132

CHAPTER 6

CONFORMALLY FLAT SPHERICALLY SYMMETRIC

SPACE-TIME IN EINSTEIN-CARTAN THEORY OF GRAVITATION

“ My own thinking is to ponder long and I hope

deeply on problems and for a long time which

I keep away for years and years and I never

let them go”.

Roger Penrose

133

1. INTRODUCTION

Amongst all theories of gravitation Einstein theory of gravitation is the most

successful one. Every physical theory involves the set of differential equations

describing dynamical behavior physical situations. Exact solutions of these equations

describe the fundamental interaction of gravitation as a result of space-time being

curved by matter and energy. Finding the exact solutions of the set of differential

equations is one of the interests of most of the researchers in the study of the theory of

relativity.

However, the general theory of relativity and its extension-the Einstein-Cartan

theory of gravitation are highly non-linear theories; it is not always easy to find exact

solutions of field equations and to understand what qualitative features solutions

might possess. Existence and uniqueness theorems have not yet been developed for

exploring the exact solutions; hence the theories of gravitation provide formidable

mathematical obstacles. However, by imposing different types of symmetry

conditions on the metric tensor, such as stationary, axsymmetry or spherical

symmetry, it is often possible to reduce the Einstein field equations to a much simpler

system of equations.

It is reported that there are over 400 new research papers on exact solutions

every year (Kramer et. al [9]). In their book entitled “Exact Solution of Einstein

Equations”, they give a unique comprehensive survey of the known solutions of

Einstein field equations for vacuum, Einstein-Maxwell, pure radiation and perfect

fluid sources. Kinnersley [8] pointed out that most of the known solutions describe

situations which are unphysical. However, certain solutions have played a very

important role in the discussion of physical problems. For example, Schwarzschild

solution is the most general spherically symmetric vacuum solution of Einstein field

equations that describes the gravitational field outside a spherical uncharged non-

rotating mass such as the Sun or black hole. Kerr solution is the generalization of the

Schwarzschild solution that describes the geometry of space-time around a rotating

massive body. Reissner- moNordstr solution describes the geometry of space-time

around a charged, spherical, non-rotating body. The natural extension to a charged

rotating body is the Kerr-Newman metric. Friedmann solution for cosmology and

134

plane wave solutions resolved some of the controversies about the existence of

gravitational radiation.

In the Einstein-Cartan theory of gravitation, Prasanna [11], Tsoubelis [13],

Kalyanshetti and Waghamode [4], Singh and Griffiths [12], Katkar [6] have presented

exact solutions. Recently, the technique of Newman-Penrose [10] tetrad formalism

has proved to be the amazingly useful in the construction of exact solutions. Using

this method it was possible to obtain many new solutions and this growth is still

continuing. The same influences, of course led us to work on exact solutions of field

equations in Einstein-Cartan theory of gravitation. Further, the methods of exterior

calculus and differential forms have been proved to be more powerful in providing

‘computational ease and conceptual gain’. We adopt Newman-Penrose tetrad

formalism and its extension by Jogia and Griffiths [3] which is especially suited for

Einstein-Cartan theory and the techniques of exterior calculus to find the exact

solutions of field equations in Einstein-Cartan theory of gravitation.

Accordingly, the cursory account of Newman-Penrose-Jogia-Griffiths (NPJG)

formalism and the Cartan’s equations of structure are attempted in the Section 2. Field

equations of Einstein-Cartan theory, when Weyssenhoff fluid is the source of

gravitation are exhibited in Section 3. By using the techniques of differential forms,

the tetrad components of Riemann curvature tensor, Ricci tensor are enumerated in

Section 4. In the next section, the field equations of gravitation are solved and the

exact solution is investigated. In the last section we have proved that the solution

obtained is of Petrov-type D. The results of this Chapter are published in the

International Journal of Theoretical Physics (Katkar and Patil [7]).

2. CARTAN’S EQUATIONS OF STRUCTURE IN EINSTEIN-CARTAN

THEORY OF GRAVITATION

The essence of geometry is summarized in the equations of structure. Katkar

[5] has derived the Cartan equations of structure in Einstein-Cartan theory of

gravitation. We summarize these results for the use in the sequel. We introduce at

each point of the space-time of EC theory of gravitation a tetrad of four vectors. The

choice of the tetrad vectors has to be compatible with the signature (-, -, -, +) of the

metric. Accordingly all the four vector fields of the tetrad may be null, or one is time-

like and three are space-like or one is null and three are space-like or two are null and

135

two are space-like. Amongst all tetrad formalisms which have been employed in the

general theory of relativity, the most prominent is the one proposed by Newman and

Penrose [10]. The Newman - Penrose tetrad consists of four null vectors. We use in

our discussion the tetrad consisting of all four null vectors.

Let the tetrad be denoted by

) , , ,( )(

iiiii mmnle ,

where ii nl and are real null vectors and ii mm and are complex conjugate of each

other. The vector fields of the tetrad satisfy the following orthonormal conditions

1 ii

ii mmnl , (2.1)

and all other inner products are zero.

The dual tetrad of vectors is defined by

)- , , ,()( iiiii mmlne

,

such that

01001000

00010010

)(

)( ii ee , (2.2)

and

)(

)( ii ee , i

jji ee )(

)( ,

where are the tetrad components of j

i .

Any vector or tensor can be uniquely expressed as the linear combinations of its tetrad

components and conversely. Thus, we can express

)()(

ii euu and i

ieuu )()( .

