85

CHAPTER 4

CONGRUENCE OF VORTEX LINES IN EINSTEIN-CARTAN THEORY

“An Experiment, like every other event which

takes place, is a natural phenomenon; but in a

Scientific Experiment the circumstances are

so arranged that the relations between a

particular set of phenomena may be studied

to the best advantage”.

James Clerk Maxwell

86

1. INTRODUCTION

Vorticity is a concept used in fluid dynamics. In the simplest way, vorticity is

the tendency for element of the fluid to “spin”. It can also be considered as the

circulation per unit area at a point in a fluid flow field. It is a vector quantity, whose

direction is along the axis of the fluid’s rotation. For a fluid having locally a rigid

rotation around an axis, vorticity is twice the angular velocity of a fluid element. An

irrotational fluid has no vorticity. The vortex line is a line which is tangent

everywhere to the local vorticity. A congruence of vortex lines is a bundle of non-

intersecting vortex lines; and a vortex tube is a surface in the fluid formed by all

vortex lines passing through a given closed curve in the fluid. The strength of a vortex

tube is the integral of the vorticity across a cross-section of the tube. In fluid

dynamics, the Helmholtz’s theorems describe the three dimensional motion of fluid in

the vicinity of vortex tubes. The three Helmholtz’s theorems are as follows:

a) the strength of a vortex tube is constant along its length,

b) a vortex tube cannot end in a fluid; it must extend to the boundaries of the

fluid or form a closed path, and

c) in the absence of rotational external forces, a fluid that is initially irrotational

remains irrotational.

Kerlick [15] has observed the effect of vorticity and pressure in the singularity

behavior of cosmological models incorporating the Dirac field as the source of the

metric and torsion. Figueiredo et al. [5] have examined the gravitational coupling of

Klein-Gorden and Dirac fields with matter vorticity and space-time torsion, in the

context of Einstein-Cartan theory. The several authors such as Obukhov and Korotky

[17], Obukhov and Piskareva [18], Ray and Smalley [22], Smalley and Krisch [23],

Smalley and Ray [24], Esposito [4] have discussed the behavior of the spin-vorticity

couplings in spinning fluid. Palle ([19], [20], and [21]) has shown that the vorticity

and acceleration play a very important role in Einstein-Cartan cosmology. Berman [2]

has studied the behavior of shear and vorticity in a combined Einstein-Cartan –Brans-

Dicke inflationary lambda- universe. The relationship between the three components

of the vorticity field on stream surface and curvatures of the streamlines (geodesic

torsion, normal curvature and geodesic and geodesic curvature) is studied by Herrera

87

[13]. Garcia de Andrade [6] has discussed the non-Riemannian acoustic space-time

structure called acoustic torsion of sound wave equation in fluids with vorticity.

The minimal coupling of electromagnetic field with torsion in electromagnetic

field equations are not gauge invariant. Hehl et al. [11] have discussed the minimal

coupling procedure for Maxwell’s field in EC space-time that leads to a spin angular

momentum tensor which is not )1(U gauge invariant. Hence, the Maxwell Lagrangian

is not minimally coupled to geometry, therefore photons are unaffected by the

presence of torsion. However, Hojman, Rossenbaum, Ryan and Shepley [14] have

proposed a hypothesis in which torsion and electromagnetic field are interacting.

There are many authors such as Benn, Dereli and Tucker [1], Hehl [12], Gracia de

Andrade and Pereira [7], Gibbons and Gielen [8] have discussed on the same problem

of )1(U gauge invariance in Riemann-Cartan spaces. Thus the wide spread conclusion

that, gauge fields do not couple minimally in RC spaces. In this Chapter our focus is

to study the role of vorticity in the study of vortex lines in the EC theory of

gravitation.

Accordingly the material of this Chapter is arranged as follows: In Section 2,

we have derived the propagation equation of the vorticity vector i and the magnitude

of the rotation tensor ij in the presence of torsion and spin tensor. It is shown

that, the additional matter sources ij (material sources) and ijE (tidal forces) are

occurred in the propagation equations in EC theory of gravitation, in contrast to

Einstein theory. In Section 3, the constraint and propagation equations of vorticity

vector i and spin vector i associated with electromagnetic field are established.

