Fibre Bundles, Connections, General Relativity, And Einstein-Cartan Theory 1110.1018v1

7/27/2019 Fibre Bundles, Connections, General Relativity, And Einstein-Cartan Theory 1110.1018v1

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a r X i v : 1 1 1 0 . 1 0 1 8 v 1 [ g r - q c ] 5 O c t 2 0 1 1

FIBRE BUNDLES, CONNECTIONS, GENERAL RELATIVITY,

AND EINSTEIN-CARTAN THEORY

M. Socolovsky

Instituto de Ciencias Nucleares, Universidad Nacional Aut onoma de MexicoCircuito Exterior, Ciudad Universitaria, 04510, Mexico D. F., Mexico

Introduction

The main purpose of this article is to present in the most natural way, that is, in the context of thetheory of vector and principal bundles and connections in them, fundamental geometrical concepts related togeneral relativity (GR) and one of its extensions, the Einstein-Cartan theory (EC). Central concepts are thecurvature tensor R , the torsion tensor T and the non-metricity tensor Q = D g as propertiesof connections in a Riemannian or pseudo-Riemannian manifold, with metric g and affine connection .(D is the covariant derivative with respect to .) GR has to do with a metric symmetric connection,the Levi-Civita connection, that only allows for R ; EC theory involves a metric but not necessarilysymmetric connection, that allows also for T ; while the theory of Weylian manifolds involves a nonnecessarily metric ( Q = 0) and non necessarily symmetric ( T = 0) connection. (In units of length [ L],[R ] = [L]2 , [T ] = [Q ] = [ ] = [L]1 , while [g ] = [L]0 .)

One of the most beautiful equations of Physics is the equality to zero of the Einstein tensor, that is, theEinsteins equations in vacuum :

G = 0 ,

whereG = R

12

g R,

with R = g R = g g R = g R . G = 0 is equivalent to Ricci atness:

R = 0 .

This however does not imply vanishing curvature; so, in GR, empty space-time can be curved . Instead, inEC theory, torsion must be zero in vacuum.

It is remarkable that G appears naturally when the Bianchi equations for the Levi-Civita connectionare expressed in terms of the Ricci tensor R and the scalar curvature R. Then, G is a purely geometricobject.

The use of the tetrads ( ec) formalism along with their duals, the coframes or anholonomic coordinates(ea ), allows us to discover how GR and EC theory have an internal or gauge symmetry (Utiyama, 1956),implemented by a connection that takes values in the Lie algebra of the Lorentz group L4 : the spin connectionab . The variations of the Einstein-Hilbert action of pure gravity or gravity coupled to Dirac matter withrespect to ab and ec lead, respectively, to the Cartan and Einstein equations, the former involving torsionand the spin of matter, and the latter involving curvature and the energy-momentum of matter.

Later, through a shift of the ecs one nds the translation gauge potential B a ; together, B a and bc ,dene a Poincare connection, extending the symmetry group of GR and EC theory to the semidirect product

P 4 D, where P 4 = T 4 L4 is the Poincare group, with T 4 the translation group, and Dthe group of generalcoordinate transformations.1
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Finally, in the last section, we discuss the problem of dening a gauge invariant eld strength for theMaxwell eld coupled to gravity, and the subsisting problem of the U (1)-gauge dependence of torsion in thesolution of the Cartan equation.

CONTENT

1. Connections in smooth real vector bundles

2. Linear connection in a differentiable manifold M

3. Total covariant derivative of a section in

4. Local expressions for the connection (covariant derivative) and the total covariant derivative

5. Local expressions for the covariant derivatives of vector elds and 1-differential forms in a manifold

6. Example: Trivial connection in R n

7. Transformation of the connection coefficients in a manifold (local concept). Provisional denition of tensors in a manifold. Tensors in arbitrary vector bundles.

8. Directional covariant derivative and parallel transport of tensors; geodesics.

9. Curvature and torsion of a connection in

10. Geometric interpretation of curvature and torsion

11. Exterior covariant derivative and curvature 2-form

12. Bianchi equation and Bianchi identities

13. The Levi-Civita connection

14. Physics 1: Equivalence principle in GR

15. Covariant components of the curvature tensor

16. Ricci tensor for the Levi-Civita connection

17. Physics 2: (Local) Einstein equations in empty space-time

18. Ricci (or curvature) scalar (with Levi-Civita connection)

19. Einstein tensor

20. Physics 2 : (Local) Einstein equations in empty space-time

21. Examples in m = D = 1 , 2, 3. Generalization to m 4 and Weyl tensor22. For the Levi-Civita connection, G ; = 0

23. Physics 3: (Local) Einstein equations in the presence of matter

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24. Tensor bundles as associated bundles to the bundle of frames of M n , F M n25. Vertical bundle of a principal bre bundle

26. Soldering form on F M n27. Linear connection in a manifold M n on F M n28. Tetrads and spin connection

29. Curvature and torsion in terms of spin connection and tetrads. Cartan structure equations;Bianchi identities

30. Spin connection in non-coordinate basis

31. Locally inertial coordinates

32. Einstein-Cartan equations

33. Lorentz gauge invariance of Einstein and Einstein-Cartan theories

34. Poincare gauge invariance of Einstein and Einstein-Cartan theories

35. Torsion and gauge invariance

Acknowledgements

References

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

1. Connections in smooth real vector bundles

Let : R m

E

M n be a smooth m dimensional real vector bundle over M n

M , a differentiable

manifold of dimension n. Let ( T M ) denote the sections of the tangent bundle of M and ( E ) denote thesections of E . E is an m + n dimensional differentiable manifold; this can be easily shown from the localtriviality condition.

A connection in is a function

: (T M ) (E ) (E ),(X, s ) (X, s ) X s

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which has the following properties:

i) X + X s = X s + X s

ii) fX s = f X s, where f C (M, R ): smooth real valued functions on M

iii) X (s + s) = X s + X s iv) X (f s ) = X (f )s + f X s (Leibnitz rule)

(T M ) and ( E ) are innite dimensional vector spaces over R , but modules over C (M, R ) as a ring,with dimensions n and m respectively. The Leibnitz rule shows that is not C (M, R )-linear in the secondentry. As will be shown below, this will be reected in the fact that under a change of local coordinates, theset of connection coefficients (Christoffel symbols) is not a tensor.

The value of the connection at ( X, s ) is called the covariant (or invariant) derivative of s in the direction of X . X s : M E , x X s(x) = ( x, ( X s)x ), with ( X s)x E x : the bre in E over x; E x is a real mdimensional vector space.

Notice that we can dene the operator

X : (E ) (E ), X (s) = X sX Lin R (( E )) and obeys the Leibnitz rule.

One summarizes these concepts in the following diagram:

R m

|E s X s

R n T M M M X

E = x M {x} E x x M E x .Note : M is a differentiable manifold; as such is a topological space. This global structure is dened in

M prior to any connection on .

2. Linear connection in a differentiable manifold M

A linear connection on M is a connection in its tangent bundle. With E = T M we have:

: (T M ) (T M ) (T M ),(X, Y ) (X, Y ) X Y

with

i) X + X (Y ) = X Y + X Y

ii) fX (Y ) = f X Y

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iii) X (Y + Y ) = X Y + X Y

iv) X (fY ) = X (f )Y + f X Y

We shall denote by Conn ( ) the set of connections in the vector (or principal, where appropiate) bundle

.

Again, X Lin R (( T M )) and obeys the Leibnitz rule.

3. Total covariant derivative of a section in

Let ( T M E ) be the set of differential 1-forms in M with values in E . T M E is a vector bundleon M :

R n m T M E M, with T M E =x M

T x M E x .

The section s (T M E ) is dened by

s : (T M ) (E ), X s(X ) := X s.As for any differential form on M , s is C (M, R )-linear i.e. s(f X ) = f s(X ); however,

(f s ) = sdf + f s.

In fact, (f s )(X ) = X (f s ) = X (f )s + f X s = f s(X ) + sX (f ) = f s(X ) + sdf (X ) = ( f s + sdf )(X ).

s is called the total covariant derivative of the section s. In detail,

s : M T M E , x ( s)(x) = ( x, ( s)x ), ( s)x = x vx : T x M E x , X x ( s)x (X x ) = x (X x )vx = x vx where x T x M , vx E x and x R .For a linear connection on M ,

Y : (T M ) (T M ), X ( Y )(X ) = X Y.

4. Local expressions for X s and s

Let (U , i ) J, i {1,...,m } be a basis of local sections of E i.e. i : U E = 1 (U ), x i (x) = ( x, ix ) {x} E x , with i = Id U , and such that if s (E ), then s = mi=1 s i i withs i C (U , R ). Let x be a local coordinate basis of ( T U ) i.e. if X (T U ) then X = X with X C (U , R ). (If the domains of the local trivializations of the bundle E do not coincide with thedomains of the atlas U of the manifold M , one can always consider their intersections.) Then, locally,

X s = X (s i i ) = X (s i i ) = X (( s i )i + s i i );

and since i (E ) then i :=

ji j

for a unique set of n m2 functions ji : U R , called the Christoffel symbols of the connection in the

atlas U . For a linear vector bundle, m = 1, and then there are only n symbols: ji .

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We then write

X s = X (( s i )i + s i ji j ) = X ( s j + s i ji )j = X

( ji + ji )s

i j = X D ji si j

where we have dened the local covariant derivative operator

Dj

i := j

i + j

i .

(Units: [ ji ] = [x ]1 ; in natural units [ ji ] = [mass ] if [x ] = [length ].)

We can also write

X s = X s j; j with D sj s j; = D ji s i = s j, + ji s i and s j, = s j .

Notice that the ordinary derivative term s j , is due to the Leibnitz rule.

Locally, we can writes = dx s.

In fact, s(X ) = ( dx s)X = dx (X ) s = dx (X ) s = X dx ( ) s = X s =X s = X s. Then

s = dx ( ji +ji )s

i j = dx ( s j j + ji si j ) = dx s j j + dx ji s

i j = ds j j + ji si j

where ji is an m m matrix of 1-forms on U given byji = dx

ji .

We can write ji s i j = ji s i j = ( s) j j and then

s = ds j j + ( s)j j = ( ds j + ( s) j ) j = (( d + ) s) j j

i.e.s = ( d + ) s (T U E ).

Then, locally,= d + .

Let s i; (x)=0 i.e.s i, (x) +

ij (x)s

j (x) = 0 .