Similarly, the tensor components of the metric tensor ijg are expressed as

)()(

jiij eeg ,

and conversely. This gives

jijijijiij mmmmlnnlg . (2.3)

2.1 Ricci’s Coefficients of Rotation

Ricci’s coefficients of rotation are defined as

136

4,3,2,1,, , )(

)(/)(

jiji eee , (2.4)

where

ljiljiji Keee

)(; )(/)( , (2.5)

and ljiljiji eee

)( , )(; )( .

Thus we have by using the definitions of covariant derivative in EC space-time

jikjikji eeee

)(

)( )( , )( ] -[ ,

jikjikkj

ki eeee )()()()( ] - [ ,

kkjij

ki

ji eee )()()( ] [ .

Using the definition ((2.10) Vide Chapter 1) we get

kk

jikjij

ki

ji eKee )(

)(

)( ] [ ,

kjijikk

kjij

ki

ji eeeKeee )(

)(

)()(

)(

)( ] [ ,

K 0 , (2.6)

where

kjiijk eeeKK )()()( , (2.7)

are the tetrad components of the contorsion tensor, and

jiji eee

)(

)( ;)(0

(2.8)

We record these tetrad components for our use in the sequel to simplify the

tensor equations.

kjiijk

jiji lmlKKKlml 1131311 ;

0311

0 , ,

kjiijk

jiji lmmKKKmml 4134311 ;

0314

0 , ,

kjiijk

jiji lmmKKKmml 3133311 ;

0313

0 , ,

kjiijk

jiji lmnKKKnml 2132311 ;

0312

0 , ,

kjiijk

jiji nmlKKKlmn 1241421 ;

0421

0 , ,

kjiijk

jiji nmmKKKmmn 4244421 ;

0424

0 , ,

kjiijk

jiji nmmKKKmmn 3243421 ;

0423

0 , ,

kjiijk

jiji nmnKKKmmn 2242421 ;

0422

0 , ,

kjiijk

jiji lnlKKKlnl 11212111 ;

0211

00 , ,

137

kjiijk

jiji mmlKKKlmm 13414311 ;

0431

00 , ,

kjiijk

jiji lnmKKKmnl 41242111 ;

0214

00 , ,

kjiijk

jiji mmmKKKmmm 43444311 ;

0434

00 , ,

kjiijk

jiji lnnKKKnnl 21222111 ;

0212

00 , ,

, ; 0432

00 jiji nmm

kjiijk mmnKKK 23424311 . (2.9)

The complex conjugates of these tetrad components are obtained by taking the

complex conjugate of tetrad vector fields. Thus for 1,3 we write equation

(2.6) as

10 .

The symbol with subscript 1 denote the tetrad components of contorsion tensor, and

the symbols with superscript zero denote the NP spin coefficients in Einstein theory

of gravitation.

The contorsion tensor ijkK in terms of its tetrad components is given by

4,3,2,1,, , )()()( kjiijk eeeKK . (2.10)

By expanding the right hand side of equation (2.10) by giving the different values to

, , , and using equations (2.9) we obtain the expression

][11][11][11 )( )( )( [ 2 kjikjikjiijk mmnnllnlnK

][1][1][1][11 { )( kjikjikjikji mlnnmnmlmmml

][11][1][1][1 )( kjikjikjikji nlmnmmmllnml

][1][1 kjikji mlmnmm

] . })( ][11 CCmmm kji , (2.11)

where CC. indicates the complex conjugate of the preceding term.

2.2. Cartan’s First Equations of Structure in Einstein-Cartan Theory of

Gravitation

To derive Cartan’s first equations of structure in Einstein-Cartan theory, we

define basis 1-form as

138

ii dxe )( . (2.12)

Now, in Einstein theory of gravitation exterior derivative of a form is obtained by

taking either the partial derivative or covariant derivative of the associated tensor.

This is possible because the Christoffel symbols are symmetric. However, in Einstein-

Cartan theory, this is not so, due to the asymmetric connection coefficients. Now we

take exterior derivative of equation (2.12) to get

ijji dxdxed )(

/ , (2.13)

where

jiij dxdxdxdx , (2.14)

are a basis of the 2-forms.

Using the definition of covariant derivative in Einstein-Cartan theory, we have

ijlji

ljilji dxdxKed - e- )()(

, .

The second term vanishes due to symmetric property of Christoffel symbol and skew-

symmetric property of basis 2-forms. Thus we have

. )( )( )( ,

ijl

ljiji dxdxeKed (2.15)

However, from equation (2.12) we have

)(ii edx

. (2.16)

Using this in equation (2.15) we get

)( )(

)(

)( )( ,

ijl

ljiji eeeKed

,

)( )( )(

)(

)(

)(

)( , l

ijlji

ijji eeeKeeed . (2.17)

Note that in the first term on the right hand side of equation (2.17) it is immaterial

whether colon ( , ) or semi-colon ( ; ) is used. Thus we have

0 Kd , (2.18)

Td 0 ,

or

Td 0 , (2.19)

where

00 , (2.20)

and

KT . (2.21)

Equation (2.19) is called the Cartan’s first equation of structure.

139

For the simplicity sake we can also write equation (2.18) as

] )[( 0 Kd ,

d ,

d ,

where

, (2.22)

0

K ,

or

0

K , (2.23)

are called the connection 1-forms.

