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**Dirac equation in spatially homogeneous cosmological models (*)**

R. PORTUGAL

Centro Brasileiro de Pesquisas Físicas, CBPF

Rua Dr. Xavier Sigaud 150, 22.290-180 Rio de Janeiro, RJ, Brasil (ricevuto il 14 Gennaio 1997; approvato il 3 Giugno 1997)

Summary. — The propagation of spin-1/2 particles in the background of the spatially homogeneous cosmological models has been analysed. By restricting the direction of propagation of the spinors, it is shown that Dirac equation reduces to a decoupled system of differential equations depending only on the time variable. A new exact solution is presented in the Joseph cosmological models.

PACS 04.20 – Classical general relativity. PACS 03.65.Pm – Relativistic wave equations.

1. – Introduction

Dirac equation is an outstanding achievement whose consequences the physicists are very far to exhaust due to the complexity of the calculations involved. The establishment of this equation was the starting point for many research problems, one of which is the study of spin-1/2 particles in the presence of gravitational fields. Brill and Wheeler [1] studied the interaction of neutrinos with the gravitational field of a spherically symmetrical body. They developed the mathematical formalism to study spinors in curved space-time and opened the road for many researchers.

The quantum mechanics of particles in curved space-times has been analysed in many articles and some important results are the creation of particles due to the expansion of space-time [2] and the Hawking radiation of a black hole [3]. These developments go in the direction of a unified theory of quantum and gravitational fields. In this context, the knowledge of exact solutions of the Dirac equation in curved space-time is demanded.

The most studied cosmological backgrounds are the Robertson-Walker ones, due to their simplicity and their accuracy in the fitting of the astronomical data of the recent eras of cosmological evolution. We can refer to Parker [4], Isham and Nelson [5], Ford [6], Andretsch and Schäfer [7], Kovalyov and Légaré [8], Barut and Duru [9] and Villalba and Percoco [10].

Recently, a great effort has been done to solve the massless Dirac equation by applying the method of separation of variables to a general diagonal metric [11, 12].

(*) The author of this paper has agreed to not receive the proofs for correction.

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This study could generalize the analysis of the propagation of spin-1/2 particles in curved space-time, since most of the background used so far depends on just one space-time variable.

There is an important class of space-times—spatially homogeneous models—that is characterized by having a three-dimensional hypersurface of space type passing in every point of the space-time [13, 14]. They have been classified from the geometrical point of view as the Bianchi cosmological models, and they are the simplest generalization of the Robertson-Walker models. The Bianchi models are spatially homogeneous but they admit anisotropies. The formalism of differential forms allows one to treat these models in a unified way. Some authors have used this formalism to study the Maxwell equations in the background of Bianchi models [15]. In the present article we use this formalism to analyse in a unified way the Dirac equations in the background of the spatially homogeneous cosmological models.

The Dirac equation has already been studied in some anisotropic Bianchi models. The Bianchi type I model is the most used one, since the flatness of the space hypersurface simplifies the analysis [9, 16, 17]. There are also exact solutions of the Dirac equation in Bianchi type VI0models [18].

Henneaux [19] has studied the problem of finding the most general spinor field with the same symmetry of the gravitational field. In the present article we study the separation of variables obeying the ansatz (15) in the background of the diagonal Bianchi models. Our purpose is to reduce Dirac equation to a decoupled system of differential equations depending only on the time variable. So, we restrict the direction of propagation of the spinor. We show that for Bianchi type II models, there are two directions of propagation allowed, for the Bianchi types III to VII there is only one particular direction of propagation and for the Bianchi types VIII and IX it is not possible to separate the variables using the ansatz (15).

We present a new exact solution to the massless Dirac equation in the Joseph cosmological models [20], which are Bianchi type V vacuum space-times.

In sect. 2, we establish the Dirac equation in the background of the diagonal Bianchi models and separate the variables to obtain a decoupled purely temporal system of differential equations. In sect. 3, we give two applications of the formalism developed here. 2. – The Dirac equation

The spatially homogeneous cosmological models have a homogeneous three-dimensional hypersurface passing in every point of the space-time. If we choose the direction of the axis of the timelike parameter to be orthogonal to these three-dimensional surfaces, we have a synchronous coordinate system and the line element is given by the following form [13]:

ds2

4 dt22 gij(t) wiwj,

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where gij is a symmetric three-dimensional matrix depending only on the t coordinate.