The propagation equation of spin density scalar is also derived. It is found that, for the

rigid flow of Weyssenhoff fluid, the evolution of spin density is triggered by the tidal

force of the fluid only. To observe the deformation and nature of the space-like

congruence we have developed a theorem in Section 4 which states that, when a

comoving observer can be employed all along any curve of the congruence if and only

if the curves in fluid are “frozen-into” the fluid. The Section 5 is concerned with

congruence of the vortex lines. We have proved the well-known result (Einstein

theory) that the vortex lines are “frozen-into” the fluid with an acceleration potential

in EC theory of gravitation. We also introduced the concept of the vortex tubes along

88

the space-like congruence of vortex lines. Finally, the general discussion and

conclusions are given in section 6.

2. PROPAGATION EQUATIONS OF VORTICITY

In this section we shall derive the propagation equation of vorticity and its

magnitude by means of Ricci identity.

If iu is the 4-velocity vector field of a self gravitating body of fluid, then the

rotation of the flow lines of the fluid at any point is determined by the tensor

)(21)(

21

/ / jijiijjiij uuuuuu . (2.1)

The vorticity vector i associated with the rotation tensor ij is defined as

lkjijkl

kljijkli uuu /2

121 . (2.2)

The scalar magnitude of the rotation tensor ij is given by

ii

ijij

212 . (2.3)

By using the completely skew-symmetric permutation tensor ijkl we have the

following useful relations:

jklljkkljijkli uuu , (2.4a)

lkijklij u , (2.4b)

)(2 ijjikl

ijkl uu , 0jij , (2.4c)

)()( kiikjm

jkkjimml

ijkl uuuu

)( ijjikm uu . (2.4d)

Since the rotation tensor ij is altered by the torsion tensor ][ij , so it is essential to

mention the relations associated with the torsion tensor ][ij as follows:

)2(][ lnm

kmnp

klpijkl

klijkl uuuQuQ , (2.5a)

)(][ lkp

klpijkl

klijkl uTuQ , (2.5b)

pklpj

ijklklj

ijkl uQuu ][ , (2.5c)

][21

kljijkli u , ][][][ jklljkkljijkl

i uuu , (2.5d)

89

][][

21

klijklij , lk

ijklij u ][ , 0][ jij , (2.5e)

)(2][ijji

klijkl uu , (2.5f)

)()(][kiikj

mjkkji

mmlijkl uuuu

)( ijjikm uu , (2.5g)

In order to derive the other forms of propagation equations from rotation tensor in EC

theory, we recall the propagation equation of rotation tensor ij as ((5.22) vide

chapter2)

kkji

kjik

kjikijjijiij uuuuu ][ ][] / [ 3

2{2

][][][][ 2}

31

ijijjik

kjikEuuuu

kli

ljkli

ljkli

ljkli

ljk uuuQhQQQ ]

31[2 ]

[]

[]

[]

[ . (2.6)

With the help of the above propagation equation we may further derive the

propagation equation for vorticity vector i and the magnitude of rotation tensor

ij . Using equations (2.2) and (2.6), the propagation equation for i is given by

)(21)(

21

kljkljijkl

kljijkli uuu

mlkmklj

ijkllkj

ijkllkj

ijkl uuuuuu ][] / [ 32{

21

21

kljijkl

lkm

mlkm

mlkmlkm uuuuuuu

21}

31

][][][

nkn

lmnkn

lmjijkl

kljijkl

kljijkl QQuukEu ]

[]

[][][ [

421

mnk

nlmnk

nlm uuuQhQ ]

31

]

[]

[ ,

kljijklm

kmljijkl

kljijkl

lkjijkli uuuuu

21

31

21

/

knknn

mljijkl

kljijkl

kljijkl QuukEu (

421

mnkkn uuuh )

31

. (2.7)

90

Using the results from (2.4), the equation (2.7) takes the form

)({)( 32

21

/ jkkji

mjjiji

jlkjijkli uuuuuuuu

)()}()( jiji

jm

kijjik

mkiikj

m uuuuuuu

knknn

mljijkl

kljijkl

kljijkl QuukEu (

421

mnkkn uuuh )

31

,

i.e. kljijklij

jlkjijklji

jii Euuuuu

21

21

32

/

mnkknknkn

nmlj

ijklklj

ijkl uuuhQuuk ) 31(

4 . (2.8)

This is the propagation equation of vorticity vector i in the presence of torsion

tensor.