2.3 Tetrad Components of Connection 1-forms in Einstein-Cartan theory

From equation (2.22) we have

,

44

33

22

11 . (2.24)

Using equation (2.6) we write this as

33

03

22

02

11

01 )()()( KKK

44

04 )( K . (2.25)

Now by giving different values to 4 ,3 ,2 ,1 , and using equation (2.9) we readily

obtain the expressions for connection 1-forms as

211

00111

0012 )()[(

])()( 411

00311

00 , (2.26a)

21

011

013 )()[(

])()( 41

031

0 , (2.26b)

21

011

023 )()(

41

031

0 )()( , (2.26c)

211

00111

0034 )()(

411

00311

00 )()( . (2.26d)

Similarly, on using equation (2.9) we obtain from equation (2.21)

140

14111

13111

1211

1 )()()( T

3411

241

231 )( ,

23111

141

131

1211

2 )()( T

3411

24111 )()( ,

23111

141

13111

1211

3 )()()( T

3411

241 )( ,

and 231

14111

131

1211

4 )()( T

3411

24111 )()( . (2.27)

Consequently, the tetrad components of the Cartan’s first equation of structure are

given by

131

011

001211

001 )()( d

231

0141

011

00 )()(

3411

00241

0 )()( , (2.28a)

141

0131

01211

002 )()()( d

23111

000 )(

24111

000 )(

3411

00 )( , (2.28b)

13111

0001211

003 )()( d

23111

000141

0 )()(

3411

00241

0 )()( . (2.28c)

The expression for 4d is obtained by taking the complex conjugate of 3d and this

is obtained by taking the complex conjugate of the spin coefficients and replacing 3

by 4 and 4 by 3 in the basis 2-form , where we have denoted

.

Similarly, from equation (2.15) for each value of 4 ,3 ,2 ,1 we find

ijljil

ijji dxdxeKdxdxed )1()1(

, 1 ,

)(

)( ,

1 ijjil

lijji eeKndxdxnd

,

) ( )(

)(

)( , 1

neeeKdxdxnd lijjil

ijji .

141

Using the relation (2.10) we get

, 1

Kndxdxnd ij

ji ,

where )( i

ienn are the tetrad components of the vector in

)0 ,0 ,1 ,0(n .

Consequently, we have

2 , 1

Kdxdxnd ijji .

By using different values to 4 ,3 ,2 ,1 , and using equation (2.9) we obtain

13111

1211 ,

1 )()( ijji dxdxnd

3411

241

231

14111 )()( . (2.29a)

Similarly, we find

1 , 2

Kdxdxld ijji ,

141

131

1211 ,

2 )( ijji dxdxld

3411

24111

23111 )()()( , (2.29b)

and 4 , 3

Kdxdxmd ijji ,

13111

1211 ,

3 )()( ijji dxdxmd

3411

241

23111

141 )()( . (2.29c)

2.4. Cartan’s Second Equation of Structure in Einstein-Cartan theory

To find the Cartan’s second equation of structure in Einstein-Cartan theory,

we start with the expression for connection 1-forms

0

K . (2.30)

Taking the exterior derivative of (2.30) we get

)(

dd .

Using the equation (2.12) we write

)( )(

ii dxedd

ijji dxdxe )( /

)( ,

ijijji dxdxeed ])()[(

21

/)(

/)(

. (2.31)

142

From equation (2.4) we find

)()(

)(/

jiji eee . (2.32)

Now taking the covariant derivative of (2.32) we get

)(/

)(

)(/

)(

)(/ ) (

kijikjjki eeeee .

Using the equation (2.31) we obtain

)()()(

)(/

)(

)(/ ) (

kijikjjki eeeeee

,

)()( /

)( )(

)(/ ) (

kjkji

jki eeeee

,

)()( )(

)(/ /

)( ) (

kji

jkikj eeeee

.

Substituting this in the equation (2.31) we get

)()( )(

)(/

)(/ ()([

21

jih

jihijh eeeeed

ijij dxdxee )])()( ,

from the Ricci theorem (vide, equation (2.24) of chapter 1) we have

)( )(/

)(

)(/

)(/

kij

kjikh

kjihkjihijh KKeReee .

Hence the above equation becomes

)](([21 )(

/

)(

)()(

k

ijk

jikhhk

jikh

k KKeeReed

) (21 )()()()(

i

ij

ji

ij

j dxedxedxedxe

,

)( )(

)(

)(

21

khijk

jih eeeeRd

ijkij

kji

hkh dxdxKKee )(

21

)()(/

. (2.33)

Using the equation (2.16) we write

))(()( )()(

)( )(

)()(/

ijkij

kjik

ijkij

kji

hkh eeKKedxdxKKee

])[( )(

)()(

kijk

ijk

ji eeeKK ,

))(()(

)(

)(/

KKdxdxKKee ijkij

kji

hkh .

Hence equation (2.33) becomes

143

)(

21

21

KKRd

. (2.34)

We define the tetrad components of curvature 2-forms as

2

1 R . (2.35)

Hence equation (2.34) on using (2.22) becomes

Kd . (2.36)

This is the Cartan’s second equation of structure in Einstein-Cartan theory of

gravitation. From equations (2.23) and (2.6) we have

0

K ,

0

K .

Using these equations in (2.36) we get

0

0

0 dKd

0

KdK

0 KKK

0 KKK .

Using Cartan’s first equation of structure we get

TKKdK 0

0

0

0 0

KKK

)( KKKK , (2.37)

where

0

0

0

0 d , (2.38)

is the Cartan’s second equation of structure in Einstein theory gravitation.

From equation (2.36) we write

Kd ,

34431221d

)( 34431221 KKKK ,

144

34431221d

)([{ 214124312122121 KKK

1312312132112

2131234 )({)}( KKKKK

)({})( 412142113

31343141343 KKKKK

232114

413143441431412 {)}( KKKKK

23323432423433212312 })()( KKKKK

424342124122421 )({ KKKK

)({}( 432342124

4232434 KKKK

]})( 34343443434313412 KKKK .

