Exact scalar and spinor solutions in some rotating universesK. D. Krori, P. Borgohain, P. K. Kar, and Dipali Das (Kar) Citation: Journal of Mathematical Physics 29, 1645 (1988); doi: 10.1063/1.527912 View online: http://dx.doi.org/10.1063/1.527912 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/29/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear spinor field in Bianchi type-I Universe filled with perfect fluid: Exact self-consistent solutions J. Math. Phys. 38, 5305 (1997); 10.1063/1.531944 An exact spinor solution in Kantowski–Sachs cosmology J. Math. Phys. 36, 1347 (1995); 10.1063/1.531124 Exact scalar and spinor solutions in the field of a stationary cosmic string J. Math. Phys. 35, 1032 (1994); 10.1063/1.530649 Spinor solution in a Kasner universe J. Math. Phys. 31, 446 (1990); 10.1063/1.528933 Some exact solutions of Einstein–Dirac–Maxwell zero mass scalar equations J. Math. Phys. 24, 1574 (1983); 10.1063/1.525851

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Exact scalar and spinor solutions in some rotating universes K. D. Krori, P. Borgohain, P. K. Kar, and Dipali Das (Kar) Mathematical Physics Forum, Cotton Col/ege, Gauhati 781 ()()J, India

(Received 27 October 1987; accepted for publication 30 December 1987)

In this paper exact solutions of the Klein-Gordon and Weyl equations in some rotating universes are presented.

I. INTRODUCTION

The study of gravitational interaction on quantum me­chanical systems is a field of investigation recently explored by a number of authors. The Klein-Gordon equation and the Dirac equation in covariant form have been used in curved space-times for such studies. 1-6 The electromagnetic equation in Godel's universe have been studied by Cohen et aC The Klein-Gordon and the Weyl equations have also been studied in the same universe by Pimentel and Macias.8

In this paper we consider the same two equations in three other well-known rotating universes, viz., Som-Raychaud­huri, Hoenselaers-Vishveshwara, and Rebouc;as universes. Exact solutions of the two equations in all three cases are presented in this paper.

II. SCALAR SOLUTIONS

The governing equation for a massive scalar field with arbitrary coupling to the gravitational field can be taken in the form

( - Va va + 5R + m2) "" = 0, (2.1 )

where 5 is a real dimensionless coupling constant and R = Rf.Lvlt'v is the Riemann scalar.

A. Som-Raychaudhuri (SR) metric

The general form of the Som-Raychaudhuri metric is given by

d~ = dt 2 - dr- - dz'2 + 2qr- dtjJ dt - (r- - ir4)dtjJ2. (2.2)

On assuming q = 1, the metric (2.2) takes the form

d~ = dt 2 - dr- - dz'2 + 2r- dtjJ dt - r-(1 - r-)dtjJ2. (2.3)

To calculate the Ricci scalar we consider the following basis one-forms:

()I=dr, ()2=rdtjJ, ()3=dz, ()o=(dt+r-dtjJ), (2.4 )

so that

gij = diag{ - 1, - 1, - 1, + n. Taking the exterior differentials of (2.4), expressing

them in terms of the basis two-forms () i 1\ () i, and comparing the expressions with the first equation of structure

d() i = _ m~ 1\ () i,

we get the following set of connection one-forms:

ml2 = ()O _ (lIr)() 2 = _ m\,

mOl = ()2 = mlo, m02 = - () 1= m2o'

(2.5)

(2.6)

We get the curvature two-forms O~ from the second equation of structure

O~ = dm~ + mik I\mki' (2.7)

and, finally, the components of the Riemann tensor R ~k/ from

(2.8)

The nonvanishing components of O~ and Rf.Lv are found to be

0\=3()11\()2= _012,

001 = () 11\ ()O = _ 001,

and Roo = 2, R II = 2, R22 = 2, and R 02 = 0. The Ricci sca­lar is found to be

R= -2. (2.10)

With Eqs. (2.2) and (2.10), Eq. (2.1) reduces to

[(1- r-)a;t - a;r - (lIr-)a~ - a~

+2a~t - (1/r)ar _p2]",,=0, (2.11)

where

p2 = m2 - 25.