Similarly, in the following, we determine the propagation equation for the

magnitude of the rotation tensor. Since we have

ijij

212 ,

ijij

ijij )(

21)( 2 .

Using equation (2.6) we obtain

kkji

kjik

kjikijjiji uuuuu

][ ][] / [2

32[2{)(

liljkijijji

kkji QkEuuuu ]

[][][][][ [2

2]

31

ijkli

ljkli

ljkli

ljk uuuQhQQ }]

31

] []

[]

[

ijkjik

ijkjik

ijij

ijjiu ] / [ 3

2

kijnkililil

lkj

ijijij uuuhQkE )

31(2)

2(

][][ ,

ijij

ijij

ijji

ijkjik

kEu 2

234)( ]/[

22

91

kijnkililil

lkj uuuhQ )

31(2 . (2.9)

The equation (2.9) represents the propagation equation of in EC theory of

gravitation. The right hand side of this equation shows the coupling of rotation tensor

with material tensor ij and tidal force ijE besides other kinematical quantities.

Since the torsion is directly related to the spin tensor through the field

equation kij

kij ukSQ , then the equation (2.8) and (2.9) are reduces to

kljijklij

jlkjijkli

jjii Euuuuu

21

21

32

/

kljijkluk

4

, (2.10)

and

ijij

ijij

ijji

ijkjik

kEu 2

234)( ]/[

22 . (2.11)

In Einstein theory, the propagation equations of vorticity vector field i and the

magnitude of the rotation tensor ij are independent of the matter sources

(Greenberg [9], [10]). It is of interesting to note that, the terms ij and ijE are the

additional matter sources appeared in equations (2.10) and (2.11) in EC theory of

gravitation.

Theorem 1: For an isometric Weyssenhoff fluid the square of the magnitude of

rotation tensor is conserved if and only if the tidal force is orthogonal to the rotation

tensor.

Proof: An isometric flow (containing rigid flow) is characterized by

0 ij and 0]/[ jiu . (2.12)

It follows from equation (2.11) that

ijij

ijij

kE 2

)( 2 . (2.13)

In case of Weyssenhoff fluid the material source ij is zero ((4.8) vide chapter 2).

With this condition the equation (2.13) yields

ijijE )( 2 ,

92

00)( 2 ijijE . (2.14)

This completes the proof.

The equation (2.14) also states that the evolution of the magnitude of ij along

vortex lines is triggered by the tidal force of the Weyssenhoff fluid.

3. ELECTROMAGNETIC FIELDS

The comprehensive description of propagation and constraint equation of

vorticity vector is given by Ellis [3] in Einstein theory. Moreover vorticity play a very

important role in EC theory of gravitation. Here we like to focus on the behavior of

vorticity by means of Maxwell’s equations. In this section we follow the evolution of

electromagnetic field by means of Maxwell’s equations and derive the constraint and

propagation equation for vorticity in EC theory of gravitation.

The usual form of Maxwell’s equations in Einstein theory is (Mason and Tsamparlis

[16])

iijj JF , , 0] , [ kijF , (3.1)

where

0 , iiJ , (3.2)

We can write the equations (3.1) and (3.2) in terms of covariant derivative with

respect to the connection ijk as

ijkijk

ijj JFQF ̂ , (3.3)

0ˆ ii J , (3.4)

02 ] []/[ lk

lijkij FQF . (3.5)

where iii Q̂ .Operating the permutation tensor ipqr on equation (3.5) and

using the identity

]

[ !3 l

rkq

jpipqr

ijkl , (3.6)

we readily obtain

0 / i

jkjkij

j TFF , (3.7)

where

][

ki

ji

jki

jk QQT , (3.8)

93

is the modified form of the torsion tensor.

The electromagnetic field tensor ijF can be expressed in terms of 4-velocity vector iu

as

ijjiij uuF ; ;

kijkjikijji uKKuu )(// ,

kijkijjiij uQuuF 2// . (3.9)

Using the decomposition of 4-velocity vector iu

jiijijijji uuhu 31

/ , (3.10)

the electromagnetic field tensor ijF may be written as

) 31()

31( ijjijijijiijijijij uuhuuhF

kijkuQ2 ,

kijkjiijij uQuuF 222 ][ . (3.11)

The dual field ijF of the electromagnetic field ijF is defined as

klijklij FF

21

)222([21

][m

klmlkklijkl uQuu ,

i.e. mklm

ijkllk

ijklkl

ijklij uQuuF . (3.12)

With the help of the results from (2.4) and (2.5), we can rewrite the equation (3.12) as

lkijkl

klijkl

lkijkl

klijklij uTuuF

][ ,

lkkijklijjiijjiij uTuuuuuF )()(2)(2 . (3.13)

The equation (3.13) is the dual of electromagnetic field tensor in terms of vorticity

vector i and torsion vector i as measured by the 4-velocity vector iu .