On using the equation (2.9) we obtain

34431221d

)()()([ 113112111

12111112

114 )([)](

)([)])( 111113

141113

1114

11141312 [)](

231411131112 ])()(

13111211 )([

)()([)]( 11211124

1114

34114113 )]()( . (2.39)

Now by giving 4,3,2,1, , and using the equation (2.6) we find only four

independent complex tetrad components of curvature 2-forms as:

231424131212 d

))(())(([ 111100

111100

121111

001111

00 )])(())((

11100

1111100 )())(([

13111

0011111

00 )])())((

145

11100

1111100 )())(([

1411111

00111

00 )])(()(

))(()[( 1111100

11100

23111

0011111

00 ])())((

))(()[( 1111100

11100

2411111

00111

00 )])(()(

))(())([( 111100

111100

))(( 111100

341111

00 )])(( , (2.40)

341313121313 d

))(())(())(([ 1110

1110

1110

12111

0 )])((

110

11110 )())(([

1311

01111

0 ])())((

110

110

11110 )()())(([

141111

0 )])((

))(())(()[( 11110

11110

110

2311

0 ])(

110

11110

110 )())(()[(

241111

0 )])((

))(())(())([( 1110

1110

1110

34111

0 )])(( , (2.41)

Similarly,

342323122323 d

))(())(())([( 1110

1110

1110

12111

0 )])((

110

11110 )())([(

146

1311

01111

0 ])())((

110

110

11110 )()())([(

141111

0 )])((

))(()([ 11110

110

2311

01111

0 ])())((

110

11110

110 )())(()([

241111

0 )])((

))(())(([ 1110

1110

34111

0111

0 )])(())(( , (2.42)

231424133434 d

))(())([( 111100

111100

))(( 111100

121111

00 )])((

11100

1111100 )())([(

))(( 1111100

13111

00 )])(

11100

1111100 )())([(

11100 )(

1411111

00 )])((

))(()([ 1111100

11100

23111

0011111

00 ])())((

))(()([ 1111100

11100

2411111

00111

00 )])(()(

))(())(([ 111100

111100

))(( 111100

341111

00 )])(( . (2.43)

147

3. FIELD EQUATIONS OF EINSTEIN-CARTAN THEORY OF

GRAVITATION

The natural generalization of Einstein theory of gravitation is the Einstein-

Cartan theory of gravitation. The relevant field equations for gravitation are given by

Hehl et. al [1,2]

ijijij tkgRR 21

, (3.1)

and

kij

lil

kj

ljl

ki

kij SkQQQ , (3.2)

where 4CG 8

k and ijt is the asymmetric energy momentum tensor defined by

khijhi

jkjk uSuuTt /)( . (3.3)

Here ijT is the stress-energy momentum tensor of matter and kijS is the spin angular

momentum tensor. We first express the tensor components of torsion tensor kijQ and

spin angular momentum tensor kijS as a linear combination of the basis vectors fields

of the dual tetrad as follows

4,3,2,1,, , )(

)()(

k

jik

ij eeeQQ . (3.4)

By giving the different values to ,, and using the relation

)(21 l

jil

ijl

ij KKQ , (3.4a)

the equation (3.4) gives

kji

kji

kji

kij nmmKKnnlKlnlKQ ][431341][121][212 )([

kji

kji mnlKKlmmKK ][214124][432342 ){()(

kji

kji mnmKKlnmKK ][143413][421142 )()(

kji

kji

kji nmlKKnnmKmnmK ][241421][141][414 )(

kji

kji

kji mmlKmmlKKlmlK ][424][423243][242 )(

].}][434 CCmnmK kji .

Now using equations (2.9) we readily get

148

kji

kji

kji

kij nmmnnllnlQ ][11][11][11 )()()(

kji

kji

kji mnllnmlmm ][11][111][11 )()[()(

kji

kji

kji nnmmnmmnm ][1][1][111 )(

kji

kji

kji mmllmlnml ][111][1][111 )()(

..])( ][11][1 CCmmmmml kji

kji . (3.5)

Similarly, we obtain the expression

kji

kji

kji

kij lmmnnllnl

kS ][11][11][11

)()()([21

kji

kji

kji nmllmmnmm ][11][1][11 )2({)(

kji

kji

kji mmllnmnmkn ][11][11][ )2()2(

kji

kji mnlmnm ][11][11 )()2(

.}.{} )( ][][1][11 ccmnmmlmmmm kji

kji

kji . (3.6)

For the classical description of spin tensor, the spin angular momentum tensor

has been decomposed in terms of the spin tensor by Hehl et. al. [2] as follows:

kij

kij uSS , (3.7)

where ijS is the skew-symmetric tensor orthogonal to the 4-velocity vector iu . This

gives

0 iij uS . (3.8)

The skew-symmetric spin tensor ijS has six independents components. These six

components can be expressed in terms of three complex tetrad components of ijS as

jiij mlSSS 130 ,

)(21)(

21

43121jiji

ij mmnlSSSS , (3.9)

jiij nmSSS 322 .