Multiplying by dx |x we obtains i, (x)dx

|x = ij (x)s j (x)dx |x T x U.The r.h.s.

ij (x)s j (x)dx |x ( ||s)i |x ,is called the innitesimal parallel transport (transfer) (Schroedinger, 1950) of the section s i by the connection

along the 1-form dx i |x (see section 8), and we see that for a covariantly constant section, it coincides withthe differential of s i at x, ds i |x . The transfer of the section, proportional to dx |x and to the section itself, just follows the values of s i along dx (when s i; = 0). When s i ; = 0, ( ||s)i |x still is the parallel transfer of s i through dx , but it fails to follow the value of the section. (A more detailed discussion can be found inCheng, 2010.)

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5. Local expression for X Y

When E = T M , i = and =

for a unique set of n3 smooth functions

: U

R . (

is not necessarily symmetric or antisymmetricin and .) Then,

X Y = X D Y , D =

+

,

D Y = Y + Y

, X Y = ( X Y ) ,

( X Y ) = X ( + )Y

.

The quantityD Y

= Y + Y Y ; D Y

is the covariant derivative of the local vector eld Y in the direction of x . Then,

( X Y ) = X D Y .

If V is a local vector eld and A is a local differential 1-form in M n , then = V A is a scalar(0-rank tensor) i.e. = or V A = V A under x x . The covariant derivative of a scalar isnaturally dened as

; = ,

and the Leibnitz rule is assumed for the covariant derivative of the product of arbitrary tensors T and S :

(T S ) ; = T ; S + T S ;

. Then,; = ( V A ) , = ( V )A + V A = V ; A + V

A;

and soV A; = V , A + V

A, V ; A = V , A + V A, V , A V A = V A, V A = V A; where

A; = A, A D Ais the covariant derivative of the local 1-form A in the direction of x .

6. Example: the trivial connection in R n

Consider M = R n ; then with X, Y (T M ) we dene

0X Y := X (Y

) = X

(Y

) .Additivity in both X and Y , and C (M, R )-linearity in X are trivial; nally, 0X (fY ) = X (fY ) =X (f )Y + fX (Y ) = X (f )Y + f 0X Y .

In particular,0 ( ) =

0 =

0 (

) = (

) = 0

and therefore0 = 0 .

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for x = x(x) in the overlap U U .Remark : The concept of covariant and contravariant indices has sense only if there exists a metric in

the manifold.

An r -contravariant and s-covariant tensor can be considered a C (M, R )-multilinear map from thetensor product of s factors of ( T M ) and r factors of ( T M ) with values in C (M, R ). On a chart U ,

T = T 1 ... r 1 ... s dx 1 . . . dx

s

x 1

. . .

x r: ( s (( T U )) ( r (T U )) C (U , R ),

V 11

x 1. . . V ss

x s

A1 1 dx1 . . . Ar r dx r T 1 ... r 1 ... s dx 1 (V

11

x 1

) . . . dx s (V s

s

x s)

x 1 (A1 1 dx 1 ) . . . x r (Ar r dx r ) = T 1 ... r 1 ... s V 1

1 11 . . . V s

s ss A1 1 1 1 . . . Ar r

r r

= T 1 ... r 1 ... s V 1

1 . . . V ss A11 . . . Ar r .

Well call rs (M ) to the C (M, R )-module of r -contravariant and s-covariant tensors on M n . Forexample, 10 (M ) = ( T M ): vector elds on M ; 01 (M ) = ( T M ): differential 1-forms on M . In general, rs (M ) = ( T rs M ): sections of the bundle of r (s) contravariant (covariant) tensors on M , with T 10 M = T M and T 01 M = T M .

Locally, given a tensor T 1 ... r 1 ... s and a connection in the manifold M n , the covariant derivative of

T 1 ... r 1 ... s in the direction of

x is given by

D T 1 ... r 1 ... s T 1 ... r 1 ... s ; =

x T 1 ... r 1 ... s +

1 1 T

1 2 ... r 1 ... s + . . . +

r r T

1 ... r 1 r 1 ... s 1 1 T 1 ... r 1 2 ... s . . .

s s T 1 ... r 1 ... s 1 s .It can be veried that T 1 ... r 1 ... s ; is an r -contravariant and s + 1-covariant tensor.

Remark : Notice that while the operators X send tensors (or sections in general) of a given order totensors of the same order, for both covariant and contravariant indices, the operators D map ( r, s )-tensorsinto ( r, s + 1)-tensors.

Tensors in arbitrary vector bundles

In U U M n consider the change of local coordinates and sections: x = x x

x and k = f 1jk j

with x = x , x = x , k = k , and j = j ; , = 1 , . . . , n ; j, k = 1 , . . . , m ; at each x U U , f and f 1 take values in GL m (R ) with j = f k j k . We study the transformation of the Christoffel symbolsij of a connection in : R m

E

M n :

i = ji j = x i = x

x

x f 1 ji j =

x x

x

f 1ji j =x x

((

x f 1 ji )j + f 1

ji

x j )

= x

x (

x f 1li + f 1

ji lj )l ) = ji f 1

lj l

i.e.

ji f 1lj =

x x

(

x f 1li + f 1

ji

lj );

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multiplying by f r l and using f 1lj f r l = r j we obtain

ri =x x

f 1ji f r llj +

x x

f r l

x (f 1 li ).

The homogeneous part in the connection coefficients gives the general law for the tensorial transformation

of an object with r (v) contravariant or upper (covariant or lower) internal indices, r, v = 1 , . . . , m , and s(t)contravariant (covariant) space-time (external) indices, s, t = 1 , . . . , n :

T 1 ... s a 1 ...a r 1 ... t b1 ...b v =x 1

x 1 x s

x sx 1x 1

x tx t

f a 1 c1 f a r c r f 1d1

b1 f 1dv

bv T 1 ... s c1 ...c r1 ... t d1 ...d v .

For example,

T a =x

x x

x f a

bT b , T ab =

x

x x

x f 1 ca f 1

dbT

cd .

8. Directional covariant derivative and parallel transport of tensors; geodesics

If c : (a, b)

M n ,

c(), with ( a, b) an open interval in R , is a smooth path in M n locally given by

c() = ( x1(), . . . , x n ()), then the covariant derivative of T 1 ... r 1 ... s along c is the tensor dened by

(DT d

) 1 ... r 1 ... s :=dx

dD T 1 ... r 1 ... s .

We have then dened the covariant derivative operator along the path c through

Dd

=dx

dD .

In detail,

(DT d

) 1 ... r 1 ... s =dx

d(

x

T 1 ... r 1 ... s + 1 1 T

1 2 ... r 1 ... s + . . . s s T 1 ... r 1 ... s 1 s )

= dd

T 1 ... r 1 ... s +dxd

1 1 T 1 2 ... r 1 ... s + . . .

dxd

s s T 1 ... r 1 ... s 1 s .

Symbolically,DT d

=d

dT + c T c T

where denotes the contractions.

For a vector eld V ,

(DV d

) =d

dV +

dx

d V

i.e.

(DV d )| = (

dV ()d +

dx ()d ()V ())

x |

where the dependence of is through c, and for a differential 1-form A ,

(DAd

) =dAd

dx

d A

i.e.(

DAd

)| = (dA ()

d dx ()

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If, in particular, V = dx

d : the tangent vector to the curve c at , i.e. V = c, then

(D cd

) =d2x

d2+

dx

d

dx

d.

Symbolically DV d = (d

d + c)V i.e.Dd =

dd + c for vectors, and

DAd = (

dd c)A i.e. Dd = dd c for

1-forms.

A tensor T is said to be parallel transported by the connection from c(0 ) to c(1) along the smooth curve c in M n (a < 0 < 1 < b ), if

DT d

= 0 f or all [0 , 1 ]

i.e. if dT ()

d= c ()T () + c ()T ().

This is a system of n r + s ordinary differential equations of rst order (ODE-1). By general theorems onODE-1, if T 0 rs (c(0 )) then there exists and is unique a parallel transported tensor T () along c, in

particular at c(1), such that T (0) = T 0 .

The parallel transport of T depends on i.e. on and on the path c. There exists a vector space isomorphism

P c : rs (c(0 )) rs (c(1 )) , T 0 P c (T 0 ) = T 1

with ( P c )1 = P c 1 where c1() = c(1 + 0 ).The equations of parallel transport for vector elds and differential 1-forms are

(DV d

) = 0dV

d=

dx

d V

and( DA

d) = 0 dAd

= dx

d A

respectively.

In particular, a curve c is a geodesic in M n with respect to the connection , if its tangent vector c isparallel transported along c:

(D cd

) = 0d2x

d 2=

dx

ddx

d.

Symbolically,D cd

= 0 c = c2 .

In more detail, x () + ()x ()x () = 0 ,

which is a system of n ordinary differential equations of second order (ODE-2) for c() (x ()). Given(x0 , vx 0 ) T M , there always exists a unique solution to this system of equations in an interval ( 0 , 0 + ), > 0 with the initial conditions c(0) = x0 and c(0) = vx 0 .

The geodesic equation is invariant under the change a + b, with a, b R , a = 0. (See also section13.)

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Notice the arbitrariness of vx 0 at x0 , and the fact that the whole geodesic is determined in the interval(0 , 0 + ) (globally) from the initial data.

For the trivial connection in R n (section 6), 0 = 0 and then the solutions of the geodesic equationare straight lines:

x () = 0 x () = a + b .

9. Curvature and torsion of a connection

Let be a connection on : R m E

M n . The curvature of is dened as follows:

R: (T M ) (T M ) (E ) (E ), (X,Y,s ) R(X,Y,s ) := ([ X , Y ] [X,Y ])(s)i.e.

R(X,Y,s ) = X ( Y (s)) Y ( X (s)) [X,Y ](s).Clearly, R(X,Y,s ) = R(Y,X,s ).

Well show that

Ris C (M, R )-linear in its three entries. This will have as a consequence that the set

of local components of Rbehaves as a tensor.i) R(fX,Y,s ) = fX Y s Y fX s [fX,Y ]s = f X Y s Y (f X s) f [X,Y ]Y (f )X s =f X Y s Y (f ) X s f Y X s f [X,Y ]s + Y (f )X s = f X Y s Y (f ) X s f Y X s f [X,Y ]s +Y (f ) X s = f ( X Y Y X [X,Y ])(s) = f R(X,Y,s );ii) R(X, fY,s ) = R(fY,X,s ) = f R(Y,X,s ) = f R(X,Y,s );iii) R(X,Y, fs ) = X Y (f s ) Y X (f s ) [X,Y ](f s ) = X (Y (f )s + f Y s) Y (X (f )s + f X s)[X, Y ](f )sf [X,Y ]s = X (Y (f ))s + Y (f ) X s + X (f ) Y s + f X Y sY (X (f ))sX (f ) Y sY (f ) X sf Y X s X (Y (f ))s + Y (X (f ))s f [X,Y ]s = f ( X Y Y X [X,Y ])(s) = f R(X,Y,s ).Locally (in a common chart for and M ),

R( , , j ) = [ , ](j ) [ , ](j ), but [ , ] = 0 and 0s = 0X s = 0 X s = 0, then

R( , , j ) = [ , ](j ) = ( (j )) ( (j )) = ( ij i ) ( ij i )= ( ij )i + ij li l ( ij )i ij li l = Rkj k with

Rkj = kj kj + ki ij ki ij .Then

R(X,Y,s ) = X Y s j R( , , j ) = X Y s j Rij i .For a linear connection in a manifold,

R = + ,with R(X,Y,Z ) = X Y Z R( , , ) = X Y Z R .