3.1. Divergence Equation of i

After the evolution of electromagnetic field tensor, we allow Ellis [3]

constraints to obtain the divergence equation for vorticity vector i in EC theory of

gravitation. On using equation (3.13) in the equation (3.7) we obtain

94

jlkkijklijjiijji uTuuuuu /])(

21)()[(

)()[( jkkjjkkj uuuu

0)21

21]()(

21 j

ikk

ij

ijkmll

jklm QQQuTu ,

)()( / / / // / / / i

jjij

jj

jiji

ji

jjij

jj

jiji

j uuuuuuuu

)(21

//// jlkljkjlkljkijkl uTuTuuuu

)()[( jkkjjkkj uuuu

0)21

21]()(

21 j

ikk

ij

ijkmll

jklm QQQuTu , (3.14)

Contracting the equation (3.14) with iu we get

kjikijjlkki

ijklii

ii

ii

ii uuQuTuuuu 2)(

21

///

0)(212 i

jkmllijklmi

ii

ikji

kij QuTuuQQuuQ ,

i.e. ii

iijlkki

ijkliii

ii

ii TTuTuuu /// )(

21)(

0)(21)( i

jkmllijklmii

i QuTuuQ ,

)()()()( /ii

iii

iii

iiii TuQ

ijkmlli

jklmjlkki

ijkl QuTuuuTuu / )(

21)(

21

,

)()()()( /ii

iii

iii

iiii TuQ

ijkimll

jklmlkiij

ijkl uQuTuuTuu )(21)(

21

/ . (3.15)

Substituting (2.2) and (2.5c) in equation (3.15) we get

)()()()( /ii

iii

iii

iiii TuQ

iii

iii TuTu )()( ,

)(2)(2)()( /ii

iii

iii

iiii TuQ . (3.16)

95

The equation (3.16) represents a divergence equation for i with the additional

couplings of torsion vectors. If the torsion vector vanishes in equation (3.16) then it

reduces to the Ellis’s constraint equation in Einstein theory.

To interpret the Maxwell’s equations physically, there is a need to assume any

specific theory of gravitation. Therefore, we use the consequences of EC field

equation kij

kij ukSQ in Maxwell’s field equations. Hence the Maxwell’s equations

(3.7) and the electromagnetic field tensor (3.13) take the form

0)( ][

/ k

ij

ijk

jkijj QQFF ,

0/ ijk

jkijj ukSFF , (3.17)

and

lkijklijjiijjiij uuuuuuF )(2)(2 , (3.18)

where ii kS is the spin vector. In order to obtain the divergence equation for the

vorticity vector, we put equation (3.18) in equation (3.17)

jlkijkl

jijjiijji uuuuuu // )(

21)]()[(

0]21)()[( i

jkmljklmjkkjjkkj ukSuuuuuu ,

)()( / / / // / / / i

jjij

jj

jiji

ji

jjij

jj

jiji

j uuuuuuuu

021)(

21

// ijkml

jklmjlkljk

ijkl ukSuuuuuu . (3.19)

Contracting the equation (3.19) with iu , the direct calculations give

021

21

/// jkmljklm

jlkiijkli

ii

iii

ii kSuuuuuuu ,

021

21

/// kljiijkl

lkjiijkli

ii

ii

iii kSuuuuuuu . (3.20)

With he aid of the following definitions

lkjijkli uu /2

1 , kljijkli kSu

21

, (3.21)

the equation (3.20) becomes

0// ii

ii

ii

ii

ii

ii uuuu ,

)(2)( /ii

iiii u . (3.22)

96

This equation represents the divergence equation of vorticity vector i in the presence

of spin vector. We see from (3.22) that the divergence of the vorticity vector is

identically zero if and only if the flow is geodesic.

