz (9) we insert condition (21), then the Dirac equation system (10) is compatible only ifH, = H 2 = H3. I.e. the metric must be an isotropic Robertson—Walker one. Therefore, we have proved that ifwe perturb the Robertson—Walker metric, the Dirac equation solution spinors shall be a small perturbation of the isotropic case solution only ifthe perturbation is isotropic. We see that the transition ofthe Dirac equation solutions from a Robertson—Walker metric to a Bianchi I metric is discontinuous. In conclusion, we have presented a series of arguments which indicate that the Dirac equation, as we use to generalize it to curved spacetime, in the case ofa Bianchi I typemetric admits as solution only two independent spinors. If we quantize the field using them as a basis, we should obtain a theory which describesparticles with only one helicity and antiparticles with the opposite helicity. In this way, the spacetime anisotropy should have the effect of orienting the spin of the matter created. This is a very significant fact, because this effect should appear regardless how small the anisotropy is. We have the purpose to investigate thoroughly the action of an anisotropicgravitational field on spin-i fields. QK=AK.
References [l]L.Parker,Phys.Rev. 183 (1969) 1057. (2] L. Parker, Phys. Rev. D 3 (1971) 346. [3] N. Bind and C. Davies, Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982). [4] B. DeWitt, Phys. Rep. 19 (1975) 297. [5] M. Henneaux, Ann. Inst. H. Poincaré 34 (1981) 329. [6] L. Chimento and M.S. Mollerach, Vacuum definition for spin ~ fields in Robertson—Walker metrics, submitted to Phys. Rev. D. [7] B. Hu and L. Parker, Phys. Rev. D 17 (1978) 933. [8] G. Bender and L Chimento, Proc. SILARG V, eds. Bressan, Castagnino and Hamity (1985). [9] S. Weinberg, Gravitationand cosmology (Wiley, New York, 1972). [10] M. Castagnino, L Chimento, D. Harani and C. Nufiez, I. Math. Phys. 25 (1984).
10
PHYSICS LETTERS A
30 March 1987
DIRAC EQUATION IN BLANCHI I METRICS L.P. CHIMENTO ‘and M.S. MOLLERACH Departamento de FIsjca, F.C.E. yN., Universidad de BuenosAires, Pab. 1, 1428 BuenosAires, Argentina Received 28 August 1986; revised manuscript received 30 December 1986; accepted for publication 9 January 1987
We present the Dirac equation generalization to spatially flat anisotropic Bianchi I metrics and we study their classical solutions. We find only two independentspinors, and we show that it is not possible to obtain the solutions from those of the Robertson—Walkercase by perturbing the metric.
The quantum field theory in curved spacetime has been a matter of great interest in recent years because of its applications to cosmology and astrophysics. The evidence of the existence of strong gravitational fields in our universe led to the study of the quantum effects of a classical gravitational field on other fields, such as creation ofparticles and antiparticles and vacuum polarization. After the appearance of Parker’s papers on scalar fields [1] and spin-i fields [2], several authors have studied this subject (see refs. [3—5]). In a recent work [61, we studied the particle creation effect for spin-i fields in Robertson—Walker metrics. Our purpose is to extend this study to the case of Bianchi I anisotropicspatially flat metrics. This is a very interesting point, as it has been shown that the creation of scalar particles by an anisotropic gravitational field leads to dissipation of the anisotropy when the universe expands [7,8]. To this effect, we shall startby finding the classical solution ofthe Dirac equation in this metric, which is the subject of this work. The generalization of the Dirac equation in curved spacetime is [9] (F~V~—m)’P(x,t)=0,
(1)
where rP=va~ya, Ti —~ 0~U’ ~/4”/4’
L
_1r ~—5LY
(2) a
,Y$117 J a
117$vj~,
are the Dirac matrices in Minkowski space and Va ~ is a set of four-vector fields called vierbein, whose components are related to the metrics by the equation yP
g~=V~av,s~,
(4)
where ,j,.~is the Minkowski metric. The generalized Dirac matrices satisfy the anticommutation relations {y~,y1}=—2g~’.
(5)
The metric of a spatially flat anisotropic Bianchi I universe is ds2 =dt2 —a~(t)dx~—a~(t)dx~—a~(t)dx~ .
Fellow ofthe Consejo Nacional de InvestigacionesCientlficas y Técnicas.