In particular, R( , , Z ) = [ , ](Z ) = Z R or R( , , ) = [ , ]( ) = R .Dening

R: (T M ) (T M ) End C M (( T M )) , R(X, Y ) : (T M ) (T M ), R(X, Y )(Z ) := R(X,Y,Z )12


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one obtains, in particular,R( , ) = [ , ],

which is the usual expression of curvature in terms of a commutator of local covariant derivatives.

If in R we contract with we obtain the antisymmetric tensorS = = S ,

and if we contract with we obtain the tensor

R = + .In general, R is non symmetric, not even when = . In this case, however, its antisymmetric partis the half of S :

R{ } =12

(R R ) =12

S .

But the denition of S does not require a symmetric connection.

In GR, where is the Levi-Civita connection (section 13) uniquely determined by the metric in apseudo-riemannian (lorentzian) manifold, it is usual to denote

R = R with R R : (T M ) (T M ) (T M ) (T M ), (X,Y,Z ) R(X,Y,Z ) = ([ X , Y ] [X,Y ])(Z ) =X Y Z R . This denition holds for any connection, like the Weyl connection (non-metric symmetric),or that corresponding to the Einstein-Cartan theory (metric non-symmetric).

Clearly, R = R . Since R( , , ) = R = R , then< dx , R( , , ) > = < dx , R > = < dx , > R = R = R

where < , > denotes the 1-form-vector contraction, which is independent of the metric.

(R: (T U ) (T U ) (T U ) (T U ), ( , , ) R( , , ).)For a symmetric connection , = (see section 13),

R = + . ( )

The torsion T of a linear connection on a manifold M n is dened as follows:

T : (T M )

(T M )

(T M ), (X, Y )

T (X, Y ) : X Y

Y X

[X, Y ].

It holds:

i) T (X, Y ) = T (Y, X )ii) T 0 = 0 for the trivial connection 0 in R n .

iii) T is C (M, R )-linear in X and Y i.e. T (fX,Y ) = T (X, fY ) = fT (X, Y ).

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iv) Locally, in a chart U (x ), J ,

T (X, Y ) = ( T (X, Y ))

x = (( X Y ) ( Y X ) [X, Y ] )

x

= X 2T Y

x

withT

=

1

2(

) =

T

=

[ ].

If X = i.e. X =

x =

x , and Y = i.e. Y = x =

x , then

T (

x ,

x

) = ( )

x = 2 T

x

i.e. T = 12 (T (

x ,

x )) .

A straightforward calculation leads to:

[D , D ]V = D (D V ) D (D V ) = ( V ; ); (V ; ); = R V 2T V ; .If is a scalar, then [ D , D ]( ) = D ( )

D ( ) = D ( ; )

D ( ; ) = ;

;

; +

; = ( ) i.e. [D , D ]( ) = 2T .So, [D , D ]( ) = 0 if T = 0.

Then, with = V W and using the Leibnitz rule, for a covariant vector (1-form) one obtains

[D , D ]W = R W 2T W ; .The generalization for a tensor T 1 ... r 1 ... s is

[D , D ](T 1 ... r 1 ... s ) = R1 T

2 ... r 1 ... s + + R r T 1 ... r 1 1 ... s R 1 T 1 ... r 2 ... s R s T 1 ... r 1 ... s 1

2T T 1 ... r 1 ... s ; .For a symmetric connection, = and therefore

T = 0

in all charts. I.e. a symmetric connection (like the Levi-Civita connection (section 13)) is torsion free. Fromthe transformation of the s, it is clear that T is a tensor. In particular one has

[D , D ]V = R V .

The modied torsion tensor is dened as

T = T

1n 1

( T T )

where T = T is the torsion vector (in fact, it is a 1-form). T is traceless i.e. T = 0.

10. Geometric interpretation of curvature and torsion

Curvature

14


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Consider the innitesimal parallelogram pqrs in M n with coordinates x , x + , x + + , x + respectively, with | |, | | 1. Let c and c, not necessarily part of geodesics, be curves which join p withr through q and s respectively, and (locally ) be an arbitrary connection in M n . Let V p T pM ; itsvariation from p to q through c is obtained from the formula of the covariant derivative of a vector alonga curve (section 8), which implies DV = d( DV d )

= dV + dx V ; if the transport is parallel thenDV = 0 i.e. dV = dx V . So through c, dV |c = V q |c V p and with dx = one obtains

V q |c = V p ( p)V p .Then,

V r |c = V q |c (q )V q |c = V p ( p)V p (q )(V p ( p)V p )= V p ( p)V p ( ( p) + ( p) )(V p ( p)V p )

= V p V p ( p) V p ( p) V p ( p) + V p ( p) = V p V p ( p) V p ( p) V p ( ( p) ( p) ( p)) ,

where we have neglected terms of order higher than two in s and s; to obtain V r |c we simply change which is equivalent to the change :

V

r |c

= V

p V

p ( p)

V

p ( p)

V

p ( ( p)

( p)

( p))

.Then the difference of the two parallel transports is

V r |c V r |c = ( ( p) ( p) + ( p) ( p) ( p) ( p))V p = R ( p)V p

R ( p)V p A where A is the innitesimal area enclosed by the curves c and c. Clearly,V r |c = V r |c if and only i f R ( p) = 0 .

Then, the curvature tensor measures the difference between the parallel transport of a vector through the paths c and c, where c (c) is a loop.

V r

|c

V r

|c amounts to a rotation, since norms of vectors do not change by parallel transport induced

by metric connections (appendix B); then one says that curvature is the rotational part of the connection .

When parallel transport is independent of the path, that is, for a vanishing curvature, the connectionis said to be integrable (or at ).

Torsion

As before, consider the points p, q and s with coordinates x , x + and x + , respectively. Considerthe innitesimal vectors at p, p x | p and p x | p ( p = , p = ); regarded as innitesimal displacements(translations) in M n , they respectively dene the points q and s. Make the parallel transport of p x | palong p : we obtain the vector at s, V s = p ( p) p; so the total displacement vector from p to r is

p

+ p

p

( p) p

;

similarly, making the parallel displacement of x | p along one obtains the vector at q , V q = p ( p) p ; and the total displacement vector from p to r is p +

p p

( p)

p .

The difference between the two vectors is

p p ( p) + p p ( p) = p p ( ( p) ( p)) = 2 T ( p) p p .15


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So, the torsion measures the failure of the closure of the parallelogram made of the innitesimal displacement vectors and their parallel transports.

Since the last expression is a translation, one says that torsion is the translational part of the connection .

11. Exterior covariant derivative and curvature 2-form

Up to here we have considered the vector bundle : R m E

M n . Now we shall consider the vectorbundle k whose sections are the k-differential forms s on M n with values in E :R

n !k !( n k )! m k T M E

kM

with k T M E = x M k T x M E x ; clearly, 0 = . s (k T M E ) with ( s)(x) = ( x, ( s)x ),

( s)x k T x M E x ; if, as before, {i}mi =1 is a basis of sections of E in U M , x U , and {x }n=1are local coordinates on U , then

{dx 1 |x . . . dxk |x ix , i = 1 , . . . , m , n k > . . . > 1 1}is a basis of k T x M E x . So,

( s)x = mi=1 n k >...> 1 1 ti1 ... k dx

1 |x . . . dxk ix ,

t i1 ... k R .

We dene the set of total exterior covariant derivative operators

{d0 , d1 , . . . , dn 1}, dk : (k T M E ) (k+1 T M E ), k = 0 , . . . , n 1,as the R -linear extension of

dk ( s) = ( dk ) s + s

with

(dk ) s + s : M k+1

T M E, x (x, (dk )x sx + x ( s)x ).For k = 0, d0 = : (E ) (T M E ).

Let us study the composition d1 d0 . If s (E ), thend1 d0 (s) = d1 (s) R(s) (2T M E ),

i.e.

R(s) : (T M )(T M ) (E ), (X, Y ) R(s)(X, Y ) : M E, x R(s)(X, Y )(x) = ( x, R(s)x (X x , Y x )) .

R(s), also denoted by 2(s), is called the second total covariant derivative of the section s. (In general,dk+1 dk = 0; compare with dk+1 dk = 0 in De Rham theory.)

Locally,

R(s) = 2 (s) = d1 ( (s)) = d1 (dx (s)) = d1 (dx (( s j )j + ji s

i j ))

= d1 (dx ) (( s j )j + ji si j ) + dx ((s j )j + ji s

i j )

= dx ( (s j

x j ) + ( ji s

i j )) = dx (d(s j

x ) j + ji

s i

x j + d( ji s

i ) j + jk ki s

i j )

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From the explicit expression of Rij and kl in terms of local coordinates, in the Bianchi equationsone has: dRij = 12 d(Rij dx dx ) = 12 Rij, dx dx dx , ik Rkj = ik dx 12 Rkj dx dx =12

ik Rkj dx dx dx , Rik kj = 12 Rik dx dx kj dx = 12 Rik kj dx dx dx ; then

12(R

ij, +

ik R

kj R

ik

kj )dx

dx

dx

= 0

i.e.

Rij, + ik Rkj Rik kj = 0 .

For a linear connection in M n , with i , j ,k = ,, ,

R, + R R = 0 .In GR, R = R , then the Bianchi equations are:

R, + R

R

= 0 .

Note 1. In section 22, well see that in the case E = T M and the connection is that of Levi-Civita(section 13), when the Bianchi equations are written in terms of the Ricci tensor (see section 16) and thescalar curvature (section 18), we obtain the vanishing of the covariant divergence of the Einsteins tensor(see section 19): G ; = 0.

Note 2 . In electromagnetism, dF = 0 in terms of the curvature tensor F (eld strength), amounts tothe homogeneous Maxwell equations. Instead, if F = dA is used, we obtain an identity.

Remark : Up to here, all the results have been independent of the existence of a metric g in themanifold M n i.e. of a non degenerate symmetric scalar product at each tangent space T x M n . This metricis introduced in the next section.