Let us choose"" in the form

"" = eia(cut+ /,p+ kz) R(r),

(2.12)

(2.13 )

where I, m, and k are constants. Substituting (2.13) in (2.11) we obtain

R "(r) + (lIr)R '(r)

+ a2[ - F/r- - r-m2 + A 2]R(r) = 0, (2.14)

with

A 2 = m2 _ k 2 + 2ml + m2/a2 - 25/a2. (2.15 )

Equation (2.14) is reduced to the confluent hypergeometric differential equation

yF" + [(al+ 1) -y]F'

- [(1 + a/)/2 - aA 2/4m]F= 0,

if we make the substitution

(2.16)

R(r) =yal12e - y12F(y), (2.17)

wherey = amr-. Equation (2.16) has two independent solutions given by

and

i - c IFI [( 1 + b - e),2 - e,y],

where

b=(I+a/)/2-aA 2/4m, e=l+al. (2.18 )

1645 J. Math. Phys. 29 (7), July 1988 0022-2488/88/071645-05$02.50 © 1988 American Institute of PhysiCS 1645

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The general solution, which satisfies the usual boundary condition that R(r) will decay when r- 00, is given by

r(1- c) U(b,e,y) = IFI (b,e,y)

r(1 + b - c)

r(e - 1) 1- c [1 b 2 ] + y IF] ( + - c), - e,y . r(b)

(2.19)

B. Hoenselaers-Vlshveshwara (HV) metric

The Hoenselaers-Vishveshwara metric is given by

ds2 = dt 2 - dr - dz'l +!A 2(e - 1)(e - 3)d¢i

+ 2(e - 1 )df/J dt, (2.20)

where e = cosh JK r and A JK = 1. On assuming A = 1, K = 1, the metric (2.20) takes the form

ds2 = dt 2 - dr - dz'l + !(e - l)(e - 3)df/J2

+2(e-l)df/Jdt. (2.21)

The Ricci scalar is calculated as in the previous case considering the following basis one-forms:

O]=dr, 02=~(e2-1)/2df/J,

03=dz, OO=dt+(e-l)df/J. (2.22)

The nonvanishing components of w~, O~, and R,.,v are found to be

w\ = OO/Yl- (e/~)02 = - w2],

WO I = ~02 = Wlo, W02 = - (lIYl)O] = w20,

012=!01/\02= _012,

001=!01/\00= -0°1,

002 = !02 /\00 = _ 0°2,

R II = 0, R22 = 0, R 02 = 0, Roo = 1.

The Ricci scalar is then

R = 1. (2.23)

Using (2.21) and (2.23) in (2.1) we get

~a 2 2 a2 a2 e - 3 a2 4 a2 rr ---2 <# + zz +-- It --- tt/>

l-e e+l e+l

+ (~ + ~) a, + /LJ ¢ = 0, e-l e+l J

(2.24)

with/L2 = m2 + s. Let us now choose ¢ in the form

¢=eia(OJt+It/>+kz)R(r), (2.25)

wherew, I, and k are constants. Substituting (2.25) in (2.24) we get the equation

R "(r) +R'(r) cothr

+ [A] +A2 cothrcschr+A3 csch2r]R(r) =0, (2.26)

with

1646

A] = m2 + 5 - a2w2 - a2k 2

,

A2 = 4a2w(w + I), A3 = - 402(12/2 + wi + ( 2).

J. Math. Phys., Vol. 29, No.7, July 1988

(2.27)

Substituting

R(r) =ym+n(1_ y)nu(y),

where

y = (1 + cosh r)/2, m = - allYl, n = - V1aw, (2.28)

in Eq. (2.26), we get the following equation:

y(1-y)U"(y) + [(1 +VlaI-2Vlaw)

- (2 - 2VlaI- 2V1aw)y] U'(y)

- [2a2/2 + a2w2 + 4a2w/- Vlal

- Vlaw - a2k 2 + m2 + 5] U(y) = 0. (2.29)

Equation (2.29) is the Gauss hypergeometric equation. The two linearly independent solutions of (2.29) are

F( a' ,b ' ;a' + b ' - e' + 1; 1 - y)

and

(1 - y)c' -a' - b'F(e' - a',e' - b ';e' - a' - b' + 1;1 - y).