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(6)
Volume 121, number 1
PHYSICS LETTERS A
30 March 1987
Using (2) and (3) for this metric we obtain the matrices 1” and the spinorial connections
~
r°=y°,T’=a~’(t)y’
(7)
cr°0, ci=~(â1/a1)p°p’
(8)
,
(in eqs. (7) and (8) there is no summation over the indices). The Dirac equation (1) can be solved using separation of variables. We propose, in analogy with ref. [10], where the solution is studied in a Robertson—Walker metric, the following form for the solutions, dt’) exp(if~OaKdt’) exp(if~OAKdt’) exp (if~ 0fl~ dt’) exp(i.ftQx
312(a,a 2 ~P(x t)— (2m) 1 2a3)” k ‘
—
(
.
9
Inserting (9) into (1) we obtain the following matrix equation, Q~+m 0 -K3 0 A~+m —K~
-K K3
—K3 —K~
aK—rn
0
0
fl~—m
—K K3
f~(:)=0,
(10)
wherefx(t) is the column matrix definedin (9) and (11)
1C1=/ç•/a1, K~=K1±iK2.
The homogeneous equation system will have a non-trivial solution when 2 —(AK+rn)(aK—rn)][K2
—
(QK+m)(PK—m)]
+K~(QK—AK)(flK—aK)=0.
(12)
[K
Solving the system, we get the following expression forfK( t), —K K2—(AK+m)(aK—m) K 2—(QK+m)(PK—m) 3 K JK(t)=
—K(AK—QK)
1
K2—(QK+m)(aK—rn)
(
exP(iJAKdt”). “ to
(13)
/
(AK—~K)K~K
K~\~.K2~(QK+rn)(aK_m) ~~~+m)
Putting (13) into (9) it will solve eq. (1) ifthe functions QK, A~,aK and JJK satisfy the following differential equations, resulting from the application of the operator (y°O~ + iK~,y~ + m) to eq. (10) and the substitution of expression (13) into it, ~K_iQK(0~C+i(H 3)
A~_iA 1_w~+i(H3 ~ 8
2—(A~+rn)(a~—m) K K~(AK—QK) _~(HKY (QK+m)=0,
K
K
)
2 —(QK+rn)(aK—m) —jH K
3(A~+m)=0,
Volume 121, number I
PHYSICS LETTERS A
2 2 K2—(A1+rn)(a1—m) a~—ia~w~+iH3 —1 (HK) K .
2
2
flx—1fix—wx
~1 ~
-
30 March 1987
2—(Q K
~
1+m)(a1—rn) —0,
=
0
‘
14
2 + rn2. with ±=H,isK1 ± iH2K2 w~= k WeH1 see=â1/a1 that inand this(HK) case there a significant difference with respect to the isotropic case (see refs. [6,10]). In that case, the form ofthe Dirac equation admits a solution with Q~=A~ and a~=/i 1,and the condition that the homogeneous system equation determinant be equal to zero makes that two equations are linearly dependent on the other two, and the subspace of solutions has dimension two. In this way, two spinors of the basis are obtained and the other two result from applying the charge conjugation operation to the formerones. As distinguished from this, in the Bianchi I metric the condition of zero determinant (12) makes that only one of the equations (10) is linearly dependent, and then we obtain only one spinor of the basis and another one by applying the charge conjugation operation to the former, 2J~.(t). (15) f~(t)=y
On the other hand, in the isotropic case it can be seen that all the spinors can be obtained from one of them by appying to this one the charge conjugation operation and transforming the resulting two spinors with the change of coordinates x~~-~x The 2.matrix which transforms the spinors is 4 S=e”~’
{
~
9]
(16)
.
It can be seen that iff~(t)is a solution of eq. (10), then also isf~(t)= (Jj~{t))”’2.(f~) 12 indicates that the change K,i—~’K 2must be made inf1(t). When we try to obtain a four-spinor base in a Bianchi I metric using this method, we find that the transformation (16) applied to the spinors obtained from (13) and (15) do not lead to independent spinors, but leave them invariant, as we shall see below. The proof follows from multiplying (13) and (15)(ax)”2 by the matrix Sand changing K~~-~K2 in the expressions of 2, (A~)~2, and (fl~)”2 which appear in (J~)’2 and j~)12 f,(t) are theandf~(t). solutionsTof hethe functions system of (Q~)’~ differential equations (14) when the changes K,~-~’K and 2 H1.-’H2 are made. Using the condition of zero determinant (12), it can be seen that ~
~
\1—.2_ A
-“K,
IA
~1-2_r
“K)
I
\1’—.2_D
_~~x’ ~aJ()
PK,
ID
~pK
\1—’2_
—ax.