Hence the expression for spin tensor ijS in terms of tetrad components becomes

][11][11][2][2 )()([2 jijijijiij mmSSnlSSmlSmlSS

]][0][0 jiji nmSnmS . (3.10)

149

Transvecting equation (3.10) in the direction of )(2

1 iii nlu , we get

)])(()()[(2

1 110202 jjjji

ij lnSSmSSmSSuS . (3.11)

Thus the condition

1102 , 0 SSSSuS iij . (3.12)

Consequently, the expression (3.10) reduces to

]2)()([2 ][1][][0][][0 jijijijijiij mmSnmmlSnmmlSS . (3.13)

Now expressing each term of equation (3.2) in terms of its tetrad components, we

obtain the tetrad components of the field equation (3.2) in the form

SkQQQ . (3.14)

For each 4 ,3 ,2 ,1 , we get four equations as

uSkQQQ

1 1 1 , (3.15a)

2 2 2 2 uSkQQQ

, (3.15b)

3 3 3 3 uSkQQQ

, (3.15c)

4 4 4 4 uSkQQQ

, (3.15d)

where

) 0 0, 1, (1, 2

1)( i

ieuu , (3.16)

are the tetrad components of the unit time-like vector iu . Now giving different values

to Greek indices in equations (3.15) we obtain the following independent equations

1214313412243234 2 ,

2SkQQSkQQ ,

1313113343321 2 ,

2SkQSkQQ ,

2334331223232 2 ,

2SkQQSkQ ,

3434134342 2 ,

2SkQSkQ ,

0 244144124 QQQ ,

150

0 -Q 421412243212143121 QQQQQ . (3.17)

Using the expressions (3.4a) and (3.9) we obtain

0 ,0 413314423324 KKKK ,

01310343231321 2 , 2 kSKkSKKK ,

2 , 2 03431323120232 kSKKKkSK ,

14313411432342 22 , 22 kSKKkSKK ,

0 ,0 241142214124 KKKK ,

0 ,0 414413143121 KKKK ,

0 ,0 424423243212 KKKK . (3.18)

Now using the equation (3.9) we get

0 , 22 ,0 101111 kS ,

0111111 2 ,0 , 22 kSkS ,

0 , 22 , 22 11111011 kSkS ,

0,2 ,0 ,02 11111

0 ,0 111 . (3.19)

Solving these equations we get

01111 ,

2 , 2

1011011 kSkS ,

222 11111 kS . (3.20)

3.1. Weyssenhoff Fluid

We consider our Einstein-Cartan space-time is filled up with matter

characterized by the Weyssenhoff fluid. The stress energy tenor of Weyssenhoff fluid

is characterized by

khijhi

jkkjjk uSuupguupt /)()( , (3.21)

where is the density and p is the pressure.

Since the spin angular momentum tensor is iu -orthogonal. Hence we have from

equation (3.21)

151

kih

hjijkkjjk uuuSpguupt/ )( . (3.22)

For the choice of flow-vector

)(2

1 iii nlu ,

we have

kjkjkjkjkjkjjk lnnlpnnlnnlllpt ())((21

10

10

10

0 ([22

1) Smmmm kjkj

10

10

110 (2{))( Snnlnnlll kjkjkjkj

0011

0001

01

0 (2) S

..)])}(11 CCnmlm kjkj . (3.23)

On using the equation (3.20) we obtain

))((21

kjkjkjkjjk nnlnnlllpt

)( kjkjkjkj mmmmlnnlp

)([{22

1 00000 S

))}(( 00000 kjkjkjkj nnlnnlllS

)(2{ 00001 S

.].))}((2 00000 CCnmlmS kjkj . (3.24)

4. CONFORMALLY FLAT STATIC SPHERICALLY SYMMETRIC SPACE-

TIME IN EINSTEIN-CARTAN THEORY

We consider a static conformally flat spherically symmetric space-time

represented by the metric

) Sin( 222222222 drdrdrdteds , (4.1)

where is a function of r alone. Now we should look for a basis 1-forms such

that the metric (4.1) becomes

43212 22 ds . (4.2)

152

In search of such we find

),(2

),(2

21 drdtedrdte

) sin(2

), sin(2

43

didredidre . (4.3)

Consequently, the equation ii dxe )( gives the null vectors of the tetrad as

, ) 1 0, 0, (1, 2

, ) 1 0, 0, (-1, 2

enel ii

. ) 0 ,sin- 1, (0, 2

m , ) 0 ,sin 1, (0, 2

ireirem ii (4.4)

The equation kiki ege )(

)( gives

, ) 1 0, 0, (-1, 2

, ) 1 0, 0, (1, 2

enel ii

) 0 ,icosec 1, (0, 2

1

rem i . (4.5)

Now from equation (2.29a) we have

13111

1211 ,

1 )()( ijji dxdxnd

3411

241

231

14111 )()( , (4.6)

where from equation (4.4), we have

411,4 , dxdxndxdxn ij

ji ,

dtdredxdxn ijji

21 , . (4.7)

Now solving equations (4.3), we have

, )(2

1 ),(2

1 43121 rededr

)(2

1 ),( icosec 2

1 21431 edtred . (4.8)

From these equations we find

122 edtdr ,

)(21 2423141321 erddr ,

153

)( cosecθ2i 2423141321 erddr ,

3422 cosecθ i erdd ,

)(21 2423141321 erddt ,

)( cosecθ2i 2423141321 erddt . (4.9)

Hence equation (4.7) becomes

12 ,

21 edxdxn ij

ji .

Hence, equation (4.6) reduces to

13111

1211

1 )()](2

1[ ed

])()( 3411

241

231

14111 . (4.10a)

Similarly, we find from equation (4.4) that

2

1 12, edxdxl ijji ,

)()1(2

1 23131, errdxdxm ijji

341 cot2

1 er .