13. The Levi-Civita connection

In Appendix A we shall prove the Fundamental Theorem of Riemannian or Pseudo-Riemannian Ge-ometry , which states that in a riemannian or pseudo-riemannian manifold ( M n , g ) there exists a uniquesymmetric and metric linear connection, the Levi-Civita connection, given by

=12

g ( g + g g ) = g with

=12

( g + g g ).Then g = g g = = .

It holds:

i) D g = g ; = 0 (and also D g = g ; = 0). A consequence of this is that for any smooth pathc : (a, b) M n the metric tensor g is parallel transported along c:

(Dgd

) =dx

dDg = 0 ,

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where dx

d = c . Then it follows that the scalar product of two parallel transported vectors along c by the

Levi-Civita connection is also parallel transported i.e. covariantly constant:

Dd

(g V W ) = (Dgd

) V W + g (DV d

) W + g V (DW d

) = 0 .

In particular, if V = W = c, then g c c (c, c) ||c||2 remains constant by parallel transport; if ||c||2 > 0,=0, < 0 the geodesic is respectively called timelike , null or lightlike , and spacelike . Since = g g ,then ; = 0.

ii)D V = D (g V ) = g D V

i.e. the covariant derivative commutes with the raising or lowering of the indices.

It can be shown that if c : (a, b) M n is a smooth path that extremizes the proper time (or pathlength) = c df 1/ 2 , with f = g dx

ddx d , then c is a geodesic of the Levi-Civita connection. Also, a

change of parameter = ( ) preserves the form of the geodesic equation if and only if is anaffine transformation , i.e. = a + b,

where a, b R and a = 0. For an arbitrary transformation one obtains

d2x

d2+

dx

ddx

d=

d2d 2

(dd

)2 dx

d.

This means that the derivation of the geodesic equation in section 8, forces the parameter to be an affine parameter i.e. a parameter linearly related, up to an additive constant, to the proper time .

It is important to mention that the fact that the connection coefficients depend on the metricfunction g , is the usual argument in the literature for denying to G.R. the character of a gauge theory of gravity. More on this below.

14. Physics 1: Equivalence principle in GR

Massive free point particles move along timelike geodesics. Massless free point particles move along lightlike geodesics; in this case can not be the proper time since (d )2 = 0, (Dirac, 1975).

15. Covariant components of the curvature tensor

Starting from the expression ( ) in section 9, a long but straightforward calculation leads to the result,for the Levi-Civita connection,

R = g R =12

( g + g g g ) + g ( ).

Clearly, R is a covariant 4-rank tensor. It has n4 components (e.g. if n = 4, 4 4 = 256).

Algebraic properties of R

i) R = 12 ( g + g g g ) + g ( ) = R(symmetry under the interchange between the rst pair of indices with the second pair of indices i.e.RAB = RBA with A = and B = ).

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ii) R = R , R = R , then R = R which can be summarized asR = R = R .

iii) R + R + R = 0 (cyclicity).

If one denesA := R + R + R ,

it can be proved that it is totally antisymmetric in its four indices. Since A = 0, this imposesm4 =

m !4!( m 4)! conditions on R . (Number of ways one can take four distinct elements among m; obviously itmust be m 4.)

Let us determine the number of algebraically independent components of R . Let S ab and Aabbe respectively a symmetric and antisymmetric tensor in m dimensions. The corresponding number of independent components are N (S ab ; m) = m

(m +1)2 and N (Aab ; m) =

m (m 1)2 . So, we have

m 1 2 3 4 5 6 . . . 10 . . .N (S

ab; m) 1 3 6 10 15 21 . . . 55 . . .

N (Aab ; m) 0 1 3 6 10 15 . . . 45 . . .

Notice that N (A ab ;m )N (S ab ;m ) 1 as m . If we write R RAB , since under or Ris antisymmetric, each index A or B contributes with m (m 1)2 independent components; but now one hasa two-index symmetric matrix RAB with A, B {1, . . . , m

(m 1)2 }, which gives 12 m(m 1)2 (

m (m 1)2 + 1) =18 m(m 1)(m(m 1) + 2) = 18 m(m 1)(m2 m + 2) independent components for R . But iii) and thenthe antisymmetry of A imposes m !4!( m 4)! =

m (m 1)( m 2)( m 3)4! conditions. Then,

N (R ; m) =18

m(m 1)(m2 m + 2) 14!

m(m 1)(m 2)(m 3) =m2(m2 1)

12.

So, we havem 1 2 3 4 5 6 . . . 10 11 . . . 26 . . .

N (R ; m) 0 1 6 20 50 105 . . . 825 1210 . . . 38025 . . .

16. Ricci tensor for the Levi-Civita connection

Dene the covariant 2-tensorR := g R R .

We contracted indices 1 and 3; contracting 1-2 and 3-4 gives zero; contracting 1-4, 2-3 and 2-4 gives R :g R = g R = R , g R = g R = R , g R = g R = R . So, up toa sign, the Ricci tensor is uniquely dened from R and g .R is symmetric : R = g R = g R = g R = R . Then,

N (R ; m) =m(m + 1)

2.

We havem 1 2 3 4 5 6 . . . 10 11 . . .

N (R ; m) 1 3 6 10 15 21 . . . 55 66 . . .

We can write

R = g g R = R

= R

= < dx

, R( , , ) > = < dx , R( , , ) > .20


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If x x = x(x) then g(x) = ( dxdx )2 g(x). Choose x(x) = x dy |g(y)|; then ( dxdx )2 = 1|g(x ) | and sog(x) = g(x )|g(x ) | which equals +1 (-1) if g(x) > 0 (< 0). From the constancy of g(x), (x) = 12g(x ) ddx g(x) = 0.

D = 2. N (R ; 2) = 1. By antisymmetry in and , the only possibilities for a non-vanishingR are R1010 , R1001 , R0110 and R0101 . We choose R0101 and it is easily veried that the unique Rwhich satises the algebraic properties of section 15 and gives R0110 =

R1010 = R1001 =

R0101 is

R = ( g g g g )R0101

g,

with g = det g00 g01g10 g11= g00 g11 g01 g10 = g00 g11 g201 . In the presence of matter, Einsteins equations

areG = const. T

where T is the energy-momentum tensor of matter. Then, for D = 2 = 1 + 1,

T = 0 .

This means that in D = 2, the unique solutions to Einsteins equations are those corresponding to thevacua i.e. T = 0.

D = 3. Since N (R ; 3) = N (R ; 3) = 3, the curvature tensor can be expressed in terms of g andR ; the most general form satisfying the symmetry properties of section 15 is

R = A(g g g g ) + B (g R g R g R + g R )with A and B numerical constants. From the denition of the Ricci tensor, R = g R = (2 A +BR )g + BR , where R is the Ricci scalar; then B = 1 and A = 12 R. Therefore,

R = R2

(g g g g ) + g R g R g R + g R .

D = m 4. In all these cases

N (R ; m) N (R ; m) := N (C ; m) =m(m + 1)( m + 2)( m 3)

12> 0

where C , the Weyl tensor , has the same algebraic properties as R but cant be obtained from g and R . One writes

R = C (g g g g ) + D (g R g R g R + g R ) + C with all traces of C vanishing i.e.

C

= 0 .

Then

R = g R = C (mg g ) + D(mR 2R + g R) = g ((m 1)C + DR ) + ( m 2)DR .Then D = 1m 2 and C =

R(m 1)( m 2) . Therefore,

R = R

(m 1)(m 2)(g g g g ) +

1m 2

(g R g R g R + g R ) + C .

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Units : [G ] = [R ] = [R ] = [L]2 , [G ][c4 ] =[t ]2

[M ][L ] =1

[force ] ; then [T ] =[M ]

[L ][t ] = [energy density ],where [L], [M ], and [t] denote the units of length, mass, and time, respectively.

24. Tensor bundles as associated bundles to the bundle of frames of M n

T rs M n is the total space of the ( n + n r + s )-dimensional real vector bundle of r -contravariant and s-covariant tensors on M n , with bre R n

r + s

= { i 1 ...i rj 1 ...j s R , ik , j l {1, . . . , n }, k = 1 , . . . , r , l = 1 , . . . , s } { }. The bundle of frames of M n , F M n , is the principal bundle with structure group GL n (R ) (the bre of the bundle) on M n (the base space), and with total space FM n consisting of the set of all ordered basis of the tangent space at each point of M n , namely

FM n =x M n

{rx (v1x , . . . , vnx ), {vkx }nk=1 : basis of T x M n }= x M n {x} {(v1x , . . . , vnx )}

x M n

(FM n )x ,

where (FM n )x is the bre over x, with dim R (FM n )x = n2 . The bundles of orthogonal frames, Lorentzframes, restricted Lorentz frames, etc. of M n , are obtained by reducing the group GL n (R ) respectively toO(n), O(n1, 1), SO 0(n 1, 1), etc. If x U U , then vkx = n=1 vk (x) x |x ; also, dim R F M n = n + n2 .The n + n2 local coordinates on F U is the set ( x , X ) with x (x, r x ) = x (x) and X (x, r x ) = vx ,,, {1, . . . , n }.

One has:n 1 2 3 4 5 . . . 10 . . .

dim R FM n 2 6 12 20 30 . . . 110 . . .

The bundle structure of F M n is represented by

F M n : GL n (R ) F M n F M n

where F is the projection F (x, (v1x , . . . , vnx )) = x and GL n (R ) F M n represents the right action of GL n (R ) on F M n given byF M n GL n (R )

F M n , ((v1x , . . . , vnx ), a) (v1x a1 1 + + vnx an 1 , . . . , v1x a1n + + vnx an n )

(v1x , . . . , vnx )a.The left action of GL n (R ) on R n

r + s, given by

GL n (R ) R nr + s

R nr + s

, (a, ) (a ) i 1 ...i rj 1 ...j s = a i 1 k1 . . . a i r k r a1l1

j 1 . . . a1l s

j s k1 ...k rl1 ...l s ,

induces the associated bundle FM n GL n (R ) R nr + s

which turns out to be isomorphic (through , see below)to T rs M n . One has the following commutative diagram:

FM n GL n (R ) R nr + s

T rs M n F rsM n Id M

n

M nwhere:

([((v1x , . . . , vnx ), )]) = ni k ,j l =1 i 1 ...i rj 1 ...j s vi 1 x . . . vi r x w

j 1x . . . wj sx ,

with [(( v1x , . . . , vnx ), )] = {((v1x , . . . , vnx )a, a 1 )}a GL n (R ) ; {w1x , . . . , wnx }is the dual basis of 24


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{v1x , . . . , vnx }i.e. wix (vjx ) = ij ; and F ([((v1x , . . . , vnx ), )]) = F (x, (v1x , . . . , vnx )) = x. Using thefact that the dual basis vectors wjx transform with a1 , it is easily veried that is well dened i.e. it isindependent of the representative element of the class [(( v1x , . . . , vnx ), )].