By means of the relations (17) and (12) it follows thatf~(1) andf~(t) transform into themselves apartfrom a phase shift. Therefore, in a Bianchi I metric the coordinate transformation x,l—~x2does not transform spinors characterizing particles or antiparticles of one helicity to the opposite as in the isotropic case. Note that in the Robertson—Walker metric this transformation not only leaves the Dirac equation invariant, but it is also a symmetry ofspacetime. In theanisotropic case the transformation leaves the Dirac equation invariant, but not the geometry. In this way, choosing the ansatz (9), we obtain only two independent spinor solutions ofthe Dirac equation (1), which are the ones obtained from substituting (13) and (15) into (9). This is another interesting feature ofthe spinor field in Bianchi type metrics [5]. On the other hand, we shall prove now that making an anisotropic perturbation to the Robertson—Walker 9
Volume 121, number 1
PHYSICS LETTERS A
30 March 1987
metric, it is not possible to obtain the Dirac equation solution spinors as a perturbation of the spinors of the isotropic case. Therefore, there is a manifest discontinuity in the solutions of the Dirac equation when we go from a Robertson—Walker metric to a Bianchi I metric, even if this is a small perturbation to the former. Ifineq. (6)wetake a=a+ö1, ö14~a,
(18)
and we consider in (9) +Qk, AX=QRW +A~, aK=aRw +a~, PK=aRW +flj~ (19) and axw are functions corresponding to isotropic case spinors (see ref. [6]) and Q~,A~,ag and where /J,~are £small perturbations, we can write the matrix equation (10) as I2K=QRW 2RW
MRwf~(t)= —M’fg,~w(t),
(20)
where MRW is the matrix analog to (10) in the Robertson—Walker metric and ~ the column matrix which solves MRwf,.,~w= 0. The matrix M’ is the first-order correction to the matrix (10). In addition, we have neglected the term M’fk because it is a second-order correction. Eq. (20) is a non-homogeneous system of four equations in fk. On the other hand, we know that the condition det MRW = 0 must be fulfilled, and also that this condition makes that two equations are linearly dependent on the other two. This fact imposes some restrictions to the coefficients of —M’f,~~ in order that the system be compatible. These conditions imply (21) However, it can be seen that if in the ansatz (9) we insert condition (21), then the Dirac equation system (10) is compatible only ifH, = H 2 = H3. I.e. the metric must be an isotropic Robertson—Walker one. Therefore, we have proved that ifwe perturb the Robertson—Walker metric, the Dirac equation solution spinors shall be a small perturbation of the isotropic case solution only ifthe perturbation is isotropic. We see that the transition ofthe Dirac equation solutions from a Robertson—Walker metric to a Bianchi I metric is discontinuous. In conclusion, we have presented a series of arguments which indicate that the Dirac equation, as we use to generalize it to curved spacetime, in the case ofa Bianchi I typemetric admits as solution only two independent spinors. If we quantize the field using them as a basis, we should obtain a theory which describesparticles with only one helicity and antiparticles with the opposite helicity. In this way, the spacetime anisotropy should have the effect of orienting the spin of the matter created. This is a very significant fact, because this effect should appear regardless how small the anisotropy is. We have the purpose to investigate thoroughly the action of an anisotropicgravitational field on spin-i fields. QK=AK.
References [l]L.Parker,Phys.Rev. 183 (1969) 1057. (2] L. Parker, Phys. Rev. D 3 (1971) 346. [3] N. Bind and C. Davies, Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982). [4] B. DeWitt, Phys. Rep. 19 (1975) 297. [5] M. Henneaux, Ann. Inst. H. Poincaré 34 (1981) 329. [6] L. Chimento and M.S. Mollerach, Vacuum definition for spin ~ fields in Robertson—Walker metrics, submitted to Phys. Rev. D. [7] B. Hu and L. Parker, Phys. Rev. D 17 (1978) 933. [8] G. Bender and L Chimento, Proc. SILARG V, eds. Bressan, Castagnino and Hamity (1985). [9] S. Weinberg, Gravitationand cosmology (Wiley, New York, 1972). [10] M. Castagnino, L Chimento, D. Harani and C. Nufiez, I. Math. Phys. 25 (1984).
10