The one-forms ]wi

, i 41, R , 3( form a basis for the three-dimensional dual space. These forms are invariant under a three-dimensional transitive Lie group which defines a Lie algebra with the structure constants Ci

jkgiven by dwi 4 21 2C i jkwjR wk. (2)

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The enumeration for all distinct groups in the context of spatially homogeneous cosmological models has been done by Taub [14] following the original work of Bianchi [21] in a pure mathematical context. There are nine types that are split into two main classes, namely: class A when SkCikk4 0 and class B when SkCikkc0.

There are alternative conventions for the choice of Ci

jk. The choice of different

conventions should not alter the physical results, but can simplify or complicate the intermediate calculations. From our experience, we follow the convention of Luminet [22] and Eardley [23]. The structure constants of all Bianchi types are listed in the appendix.

In general, the three-dimensional metric gij does not need to be diagonal. In the

present article we restrict ourselves to analysing the models with diagonal metric. In this case the line element (1) reduces to

ds2_{4 dt}2_{2 R}12(w1)22 R22(w2)22 R32(w3)2,

(3)

where the metric components R1, R2and R3depend on t only. We use a tetrad basis of

one-forms ]s2, a 40, R , 3( given by

s0_{4 dt ,} si

4 Riwi ( no sum ) .

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The Dirac equation in curved space-time is [24]

(gaHam¯m2 gaGa1 im) C 4 0 ,

(5)

where the Ham establish the connection between tetrad indices (Latin indices) and

world indices (Greek indices). The components Hamare obtained from

sa_{4 H}amdxm,

(6) where Ha

mand Hamare inverse components. The quantities Gaare given by

Ga4 2

1 4Gbcag

b_gc

(7)

for non-charged spinors. ga _{are the Dirac matrices of Minkowski space-time and G} abc

are the Ricci rotation coefficients defined by dsa

4 GabcsbR sc

(8)

and are antisymmetric in the first two indices Gabc4 2Gbac. Using eqs. (2), (4) and (8)

we can obtain the components of the Ricci coefficients:

G0 ii4 R n i Ri (9) and Gijk4 2 1 2

u

Ri RjRk hipCjkp1 Rj RiRk hjpCkip1 Rk RiRj hkpCjip

v

, (10)

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where hij4 diag (21 , 21 , 21 ). Using eq. (7) we can find Gi: Gi4 2 1 2 i 41

!

3 Rni Ri g0_gi 2 1 4Gjkig j_gk_. (11)

We have already discussed the choice of a convention for the structure constants. Once this convention has been established, it is possible to have alternative choices for the coordinate system. Differently from the choice of Ci

jk, the coordinate system does

not alter the form of the spinor. The basis wi _{can be expressed as}

w(i)

4 h(i)jdxj

(12)

and the wi_{’s are listed in the appendix for all Bianchi type models with the respective}

dual basis ji. We use parenthised indices to distinguish tetrad indices from coordinate

indices when necessary. The components h(i)

j are time-independent but they are

related to the time-dependent components H(i)jby eqs. (4) and (6),

H(i)

j4 Rih(i)j ( no sum ) .

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Using eqs. (11) and (13), Dirac equation (5) reduces to

u

¯t1 g0_gj_h ( j )l Rj ¯l1 1 2

!

i R n i Ri 1 img01 1 4 Gkjig 0_gi_gk_gj

v

C 40 . (14)

Equation (14) is the Dirac equation for the diagonal Bianchi models. Our main interest here is to try a separation of variables of the form

C 4 e ik Q x ( 2 p)3 /2_k_R 1R2R3 f (k , t) (15)

to obtain a differential equation depending only on the time coordinate. Note that the dependence on space coordinates of eq. (14) is restricted to the second term (aside the spinor itself), since the inverse components h( j )kis the unique term that can depend on

space coordinates. Using the ansatz (15), eq. (14) reduces to

u

¯t1 iklh( j )laj Rj 1 img01 1 4Gkjig 0_gi_gk_gj

v

f (k , t) 40 . (16)

