1

CHAPTER – 1

EINSTEIN-CARTAN THEORY

“[My father] knew more than I did about Lie groups, and it

was necessary to use this knowledge for the determination

of all bounded circled domains which admit a transitive group. So we wrote an article on the subject together. But in

general my father worked in his corner, and I worked in

mine”.

Elie Cartan

2

1. INTRODUCTION

Newton and Einstein were genuine free thinkers and had a great curiosity

about nature. Both of them lived by a deep faith – a faith that there are laws of nature

to be discovered. Their life long pursuit was to discover them. Einstein realism and

his optimism are illuminated by his remark “Subtle is a Lord, but malicious He is

not”. It means that “Nature hides her secret because of her essential loftiness, but not

by means of ruse”. Both of them spend their lives to understand the secret of nature.

Newton (1642-1727) had belief that the natural phenomenon take place

according to some definite laws and those laws are well understood. He defined a

frame of reference with respect to which he studied laws of motion and the law of

gravitation, and hence supplied the reason why planets move according to the

Kepler’s laws. It is because of this law of gravitation; the Moon goes around the

Earth, Earth around the Sun, the Sun around the centre of our galaxy and so on. Even

today this law forms a basis of the calculations of the flight trajectories of the

satellites sent out from the Earth. More practically our entire daily life depends upon

this force. It works beautifully in the domain where the velocities of particles are very

small as compared to the velocity of light, but it breaks down at speed approaching

the velocity of light.

Einstein (1879-1955) developed a new mechanics in1905 called special theory

of relativity. This theory describes the motion of the fast moving particles. However,

this theory is considered as restricted theory because it does not contain gravitational

force and deals with only special types of observers called inertial observers.

Einstein’s general theory of relativity is the geometric theory of gravitation published

in 1915. The theory deals with all types of motion and explains the phenomena of

gravity. Einstein formulated the field equations of gravitation in the form

ijijij kTgRR 21 .

These are ten non-linear partial differential equations. These equations explain space-

time grips mass, telling it how to move and mass grips space-time telling it how to

curve (John Wheeler). The theory predicts that the ray of light bends in the

gravitational field of the Sun; the perihelion of the planet Mercury shifts in the

gravitational field of the Sun and the gravitational red shift. In spite of these

3

spectacular successes, the theory has defects in the sense that it does not explain the

phenomenon of intrinsic spin of matter and singularities can not be prevented.

Einstein-Cartan Theory of Gravitation

The Einstein’s general relativity is essentially a classical theory of gravitation

which does not take into account the quantum effects. According to relativistic

quantum mechanics mass and spin are two fundamental characters of an elementary

particle. In Einstein’s general relativity mass plays a dynamical role but not the spin.

However spin is an intrinsic characteristic of gravitating matter. In order to

incorporate spin in the theory of gravitation, Trautman [25, 26] formulated Einstein-

Cartan theory. This theory postulates that the spin of matter is the source of torsion of

the space-time geometry. According to Will [29] there are three reasons for

introduction of torsion into the gravitational theory:

a) as a means to incorporate quantum mechanical spin in a consistent way,

b) as a byproduct of attempts to construct gauge theories of gravitation,

c) as a possible route to a unified theory of gravity and electromagnetism.

In Einstein-Cartan theory affine connection compatible with the metric tensor

is not necessarily symmetric in general; and the antisymmetric part of the connection

is coupled with the intrinsic spin of matter. According to Hehl [17,18] the Einstein-

Cartan theory is the simplest and the most natural modification of the original

Einstein theory of gravitation. In Hehl’s opinion Einstein-Cartan theory is an even

more beautiful theory than Einstein’s general relativity because of its relation to the

Poincare group. In Einstein’s theory of gravitation, mass directly influences the

geometry but spin has no such dynamical effect. Under general assumption,

singularity cannot be prevented in Einstein’ general relativity (Hawking-Penrose

theorem, vide Hawking and Ellis [16]); however the singularity can be prevented in

the Einstein-Cartan theory by direct influences of spin on the geometry of space-time

(Trautman [27], Hehl et al. [18, 19]).

Today, several people agree with the line of thinking that torsion could have

played some specific role in the dynamics of early universe. In fact, Minkowski [24],

De Sabbata and Sivaram [7], Fennelly and Smalley [9] and de Rits et al. [5] have

argued that the presence of torsion naturally gives repulsive contribution to the energy

4

momentum tensor so that a cosmological model becomes singularity free. This

features, essentially, depends on spin alignments of primordial particles which can be

considered as the source of torsion (Dobato and Maroto [4]). If the universe

undergoes one or several phase transition, torsion gives rise to topological defects

(e.g. torsion walls) which today can act as intrinsic angular momentum for cosmic

structures as galaxies ( Garcia de Andrade [10, 11, 12], Kuhne [22, 23], Capozziello

et al. [2]). Further, Capozziello and Stornaiolo [3] have argued that the presence of

torsion in an effective energy-momentum tensor alters the spectrum of cosmological

perturbations giving characteristic length for large scale structures. Furthermore, a

torsion field appears in a superstring theory if we consider the fundamental modes: a

symmetric and antisymmetric one (De Sabbata [6], Hammond [15]). All these

arguments and several more, do not allow neglecting torsion in any comprehensive

theory of gravitation. Any theory of gravity considering twistor needs the inclusion of

torsion. Hence we intend to study the role of torsion in the study of relativistic

kinematics in Einstein-Cartan theory of gravitation.