3.2. Propagation Equation of i

In this section, we employ the Ellis [3] method to obtain the propagation

equation of vorticity vector in the presence of spin tensor. It is necessary to notice that

it does not contain any material or geometric sources from the fluid. The propagation

equation of i is obtained by contracting the equation (3.19) with mih

jj

ijij

mi

ij

jijj

jj

ijij

mi uuhuuuuh / / / / / / ()(

0)(21) // / / jlkljk

ijklmi

ij

jijj uuuuhuu . (3.23)

After simplification we have

ij

jmi

mjj

jmj

ij

jmi

mjj

jmj uhuuuhuu / / / / / /

021

21

/ / jlkijklm

iljkmjkl uuhuu ,

lkjmjklmmj

jjjm

jiim

j uuuuh // / 21)()()(

lkjijklm

i uuh / 21

. (3.24)

Substituting (3.10) in equation (3.24) we get

lkjmjklmmjjm

jjjm

j uuuh // 21)()()(

kljijklm

i uh 21

. (3.25)

Using equation (2.4c), the equation (3.25) becomes

lkjmjklmmjjm

jjjm

j uuuh // 21)()()(

jijjim

i uuuh )( ,

lkjijkliijji

jjji

j uuuh // 21)()()( . (3.26)

97

which shows that how vorticity vector get propagated in EC theory of gravitation.

Note that there is spin contribution, which means that the spin affect vorticity

evolution. The constraint equation (3.22) and the propagation equation (3.26) are

independent of the material or geometric source of matter.

3.2. Propagation Equation of Spin Vector

The propagation equation (3.26) of vorticity vector i can be written as

)()(/iijji

jj

jij

jiii uuuuu

lkjijkl uu /2

1 ,

)()(/iijji

ji

jij

jiii uuuuu

lkjijkl uu /2

1 . (3.27)

Substituting the equation (2.10) in equation (3.27) we get

kljijklij

jlkjijkli

jjii Euuuuu

21

21

32

/

)(4 /

jjij

jj

ijj

iklj

ijkl uuuuuuk

lkjijklii uu /2

1)( ,

)()(32

/ iijji

jj

jii

jjii uuu

kljijkl

kljijkl ukEu

421

. (3.28)

With the aid of the decomposition (3.10) of the 4-velocity vector iu the equation

(3.28) becomes

))( 31(

32

jj

jii

jij

ij

jj

ijij

ii uuhuu

kljijkl

kljijklii ukEu

421)( ,

jj

iijij

jij

i uu 32

kljijkl

kljijkl ukEu

421

, .

98

)2

(21

32)( klklj

ijklijij

ij

jij

kEuh . (3.29)

This equation describes the propagation equation of spin vector in EC the. The

expression (3.29) can be recast in terms of the spin- density scalar ii2 as

follows:

ii )( 2

kljijklj

jiiji

jji

j Euuu 21

32[

ikljijkluk

]4

,

)2

(21

32

klkljiijkl

iiji

ijkEu ,

)2

(21

32)( 22

klkljiijklji

ijkEu .

It follows from (2.5e) that

][22 )2

(21

32)( kl

klklji

ijkE . (3.30)

This relation shows the evolution of the spin density scalar 2 .

Theorem 2: The flow of Weyssenhoff fluid is rigid then spin density is conserved if

and only if 0][ klklE .

Proof: The rigid flow is characterize by

0 ij . (3.31)

With these condition the equation (3.30) reduces to

][2 )2

(21)( kl

klklkE . (3.32)

For the Weyssenhoff fluid we have 0ij

00)( ][2 klklE . (3.33)

Hence the proof of the theorem is completed.

4. COMOVING OBSERVERS

The vorticity vector field i is defined in terms of 4-velocity vector iu and is

orthogonal to iu .The space-like vector i give rise to a space-like congruence of

99

vortex lines. To measure the deformation of the curves of the congruence, a comoving

observer ( ii uw ) can be employed at any point P on a curve of the congruence.

In this section we develop a theorem to determine the nature of the curves of the

congruence of vortex lines in the presence of torsion tensor.

Theorem 3: A comoving observer can be employed all along any curve of the space-

like congruence if and only if the curves in fluid are “frozen-into” the fluid.

Proof: (i) Let us suppose the curves in fluid are “frozen-into” the fluid. We show that

the comoving observers can be employed all along the curve of the congruence.

Since 0iiuh , we can choose ii uw at some given point P and at another

point iii uw on a curve of the congruence where i satisfy the conditions

0 iih and 02

i

ii

i w .