A section of the bundle of frames of M n i.e. a smooth function s : M n

F M n with F

s = Id M n ,

trivializes F M n and therefore all the tensor bundles associated with it (in particular the tangent bundleof M n ). The same occurs for any of the reductions of F M n (bundle of Lorentz frames, restricted Lorentzframes, etc.).25. Vertical bundle of a principal bre bundle

Let be a principal bre bundle (p.f.b.), = ( P r + s , B s , , G r , , U ) : Gr P r + s

B s , whereB s B (base space) and P r + s P (total space) are differentiable manifolds of dimensions s and r + srespectively, Gr G is an r -dimensional Lie group with right action on P , P G P , ( p, g) pg, and U is a system of local trivializations 1(U ) U G with 1 = .

For each p P there exists a canonical vector space isomorphism p between G= Lie (G) : the Liealgebra of G, and V p = T pP ( p)

: the tangent space to the bre over ( p) at p, the vertical space at p:

p : G V p , A p(A) A p,with

A p : C (P, R ) R , f ddt

f ( petA )|t =0 .We used the fact that T pP ( p) T pP ; if Ai , i = 1 , . . . , r is a basis of G, then p(Ai ) is a basis of V p; ingeneral, neither Ai nor p(Ai ) are canonical basis.

Given p, p P , since p : G V p and p : G V p are isomorphisms, there is a canonical vector spaceisomorphism ( absolute teleparallelism ) between V p and V p , for all p, p P :

V pp

1p

V p .Remark : This result, namely, the existence of p at each p P , is independent of any connection.

This implies the triviality of the vertical bundle V of the p.f.b. :

V : R r V 2r + s V

P,with V 2r + s = p P V p = p P { p}V p and V ( p, v p) = p.

In fact, V admits r independent global sections i : P V 2r + s , i ( p) = ( p, p(Ai )); then there is thefollowing vector bundle isomorphism:R r R r

| |V 2r + s

P R rV 1P Id P

with ( p, v p) = ( p,ri=1

i p(Ai )) = ( p, (1 , . . . , r )). is not canonical since it depends on the basis Ai

of G.

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Let be a connection on , i.e. (T P G) with : P T P G, p ( p) = ( p, p), p : T pP G, v p p(v p) = 1 p (ver (v p)). Since ker ( p) = H s p H p: the horizontal vector space at p,then p is |H p| = 1. However, p|V p : V p G, p|V p (v p) = 1 p (v p) is a vector space isomorphism i.e. p|V p = 1 p .

In other words, if is a connection on , then at each p P , gives an inverse of p. Therefore, for theisomorphism between V p and V p , one has

V pp |V p

1 p |V p V p .

In particular, we are interested in the case P = F M n , the frame bundle of a differentiable manifold M n ,where (P,B,,G,, U ) = ( F M n , M n , F , GL n (R ), , U ); p = ( x, r x ) FM n , and r x = ( v1x , . . . , vnx ). Itsvertical bundle is isomorphic to the product bundle F M n R n

2:

R n 2 R n 2

| |V F

M n c

F M n

R n 2

V 1F M n Id F M n

where c is the canonical isomorphism determined by the canonical basis of gln (R ) = R (n) given by the n2matrices ( Aij )kl = ik jl . The similar result holds for the reductions of GL n (R ) to O(n), SO 0(n 1, 1), etc.mentioned in section 24.

In particular, for the case n = 4 and G = SO 0(3, 1), with dim R (SO 0 (3, 1)) = dim R (so(3, 1)) =dim R (o(3, 1)) = 6, case relevant in GR, F M 4 is the bundle of Lorentz frames F LM 4 , and we have thevector bundle isomorphism

R 6 R 6

| |V F LM 4

Lc

F L M 4 R 6V | |1F L M 4

Id

F L M 4with dim R (F L M 4) = 4 + 6 = 10 and dim R (V F LM 4 ) = 16. In this case, the canonical basis of so(3, 1) (or of o(3, 1)) is the set of matrices

{0 0 0 00 0 0 00 0 0 10 0 1 0

,

0 0 0 00 0 0 10 0 0 00 1 0 0

,

0 0 0 00 0 1 00 1 0 00 0 0 0

,

0 1 0 01 0 0 00 0 0 00 0 0 0

,

0 0 1 00 0 0 01 0 0 00 0 0 0

,

0 0 0 10 0 0 00 0 0 01 0 0 0

}, respectively {l23 a1 , l31 a2 , l12 a3 , l01 b1 , l02 b2 , l03 b3}, where the rst three

matrices generate rotations around the axis x, y, and z, and the second three matrices generate boosts alongthe same axis, respectively. The derivation of the canonical basis is as follows: one starts from the denitionof the Lorentz transformations : L := T L , with L = ( 00 , 11 , 22 , 33 ) = (+1 , 1, 1, 1)(or (1, +1 , +1 , +1)) and ab = 0 i f a = b; if () is a smooth path through the identity (0) = I ,the corresponding tangent vector at I , (0) = L, obeys the equation LT = L. The generators a iand bi obey [a i , a j ] = ijk ak , [bi , bj ] = ijk ak , [a i , bj ] = ijk bk . If l =

3i =1 ( i bi + i a i ) and l =

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3i=1 ( i bi + i a i ) are in o(3, 1) ( s, s R ), then is Lie bracket [ l, l] = l =

3i =1 ( i bi + i a i ) o(3, 1)

with 1 = 2 3 + 2 3 3 2 3 2 , 2 = 3 1 + 3 1 1 3 1 3 , 3 = 1 2 + 1 2 2 1 2 1 , 1 = 2 3 + 2 3 + 3 2 3 2 , 2 = 3 1 + 3 1 + 1 3 1 3 , 3 = 1 2 + 1 2 + 2 1 2 1 .26. Soldering form on F M n

Given the differentiable manifold M n

, the soldering or canonical form on F M n is the Rn

-valueddifferential 1-form on FM n i.e. (T F M n R n ) dened as follows:

: F M n T FM n R n , (x, r x ) ((x, r x )) = (( x, r x ), (x,r x ) ),(x,r x ) : T (x,r x ) FM n R n , v(x,r x ) (x,r x ) (v(x,r x ) ) = r 1x F ( x,r x ) (v(x,r x ) )i.e.

(x,r x ) = r1x dF |(x,r x ) ,where F is the projection in the bundle F M n (section 24) and r x is the vector space isomorphism

r x : R n T x M , (1 , . . . , n ) r x (1 , . . . , n ) =ni =1

i vix

with inverse

r1x ( ni=1 i vix ) = ( 1 , . . . , n ). We have the commutative diagram

R n Id R n(x,r x ) r 1xT (x,r x ) F M n

d F |( x,r x ) T x M n

Notice that dim R (T (x,r x ) F M n ) = n + n2 . Since dF |(x,r x ) is onto, (x,r x ) is a vector space epimorphism, withker ((x,r x ) ) = V (x,r x ) , the vertical space of the bundle FM n at ( x, r x ), with dim R ker ((x,r x ) ) = n2 . Theexistence of is independent of any connection. Also, it is clearly a global section of the bundle T F M n R n .

(e ) j = j , , j = 1 , . . . , n is the canonical basis of R n , then

(x,r x ) =n

=1(x,r x ) e

where (x,r x ) T (x,r x ) F M n .

In local coordinates on F U , =

n

=1(X 1) dx

where (X 1)

(x, r x ) = ( X

(x, r x ))

1. [In fact, if v(x,r x ) T (x,r x ) FU , then v(x,r x ) =

n=1

x |(x,r x ) +n, =1 X |(x,r x ) with dF |(x,r x ) (v(x,r x ) ) =

n=1

x |x T x U ; then (x,r x ) (v(x,r x ) )

= r1x dF |(x,r x ) (v(x,r x ) ) = r 1x (n=1

x |x ) =

n=1

r 1x ( x |x ) =n, =1

(X (x, r x ))1e ;

on the other hand, (x,r x ) (v(x,r x ) ) = (n=1

(x,r x ) e )(

n =1

x |(x,r x ) +

n, =1

X |(x,r x ) )

= ( n, =1 (X 1) (x, r x )dx |(x,r x ) e )(n

x |(x,r x ) +

n, =1

X |(x,r x ) )

= n,, =1 (X (x, r x ))1 e = n, =1 (X (x, r x ))1 e .]

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Thus, a local section on F M n , s : U FU , x s (x) = ( x, r x ), gives rise to a set of n localdifferential 1-forms on F U .If is a connection on F M n , and H (x,r x ) is the horizontal space at ( x, r x ), then

(x,r x ) |H ( x,r x ) = r 1

x dF |(x,r x ) |H (x,r x )is a vector space isomorphism, since dF |(x,r x ) is a canonical isomorphism between H (x,r x ) and T x M n :

R n Id R n(x,r x ) |H ( x,r x ) r 1xH (x,r x )

d F |( x,r x ) T x M nWe emphasize that (x,r x ) |H ( x,r x ) depends on both the frame at x (rx ) and the connection .

Any connection on the frame bundle F M n , together with the canonical soldering form , trivializesthe tangent bundle of F M n . This fact is known as absolute parallelism . The canonical bundle isomorphism(only depending on ) is given through the following diagram:

R n + n 2 R n + n 2

| |(T FM n )2( n + n

2 ) c(FM n )n + n2

R n + n2

F 1(F M n )n + n

2 Id

(F M n )n + n2

withc((( x, r x ), v(x,r x ) )) = (( x, r x ), ((x,r x ) |H ( x,r x ) (x,r x ) |V ( x,r x ) )(hor (v(x,r x ) ),ver (v(x,r x ) ))) ,

where v(x,r x ) T (x,r x ) F M n and gln (R ) = R (n) = R n2.

Absolute parallelism in the bundle of Lorentz frames F LM n is given by the diagramR n ( n +1)2 R

n ( n +1)2

| |(T F L M n )n (n +1)

Lc(F L M n )

n ( n +1)2 R

n ( n +1)2

L 1(F L M n )

n ( n +1)2

Id

(F L M n )n ( n +1)

2

withLc ((( x, ex ), v(x,e x ) )) = (( x, e x ), ((x,e x ) |H ( x,e x ) L(x,e x ) |V ( x,e x ) )(hor (v(x,e x ) ),ver (v(x,e x ) ))) ,

ex = ( e1x , . . . , enx ), and H (x,e x ) = ker (L(x,e x ) ).

In particular, for the n = 4 case:

R 10 R 10

| |(T F L M 4)20

Lc(F L M 4)10 R 10L 1

(F L M 4)10 Id (F L M 4)10.