Though this Chapter is introductory, some definitions of Einstein theory of

gravitation are extended to Einstein-Cartan theory of gravitation and the role of

torsion tensor is examined. In Section 2, non-Riemannian space-time is described

through the antisymmetric affine connection ijk . Ricci identities and Bianchi

identities for torsion are also exhibited. In the Section 3, and 4, some definitions in

non-Riemannian space-time are cited. The field equations of Einstein-Cartan theory

of gravitation is presented in the Section 5. The consequences of torsion tensor on the

definitions given in Section 3 and 4 are also summarized. Decomposition of the

torsion tensor in the three irreducible tensors viz.,T -torsion, A -torsion and V -torsion

is displayed in the same section. A cursory account of time-like and space-like

congruence in the Einstein theory of gravitation is given in the last section. 2. DEFINITIONS AND NOTATIONS

In Einstein-Cartan theory of gravitation, a space-time is described by a non-

Riemannian geometry. The non-Riemannian part is defined by the torsion tensor i

jkQ and is defined by

5

)(21 i

kjijk

ijkQ , (2.1)

where ijk are the affine connections which are different from Christoffel

symbols ijk , where

ikji

jk ,

and

ikj

ijk .

We see from equation (2.1) that

ikj

ijk QQ . (2.2)

The covariant derivative of a vector field iA with respect to the affine connection ijk

is defined by

kjikjiji AAA , / , (2.3)

where colon ( , ) denotes the partial derivative and ( / ) denotes the covariant

derivative with respect to affine connection ijk . Similarly, for contra-variant vector

field, we have

ijk

kij

ij AAA , / . (2.4)

This definition can be extended to any tensor of any rank. The covariant derivative of

iA ( iA ) with respect to the Christoffel symmetric symbol ijk is denoted by semi-

colon ( ; ) and is defined by

kjikjiji AAA

, ; , (2.5)

ijk

kij

ij AAA

, ; . (2.6)

At every point of the Einstein-Cartan space-time, there exists a Lorentz metric

ijg which satisfies the metric postulates

0/ kijg .

Using the definition (2.3) we have

0, hkjih

hkihjkij ggg ,

hkjih

hkihjkij ggg , . (2.7)

6

By cyclic permutations of the indices kji ,, in equation (2.7) twice in turn, we obtain

two more equations

hikjh

hijhkijk ggg , , (2.8)

and

hjikh

hjkhijki ggg , . (2.9)

Adding equations (2.8) and (2.9) and subtracting equation (2.7) we obtain

)()()(],[2 hki

hikhj

hkj

hjkhi

hji

hijhk gggkij ,

where ],[ kij is the Christoffel symbol of first kind. Now the definition

],[ kijg lklij ,

gives

)(21)(

21)(

21 h

kjhjkih

lkhki

hikhj

lklji

lij

lij gggg .

We write this equation as

)(21)(

21 h

kih

ikhjlkl

jilij

lij

lij gg

)(21 h

kjhjkih

lk gg .

Using the definition (2.1) we obtain

hjkih

lkhikhj

lklij

lij

lij QggQggQ ,

][ lijjki

lkikj

lklij

lij QQgQg

][

lij

lij

lij

lij QQQ ,

lij

lij

lij K , (2.10)

where

lij

lij

lij

lij QQQK

, (2.11)

is the contorsion tensor satisfying the condition

0)( jkiK . (2.12)

Using equation (2.10) in (2.1) we get

)(21 l

jil

ijl

ij KKQ . (2.13)

7

The relation between the covariant derivative with respect to the connection and the

Christoffel symbol is obtained from equations (2.3), (2.5) and (2.10)

ljiljiji KAAA

; / , (2.14)

and

ijl

lij

ij KAAA

; / . (2.15)

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

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

grad ; , / iii . (2.16)

2.1. Ricci Identity in Einstein-Cartan Theory

Let iA be the arbitrary vector field in the space-time of Einstein-Cartan theory

of gravitation. Then by definition of the covariant derivative, we have

ijl

lij

ij AAA , / . (2.17)

Since ijA / is a tensor field of mixed rank two. Hence we can differentiate it

covariantly in Einstein-Cartan theory as

lkj

il

ikl

lj

ijkk

ij AAA

xA

/ / / // )()( ,

lkj

ilh

hl

iikl

ljh

hj

lijl

lj

i

ki

jk AxAA

xAA

xA

xA

/ ,

i

klljh

hj

likl

ijlk

lijlk

l

kj

ii

jk AxA

xA

xA

xxAA

2

/

lkj

ilh

hl

ilkj A

xA

. (2.18)

Now interchanging kj and in the equation (2.18) we get

2

/

i

jllkh

hk

lijl

iklj

liklj

l

jk

ii

kj AxA

xA

xA

xxAA

ljk

ilh

hl

iljk A

xA

. (2.19)