The components h( j )lcan be read as the coefficients of the ¯lterms of the vector basis

given in the appendix. By inspection, we can see that Bianchi type VIII and IX models do not admit a separation of variables of the form (15), since the second term of eq. (16) can be independent of the space coordinates for no choice of k. On the other hand, for Bianchi types III to VII we must impose k 4 (k1, 0 , 0 ), for eq. (16) to be purely

temporal. For Bianchi type II, we must impose k 4 (k1, 0 , k3) and for Bianchi type I, k

can be arbitrary since the h( j )l vanish. The last result was expected, since Bianchi I

models admit three independent Killing vectors of the form ¯i. Therefore, a spinor of

the form (15) can propagate in all directions in a Bianchi I model. In the other models the spinor is restricted to propagate in two directions (Bianchi II), or in one direction (Bianchi III to VII).

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In order to continue in a unified way, we impose k 4 (k1, 0 , 0 ). The Bianchi models

of type II to VII admit this kind of momentum. For these models, the second term of eq. (16) reduces to iek1a3/R3, where e 41 for Bianchi II and e421 for Bianchi III to

VII. Using eq. (10), we can put the fourth term of eq. (16) in the form iA(t) g5

1 B(t) a3, where A(t) and B(t) are given by

Bianchi II : A(t) 4 R1 4 R2R3 , _{B(t) 40 ,} (17) Bianchi IV : A(t) 4 R1 4 R2R3 , _{B(t) 42} 1 R3 , (18) Bianchi V : A(t) 40, B(t) 42 1 R3 , (19) Bianchi VIh: A(t) 4 1 4

g

R1 R2R3 2 R2 R1R3

h

, _{B(t) 42}k2h R3 , (20)

Bianchi VIIh: A(t) 4

1 4

g

R1 R2R3 1 R2 R1R3

h

, _{B(t) 42}kh R3 . (21)

The expressions for Bianchi type III and VI0 can be obtained from VIh using h 421

and h 40, respectively, and the type VII0 from VIIh using h 40. Now we can put

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Let us call C6 4 iA 6

g

B 1 iek1 R3

h

. (26)

Equations (24) and (25) yield the following equations:

u

¯2t2 C n 6 C6(¯t1 im) 1 C 62 1 m2

v

f6 I 4 0 (27) and f1 II4 2 1 C6(¯t1 im) f 6 I . (28)

Equation (27) can be put in the form

u

¯2t1 1 2 C n n C 2 3 4 C n 2 C2 2 C 2 2 imC n C 1 m 2

v

g

fI kC

h

4 0 , (29)

where we have taken out the 6 sign. The problem of finding the solutions for the Dirac equation in a spatially homogeneous model satisfying the ansatz (15) now reduces to find the solutions for an equation of the form (29). This differential equation is rather complicated in the general case even when m is zero (neutrinos). The function C(t) must obey eq. (26), where A(t) and B(t) are given by eqs. (17)-(21). Most of the known exact solutions of Einstein equations [25] yield a differential equation

(

eq. (29)

)

that has no known exact solution. In the following section we give two applications where eq. (29) can be integrated.

3. – Two examples: Bianchi VI0and V

As an application of results obtained so far, let us consider the line element ds2 4 dt22 t 2_dx2 ( m 2n)2 2 dy2 t2( m 1n)e2 x 2 e2 x_dz2 t2( m 1n) (30)

studied in ref. [18], where solutions for the Dirac equation have been presented both for neutrinos and massive spinors using the coordinate system of eq. (30). With the coordinate transformation

.

`

/

`

´

x 4x1_, y 4 e x1 k2(x 2 1 x3) , z 4 e 2x1 k2 (x 2 2 x3) , (31)

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the line element (30) can be put in the form ds2 4 dt22 t 2_{( dx}1₎2 ( m 2n)2 2 (x2_dx1 1 dx3)2 t2( m 1n) 2 (x3_dx1 1 dx2)2 t2( m 1n) . (32) Choosing

.