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27. Linear connection in a manifold M n on F M nA gln (R )-valued linear connection on M n is locally given by

U = dx E

where (E

) =

with ,,, = 1 , . . . , n is the canonical basis of R (n) = gln (R ) = Lie (GL n (R )),and are the Christoffel symbols (section 4).

On F U , the connection F U such that U = (F U ) with : U FU the local section given byx (x ) = ( x , )) is given byF U = ( X 1) (dX + X dx ) E .

(Kobayashi and Nomizu, 1963; pp 140-143)) Clearly, U (T U gln (R )) and F U (T FU gln (R )).

Real-valued connection 1-forms U and F U are dened by

U = U E and F U = F U E .

The horizontal lift of a local vector eld x by the connection in F M n is then given by

(

x )=

x X

X

.

In fact, F U (( x )) = ( X 1) (dX + X dx

)( x X X )= ( X 1) ( X dX ( X ) + X

dx

( x ))

= ( X 1) ( X + X ) = ( X 1) ( X + X ) = 0 .28. Tetrads and spin connection

1. At each chart U M n we can take as a basis of ( T U ) the local vector elds (Vielbeine)

ea = ea , a = 1 , . . . , n

with r x = ( e1x , . . . , enx ) (FU )x . Since the n n matrices ( ea (x)) GL n (R ), there exist the inversevector elds e1a ea = ea dx : 1-forms with ea = ( ea )1 GL n (R ) andea e b = ba and e a

e a = .

Then e a ea = e a ea = i.e. = e a ea .

In general, the ea s are called non-coordinate basis and the ea s anholonomic coordinates . For n = 4, theVierbeine ea are called tetrads .

2. While [ , ] = 0, the Vielbeine have non-vanishing Lie brackets. In fact, applying the commutator[ea , eb] to a function f C (U, R ), one easily obtains

[ea , eb] = cab ec

with cab = ec(ea ( eb ) eb ( ea )) = cba .

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3. For a local vector eld, V = V = V e a ea = V a ea , with

V a = e a V , V = ea V a .

4. At each x M n , the bre of the co-frame bundle of M n ,

F M n : GL n (R ) (FM n ) M n ,is the set

(F M n )x = {ordered basis of T x M }= {(f 1x , . . . , f nx ), f ax = f a (x)dx |x}.Again, ( f a (x)) GL n (R ) and, locally, ( f a )1 = f a with

f a f b = ba and f a f a = .

Also,f a = f a dx and dx = f a f a .

From the duality relation dx ( ) = we obtain = f a f a (e beb) = f a e bf a (eb); imposing the duality

relation between f s and es,f a (eb) = ab

one obtainsf d = e d and f a = ea .

Then,f a = e a dx and dx = ea f a .

(Another usual notation for f a is a .)

5. Given an ( r, s )-tensor in M n ,

T = T 1 ... r 1 ... s 1 . . . r dx 1 . . . dx s ,

we obtainT = T 1 ... r 1 ... s e1

a 1 ea 1 . . . e r a r ea r f b1 1 f b1 . . . f bs

s f bs = T a 1 ...a rb1 ...b s ea 1 . . . ea r f b1 . . . f bs

withT a 1 ...a rb1 ...b s = e 1

a 1 . . . e r a r f b1 1 . . . f bs

a s T 1 ... r 1 ... s .

For example,

T = T dx = T e a ea dx = T a ea dx = T f b f b = T b f b = T e a ea f b f b

= T a bea f b.

6. Let g = g dx

dx

be a metric in M n

. g has a signature, given by the diagonal metric ab , equalto ab in the euclidean case ( = E ), or with -1s and +1s in the general riemannian case; the lorentziancase, relevant for GR, has = L (section 25. ). g being a symmetric matrix, at any point x M n itcan be diagonalized to ab . The metric and its signature distinguish the subset of Vielbeine which obey thefollowing orthonormality condition:

g(ea , eb) = ab .

In detail,

g dx dx (ea , eb ) = g ea eb dx ( )dx ( ) = g ea eb = g ea

eb

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i.e.g ea eb = ab (a).

This relation can be easily inverted, namely, g ea eb e a e b = g = g i.e.

ab e a e b = g or g = ab ea eb. (b)

The unique solution of g (x) = ab e a (x)e b(x) = is e a (x) = a . Such a frame is called inertial at x.

Notice that formula ( b) is fundamental: since it holds in all charts in M n , we have, up to topologicalobstructions, trivialized the metric everywhere i.e. globally, at the expense of the x-dependence of thecoframes ea .

It is usual the rough statement that the duals of the Vielbeine are the square roots of the metric. Inparticular, for the lorentzian n = 4 case, det (g ) = (det (e a ))2 . Also, equation ( a) allows to interpretethe n n matrices ec as the matrices which diagonalize the metric g to the Lorentz metric ab . (b) saysthat the ea s are more fundamental than the metric.

7. Equation ( b) in the last subsection appears naturally when describing spinor elds in curved space-times. If (x) are the Dirac matrices in M n , then

{ (x), (x)}= 2 g (x)I.The solution

(x) = E a (x) a ( )

with a the at Dirac matrices obeying { a , b}= 2 ab I , leads toab E a (x)E b(x) = g

which says that the E a s are the duals of the Vielbeine ea . It can be proved that the solution ( ) isunique (ORaifeartaigh, 1997).

8. It is clear that through ( b) in subsection 6., the n2 quantities involved by a Vielbein determineuniquely a metric g ; however, the set of n (n +1)2 components of a metric determines a Vielbein only up toan equivalence relation i.e. it determines a class of Vielbeine, whose elements are related by a group G of n (n 1)2 elements.

Let ea = ha cec ; then

ab = g ha cec hbd ed = g ec ed

ha chbd = cd ha chbd = ha ccd hbd i.e.

= hh T .

So, if = L then h Ln = O(1, n 1); if = E then h On = O(n); etc.In the following we shall restrict to the case of orthonormal frames in the sense dened in 6., so one has

a principal G-bundle over M n :G

n ( n 1)2 F G M n ,

with G = Ln or On , and F G F M n . The interest in GR is for G = L4 and one has the lorentzian bundle

L4 F L4 M 4

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With this denition, the equations (e) and (f) read

D ea = 0 and D e b = 0respectively. (We also denote D T 1 ... r a 1 ...a t 1 ... s b1 ...b u = T 1 ... r a 1 ...a t 1 ... s b1 ...b u ; .)

From the Lorentz transformations of the tetrads ea

and their inverses, ea

= ec

h1

ca

we obtain:

e c ea = er h1r

c (ha d ed

) = er h1r

c (ha d )ed

+ er h1r

cha d ed

= (ha d)h1dc

+ ha d(e r ed )h1r

c ,

ande cea = e

r h1rcha ded

= had (e

r ed )h1r

c.

Then

ca = had (e

r ed + e

r ed )h1r

c+ (ha d )h1d

c= ha d rd h1r

c+ (ha d )h1d

c.

I.e. = hh1 + ( dh)h1

or, equivalently, = h1h h1dh.

So, the 1-form a b := ab dx is not a 1-1 L-tensor, since its transformation has an inhomogeneous term.Notice that from (c), has the structure

= e1( + ) e,

and from (d), the structure of is = e( + )e1 .

It can be easily shown that for a metric connection , for which

Dg = 0 ,

the spin connection with lower Lorentz indices

bc = ac ab

is antisymmetric in these indices. In fact, using (e) and (f),

0 = g ; = ( ab e a e b); = ab ; e a e b + ab e a ; e b + ab e a e b;

= ab ; e a e b = ( ca cb cb ac )e a e b = (ba + ab )e a e b,then

0 = (ba + ab )e a e bec ed = (ba + ab ) ac bd = (dc + cd )i.e.

dc = cd .Thus, we see that it is the condition of metric compatibility which reduces the Lie algebra of the gauge groupfrom gln (R ) to o(1, n 1) (or o(n)), where the 1-form bc = bc dx takes values. The reduced gauge group

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can be O(1, n 1) (or O(n)) or, for the case n = 4, SL (2, C ) if ha c L+ = SO 0(1, 3) at each x M 4(Randono, 2010).Up to here the content of this section does not depend on the symmetry properties of in its lower

indices. If in particular the Levi-Civita connection of section 13. is inserted in ( b), using (c) (valid for anyconnection with local coefficients ), we obtain the spin connection coefficients ca in terms of the

Vielbeine, their derivatives, and the Lorentz metric ab :

ca = ec ea +

12

e cea fd ef ed hk ( (e h e k ) + (e h e k ) (e h e k )) .So, we have the result analogous to the dependence of the Levi-Civita connection on the metric: the depen-dence of the spin connection on the tetrads.

10. Explicitly, on each chart ( U, x ) in M , the (metric) spin connection with values in so(3, 1), isconstructed as follows:

= 12 ab dx lab (T U so(3, 1)), x (x) = ( x, x ) with x = 12 ab (x)dx |x labT x U so(3, 1)

i.e.

x : T x U so(3, 1), vx x (vx ) =12

ab (x)dx |x (vx )lab =12

ab (x)vx lab =12

ab (x)lab

with ab (x) = ba (x) := ab (x)vx and x (vx ) = 01 l01 + . . . 31 l31 .For later use, consider the connected component of the Poincare group P 4 , the semidirect sum of the of the translation group T 4 and the connected component of the Lorentz group L4 :

P 4 = T 4 SO 0(3, 1), (a , )(a, ) = ( a + a, )

with Lie algebra

p4 = R 4 so(3, 1), ( , l)( , l ) = ( l l , [l, l]) = ( , l )( , l).(P 4 is a subgroup of the affine group A4 ; for arbitrary n, An = R n GL n (R ) with Lie algebra an =R n R (n).)

Also, on each chart ( U, xn ) in M , the tetrad (1-form) ea with values in Lie (T 4) = R 4 i.e. ea(T U R 4) is constructed as follows:ea = ea dx

: U T U R 4 , ea (x) = ( x, eax ), eax = ea (x)dx |x T x U R 4 ,

eax : T x U R

4, e

a (x)dx

|x (vx ) = x with x = ea (x)v

x R .

11. The Lorentz bundle L4 F L4 M 4 in subsection 8. extends the symmetry group of GR, thegroup of general coordinate transformations of M 4 , D, to the semidirect sum GGR = L4 D,

with composition law given by(h, g)(h, g) = ( h (ghg 1), gg).

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In fact, it is easy to verify that Dhas a left action on L4 at each bre F L4 x of the bundle, given by thecommutative diagramF L4 x

h

F L4 xg gF L4 x

h

F L4 xwhich denes h = ghg1 Lg (h)(conjugation action), with g(x, (e1(x), . . . , e4(x))) = g(ea (x)) = g(ea (x) x |x ) = ea

(x) x |x , ea (x) =

x x ea

(x). The action is left since h h = ghg1 = g(ghg1)g1 = ( gg)h(gg)1 i.e. Lg g(h) =

Lg Lg(h). (See the extension to GGR = P 4 D in section 34 .)29. Curvature and torsion in terms of spin connection and tetrads. Cartan structure

equations; Bianchi identities

In what follows we shall designate by k (Lrs ) the real vector space of k differential forms on M withvalues in the ( r, s )-Lorentz tensors.