Subtracting equation (2.19) from equation (2.18) we get

]-)()([/ /

ijh

hkl

ikh

hjl

iklj

ijlk

likj

ijk xx

AAA

8

lkj

ljk

ilh

hl

i

AxA

,

-/

/ / lkj

ljk

il

ikjl

likj

ijk ARAAA , (2.20)

where

hkl

ijh

hjl

ikh

iklj

ijlkkjlh

ihikjl xx

RgR

)()( , (2.21)

is the Riemann curvature tensor in Einstein-Cartan theory satisfying the only

properties 0)( jkhiR and 0)( jkhiR . On using equation (2.10) we write (2.20) as

/

/ /

hjk

hkj

ih

ikjh

hikj

ijk KKARAAA , (2.22)

ih

hkj

ikjh

hikj

ijk ARAAA /

/ / 2Q- . (2.23)

Similarly, one can also determine

/

//

hjk

hkjhi

hkjihkjijki KKARAAA , (2.24)

2 /

// hih

kjh

kjihkjijki AQRAAA . (2.25)

Now we use the expression (2.10) in the equation (2.21) to find the relation between

the curvature tensor in Einstein theory of gravitation and Riemann tensor in Einstein-

Cartan theory of gravitation, we obtain

ikh

ikhj

ijh

ijhk

ikjh K

xK

xR

lkh

lkh

ijl

ijl

ljh

ljh

ikl

ikl KKKK ,

ikl

ljh

ljh

ikl

ikhj

ijhk

ikjh

ikjh KKK

xK

xRR ˆ

lkh

ijl

ijl

lkh

lkh

ijl

ljh

ikl KKKKKK , (2.26)

where

lkh

ijl

ljh

ikl

ikhj

ijhk

ikjh xx

R ˆ

, (2.27)

is the curvature tensor in Einstein space-time. Now to simplify equation (2.26), we

use the definition

ikl

ljh

lkh

ijl

lkj

ilh

ijhk

ikjh KKKK

xK

/ )( .

Using the equation (2.10) we write this as

9

ikl

ljh

lkh

ijl

lkj

ilh

lkj

ilh

ijhk

ikjh KKKKKK

xK

/ )(

ikl

ljh

lkh

ijl KKKK ,

l

kji

lhi

kjhl

jhi

kli

jll

khi

lhl

kji

jhk KKKKKKKx

/

ikl

ljh

lkh

ijl KKKK . (2.28)

Interchanging kj and in equation (2.28) we get

l

jki

lhi

jkhl

khi

jli

kll

jhi

lhl

jki

khj KKKKKKKx

/

ijl

lkh

ljh

ikl KKKK . (2.29)

Subtracting equation (2.29) from (2.28) we get

l

khi

jli

kll

jhl

jhi

kli

jll

khi

khji

jhk KKKKKx

Kx

ikl

ljh

lkh

ijl

lkj

ilh

ijkh

ikjh KKKKKKKK

/

/

ijl

lkh

ljh

ikl

ljk

ilh KKKKKK .

Substituting this in the equation (2.26) we get

)(ˆ /

/

ljk

lkj

ilh

ijkh

ikjh

ikjh

ikjh KKKKKRR

ikl

ljh

lkh

ijl KKKK . (2.30)

Equation (2.30) gives the relation between the components of curvature tensors

relative to Einstein-Cartan theory and Einstein theory of gravitation. In fact it clearly

depicts the influence of spin and torsion on the space-time curvature.

2.2. Curvature Tensor of First Kind

Curvature tensor of first kind is defined as

ikjlhikjlh RgR .

Using equation (2.21) we write this as

pklhpjhklj

pjlhpkhjlkkjlh g

xxg

xxR , ,

pklhjp

pjlhkp , , ,

where

10

)(21

, , , , hjlljhjlhhjl ggg ,

hkppkhhpk gx , , .

Using these equations we obtain

pklpjh

pjlpkhhjkllkjhljkhhkjlkjlh ggggR , , , , , , )(

21 . (2.31)

From this equation we easily prove that

kjhlkjlh RR ,

jklhkjlh RR ,

and

lhkjkjlh RR . (2.32)

Also by cyclic permutation of indices hlj , , twice in turn in equation (2.31) and

adding the results we obtain

)()( hjpjhpp

kljlpljpp

khkhjlklhjkjlh KKKKRRR

)( lhphlpp

kj KK . (2.33)

From equations (2.32) and (2.33) we see that in Einstein-Cartan theory, the Riemann

curvature tensor is skew-symmetric in the first and second pair of indices but not

symmetric in the pair of indices. Further, it does not satisfy the cyclic property. Hence

it has 36 independent components.

From equation (2.31) on using (2.10) we readily obtain the relations

jlppkh

pkljhp

pjlkhpkjlhkjlh KKKKKRR ˆ

jhppklkhp

pjlklp

pjh KKK . (2.34)

2.3. Ricci Tensor

From equation (2.30) we have

lkh

ijl

ilh

lkj

ihjk

ikjh

ikjh KKKKKRR

][

][ 2ˆ

ikl

ljh KK , (2.35)

where ik

ik AA / .

i.e. ikjh

ikjh

ikjh RR ˆ , (2.36)

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where

ikl

ljh

lkh

ijl

ilh

lkj

ihjk

ikjh KKKKKKK

][

][ 22 . (2.37)

Contracting ik with in equation (2.35) and using the definition

jhk

kjh RR , (2.38)

we get

jhjhjh RR ˆ , (2.39)

where

kkl

ljh

lkh

kjl

klh

lkj

khjkjh KKKKKKK

][ ][ 22 , (2.40)

and jhR is the symmetric Ricci tensor in Einstein theory of gravitation.