By virtue of these equations, the expressions (2.29b) and (2.29c) reduce to

141

131

1211

12 )2

1( ed

3411

24111

23111 )()()( , (4.10b)

131111

1211

3 ])1(2

1)[()( errd

241

231111

141 ])1(

21)[( err

34111 ]cot

21)[( er . (4.10c)

The expression for 4d is obtain from (4.10c) by taking the complex conjugate of the

coefficient terms and interchanging 3 and 4. Thus as an illustration, we obtain

154

)[()( 11113

112

114 d

141 ]) 1(2

1 err

241111

231 ])1(

21)[( err

34111 ]cot

21)[( er . (4.10d)

Now using equations (3.20) we rewrite equations (4.10) as

)θ(θ kSedd 23130

1221 22

1

341

24140 222 θ kS)θ(θS k , (4.11a)

341231313 cot2

1)()1(2

1 ererrd , (4.11b)

and

341241414 cot2

1)()1(2

1 ererrd . (4.11c)

Similarly, by virtue of the equations (4.20) the tetrad components of the Cartan’s first

equations of structure (2.28) reduce to the form

130

00012001 )2()( θkSd

230

0140

000 )2()2( θkSθSk

341

00240

0 )22()2( θkSθSk , (4.12a)

140

0130

012002 )2()2()( θSkθkSd

240

000230

000 )2()2( θSkθkS

341

00 )22( θkS , (4.12b)

1401300012003 )()( θθd

340024023000 )()( θθ . (4.12c)

Now equating the coefficients of corresponding basis 2-forms of equations (4.11) and

(4.12) we readily get

e2

10000 ,

155

0 ,0 ,0 0000000 ,

0 ,0 ,0 0000000 ,

0 ,)1(2

1 01000 err ,

0 ,) 1(2

1 01000 err ,

cot2

1 100 er .

Solving these equations we get

0000000 ,

err 100 )1(2

1 ,

cot22

1 100 er ,

e22

100 . (4.13)

4.1. The Tetrad Components of Connection 1-forms

Now by virtue of equations (3.20) and (4.13) the tetrad components of

connection 1-forms (2.26) become

40

30

2112 2 2)(

21 θSkθkSe ,

41

11013 ]2 )1([

21 2 θkSrerθkS ,

41

12023 ] 2)1([

21 2 θkSrerθkS ,

)(cot2

1)( 2 43121134 θerθkS . (4.14)

Similarly, on using equations (3.20) and (4.13), we determine from equations (2.40)

to (2.43) the tetrad components of the curvature 2-forms as

)([2 23130231424131212 θθSekd

]2)( 341

24140 SθθS ,

156

)([2 231320

2341323121313 θθSkd

]2)( 3410

241400 SSθθSS ,

)([2 231320

2342323122323 θθSkd

]2)( 3410

241400 SSθθSS ,

)([4 231310

2231424133434 θθSSkd

]2)( 3421

241410 SθθSS . (4.15)

Now to find the explicit expressions for the tetrad components of curvature 2-

form we take the exterior derivative of connection 1-form 12 and find

3 0

2112 , 2)( ,)(

21

ii

i

idxkSdxed

)(2

1 , 2 214

0 ddedxSk ii

40

30 2 2 dSkdSk .

Using equations (4.8) and (4.11), a straight forward calculations lead to

))](1(2,[ 23131000

12212 θθrrSSSkeed r

))](1(2 ,[ 2414100 0 rrSSSke r

34100

1 ]4)( cot[ SSSrke . (4.16a)

Similarly, we obtain

rr Ske)θ(θSkSSked ,[2),( 1

231320

2120013

)](2)1()(2

241400

211

12

SSkrerkSre

341

1110

2 }]2)1({cot214[ kSrererSSk . (4.16b)

rr Ske)θ(θSkSSked ,[2),( 1

231320

2120023

)](2)1()(2

241400

211

12

SSkrerkSre

3410

21

11 ]4}2)1({cot21[ SSkkSrerer , (4.16c)

and

157

)θ(θSSkSSked r2313

10212

1134 4),(2

3421

222241410

2 )8(4 Skre)θ(θSSk . (4.16d)

We record below the wedge product of connection 1-forms

131

10

1200

22413 ]2)1([2 kSrerkSθSSk

241

10 ]2)1([ kSrerSk

341

1 ]2)1([21 kSrer , (4.17a)

00

21320

21201312 2[2 SSkSkθekS

141

1 }]2)1({21 kSrere

241

1 ]2)1([21 kSrere

341

10 ]2)1([ kSrerkS , (4.17b)

1310

1210

23413 cot2 erkSθSSk

141

11

10 }]2)1({cot[ kSrerkSerkS

241

11 ]2)1([ kSrerkS

341

11 ]2)1([cot21 kSrerer , (4.17c)

141

11202312 ]2)1([

21

kSrereθekS

241

100

22320

2 }]2)1({212[2 kSrereSSkθSk

341

10 ]2)1([ kSrerkS , (4.17d)

and

141

11

1210

23423 ]2)1([2 kSrerkSθSSk

2310 cot erkS

24101

11 ]cot}2)1({[ erkSkSrerkS

341

11 ]2)1([ cot21

kSrerer . (4.17e)

158

4.2. Tetrad Components of Curvature 2-form

On using equations (4.16) and (4.17) in the equation (4.15) we obtain

13100

1200

2212 ]2,[)4( θSkSrSekSSke

23100

14100 ]2,[]2,[ θSkSrSekθSSkSek r

24100 ]2,[ θSSkSek r

341100

1 )]1(4)(cot[ rerkSSSekr , (4.18a)

12100013 ]22,[ θSkSSeSek r

)(21

,[]2cot[ 221

130

10 eSkekSerkS r

1400

21

210

11 )](2cot)2( SSSkerkSrekS

21

22121 2)2(

21

,[ SkreSke r

341

10

241 ]2)1([] kSrerkSekS , (4.18b)

12100023 ]22,[ θSkSSeSek r

1421

21

2121 ]2)2(

21

,[ SkekSreSke r

)(21

,[]2cot[ 221

230

10 eSkekSerkS r

2410

1100

21

2 ]cot)2()(2 erkSrekSSSSk

341

10 ]2)1([ kSrerkS , (4.18c)

and

131

10

121134 ]2)1([],(2 kSrerkSθSSke r

141

10 ]2)1([ kSrerSk

231

10 ]2)1([ kSrerkS

241

10 ]2)1([ kSrerSk

341221

2 )]2(4[ reSk . (4.18d)

159

4.3. Tetrad Components of Riemann Curvature Tensor

We now recall equation (2.35) and write it explicitly as

2323

1414

1313

1212 RRRR

3434

2424 RR , (4.19)

where

4 ,3 ,2 ,1, , .