The general solution, which satisfies the general boundary condition that R(r) will decay when r- 00, is

F(a' b "e'. ) = r(e')r(e' - a' - b ') , ,;Y r(e' - a')r(e' - b')

XF(a',b ';a' + b' - e' + 1;1 - y)

+ (1 _ )c' -a' - b' r(e')r(a' + b' - e') y r(a')r(b')

XF(e' - a',e' - b ';e' - a' - b' + 1;1 - y), (2.30)

with

a' = ! [ 1 - 2v2al - 2Vlaw + ~ 1 + 402w2 + 402k 2 - 4/L2 ] ,

b' =![ 1 - 2Vlal- 2V1aw - ~1 + 402w2 + 402k 2 - 4,lP] ,

e' = 1 - Vlal- 2Vlaw. (2.31)

c. Rebou~as metric

The Rebouc;as metric is

d~ = dt 2 - dr - dz'l + 4 cosh 2rdf/J dt

+ (3 cosh22r + 1 )df/J2, (2.32)

The Ricci scalar is calculated as before considering the following basis one-forms:

0 1 = dr, 0 2 = sinh 2rdf/J, 0 3 = dz,

0° = dt + 2 cosh 2r df/J. (2.33)

The nonvanishing components of w~, O~, and Rij in this case are found to be

WO I = 20 2 = Wlo, W02 = - 20 1= w20,

Wl 2 = - 2 coth 2r02 + 200 = _ W

21'

0 12=80 1/\0 2= _012

,

0° I = 40 I /\ 0 ° = - 0°1,

002 = 40 2 /\ 0° = _ 0°2,

Roo = 8, R II = 4, R22 = 4.

The Ricci scalar for the metric is found to vanish, i.e.,

Krori et al. 1646

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R = 0. (2.34)

Using (2.32) and (2.34) in (2.1) we get

[a 2 1 a 2 + a 2 + 3 cosh

2 2r + 1 a 2

" + sinh2 2r H zz sinh2 2r tt

_ 4 ~osh 2r a:~ + 2 coth 2ra, + m2] '" = 0. (2.35) smh2 2r

Let us choose'" as before in the form

"'=eia(D)'+l~+kz)R(r), (2.36)

where a, ll), I, and k are constants. Putting (2.36) in (2.35) we get

R "(r) + 2 coth 2rR '(r) + [ - (a2/2 + 4a2ll)2)csch2 2r

+ 4a2ll)1 coth 2r csch 2r + (m2 _ 3a2ll)2 _ a2k 2)]

xR(r) = 0. (2.37)

Equation (2.37) is reduced to Gauss' hypergeometric differential equation,

y(l- y)U"(y)

+ [( - all) - al12 + 1) - (2 - loll)YU'(y)

_ Ha2ll)2 - 4all) - a2k 2 + m2 ] U(y) = 0, (2.38)

if we make the following substitutions:

R (r) = yP(l - y)qU(y),

with

y =!(l + cosh 2r), p = - (aI2)(ll) + 112),

q = - (a/2)(ll) - 1/2). (2.39)

The general solution ofEq. (2.38), which satisfies the usual boundary condition that R(r) will decay when r-+ 00,

is

F(bcd ) = r(d)r(d-b-c) ",y r(d-b)r(d-c)

XF(b,c;b + c - d + 1;1 - y)

+ (1- )d-b-c r(d)r(b+c-d) y r(b)r(c)

XF(d-b,d-c;d-b-c+ 1;I-y), (2.40)

where

b = ~{1 - loll) + ~1 - m2 + 3a2ll)2 + a2k 2} ,

c = ~[1 - 2all) - ~1 - m2 + 3a2ll)2 + a2k 2] ,

d = 1 - all) - a112.

III. SPINOR SOLUTIONS

(2.41)

The Weyl equations for a massless spin-! field are

YZVa '" = 0, (3.1)

(1 + r)'" = 0, (3.2)

where

and

(3.3)

1647 J. Math. Phys., Vol. 29, No.7, July 1988

The yz are generalized Dirac matrices and are given in terms of the fiat space-time gammas y(a) ,and h(a) a are vier­beins defined by the relations

h ah P7J(a)(b) - ,.a/J (a) (b) -~ .• (3.4)

A.SR metric

First we consider the Weyl equations in the SR metric. For this metric we shall use the following set ofvierbeins:

h(l) a = 8al , h(2) a = r[8a

o - (l/~)8a2]'

h(J) a = 8a3, h(o) a = 8a

o•

Now we have

Va'" = [aa - Ta ] "',

where

and

T - ly(a)y(b)h Vh a - -" (a) (b)v,a'

h(b)v,a = aah(b)v - r!.h(b),t.

Using Eqs. (3.6)-(3.8) we obtain

T. = _ !yl0)yl2),

T2 = (r/2)yluyl0) + !(~ _ 1)yl2)yI.),

T3 = 0, To = !yl2)yl1l.

We now choose '" in the form

'" = e - ;a(." + 1~ + kz) (71. (r») , 7J2(r)

where 71. and 712 are two-component spinors.