3. DIVERGENCE, CURL AND LAPLACIAN IN EINSTEIN-CARTAN

THEORY

3.1. Divergence of a Vector

Divergence of a contravariant vector iA is defined as the contraction of its

covariant derivative with respect to connection. Thus we have from equation (2.15)

ijl

lij

ij KAAA

; / . (3.1)

Contracting equation (3.1) we obtain divergence of iA as

iil

lii

ii

i KAAAA ; / div , (3.2)

or

iil

lii

ii KAAg

xgA

/ )(1

, (3.3)

lili

ii

ii AQAg

xgA

/ 2)(1

. (3.4)

Similarly, the divergence of a covariant vector iA is defined as

jiij

i AgA /div

)( ;

ljilji

ij KAAg

iil

lii KAA

; ,

12

iil

liii KAAg

xgA )(1div

, (3.5)

liil

iii AQAg

xgA

2)(1div

. (3.6)

From equations (3.4) and (3.6) we have

ii AA divdiv .

3.2. Curl of a Vector

Curl of a covariant vector iA is defined as

ijjii AAA //curl . (3.7)

Using the definition (2.14) we obtain

)(curl ; ;

lij

ljilijjii KKAAAA ,

)()curl(curl lij

ljilii KKAAA , (3.8)

where

ijjiijjii AAAAA , , ; ; )curl( . (3.9)

On using equation (2.13), equation (3.8) can also be written as

ll

ijii AQAA 2)curl(curl . (3.10)

3.3. Laplacian

Laplacian of a scalar invariant is defined as

.2

j

iij

xg

/

kk

jiij

ji

ij

xKg

xg

;

,

kiiK

22 )( , (3.11)

)()( 22 ki

iki

i QQ ,

kiiQ

22 2)( . (3.12)

This exhibits the role of torsion field on the Laplacian scalar field.

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4. GEODESIC EQUATION AND LIE DERIVATIVE IN EINSTEIN-CARTAN

THEORY

Let C be a curve in a space-time of Einstein-Cartan theory of gravitation and it a unit tangent vector to the curve C . Then we have

dsdxt

ii ,

where s is the arc length of the curve. Geodesic is defined as the curve of unchanging

direction. It is also defined as the curve of extremum distance. Both these definitions

in the space-time of Einstein-Cartan theory of gravitation reduce to the geodesic

equation

0 2

2

ds

dxds

dxKds

dxds

dxds

xd kji

jk

kji

jk

i

. (4.1)

Using the relation (2.11) we write this equation as

0)(

2

2

ds

dxds

dxQQds

dxds

dxds

xd kji

jkijk

kji

jk

i

,

02

2

2

ds

dxds

dxQds

dxds

dxds

xd kji

jk

kji

jk

i

. (4.2)

4.1. Lie Derivative

(a) Lie Derivative of a Covariant Vector

Let iA be an arbitrary vector. In Einstein-Cartan space-time the Lie derivative

of the covariant vector iA with respect to a vector i is defined as

kik

ikii AAA / / £

. (4.3)

This on using equations (2.14) and (2.15) we get

kliklkil

kik

kkii AKKAAA

)( £ ; ; ,

kliklii AQAA

2££ , (4.4)

where

kik

iki

kik

ikii AAAAA , , ; ; £

, (4.5)

is the Lie derivative of iA in Einstein space-time.

14

Similarly, the Lie derivative of a contra-variant vector in Einstein-Cartan space-

time is given by

klikl

ilk

ii AKKAA

)(££ , (4.6)

klikl

ii AQAA

2££ , (4.7)

where

ik

kkik

ik

kkik

i AAAAA , , ; ; £

, (4.8)

is the Lie derivative of iA in Einstein space-time.

(b) Lie Derivative of the Metric Tensor

The definition of Lie derivative can be extended to any tensor of any rank. For

the metric tensor ijg we obtain the expression as

kjikijkijij KKgg

)(2££ , (4.9)

i.e. kjkiikjijij QQgg

)(2££ . (4.10)

(c) Lie Derivative of a Mixed Tensor of Rank Two

We record the expression for the Lie derivative of an arbitrary mixed tensor of

rank two as

)()(££ lkj

ljk

kil

ikl

ilk

klj

ij

ij KKTKKTTT

, (4.11)

where

kj

ik

ik

kj

kikj

ij TTTT ; ; ;£

, (4.12)

is the expression for the Lie derivative of ijT in Einstein space-time. On using

equation (2.13) we write equation (4.11) as

kil

ljk

klj

ilk

ij

ij TQTQTT

22££ . (4.13)

From above equation it is clear that the Lie derivative of any tensor in Einstein space-

time can be expressed covariantly on any manifold, which do not depend on

Christoffel symbols, either in terms of ordinary partial derivative or covariant

derivative. However, it is evident from equations (4.4), (4.7), (4.10) and (4.13) that it

is not true in Einstein-Cartan space-time.

15

5. FIELD EQUATIONS IN EINSTEIN-CARTAN THEORY

The Einstein-Cartan theory of gravitation describes the gravitational field

through the curvature and torsion. The set of field equations are given by Hehl et al.