Given the Vielbeine ea

and the spin connection ab on the chart ( U, x

) on M , we have the differentialformsea = e a dx 1(L1) and a b = ab dx

.

(a b / 1(L11) since a b is not an L11-tensor, but a connection on the Lorentz bundle L4 F L4 M 4 .)Then we have the 2-formsT a = dea + a b eb =

12

T a dx dx 2(L1), ( )

withT a = e a e a + ab e b ab e b = T a ,

andRa b = da b + a c c b =

1

2Ra b dx dx 2(L1

1), ( )

withR ab =

ab ab + ac cb ac cb .

( ) and ( ) are known as the Cartan structure equations . As we shall show below, T a and R a b are,respectively, the torsion and curvature 2-forms of section 9.

For dea one has

dea = d(e a dx ) = e a dx dx = a dx dx a

witha = ( de

a ) =1

2(e, a

e, a ) =

a .

Also,

abc = ( dea )bc = eb ec a =

12

e a (ec eb eb ec ) =12

e a (eb,c ec,b ) = acb.Comparing with abc of 28 .2, we have

abc = 12

abc

and therefore[eb, ec] = 2abcea = 2(dea )bcea .

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So abc also measures the non-commutativity of the Vielbeine. By the Jacobi identity,

f ad dbc +

f bd

dca +

f cd

dab = 0 .

It is easy to show that for a metric connection, the curvature tensor with lower Lorentz indices

Rab = ad Rd b

is antisymmetric i.e.Rab = Rba .

In fact, Rab = ad (dd b + d c c b) = ad dd b + ad d c c b = dab + ac c b = (dba + ca c b) =(dba c b ca ) = (dba cb c a ), while Rba = bcRc a = bc (dc a + cd d a ) = dba + bd d a =dba db d a .

Symbolically we writeT = de + e, R = d + .

We notice that torsion is related to the tetrads as curvature is related to the spin connection .On the other hand, while curvature involves only , torsion involves both e and (not only e). This

is related to the fact that the Poincare group is the semidirect (not direct) sum of R 4 (translations) andSO 0 (3, 1) (spacetime rotations).

A manifold equipped with a metric g and a connection compatible with the metric but with non-vanishing torsion, is called an Einstein-Cartan manifold . The metric induces the Levi-Civita connection,(LC )

(section 13 ) with

= ( LC )

+ contortion tensor.

In the Einstein-Cartan (E-C) theory of gravity, the 1-forms

{ea , ab

}are called gauge or gravitational potentials , respectively translational and rotational , while the 2-forms

{T a , R a b}are called gauge or gravitational eld strengths , respectively translational and rotational . (See, however,section 34 .) At a point, it is always possible to set e pt = 1 and pt = 0, i.e. respectively e a = a (16conditions) and ab = 0 (24 conditions). (Hehl, 1985; Hartley, 1995.) The total number of conditions,40, coincides with that for making zero the Christoffel symbols in the case of the Levi-Civita connection(|{(LC )

}|= 40).

Comment . Together with the comments in section 10 , we have the following relations:

curvature spin connection spacetime rotations,torsion tetrads spacetime translations.

On the other hand, from Noether theorem, we have the relations:

spacetime rotations angular momentum,spacetime translations energy momentum.

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Naively, one should then expect the following relations:

curvature angular momentum,torsion energy momentum.

However, in Einstein-Cartan theory, based on a non-symmetric metric connection , the sources of cur-vature and torsion are respectively energy momentum and spin angular momentum i.e.

curvature energy momentum,torsion spin angular momentum.

This crossing of relations is due to holonomy theorems (Trautman, 1973).

These facts can be better understood as follows: In (special) relativistic eld theory (r.f.t.), elds belongto irreducible representations of the Poincare group P 4 , which are characterized by two parameters: massand spin. Invariance under translations ( T 4 ) and rotations ( L4) respectively leads, by Noether theorem, tothe conservation of energy-momentum ( T ) and angular momentum: orbital + intrinsic (spin, with densityS ). On the other hand, differential geometry (d.g.), through holonomy theorems, relates curvature ( R )with the Lorentz group and torsion ( T ) with translations (section 10 ). Finally, Einstein (E) equationsmake energy-momentum the source of curvature, while Cartan (C) equations makes spin the source of torsion.This is summarized in the following diagram:

L4d.g. r.f.t.R S E C T e T

r.f.t. d.g.T 4

In a formulation of the Einstein-Cartan theory based on tetrads and spin connection, the Einstein equations

are obtained by variation with respect to the tetrads ( e ), related to translations, and the Cartan equationsby variation with respect to the spin connection ( ), related to rotations. (See section 32 .)

Locally, as differential 2-forms with values in so(3, 1) and Lie (T 4) = R 4 , R and T a are respectivelygiven as follows:

R =12

Rab dx dx lab (2U so(3, 1)) , Rab = ad Rd b , R (x) = ( x, R x ),

R x =12

Rab (x)dx |x dx |x lab 2x U so(3, 1), R x : T x U T x U so(3, 1), Rx (vx , wx )

=14

Rab (x)(vx wx vx wx )lab ;

T a = T a dx dx (2U R 4), T a (x) = ( x, T ax ), T ax = T

a (x)dx

|x dx |x ,T ax : T x U T x U R 4 , T ax (vx , wx ) =

12

T a (x)(vx w

x v x wx ) .

From the denition of T , we have, since d2 = 0, dT = d e de = d e (T e) =d e T + e, i.e. dT + T = ( d + ) e, that isdT + T = R e 3(L1). ()

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In Lorentz components,dT a + a b T b = Ra b eb. ( )

For R one has dR = d d = ( R ) (R ) = R R , i.e.dR + R R = 0 3(L11). ( )

In components,dR a b + a c R cb R a c cb = 0 . ( )

() and ( ) (or ( ) and ( )) are the so called Bianchi identities. (Compare ( ) with the correspondingequation in section 12. )

Dening the covariant exterior derivative operator acting on Lorentz tensors-valued differential forms

D = d + we have the equations

T = De, DT = R e, DR = R .

Though is not a Lorentz tensor, one has R = D.It is easy to verify that T a is nothing but twice the torsion tensor of section 9. :

ea T a = ea ((dea ) +( a b eb) ) = ea ( e a e a + ab e bab e b) = ( ea e a + ea e bab )

(ea e a + ea e bab ) = = 2 T .A similar calculation leads to

ea e bRa b = ea e b(da b + a c c b) = ea e b( a b a b + a c c b a c c b ) = R .For the Ricci tensor and the Ricci scalar of sections 16 and 18 respectively (but now not restricted to

the Levi-Civita connection) we have

R = R = ea e bR a b and R = R = ea e bR a b g = ea eb Rab .

We summarize the above formulae in the following table:

1(L1) 1(L11) 2(L1) 2 (L11) 3(L1) 3 (L11) Componentse ea abT = de + e T a = dea + ab eb

R = d + Rab = d

ab +

ac

cb

dT + T = R e dT a + ab T b = Rab ebdR + R = R dR ab + ac R cb = R ac cb30. Spin connection in non-coordinate basis

The Christoffel symbols for a metric connection with torsion is given in Appendix B:

= ( LC ) + K

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40/60

and( LC )

(x) =

12

( (g )(x) + (g )(x) (g )(x)) . (b)So,

T (x) = 0 (g )(x) = 0 and ( LC ) (x) = 0 ,

i.e. the vanishing of the torsion at x is a sufficient condition for having a local inertial system at x.

However, the condition is not necessary : in fact,

T ( p) + T ( p) = T ( p) + T ( p) = 0

implies that T is also antisymmetric in its second and third indices, and then it is totally antisymmetric,since T = T = T = T .

A calculation gives:

n = 2:T 001 = T

101 = 0

n = 3:T 001 = T

002 = T

101 = T

112 = T

202 = T

212 = 0 ,

T 012 = T 210 = T

102

n = 4:

T 001 = T 002 = T

003 = T

101 = T

112 = T

113 = T

202 = T

212 = T

232 = T

303 = T

313 = T

323 = 0 ,

T 012 = T 210 = T

102 ,

T 013 = T 310 = T

103 ,

T 023 = T 320 = T 203 ,

T 123 = T 231 = T

312

In each case, the number of independent but not necessarily zero components of the torsion tensor coincideswith the number of independent components of the totally antisymmetric torsion tensor with covariantindices, number which results from the condition that the denition of geodesics as world-lines of particles(parallel transported velocities, section 8) to coincide with their denition as extremals of length. This lastfact can be seen as follows:

As world-lines, geodesics are dened in section 8, the equation being

d2x

d 2+ ( )

dx

ddx

d= 0 ( )

where only the symmetric part of contributes:

( ) = ( LC ) g (T g + T g ) = ( LC )

(T + T ) = ( LC )

2T ( )

with g T g = g T = T ; notice that the covariant form of the torsion tensor, T , is anti-

symmetric in the rst two indices: T = T . (With this denition of T , the covariant form of thecontortion tensor isK = g K = T T + T ,

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which is antisymmetric in the last two indices i.e. K = K .)On the other hand, the equation of geodesics dened as extremals of arc-length:

0 = ds = (g dx dx ) 12 ,turns out to be (Carroll, 2004, pp.106-109)

d2x

d 2+ ( LC )

dx

ddx

d= 0 . ( )

Then, for the denitions ( ) and ( ) to coincide, T ( ) must vanish i.e. T = T g T =g T T = T i.e. T must be 1-3 antisymmetric; but this implies that T is also 2-3antisymmetric: T = T = T = T . Since, by denition, T is antisymmetric in the rsttwo indices, it then turns out to be totally antisymmetric; in n dimensions, its number of independentcomponents is n3 =

n (n 1)( n 2)6 N . Some values are:

n 2 3 4N 0 1 4

The set of allowed non-vanishing components of the torsion tensor still leads to physical (geometrical)effects in the sense of section 10 . The non-closure of a parallelogram with innitesimal sides and ismeasured by the vector

= 2 T = T (

). For n = 4 its components are:

0 = T 012 ( 1 2 2 1) + T 013 ( 1 3 3 1) + T 023 ( 2 3 3 2),

1 = T 123 ( 2 3 3 2) + T 102 ( 0 2 2 0) + T 103 ( 0 3 3 0),

2

= T 231 (

3 1

1 3

) + T 210 (

1 0

0 1

) + T 203 (

0 3

3 0

), 3 = T 310 (

1 0 0 1) + T 320 ( 2 0 0 2) + T 312 ( 1 2 2 1),which can be distinct from zero.