The 4-velocity of the observer is related to the natural transport law for iw . So we

recall the natural transport law (2.14) from chapter 3 for iw

kmikm

ijj

ij

kjk

jij

i whQhwhwwwhwhwdsD

/ / 2

ij

kmjkm

ij

kmjkm hhwhQwwwhQ 22 . (4.1)

where jij

i hAA / .

At any point of the curve of the congruence, the natural transport law therefore

assumes the form

))()(()()( / / ii

jjkkj

kkki

kii uuuhuhu

dsD

)(2)( kkmikm

ikkk uhQhuh

))()((2 iijj

kkmjkm uuuhQ

ij

kkmjkm hhuhQ )(2 . (4.2)

With the use of projection tensor jiijij uugh , jij

i uAA / , the equation (4.2) can

be written as

ikjmkmj

jmjm

ijj

ijij

i uuuhQuhQhuhuhhdsD 22)( i

),,;(2 QhuAhhuhQ iijkmkmj , (4.3)

100

where

])()[(),,;( /kjj

kjjjjii uhhuQhuA

)]([ // / kj

kjj

kji

kii

kk hhuhuhhh

])(2)(2[ kjjmkmj

jjktktj

i uhQuuhQ

ijmkmj

mikm

k hhhQhQ 22[

)]222( jmjmk

jmkmj

jmkmj

i uhQhQuhQu . (4.4)

In the expression of iA , there is no separate term of torsion without . If 0 at any

point on the curve then equation (4.4) imply that 0iA .

Since the curve is a material curve in the fluid, the material curves are

sometime said to be “frozen-into” the fluid i.e. it consist of, at all time, the same fluid

particles. Thus any two neighboring fluid particles lie in the instantaneous direction of

the unit tangent vector ih to curve . If ) ( l is the distance measured by iu between

the neighboring particles on curve at any instant, then ihl) ( is a relative position

vector lies in the rest space of iu . As if ix is a relative position vector then it

satisfies the following equation (Vide equation (2.29), Chapter 2)

jij

jij xAxh )( , (4.5)

where,

lkjlkilk

lj

kiij uQhuhhA

/ 2 . (4.6)

If we substitute ihl) ( instead of ix in equation (4.5), then

jij

jij hlAhlh ) ( ]) [( ,

jij

jjij hlAhlhlh ) (]) () [( ,

jklijkl

kijk

ij

jij

jij huuuQuQuhhh

llh )22(

) ( /

. (4.7)

On contracting equating (4.7) with ih we have

jikjki

jj hhuQuh

ll 2) () (

. (4.8)

By using equation (4.8), the equation (4.7) becomes

jkjk

ijj

ijij huQhuhuhh i 2

101

iljkjkl

iljkjkl hhhuQuuhuQ 22 . (4.9)

From equations (4.3) and (4.9) we obtain

),,;()( QhuAdsD ii . (4.10)

This system of first-order differential equations for the components of is

homogeneous. The equation (4.10) is the first order differential equation. Its solution

subject to the initial condition 0i at the point P on curve is 0i .Therefore

comoving observers can be employed all along the curve of the congruence.

(ii) Conversely, let us assume that the comoving observers, ii uw with 0i can

be employed all along the curve of the congruence and show that the curves in a fluid

are “frozen-into” the fluid.

From equations (4.3) and (4.4) with 0i , the natural transport law therefore

becomes

jkjk

ijj

ijij huQhuhuhh i 2

iljkjkl

iljkjkl hhhuQuuhuQ 22 . (4.11)

We define

0

])2(exp[ dhhuQujhf ljkjkl

j , (4.12)

where the integration is performed along the world line of iu from some arbitrary

initial proper time 0 ; then

ljkjkl

jj hhuQuh

ff 2

. (4.13)

The equation (4.11) may be written as

iljkjkl

jj

jij

ji hhhuQuhhuhh )2(/ j

iljkjkl

jkijk uuhuQhuQ 22 .

With the aid of the equation (4.13), this equation reduces to

iljkjkl

jkijk

ijij

jij uuhuQhuQh

ffhuhh 22

/

,

))(22() ( /

jilkjkl

kijk

ij

jjij fhuuuQuQuhfhfh .