/

´

w1_{4 x}3dx1_{1 dx}2, w2 4 x2dx1 1 dx3, w3 4 2 dx1, (33)

we can see that the line element (32) can be put in the form (3) with R14 R24 1 /t( m 1n)

and R34 t/( m 2 n). The structure constants of this model are C2314 C1324 1, therefore it

is of type Bianchi VI0(see the appendix). The function C(t), given by eqs. (26) and (20), is

C(t) 4 ik1( m 2n)

t .

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Substituting this value in eq. (29), we obtain

g

¯t1 ( 1 O4)1k12( m 2n) 2 t2 1 im t 1 m 2

h

₍ kt fI) 40 . (35)

It can be verified that eq. (35) is equal to eq. (20) of ref. [18]. Therefore, using the formalism developed here, we obtained the same Dirac equation by a suitable choice of the basis wi_{. The explicit form of the spinor will differ for these coordinate systems,}

since in one case it will be given in coordinates (x , y , z) and in the other case in (x1_{, x}2_{, x}3_{). But the temporal part will be the same. Equation (35) has been solved in}

ref. [18].

Let us consider a second application. By inspection of eqs. (17) to (21) we can see that Bianchi type V furnishes the simplest Dirac equation since A(t) 40 and B(t) has the same form of the term iek1/R3of eq. (22). The vacuum Bianchi V models have been

studied by Joseph [20]. The line element (1_{) can be put in the form [25]}

ds2 4 ( sinh 2 at)

k

dt2 2 ( tanh at)k3(w1₎2 2 ( tanh at)2k3(w2₎2 2

g

w 3 a

h

2

l

, (36)

where w1_{, w}2_{and w}3_{are listed in the appendix. Since this line element has a coefficient}

in the dt2 _{term, it is interesting to introduce the coefficient R}

0(t)2 in the general line

element (3) to obtain the corresponding Dirac equation of the form (29). The equations that change are eqs. (4), (9), (11), (14), (16), (22) and the subsequent ones. Equation (16) changes to

u

¯t1 iklR0h( j )laj Rj 1 imR0g01 R0 4 Gkjig 0_gi_gk_gj

v

f (k , t) 40 (37)

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and eq. (29) to

u

¯2t1 1 2 C n n C 2 3 4 C n 2 C2 2 C 2 2 imR0C n C 1 m 2 R021 im R n 0

v

g

fI kC

h

4 0 , (38) where C is given by C6 4 iR0A 6R0

g

B 1 iek1 R3

h

. (36)

For the line element (36) we have

g

¯2t2 (ik1a 1a)21 m2sinh 2 at 1

iam cosh 2 at ksinh 2 at

h

( fI) 40 (40) and fII4 1 a 1ik1a (¯t1 im sinh 2 at) fI.

For neutrinos, the solutions are

fI4 c1e(a 1ik1a) t1 c2e2(a 1 ik1a) t,

(41)

where c1 and c2 are the integration constants. The four independent solutions of the

form (15) are

C6 I 4

ei(k1x 1ak1t) 1at

( 2 p)3 /2_{( sinh 2 at)}3 /4

u

1 0 61 0

v

, C6 II4

ei(k1x 2ak1t) 2at

( 2 p)3 /2_{( sinh 2 at)}3 /4

u

0 1 0 61

v

.

This spinor cannot be normalized due to the term e6at_{. This fact indicates that this}

solution is non-stable. A similar situation happens with a scalar field f that has a Lagrangian with a negative mass squared, 2m2_{. In this case the scalar field has a term}

e6km22 k2t

. The usual interpretation is that the solution f 40 is not stable and the spontaneous symmetry breaking occurs after a time of order m21_{(see ref. [26]).}

4. – Conclusion

We have analysed the Dirac equation in the background of the spatially homogeneous cosmological models. We have used the method of separation of variables through the ansatz (15). The dynamics of the spatially homogeneous models is

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characterized by functions depending only on the time variables. For this reason we have chosen a spinor of the form of eq. (15). With this choice it is possible to reduce the Dirac equation to a system of decoupled differential equations, depending only on the time variable.