[18, 19] in the form

ijijij tkgRR 21

, (5.1)

and

kij

lil

kj

ljl

ki

kij SkQQQ , (5.2)

where 4

8c

Gk and k

ijS is the spin angular momentum tensor. These equations tell

how geometry of space-time is influenced by matter and spin. Obviously, in the

presence of spin of the gravitating matter the geometry is non-Riemannian. The first

set of field equations reads the same as in Einstein theory of gravitation, but neither

the Ricci tensor nor the energy-momentum tensor ijt is symmetric. Both the sides of

this equation are not divergence free, in contrast to Einstein theory of gravitation. The

second set of field equations connects the torsion tensor with spin of matter. In the

first set of field equations ijt is the asymmetric canonical energy momentum tensor

defined by

))(2( kijjkiijklklk

ijij SSSQTt , (5.3)

where ijT is the stress-energy momentum tensor of matter, and

lklkk Q 2ˆ . (5.4)

The field equation (5.2) is an algebraic in character relating to spin angular

momentum tensor. Therefore, one can obtain the torsion tensor in terms of spin

angular momentum tensor as

)21

21( l

ilk

jl

ljk

ik

ijk

ij SSSkQ . (5.5)

5.1. The Spin Tensor:

For the classical description of the spin tensor, Hehl et al. [19] have

decomposed the spin angular momentum tensor as

kij

kij uSS , (5.6)

16

where iu is the time-like 4-velocity vector; and ijS is the spin tensor antisymmetric in

character.

i.e. jiij SS . (5.7)

This spin tensor is orthogonal to the 4-velocity vector.

i.e. 0jijuS . (5.8)

The condition (5.8) is usually known as Frenkel condition. With the help of this

condition the field equation (5.2) or (5.5) gives an algebraic coupling between the

spin tensor and torsion tensor as

kij

kij uSkQ . (5.9)

Thus the torsion contribution to Einstein-Cartan field equation is entirely described by

the spin tensor. Contracting the index kj with in the equation (5.9) we obtain

0iQ , (5.10)

where kiki QQ 2 .

The square of the spin scalar is defined as

0212 ij

ijSSS . (5.11)

Hence, for the time-like unit vector field iu , we obtain from equation (2.14) that

)( ;/ ljiiljjill

jiji QQQuuu . (5.12)

On using (5.8) and (5.9) we readily obtain

ijjiji kSuu ; / . (5.13)

Multiplying equation (5.12) by ju we get

jlljiiljjil

jji

jji uuQQQuuuu )(;/ ,

we denote

ij

ji uuu / and i

jji uuu ; .

Therefore,

jllijii uuQuu 2 . (5.14)

This proves that the acceleration vector has invariant status in both Einstein and

Einstein-Cartan theory of gravitation.

Using equation (5.8) and (5.9) we get

17

ii uu . (5.15a)

This provides that the acceleration vector has invariant status in both Einstein and

Einstein-Cartan theories of gravitation.

It is also evident from equations (5.13) and (5.8).

Now we are in a position to use the consequences of the Einstein-Cartan field

equations (5.8) and (5.9) and summarize all above results viz. (3.4), (3.10), (3.12),

(4.2), (4.4), (4.7), (4.10) and (4.13) in the following:

ii

ii AA ; / , (5.15b)

ll

ijii AukSAA 2)curl(curl ,

for ii uA we have

ijii kSuu 2)curl(curl , (5.15c)

)( 22 , (5.15d)

0 2

2

ds

dxds

dxds

xd kji

jk

i

, (5.15e)

dsdxu

ii if is the unit tangent vector to the geodesic.

For covariant vector iA ,

kl

likii AukSAA

2££ . (5.15f)

If ii u , then

iuiuAA ££ . (5.15g)

In particular

0££ iuiuuu . (5.15h)

For contravariant vector iA ,

klikl

ii AukSAA

2££ . (5.15i)

If for ii u , this becomes

i

u

i

uAA ££ . (5.15j)

In particular if ii uA ,

18

0££ i

u

i

uuu , (5.15k)

kkjiijij Sukgg

)( 4££ . (5.15l)

However, for ii u , we have

ijuijugg ££ . (5.15m)

Finally,

kil

ljk

klj

ilk

ij

ij TukSTukSTT

22££ . (5.15n)

This for ii u gives

iju

iju

TT ££ . (5.15o)

We observe that the Lie derivative with respect to flow vector iu can be expressed on

any manifold which do not depend on any specific connection. It can be expressed

either in terms of partial derivative or covariant derivative with respect to the

Christoffel symbol or with respect to connection.