(3.5)

(3.6)

(3.7)

(3.8)

(3.9a)

(3.9b)

Using the standard representations of gamma matrices Eq. (3.2) is reduced to

(R.(r») 7J.(r) = 7J2(r) = \R2(r) .

On using Eqs. (3.6)-(3.10) in Eq. (3.1) we get

[ all] . [ 1 ] a, +-;-all)r- 2r R 2 =la k+ll)+2; R.,

[a, - a: + all)r - ;r] R. = ia[ ll) - k - ~] R 2•

Eliminating RI from (3.11) and (3.12) we get

~R;-rR2

(3.10)

(3.11 )

(3.12)

+ [(~ - a/- a2/ 2) + A, 2~ - a2ll)2""]R2 = 0, (3.13)

where

A, 2 = 2a2ll)/- all) + a2ll)2 - a2k 2 + ak -l. (3.14)

This equation can be reduced to the following confluent hypergeometric differential equation:

yF" + (2k.-y)F' - (k.-A, 214all)F=0, (3.15)

if we make the substitutions

(3.16)

where y = all)~, and k. is a constant. Equation (3.15) has two independent solutions given by

IF. (k. - A, 214all),2k.,y)

and

Krori etal. 1647

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The general solution, which satisfies the usual boundary condition that R(r) will decay when r- 00, is given by

U(k l - ~: ,2kl ,y)

r(l - 2kl ) (A 2 ) ----'---:::--- IFI kl - 4a,.' ,2kl>Y r(1 - kl - A 214a(() W'

+ r(2kl -1) 1-2k,

r(kl -A 214a(() y

X IFI (I-kl - ~: ,2-2kl ,y). (3.17)

B. HVmetrlc

For the HV metric we choose the following set of vier­beins:

h(o) a = ~ao, h(!) a = ~al> h(3) a = ~a3'

h(2)a=[ -\12 ~a2+ ~~ao]. v?=T \j c+ 1

Using Eqs. (3.6)-(3.8) in this case we obtain

TI = - (\I214)rO)r21,

T2 = (v?=T/4)ri)rO) - (\I214)r2)ri),

T3 = 0, To = (\I214)r2)ri)·

As before, we choose

tP = e- ia(IlJ'+ 1.,,+ lcz) ('TIl (r») , 'TI2(r)

(3.18)

(3.19)

(3.20)

where 'TIl and 'TI2 are two-component spinors. Using standard representation of gamma matrices, Eq. (3.2) is reduced to

(RI(Y») 'TII(Y) ='TI2(Y) =\R2(Y) . (3.21)

UsingEqs. (3.6)-(3.8) and (3.18)-(3.21) inEq. (3.1) we get

[a + Y'lal _ \I2(c-l) a(()- c ] R2

r v?=T v?=T 2v?=T

= ia[ (() + k + ~] R I, (3.22)

and

[a _ Y'lal + \I2(c - 1) a(() _ c ] RI

r v?=T v?=T 2v?=T

= ia[ (() - k - !] R 2. (3.23)

Eliminating R2 from (3.22) and (3.23) we get

R ~ -cothrR;

+ [A + B coth rcsch r + D csch2 r]R I = 0, (3.24)

where

A = - a2(()2 - a2k 2 - (l1\12)ak + I. B = 4a2(()2 + 4a2(()1 + Y'la(() + Y'lal, (3.25)

D = - 4a2(()2 - 4a2(()1_ 2a2[2 - Y'la(() +~.

y(1 - y) U" (y) + [(! - Y'lal- 2Y'la(()

- (2 - 2Y'lal- 2Y'la(()y] u' (y)

- (A + 2a2[2 - Y'lal + a2(()2 - Y'la(()

+ 4a2(()1- a2k 2 - (l1\12)ak )U(y) = 0,

if we make the following substitutions:

RI(r) =ym+n(1_ y)mu(y),

m = - (lI\l2)al + ~, n = - V2a(() - !, y =!(1 + cosh r).

(3.26)

(3.27)

The two linearly independent solutions of (3.26) are

F( a,b;a + b - c + 1; 1 - y)

and

(1 - y) C - a - b F( c - a,c - b;c - a - b + 1; 1 - y).