102

It follows directly from (4.6) that

)()( ji

jji

j fhAfhh . (4.14)

Comparison of equation (4.14) with (4.5) shows that ifh is a relative position

vector, where is a constant. Hence if ifh links two neighboring fluid particles at

one instant, it does so at all later times. The same fluid particles therefore always lie

on as the fluid involves and hence we conclude that the curves in the fluid in

Einstein-Cartan theory are “frozen-into” the fluid.

This completes the proof of the theorem.

If we define the reduced form of the natural transport law (4.11) in the form

jkijk

ijj

ijij

i huQhuhuhhL 2)(

lijkjkl

ljikjkl uuhuQhhhuQ 22 , (4.15)

then by using the proof of the above theorem, it is clear that comoving observers can

be employed all along a given curve , and then will be “frozen-into” the fluid if

and only if 0iL . The nature of the space-like congruence of vortex, magnetic and

electric field lines has been specified with the expression of iL and can be calculated

with the help of the propagation equation for ih .

As the torsion tensor is linked up with the spin angular momentum tensor

through the EC field equation kij

kij ukSQ , then the expression (4.15) of iL becomes

jkijk

ijj

ijij

i huQhuhuhhL 2)( , (4.16)

and

jj uh

ff

. (4.17)

5. CONGRUENCE OF VORTEX LINES

5.2. Comoving Observer

In this section we consider a congruence of vortex lines in an electromagnetic

fluid with 4-velocity iu . The vorticity vector i of the fluid associated with iu at any

point is defined by (2.2). Since, i is a space-like vector then the unit tangent vector

103

to a vortex line at any point is i

ih . In EC theory, the vorticity vector i and the

spin vector i do not in general separately propagate, but they propagate

simultaneously along the vortex lines. Since 0 iiu , the unit space-like vector

associated with the spin vector i is

i

ih . In order to determine iL defined by

(4.16), we consider the propagation equation (3.26) as

lkjijkliijji

jjji

j uuuh // 21)()()( .

This equation may be written as

) (21

/ // iji

jji

jlkjijkliji

jji

j uhuuuh . (5.1)

The equation (5.1) with i

ih and

i

ih yields

lkjijkliji

jji

j uuhhuhh // 21)()()(

)]( )()([ / ijij

jij hhuhh ,

lkjijkliji

jjji

j uuhhuhhh // 21)( )() (

)]( )() ([ / ijij

jjij hhuhhh ,

lkjijkliji

jjji

j uuhhuhhh // 21 )(

] )([ / iji

jjji

j hhuhhh

,

lkj

ijkliiijij uuhhuhh /

2

1

] )([

iijjij huhhh

. (5.2)

Since

jj uh

ff

,

with this result the equation (5.2) becomes

104

lkj

ijklijj

iijij uuhuhhuhh /

2

1)(

] )([

iiijj

jij huhuhhh

. (5.3)

Contracting equation (5.3) with ih we have

0])([21)(

/

ii

jjlkji

ijkliii

i uhuhuuhuhhu ,

021 /

lkji

ijkl uuh

,

lkjiijkl uuh /2

1)1(

. (5.4)

From equations (4.16) and (5.3) we find that

) (21 /

iilkj

ijklii hLuuhL

,

lkjijklii uuhL /

21)-(1 )1(

. (5.5)

Substituting equation (5.4) in (5.5) we have

lkjmjkl

lkjijkl

imm uuuuhhL //

21

21)1(

. (5.6)

The equation (5.6) can be written in the desire form as

lkjijklm

im uuPL /

21) 1 (

,

lkjijklm

im uuPL /

)(21

, (5.7)

where jijiijij hhuugP .

The expression (5.7) shows that, the comoving observer can be employed all along a

vortex line and the vortex lines are “frozen-in” the fluid in EC space-time if and only

if

0/ lkjijklm

i uuP . (5.8)

In case of “dust” distribution ( 0 iu ), equation (5.7) is satisfied and all observers

employed along a vortex congruence can be comoving. Hence, we conclude that the

vortex lines in dust distribution are material lines (i.e. the vortex lines consist of the

same substratum particles at all times) in EC space-time.