Now we will use the Einstein-Cartan field equations (5.9) and (5.8) to simplify

the Riemann curvature tensor and the Ricci tensor in terms of spin tensor. Using

equations (2.11) and (2.13) in the equation (2.37) and simplifying with the help of

(5.9) and (5.8) we obtain

hi

kjkji

hji

khhki

ji

hkji

kjh uSuSuSSuuSk ]/[ ]/[

]

/[]/[ ][ [2

ljkil

hhki

ji

hkjikjh SuukSSkSSkSSu ][

][

] /[

]][

kl

jhi

l uSukS . (5.16)

(For detail see Appendix I)

Contracting ik with in (5.16) and after simplifying we get

ihijji

ihj

iihhi

ij

ihijjh uSuSuSSuuSk / ]/[

]

/[]/[ ][ )(

21[2

]2 hj

ihij uukSSkS . (5.17)

Hence from equations (2.35), (5.16) we simplify the Ricci theorem for the time-like

vector iu and hence obtain

ikj

ihkjj

ikh

hikjh

hikj

ijk kSuSuSkuRuuu ]/[ ][]

/[

/ / 2)(2ˆ

kji

kl

ji

l SkuuSkS 22 ][ . (5.18)

19

5.2. The Decomposition of Torsion Tensor

A torsion tensor ijkQ can be decomposed in several ways. For a systematic

study of the torsion tensor, its decomposition has been given by Adamowicz and

Trautman [1], Hehl et al. [18] and Tsamparlis [28]. The decomposition of the torsion

tensor into three irreducible tensors is given by Hehl et al. [20, 21] in the form

ijkV

ijkA

ijkT

ijk QQQQ . (5.19)

The torsion tensor is skew-symmetric in the first two indices in the 4-dimensional

space-time of Einstein-Cartan theory, the torsion tensor has 24 independent

components, of which ijkT Q has 16 components, ijk

AQ has 4 components and ijkV Q has

the remaining 4 components, and are called T -torsion, A -torsion and V -torsion

respectively. These three irreducible parts of the torsion tensor are defined as

)(31

][ kjiijkV gQQ , (5.20)

][ijkijkA QQ

)(!3

1kjiikjjikkijjkiijk QQQQQQ .

Because of the skew-symmetric property of the torsion tensor this reduces to

)(31

kijjkiijkijkA QQQQ . (5.21)

This totally antysymmetric torsion tensor is also called as the axial torsion tensor.

Finally,

ijkV

ijkA

ijkijkT QQQQ .

Using the definitions (5.20) and (5.21), this reduces to

)22(31

][][ kjiijkijkijkT gQQQQ . (5.22)

The trace of these tensors are defined as

ijkVjkk

ikV QgQ ,

ik

ikV QQ

21 , (5.23)

0 kik

AQ , (5.24)

ijkTjkk

ikT QgQ ,

20

)]3(212[

31

ijk

kijk

ik QgQQ

)232[

31

ik

ikk

ik QQQ

)21[

ik

ik QQ ,

0 kik

T Q . (5.25)

6. KINEMATICAL PARAMETERS

6.1 Einstein Theory of General Relativity (Time-like Congruence)

Let )(sxx ii be the world line of a particle in a 4-dimensional space-time of

general relativity. The tangent vector field to the world-line at any point is given by

dsdxu

ii . (6.1)

This 4-velocity vector field satisfies

1 iiuu . (6.2)

The gradient of the velocity vector field is decomposed in the following way

((Greenberg [13], Ellis [8])

ji

ijijijji uuhu ˆ31ˆˆ ; , (6.3)

where the semi-colon ( ; ) indicates the covariant differentiation with respect to the

Christoffel symbol , and

jji

i uuu ; , (6.4)

is the acceleration vector field. The anti-symmetric space-like part of the gradient

tensor is called the rotation tensor of the flow of the fluid and is defined as

)(21)(

21ˆ ; ;

jijiijjiij uuuuuu . (6.5)

This can be written as

] ; [ ˆ lklj

kiij uhh , (6.6)

where

21

jiijij uugh , (6.7)

is the 3-dimensional projection operator. This tensor satisfies the following properties

3 , 0 ijij

ijij

jij hhghuh ,

ki

ki

ki

kj

ji uuhhh . (6.8)

The space-like symmetric traceless part of the gradient tensor is called the shear

tensor and is defined as

ijjijiijjiij huuuuuu ˆ31)(

21)(

21ˆ ; ; , (6.9)

where

iiu ; ˆ , (6.10)

is the volume expansion (or contraction) scalar.

We also write equation (6.9) into the compact form as

ijlklj

kiij huhh ˆ

31ˆ ) ; ( . (6.11)

These kinematical quantities satisfy the following conditions

0ˆ , 0ˆ jij

jij uu ,

ijij

ijij ˆˆ

21ˆ , 0ˆˆ

21ˆ 22 ,

0ˆˆ , 0ˆˆ ijij

ijij

ijij

ijij hhgg ,

ijk

jikijk

jik hh ˆˆ , ˆˆ . (6.12)

It is useful to introduced the pseudo rotation vector as

kljijkli u ˆ

21ˆ , (6.13)

where ijkl is the completely skew-symmetric permutation tensor.

6.2. Propagation Equations of the Kinematical Parameters:

From Einstein’s field equations of gravitation, we have

ijijij TkgRR ˆ21ˆ , (6.14)

where the symmetric energy-momentum tensor ijT satisfies the conservations laws

0; ijjT . (6.15)

22

For the time-like unit vector field, the Ricci identities give

lkjiljkikji uRuu ˆ

; ; ; ; . (6.16)

Multiplying the equation (6.16) by ku we get

klkjil

kjki

kkji uuRuuuu ˆ

; ; ; ; ,

klkjil

kjkiji uuRuuu ˆ)( ; ; ; . (6.17)

However, we know

kkii uuu ;

,

kjki

kjkiji uuuuu ; ; ; ; ; ,

kjkiji

kjki uuuuu ; ; ; ; ; .