The general solution, which satisfies the usual boundary conditions that RI (r) will decay when r- 00, is

F(a,b;c;y) = r(c)r(c - a - b) r(c - a)r(c - b)

XF(a,b;a + b - c + 1;1 - y)

+ (1_y)c-a-br(C)r(a+b-c) r(a)r(c)

XF(c - a,c - b;c - a - b + 1;1 - y), (3.28)

where

a = ! [1 - 2..j2al - 2..j2a(()

+ ~! + 4a2(()2 + 4a2k 2 + 2..j2ak ],

b =![t - 2..j2al- 2..j2a(() (3.29)

- ~! + 4a2(()2 + 4a2k 2 + 2..j2ak ].

c = ! - ..j2al - 2..j2a(().

c. Rebouc;as metric

For the Rebou'!as metric we choose the following set of vierbeins:

h(!) a = l)a l , h(3) a = ~a3' h(o) a = l)ao,

h(2) a = (l/v?=T) [2cl)ao - ~a2]'

We have as before

VatP= [aa - TaW,

with

T - 1 (a) (b)h vh a - - 4Y Y (a) (b)v,a and

h(b)v,a = aah(b)v - r~h(b)A' whence we get

TI = - y(O)y(2),

T2 = v?=T y(l)y(O) + 2cy(2)y(l>,

(3.30)

(3.31 )

Equation (3.24) will be reduced to Gauss' hypergeometric T3 = 0, To = y(2)y(l). equation, As before we consider

1648 J. Math. Phys., Vol. 29, No.7, July 1988 Krori et al. 1648

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'" = e- ia(<>.J' + 14> + kz) (111 (r») 112(r)

(3.32)

where 111 and 112 are two-component spinors. We get from (3.2)

(R1(r») 111(r) = 112(r) = \R2(r) .

Using (3.31)-(3.33) in (3.1) we get

[a + a/ _ 2caw _ c ] R2 , rcr=T rcr=T rcr=T

=ia[W+k+ !]R 1,

[a _ a/ + 2caw _ c ] RI , rcr=T rcr=T rcr=T

= ia[ W - k - !] R 2•

Eliminating R2 from (3.34) and (3.35) we get

(3.33)

(3.34)

(3.35)

[a;, - 2 coth 2ra, + (3 - a2[2 - 4aw - 4a2(

2 )csch2 2r

+ (20/ + 4a2w/)coth 2r csch 2r

(3.36)

Equation (3.36) is now reduced to Gauss' hypergeometric equation,

y(1-y)U"(y)

+ [( ~ - ~ - aw) - (1- 2aW)y] U'(y)

[a2w2 a2k 2 ak 1 ] - 4-4-2-4" U(y) =0,

if we make the following substitutions:

RI (r) = yP(1 - y)qU(y),

where

1649 J. Math. Phys., Vol. 29, No.7. July 1988

(3.37)

y =!(1 + cosh 2r),

p = - aw/2 -a//4 +!, q = - aw/2 + a/ /4 +!.

(3.38)

The general solution ofEq. (3.37), which satisfies the usual boundary condition that RI (r) will decay when r ..... 00, is

F(a' b' c'. ) = r(c')r(c' - a' - b') , ,~ r(e' - a')r(c' - b')

xF(a',b ';a' + b' - e' + 1;1 - y)

+ (1 _ )c'-a'-b' r(e')r(a' + b' - c') y r(a')r(b')

XF(e' - a',e' - b ';e' - a' - b' + 1;1 - y), (3.39)

with

a' = - aw + !~3aZwZ + aZk Z + 10k + 1

b' = - aw - !~3aZw2 + aZk Z + 10k + 1

e' = ! - a/ /2 - aw.

ACKNOWLEDGMENTS

(3.40)

The authors express their profound gratitude to the Government of Assam for all facilities provided at Cotton College, Guahati 781001, India, to carry out the investiga­tions reported in this paper.

'J. Audretsch and G. Schafer, Gen. Relativ. Gravit. 9, 243 (1978). 2E. Fischback, B. S. Freeman, and W. Cheng, Phys. Rev. D 23, 2157 (1981).

3L. Parker and L. O. Pimentel, Phys. Rev. D 25,3180 (1982). 40. Y. Li and H. Y. Guo, Acta. Phys. Sin. 31,1501 (1982). 5M. Parsic, Phys. Lett. A 90, 175 (1982). 6p. Bureev, Gen. Relativ. Gravit. 17,799 (1985). 7J. M. Cohen, C. V. Vishveshwara, and S. V. Dhurandhar, J. Phys. 13,933 (1980).

8L. O. Pimentel and A. Macias, Phys. Lett. A 117, 325 (1986).

Krori etat 1649

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