We have showed that Bianchi type VIII and IX models do not admit this kind of separation of variables, since too many components of the Killing vectors depend on space coordinates. In this case the spatial dependence of the spinor cannot be eikx. For the other Bianchi models, it is possible to have solutions of the form of eq. (15), if we impose restrictions on the direction of propagation of the spinor. For Bianchi types III to VII, the spinor is allowed to propagate in only one direction. The direction is determined by the spatial dependence of the space-like Killing vectors. In the present article the direction is k 4 (k1, 0 , 0 ), since the first component of all space-like Killing

vectors does not depend on the spatial variables. For the Bianchi type II model, there are two directions allowed and for Bianchi type I, there are no restrictions on the direction of propagation.

We have made two applications. In the first one, the Dirac equation was analysed in the background of a Bianchi type VI0 model that has been widely studied in the

literature. In the second one, the background used was the vacuum Bianchi type V model studied by Joseph [20]. In this case, the spinor found cannot be normalized. This probably indicates that the Dirac equation yields non-stable solutions when the spinor is in the presence of an external gravitational field.

* * *

The author would like to thank F. R. CARVALHO and N. F. SVAITER for helpful discussions during this work.

AP P E N D I X

The structure constants (Ci

jk), the vector basis (ji) and the invariant 1-form

basis [22] (wi_{) of the Bianchi models.}

The structure constants (Ci

jk4 2Cikj): CLASS A (Cl il4 0 ): Bianchi I: Ci jk4 0 , Bianchi II: C1 234 1 , Bianchi VI0: C2314 C1324 1 , Bianchi VII0: C2314 C3124 1 , Bianchi VIII: C1 234 C3124 C2134 1 , Bianchi IX: C1 234 C3124 C1234 1 . CLASS B (Cl ilc0 ):

Bianchi III: Bianchi VI₂₁, Bianchi IV: C1 314 C2314 C3214 1 , Bianchi V: C1 314 C3224 1 , Bianchi VIh: C2314 C1324 1 , C3114 C3224k2h , h E 0 , Bianchi VIIh: C2314 C3124 1 , C3114 C3224kh , h D0.

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The vector and 1-form basis: Bianchi type I :

j14 ¯1, w14 dx1,

j24 ¯2, w24 dx2,

j34 ¯3, w34 dx3.

Bianchi type II:

j14 ¯2, w14 2x3dx11 dx2,

j24 ¯3, w24 dx3,

j34 ¯11 x3¯2, w34 dx1.

Bianchi type IV:

j14 ¯2, w14 (x32 x2) dx11 dx2, j24 ¯3, w24 2x3dx11 dx3, j34 2¯11 (x32 x2) ¯22 x3¯3, w34 2dx1. Bianchi type V: j14 ¯2, w14 2x2dx11 dx2, j24 ¯3, w24 2x3dx11 dx3, j34 2¯12 x2¯22 x3¯3, w34 2dx1. Bianchi type VIh: j14 ¯2, w14 (x32 ax2) dx11 dx2, j24 ¯3, w24 (x22 ax3) dx11 dx3, j34 2 ¯11 (x32 ax2) ¯21 (x22 ax3) ¯3, w34 2dx1.

Bianchi type VIIh:

j14 ¯2, w14 (x32 ax2) dx11 dx2,

j24 ¯3, w24 2(x21 ax3) dx11 dx3,

j14 2¯11 (x32 ax2) ¯22 (x21 ax3) ¯3, w34 2dx1.

Bianchi type VIII:

_dx1 2 ex3dx2_.

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Bianchi type IX:

j14 ¯2, w14 dx21 cos x1dx3,

j24 cos x2¯12 cotan x1sin x2¯21

sin x2

sin x1¯3, w 2

4 cos x2dx1

1 sin x2sin x1_dx3_,

j34 2sin x2¯12 cotan x1cos x2¯21

cos x2 sin x1¯3, w 3 4 2sin x2dx1 1 cos x2sin x1_dx3_. R E F E R E N C E S

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