Now by giving different values to 4 ,3 ,2 ,1, in equation (4.19) and then equating

the corresponding coefficients of basis 2-forms of equations (4.18) and (4.19) we

readily obtain the tetrad components of Riemann curvature tensor as:

)4( 0022

1212 SSkeR ,

)2,( 1001312 SkSSekR r ,

)2,( 1002312 SkSSekR r ,

)]1(4)([cot 1001

3412 rSSSekrR ,

]22,[ 10001213 SkSSeSekR r ,

]2cot[ 01

01313 kSerkSR ,

)2()(21

,[ 11

2211413 rekSeSkeR r

)](2cot 002

121

0 SSSkerkS ,

ekSSkreSkeR r 12

12212

12413 2)2(21

, ,

]2)1([ 11

03413 kSrerkSR ,

]22,[ 10001223 SkSeSSekR r ,

21

21

21211423 2)2(

21

, SkekSreSkeR r ,

]2cot[ 01

02323 kSerkSR ,

)(2)(21

, 002

1222

12423 SSSkeSkeR r

cot)2( 10

11

erkSrekS ,

160

]2)1([ 11

03423 kSrerkSR ,

),(2 111234 SSkeR r ,

]2)1([ 11

01334 kSrerkSR ,

]2)1([ 11

02334 kSrerkSR ,

)2(4 1221

23434

reSkR , (4.20)

and

013232313 RR

The complex conjugates of these equations are obtained by interchanging the

suffixes 3 and 4 and taking the complex conjugates of the right hand sides of the

respective equations.

4.4. Tetrad Components of Ricci Tensor and Ricci Scalar

The tetrad components of Ricci tensor and Ricci scalar are defined by

RRRR and , , (4.21)

where is defined in equation (2.2).

34431221 RRRRR , (4.22)

1314141311 RRR .

Similarly, we obtain

14231324121212 RRRR ,

1334121313 RRR ,

23142413121221 RRRR ,

2423232422 RRR ,

2334122323 RRR ,

3413131231 RRR ,

3423231232 RRR ,

2313132333 RRR ,

34342314132434 RRRR , (4.23)

and the Ricci curvature scalar is given by

161

43342112 RRRRR . (4.24)

Using equations (4.20), we find from equations (4.23) the expressions for Ricci

tensors

)(4)(cot)( 002

12

00122

11 SSSkSSekreR ,

)(4)22( 002

12212

12 SSSkreR ,

) 1(2,1

00013 rerkSekSSkeR r ,

)(4)22( 002

12212

21 SSSkreR ,

)(cot)(4)( 001

002

1222

22 SSekrSSSkeR ,

)]1(2,[ 100023 rerSeSSekR r ,

)]1(,[ 10031 rrSSkeR r ,

)]1(,[ 10032 rerSSekR r ,

)242( 21234 reR ,

033 R , (4.25)

and

)](4)2( 3[2 002

12212 SSSkreR . (4.26)

5. EXACT SOLUTION OF FIELD EQUATIONS

The Einstein-Cartan field equations of gravitation are

ijijij tkgRR 21

, (5.1)

where ijt is the canonical energy momentum tensor. For Weyssenhoff fluid, we have

from equation (3.24) after using equations (4.13)

))((21

jijijijiij nnlnnlllpt

)( jijijiji mmmmlnnlp

)()( 00 jijijiji nmlmeSnmlmeS . (5.2)

The tetrad components of the energy-momentum tensor are given by

162

jiij eett

)(

)( . (5.3)

This gives

)(21

2211 ptt ,

)(21

1221 ptt ,

eStt 03231 ,

ptt 4334 ,

0332313 ttt . (5.4)

Now the tetrad components of the Einstein-Cartan field equations are given by

4 ,3 ,2 ,1, , 2

tkRR . (5.5)

This gives

12121111 21 , tkRRktR ,

21211313 21 , tkRRktR ,

23232222 , tkRktR ,

32323131 , tkRktR ,

34343333 21 , tkRRktR . (5.6)

On using the equations (4.25), (4.26) and (5.4) in the equations (5.6), the independent

field equations for gravitation in the Einstein-Cartan theory are

)(4)(cot)( 002

12

00122 SSSkSSekre

)(2

pk , (5.7a)

)(2

)24( 212 pkre , (5.7b)

03,1

000 rSSrS , (5.7c)

)(4)(cot)( 002

12

00122 SSSkSSekre

)(2

pk , (5.7d)

163

0,1

000 rSSrS , (5.7e)

pkSSSkre )(4)22( 002

12212 , (5.7f)

We see from equations (5.7) that the equations (5.7c) and (5.7e) are consistent

provided 00 S or 0 . Here 0 does not give any information, hence we

assume 0 and 00 S . Consequently equations (5.7) reduce to only three

independent equations.

)(2

4)( 21

222 pkSke , (5.8)

)-( 2

)24( 212 pkre , (5.9)

and

kpSkre 21

2212 4)22( . (5.10)

Adding equations (5.8) and (5.9) we get

kSkre 21

2212 4)42( . (5.11)

Subtracting equation (5.9) from equation (5.8) we get

pkSkre 4)43( 21

2122 . (5.12)

Equations (5.10) and (5.12) give

021 r . (5.13)

This is a non-linear differential equation. To solve this equation, we make the

following substitution

ey ,

yy

1 ,

and yy

yy

1)(1 2

2 .