Substituting this in the equation (6.17) we get

klkjil

kjkijiji uuRuuuu ˆ)( ; ; ; ; .

On using equation (6.3) we write this as

kj

kjkiikikikjiji uuhuu ; ; ˆˆ)( ˆ

31ˆˆ()(

klkjilj

kkj uuRuuh ˆ) ˆ

31

. (6.18)

Contracting the equation (6.18) with ijg we get

kj

kj

kjk

jjk

jk

jk

ii huuhu ; ˆ

31ˆˆ)( ˆ

31ˆˆ(ˆ

klklj

k uuRuu ˆ) .

This on simplifying gives

ˆˆ31)ˆˆ(2ˆ 222

; kl

kli

i uuRu . (6.19)

This is the well known Raychaudhuri equation.

Now to find the propagation equation for shear tensor, we differentiate equation

(6.9) covariantly in the direction of the flow vector ku to get

kkij

kkjijiijji

kkij uhuuuuuuuu ; ; ; ; ; ) ˆ(

31)(

21ˆ ,

]2)()[(21ˆ ; ;

jijijiijjiij uuuuuuuu

23

)( ˆ

32ˆ

31

jiij uuh .

Using the expression (6.18), we obtain from the above equation

)( ; ; 22)()[(21ˆ jijiijjiij uuuuuu

) ˆ31ˆˆ)( ˆ

31ˆˆ( j

kkj

kj

kjkiikikik uuhuuh

) ˆ31ˆˆ)( ˆ

31ˆˆ( i

kki

ki

kikjjkjkjk uuhuuh

)( ˆ

32ˆ

31])ˆˆ( jiij

klkijlkjil uuhuuRR ,

ijk

jikijk

jikjijijiij huuuuu ˆ31ˆˆˆˆ

32ˆˆ[ˆ )() ; (

kllijkji

kkji

kkjiij uuRuuuuuuh ˆ] ˆ

31ˆˆ)ˆ(

91

)()()(2 .

With the help of (6.19), we get from the above equation

kjikij

kjikjijijiij uuuuu )() ; ( ˆˆˆˆ

32ˆˆ{ˆ

} ˆ31ˆˆ ])ˆˆ(2[

31

)()()( ; 22

jik

kjik

kjiiji

i uuuuuuhu

klkl

kllijk uuRuuR ˆ

31ˆ . (6.20)

This is the required propagation equation for shear tensor. Similarly, the propagation

equation for the rotation tensor is obtained in the form

kkji

kjik

kjikijjijiij uuuuu

][ ][] ; [ ˆˆˆˆˆˆˆ32[ˆ

] ˆ31ˆ ][][ ji

kkji uuuu . (6.21)

(For detail see appendix 2)

These propagation equations will be used to find the propagation equation of

kinematical parameters in the Einstein-Cartan space-time in the latter chapter.

6.3. Space-like Congruences in Einstein Theory

A space-like vector to a space-like congruence at a point P is defined by

24

dsdxh

ii . (6.22)

This vector satisfies the normalizing condition 1iihh , The connecting vector

ix of two particles of neighboring curves of the space-like congruence satisfies

0£h

ix . (6.23)

We now introduce an observer at point P to observe the given space-like curves with

4-velocity iw such that iw is orthogonal to ih at point P ,

0 , 1 ii

ii hwww .

The direct calculations give

jij

jij

jij xBxAxP

ˆˆ)(

, (6.24)

where

0 ,0 with , jij

jijjijiijij hPwPhhwwgP ,

and

lkl

jkiij hPPA ; , (6.25)

jkk

ikij whwPB )~(ˆ , (6.26)

We have introduced the notations jij

ijij

i whhhww ;

; ~ and . In (6.24) the

presence of the second term ijB is crucial (Greenberg [14]). Except at the given

point P , the motion of the observer employed along the curve has still to be

specified. The 4-velocity iw of the observer used along are related by

kik

kik hPwP ~

. (6.27)

The equation (6.27) represents the necessary and sufficient condition for 0ˆ ijB .

Hence, the equation (6.27) specified the laws for iw as follows

kk

ik

iik

kk

ik

iik hhhwwwhhww ~)()( ,

ijj

ij

jii hwhwwhhw ~~ , (6.28)

which is the Greenberg’s natural transport law for iw in Einstein theory. With 0ˆ ijB ,

the equation (6.24) reduces to

jij

jij xAxP

ˆ)( . (6.29)

25

The projection tensor ijA decompose into its irreducible parts in the following way

ijijijij PA ˆˆ21ˆˆ , (6.30)

where the expansion is given by

jiji

iiji

ijii wwhhhPA ; ; ; .

ˆˆ . (6.31)

The shear tensor ij and rotation tensor ij are defined as follows

ijlkl

jkiijk

kijij PhPPPAA ˆ

21.ˆ

21ˆˆ

) ; ( )( .

)()( ; )()();(

~ˆji

kk

kjikjijijiij hwwhwwhwhhhh

ijjik

k Pwwwh ˆ21~ , (6.32)

] ;[ )(ˆˆ

lklj

kiijij hPPA ,

][][ ;][][];[~ˆ

jil

ll

ijljijijiij hwwhwwhhhwhh , (6.33)

where ij , and ij are called, respectively, the rotation tensor, the expansion and

the shear tensor of the curves of the congruence, as measured by the observer iw .