105

A potential function r (acceleration potential) in EC theory of gravitation is

defined by

jjii rhu , )(log , (5.9)

with this definition we have

lmmkj

ijkllkj

ijkl rhuuu / , / ))(log(21

21

mlkm

km

ljijkl ruuuuu ,// ))(log([

21

])((log / , lm

mk rh

])((log)(log[21

/ , ,/ lkm

mlkjijkl ruruu

l

kmmlkj

ijkl

rr

ur

ruu

/

, ,/2

1 ,

l

kj

ijklilkj

ijkl

rr

urruu

/

,/ 2

121

. (5.10)

For a scalar function , the covariant derivative with respect to the connection is

same as the covariant derivative with respect to the Christoffel symbol i.e.

grad ; , / iii . Hence the equation (5.10) becomes

][121

21

/ ,/ ,2/ lklkjijkli

lkjijkl rrrr

ru

rruu

rr

urr lk

jijkli / ,

21

rrKr

urr p

plklk

jijkli ,

; ,

21

,

rr

Kurruu pp

lkjijkli

lkjijkl ,

/ 21

21

. (5.11)

Since we have

lij

lij

lij

lij QQQK

, kij

kij ukSQ and klj

ijkli Suk 21 ,

then equation (5.11) gives

106

rr

ukSurruu pp

kljijkli

lkjijkl ,

/ 21

21

)( ii

rr

,

ilkj

ijkl hrruu )(

21

/

. (5.12)

Since 0 jij hP , then from equations (5.7) and (5.12) we have 0iL . Hence we

conclude that the vortex lines are “frozen-into” the fluid with an acceleration potential

in EC theory of gravitation. This is well-known result in Einstein theory discussed by

Ellis [3] and Tsamparlis and Mason [25].

5.2. Vortex Tubes

In this section, we introduce the concept of the flux tube along the space-like

congruence. A flux tube is defined as the surface in the fluid formed by curves

passing through a given closed curve in the fluid. Further, A be the cross-section

area of the flux tube subtended by the space-like curves as they pass through the

screen. The terms 0 and 0 ij leave the cross-section area A invariant, and

then we define

AA )( . (5.13)

The equation (5.9) with the expression of (equation (2.22) from chapter 3) may be

expressed as

jiji

ii wwhh

AA

// )(

. (5.14)

We first consider the Helmholtz theorem on vortex tubes in Newtonian theory.

As the torsion is connected with the covariant derivative of a vector, it vanishes in

Newtonian theory. In Newtonian theory, it follows from (5.14) with i

ih , that

iiA

dsD

, 1)ln(

, (5.15)

where AAA

dsD

)()ln( and A is the cross-sectional area of the vortex tube. But

107

0 , ii as shown by Greenberg [10] then A is constant along the vortex tubes; it

therefore represents a characteristic of the tube and is called the strength of vortex

tube. This is called the first Helmholtz theorem.

To obtain relativistic generalization of the Helmholtz theorem in EC theory,

we consider a comoving observer at any point P on one of the vortex lines of vortex

tube. With the aid of (5.14) and i

ih , we have

ji

ij hA

dsD

/ 1)ln(

, (5.16)

where A is the cross-sectional area of the vortex tube at P as measured by iu .

With the help of the constraint equation (3.22) and (5.16) we have

)2(1)ln( /i

iii

ii uuA

dsD

. (5.17)

For a dust distribution the equation (5.17) reduces to

iiA

dsD

/ 1)ln(

. (5.18)

It follows from (5.18) that the cross-sectional area A of the vortex tube is not

conserved and hence the vortex tube cannot measure the strength of the tube. The

relativistic analogues of the second and third Helmholtz theorem depend on the fluid

or flow chosen.

6. DISCUSSION AND CONCLUSIONS

The derivation of divergence and propagation equations of vorticity vector and

its magnitude in EC theory shows some additional material and geometric sources in

contrast to Einstein theory of gravitation. Especially the matter sources (material

source ij and electric type source ijE ) are also related to propagation equation of spin

vector. But these sources are no longer symmetric in the propagation equations of

vorticity and spin vector. It is shown that, for an isometric Weyssenhoff fluid the

square of the magnitude of rotation tensor is conserved if and only if the tidal force is

orthogonal to the rotation tensor.

The theorem 1 shows that, the comoving observer can be employed all along

the vortex lines and the curves will be “frozen-into” the fluid if and only if 0iL .

108

Especially we have proved this theorem for the case of dust distribution. Moreover,

we have proved that the vortex lines are “frozen-into” the fluid with an acceleration

potential in EC theory of gravitation. This well-known result is discussed by Ellis [3]

and Tsamparlis and Mason [25] in Einstein theory.

109

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