Hence equation (5.13) reduces to the ordinary differential

01 yry . (5.14)

A solution of this equation is given by

dcry 2 , (5.15)

164

where dc and are arbitrary constants. Consequently, the solution of the equation (5.13)

is given by

12 )log( dcr . (5.16)

Hence the static conformally flat spherically symmetric Einstein-Cartan space-time

(4.1) now takes the form

) Sin()(

1 222222222

2 drdrdrdtdcr

ds

. (5.17)

The same solution is also obtained by Kalyanshetti and Waghamode [4]. Using

equation (5.16) and solving the equations (5.11) and (5.12) for the density and

pressure p , we get

)(4 21

2Skcdk , (5.18)

)2(4 21

222 Skcdrckp . (5.19)

Now we solve equations (5.18) and (5.19) for dc and we obtain

212

1 ]122[2

kSprkc , (5.20)

and

2

121

21

]122[

)4(2 kSp

kSkrd

. (5.21)

Consequently, the metric (5.17) becomes

] Sin[)163()122(4 222222222

1

21

22

drdrdrdt

kSpkSp

krds

. (5.22)

If 0p and 0 , then the metric (5.22) reduces to the empty model

] Sin[16

3 22222222

122

2 drdrdrdtSrk

ds . (5.23)

6. PETROV TYPE D SOLUTION

The free gravitational field is characterized by the Weyl conformal curvature

tensor hijkC . It is completely trace free. It means that the contraction with each pair of

indices vanishes. In the Einstein-Cartan theory gravitation, the trace free part of the

curvature tensor has 20 independent components. These 20 components are

represented by five complex tetrad components of the Weyl tensor )4,3,2,1,0( AA ,

165

nine components of a Hermitian 33 matrix )2,1,0,( BAAB and a real scalar .

These components are exhibited by Jogia and Griffiths [7], Katkar [6], and also by

Katkar and Patil [10]. We recall these results here for ready references.

13130 C ,

)(21

131212131 CC ,

13242 C ,

)(21

241212243 CC ,

24244 C ,

131400 Ci ,

)(2 1312121301 CCi

,

132302 Ci ,

)(4 1234341211 CCi

,

)(2 2412122421 CCi

,

232422 Ci , (6.1)

)(4

)(2 3412123422 CCii

,

where the tetrad components of the Weyl curvature tensor are given by

)(21

RRRRRC

)(6 R . (6.2)

Here

jiij eeg

)(

)( .

Its components are given in (4.2). From equation (6.2) we find

61212121212RRRC ,

1312131213 21 RRC ,

166

2312231223 21 RRC ,

12341234 RC ,

3113121312 21 RRC ,

13131313 RC ,

1113141314 21 RRC , (6.3)

01323 C ,

6

)(21

341213241324RRRRC ,

1313341334 21 RRC ,

3223122312 21 RRC ,

02313 C ,

6

)(21

342123142314RRRRC ,

23232323 RC ,

2223242324 21 RRC ,

2323342334 21 RRC ,

34123412 RC ,

3134133413 21 RRC ,

3234233423 21 RRC ,

6

)(21

433434343434RRRRC .

In the Section 5, we have seen that the field equations (5.7) are consistent

provided 00 S . Consequently, from equations (4.20), (4.25), (4.26) and (6.1) we

obtain after simplification

167

0 ,0 2102014310 ,

21

21 , 12 4

3)( SKSSke r ,

)2( 111 , 12200

rSSSike r ,

)23(2

111 , 111

rSSSiker

,

)2(2

111 , 1

rSSSiker

. (6.4)

This is the Petrov-type D solution. We note that if 01 S , the solution is conformally

flat solution.

7. DISCUSSION AND CONCLUSIONS

The exact solution of Einstein-Cartan field equations for static, conformally

flat spherically symmetric space-time is obtained and it is proved to be Petrov-type D.

168

R E F E R E N C E S

1. Hehl, F. W., Von der Heyde, P. and Kerlick, G. D.: 1974, Phys. Rev. D. 10, 1060.

2. Hehl, F. W., Von der Heyde, P., Kerlick, G. D. and Nester, J. M.:1976,

Rev. Mod. Phys., 48, 393.

3. Jogia, S., Griffiths, J. B.: 1980, Gen. Rel. and Grav., 12, No. 8, 597.

4. Kalyanshetti, S. B., Waghamode, B. B.: 1983, Phys. Rev. D., 27, 92.

5. Katkar, L. N.: 2008, Int. J. Theor. Phys., 48, No.3, 874.

6. Katkar, L. N.: 2010, Astro. Phys. Space Sci., 326, 19.

7. Katkar, L. N. and Patil, V. K.: 2011, Int. J. Theor. Phys.,

DOI 10.1007/s10773-011-0980-y. (Online published).

8. Kinnersley, W. M.: 1975, Eds. G. Shiviv and J. Rosen (New York: John Wiley

and Son), 109.

9. Kramer, D., Stephani, H., Herlt, E. and MacCallum, M.:1980, Ed. E. Schmutzer,

Cambridge: University Press.

10. Newman, E. T., Penrose, R.: 1962, J. Math. Phys., 3, 566.

11. Prasanna, A. R.: 1975, Phys. Rev. D., 11, 2076.

12. Singh, P., Griffiths, J. B.: Gen. Rel. and Grav. 22, No.3, 269 (1990)

13. Tsoubelis, D.: 1979, Phys. Rev. D., 20, 3004.