These kinematical quantities satisfy the following conditions

ijij ˆˆ

21ˆ 2 , ij

ij ˆˆ21ˆ 2 ,

0ˆ , 0ˆ jij

jij hw ,

0ˆ , 0ˆ jij

jij hw . (6.34)

Finally, (6.26) and (6.30) imply the following useful identity:

i

kjkjijiijijijji wwhhhwhPh ; ;

~ˆˆ21ˆ

jik

kjik

k hwwhwwwh ~ . (6.35)

26

APPENDIX I

The Riemann curvature tensor in EC theory and Einstein theory 0f gravitation

can be related as

ikjh

ikjh

ikjh RR ˆ , (AI.1)

where

ikl

ljh

lkh

ijl

ilh

lkj

ihjk

ikjh KKKKKKK

][

][ 22

ikl

ljh

lkh

ijl

ilh

lkj

ikhj

ijhk KKKKKKKK

][

2 .

Using equation (2.11), the above equation gives

)()(

ikh

ikh

ikhj

ijh

ijh

ijhk

ikjh QQQQQQ

lkh

ijl

ijl

ijl

ilh

ilh

ilh

lkj QQQQQQQQ

)(()(2

))(()

ihl

ikl

ikl

ljh

ljh

ljh

lkh

lkh QQQQQQQQ ,

)()([

hikk

ih

ikhjh

ijj

ih

ijhk

ikjh uSuSuSuSuSuSk

ji

li

jlhill

ih

ilh

lkj uSuSkuSuSuSukS

()(2

jl

hl

jhhlkk

lh

lkhl

ij uSuSkuSuSuSuS

())(

)])(

likk

il

iklh

lj uSuSuSuS

hijk

ihjkj

ihkjh

ik

ijhk uSSuuSSuuSk

[

ihkjk

ihjkh

ij

ikhj

ijhk SuuSSuuSSu

hlk

ijlk

lh

ijl

ihkj

ikhjh

ikj uSukSuSukSSkSSuuS

2

ikjhkh

ijh

lkj

ilk

lhj

il SkSSkSuSukSuSukS

]

k

ilh

lj

iklh

ljk

ilj

lh

iklj

lh uSukSuSukSuSukSuSukS

hijk

ihjkj

ihkjh

ik

ijhk uSSuuSSuuSk

[

ihkjk

ihjkh

ij

ikhj

ijhk SuuSSuuSSu

ikjhkh

ij

ihkj

ikhjh

ikj SkSSkSSkSSuuS

2

ki

lhl

jhlkj

il

iklj

lhk

lh

ijl uSukSuSukSuSukSuSukS

,

hi

kjkji

hji

khhki

ji

hkji

kjh uSuSuSSuuSk ]/[ ]/[

]/[ ]/[ ][ [2

27

ljkil

hhki

ji

hkjikjh SuukSSkSSkSSu ][

][

] /[

]][

kl

jhi

l uSukS . (AI.2)

APPENDIX I1

The propagation equation for rotation tensor is derived by differentiating the

expression of rotation tensor (6.6) covariantly in the direction of the flow vector ku

kkiijiijji

kkij uuuuuuuu ; ; ; ; )(

21ˆ ,

]2)()[(21ˆ ][ ; ; jiijjiij uuuu . (AII.1)

With the help of the expression

klkjil

kjkijiji uuRuuuu ˆ)( ; ; ; ; ,

the equation (AII.1) becomes

]ˆˆ2[21ˆ ; ; ; ; ][ ; ;

klkijl

klkjil

kikj

kjkijiijjiij uuRuuRuuuuuuuu ,

]2[21ˆ ; ; ; ; ][ ; ;

kikj

kjkijiijjiij uuuuuuuu ,

kikj

kjkijijiij uuuuuuu ; ; ; ; ][] ; [ 2

121ˆ . (AII.2)

Using the decomposition of iu

jiijijijji uuhu ˆ31ˆˆ ; ,

the equation (AII.2) gives

kj

kjkiikikikjijiij uuhuuu ][] ; [ ˆˆ)( ˆ

31ˆˆ(

21ˆ

ki

kikjjkjkjkj

kkj uuhuuh ˆˆ)( ˆ

31ˆˆ(

21) ˆ

31

) ˆ31

ikk

i uuh . (AII.3)

Simplification of (AII.3) gives

jk

ikijk

jikk

jikjijiij uuuuu ˆˆ ˆ31ˆˆˆˆ(

21ˆ ][] ; [

28

ijijjk

ikijk

jikk

jik uu ˆ ˆ31ˆ ˆ

31ˆˆ ˆ

31ˆˆˆˆ

jik

ijkk

ijkjiij uuh ˆ ˆ31ˆˆˆˆ(

21) ˆ

31 ˆ

91

2

jiik

jkjik

ijkk

ijkik

jk uuuu ˆ ˆ31ˆˆ ˆ

31ˆˆˆˆˆ

) ˆ31 ˆ

91ˆ ˆ

31 2

ijijji uuh ,

kkji

kjik

jkikijjijiij uuuuu

][ ][] ; [ ˆˆˆˆˆˆˆ32{ˆ

} ˆ31

][][ jik

kji uuuu . (AII.4)

29

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30

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