In summary, the necessary and sufficient condition for erecting a locally inertial coordinate system at apoint x in a U 4-space, is that the symmetric part of the contortion tensor vanish up to terms of order ( x )2 ,where x (x) = 0. (Socolovsky, 2010.)

In the above sense, the weak equivalence principle , which only refers to the free motion of point-likeand therefore classical particles, still holds in a U 4-space (U 4 = ( M 4 , g, ) with a metric connection). (Asimilar result was recently found by Fabbri (Fabbri, 2011).)

32. Einstein-Cartan equations

(We owe this derivation to L. Fabbri, 2010.)

A. Pure gravitational case (vacuum)

We start from the curvature 2-form R ab with components R a b given in section 29 ; we dene the Riccitensor

Rb := Ra b ea

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and the Ricci scalarR = Rb bcec .

ThenR = bcR a b ea ec = R(a b, ec).

The gravitational action is given by the Einstein-Hilbert lagrangian density eR,

S G = d4 x eRwith e = detg .

A.1. Variation with respect to the spin connection: = (Cartan equations )Varying R a b w.r.t. , using ( cd ) = (cd ), and adding and subtracting ( ab ) one obtains

R a b = D (ab ) D (ab ) + 2 ab T where

D (ab ) = (

ab )

cb (

ac ) +

ac (

cb )

(

ab )

and

D (ab ) = (ab ) cb (ac ) + ac (cb ) (ab )are covariant derivatives since cd is a tensor ( cd is a connection, not a tensor, but the difference of twoconnections is a tensor). Then,

S G = d4x eea bcec R a b = d4x e(D ((ab )ea bcec ) D ((ab )ea bcec )+2 ea bcec abT

) = d4 x e(D V + 2( ac )T ac )

with V the 4-vector given byV = ( ac )(ea ec ea ec ),

where we have used the Leibnitz rule for D , D ea = 0, D bc = 0, and raised Lorentz indices with ab .For D V one hasD V = ( DLC ) V + K V

= V + ( LC ) V + K V

,

thenD V = ( DLC ) V + K V = e1(eV ), 2T V

where we have used ( LC ) = 1 (appendix C), appendix B, and the denition of the torsion 1-formT = T (section 9). Neglecting the surface term d4x (eV ) , one obtains

0 = S G = d4x e(T V + ac T ac ) = d4x e( ac )(ec T a + ea T c + T ac )with T a = T ea , which, due to the arbitrariness of ac leads to the Cartan equations for torsion (invacuum):

T ac + ea T c ec T a = 0 . (i)

Multiplying ( i) by e a e c one obtains the Cartan equations in local coordinates:

T + T T = 0 . (ii )

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Proposition 1 : Torsion vanishes.

Proof. Taking the trace in ( ii ) leads to T + T 4T = 2T = 0; then T = 0 and from ( ii )again,T = 0 . qed

Note : The above result holds in n dimensions for n = 2: the trace gives (2 n)T = 0. For n = 2, T is arbitrary with independent components T 001 = T 1 and T 101 = T 0 . Also, notice that ( i) (or ( ii )) is not adifferential equation, but an algebraic one; this is the mathematical expression of the fact that in E-C theorytorsion does not propagate .

Proposition 2 : LetT +

T T = S (iii )

with S = S , and a constant. (( iii ) corresponds to a non-vacuum case and will be used in part B.)Then,T = (S

12 n

( S S )) ( iv)

where n = 2 is the dimension of the manifold and S = S . In particular, for n = 4,

T = (S +

12

( S S )) . (v)

Proof. Again taking the trace in (iii ), (2 n)T = S , then T = 12n S and T is (iv). qed

For n = 2, the unique solution of ( iii ) is S = 0: in fact, = 2 and then S = 0; so S 0 = S 101 = 0 andS 1 = S 010 = 0.

A.2. Variation with respect to the tetrads: = e (Einstein equations )Again from S G ,

e S G = d4x ((R)e + Re ) = d4x (Ra b bce((ea )ec + ea ec ) Ree d ed )= d4x (2R a Re a )eea = 0

where we used e = ee d ed (appendix C), and from the arbitrariness of ea , we obtain the Einstein equations for curvature (in vacuum):R a

12

Re a = 0 ( vi)

orGa = 0 ( vii )

withGa = Ra

12

Re a . (viii )

(R a = ac Rc .) Since in vacuum R = 0 (section 18 ), (vi) amounts to

R a = 0 . (ix )

Of course, multiplying ( vi) by ea we obtain Einstein equations in local coordinates (section 20 ).

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In summary, for the pure gravitational case, Einstein theory=Einstein-Cartan theory ; this is a conse-quence of the form of the Einstein-Hilbert action S G . From the form of R, gravity has been expressed as aninteracting gauge theory (see section 33 ) between the spin connection a b and the coframes eld ea ; botha b and ea are pure geometric elds, which live in the frame and coframe bundles L4 F L4 M 4 andL4 F L4 M 4 respectively.

B. Minimal coupling to Dirac elds

The Dirac-Einstein action is given by

S D E = k d4x eLD E = k d4x e ( i2 ( a (D a ) (D a ) a ) m)where

D a = ( ea i4

abc bc ) = ea ( i4

bc bc) = ea D

andD a = ea +

i4

abc bc = ea ( +i4

bc bc) = ea D

are the covariant derivatives of the Dirac eld and its conjugate = 0 with respect to the spinconnection, which give the minimal coupling between fermions and gravity; they are obtained through thereplacement

da D a i.e. ea = ea D a which amounts to the comma goes to semicolon rule for tensors but here adapted to spinor elds. bc =i2 [

b, c], and the a s are the usual numerical (constant) Dirac gamma matrices satisfying { a , b}= 2 ab I , 0= 0 and j = j . k = 16 Gc4 (16 in natural units). Then the action is

S D E = k d4x e ( i2 ( + 18bc [ b, c]) i2( 18bc [ b, c]) m)where = ea a = (x).

B.1. Variation with respect to the spin connection: =

S D E =k8 d4x e { , bc}bc = k2 d4x e S bc bc

with S bc = ea S abc , where

S abc =14

{ a , bc}is the spin density tensor of the Dirac eld. S abc is totally antisymmetric and therefore in 4 dimensions ithas 4 independent components: S 012 , S 123 , S 230 and S 301 .

Combining this result with the corresponding variation for the pure gravitational eld (part A), we

obtain 0 = (S G + S D E ) = d4x e ac (T ac + ea T c ec T a + k2 S ac )and therefore

T ac + ea T c ec T a =

k2

S ac ,

the Cartan equation . Multiplying by e a e c one obtains

T + T T =

k2

S

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46/60


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as follows:GAn (R ) A n A n , (

g 0 1 ,

1 )

g + 1 .

Then, one has the following diagram of short exact sequences (s.e.s.s) of groups and group homomorphisms:

0

R n

GA

n(R )

GL

n(R )

0

Id 0 R n

| P n || L

n 0

with ( ) = I n 0 1 and (g 0 1 ) = g. is 1-1, is onto, and ker ( ) = Im () = {

I n 0 1 ,

R n }.We have also restricted and (respectively | and |) to the connected components of the Poincare ( P n )and Lorentz ( Ln ) groups in n dimensions. Both s.e.s.s split, i.e. there exists the group homomorphism : GL n (R ) GAn (R ), g (g) =

g 00 1 and its restriction | to Ln , such that = Id GL n (R ) and

| | = IdLn . So GAn (R ) = R n GL n (R ),P

n = R n

Ln

with composition law( , g)(, g ) = ( + g, g g).

As a consequence, the factorization of an element of GAn (R ) (P n ) in terms of elements of R n and GL n (R )(Ln ) is unique:

g 0 1 = ( )(g) (or |( )|(g)). The dimensions of GAn (R ), GL n (R ), P n and Ln are,

respectively, n + n 2 , n 2 , n (n +1)2 andn (n 1)2 (20, 16, 10 and 6 for n = 4).

The above s.e.s.s pass to s.e.s.s of the corresponding Lie algebras:

0 R n

gan (R )

gln (R ) 0Id

0 R n | pn

||ln 0

with gln (R ) = R (n), gan (R ) = R n gln (R ) with Lie product

( , R )(, R ) = ( R R , [R , R]),where [R , R] is the Lie product in gln (R ) and [ , ] = 0 in R n , ( ) = ( , 0), (, R) = R, and (R) =(0, R). , and (and their corresponding restrictions |, | and |) are Lie algebra homomorphisms, with = Id gl n (R ) and | | = Id ln . The s.e.s.s split only at the level of vector spaces i.e. if ( , R ) gan (R ),then ( , R ) = () + (R), but ( , R ) = ()(R).

Let us denote:

M n : n-dimensional differentiable manifold

F M n : frame bundle of M n : GL n FM n F M n

AM n : affine frame bundle of M n : GAn AM n AM n

GL n : general linear group in n dimensions (section 24 ), dim R GL n = n2

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GAn : general affine group in n dimensions, dim R GAn = n + n2

F LM n : bundle of Lorentz frames of M n : Ln F L M n F |M n (section 25 )

F P M n : bundle of Poincare frames of M n :

P n

AP M n

A |

M n

One has the following diagram of bundle homomorphisms:

AM n GAnAP M n P n

| | ||F L M n Ln

FM n GL nA A | F | F AM n AP M n

| |F L M n F M n

A A | F | F M n Id M n

Id

M nId

M n

where is the bundle homomorphism

AM n GAn

F M n GL nA F AM n

FM n

A F M n Id M n

between the bundle of affine frames and the bundle of linear frames over M n , with

(x, (vx , r x )) = ( x, r x ), (x, r x ) = ( x, (0x , r x )) , 0 T x M n

,

AM n = x M n ({x} AM nx ), AM nx = {(vx , r x ), vx Ax M n , rx F x},where Ax M n is the tangent space at x considered as an affine space (Appendix E); and

A ((x, (vx , r x )) , (, g)) = ( x, (vx + r x , r x g))

is the action of GAn on AM n , with r x = na =1 eax a . |F , . . . , |A , . . . , |, . . ., etc. are restrictions; inparticular |(a) = (a) and |(e) = (e).

A general affine connection (g.a.c.) on M n is a connection in the bundle of affine frames AM n . If A isthe 1-form of the connection, thenA (T AM

n

gan )i.e.

A : AM n T AM n gan , (vx , r x ) ((vx , r x ), A (vx ,r x ) ), A(vx ,r x ) : T (vx ,r x ) AM n gan ,V (vx ,r x ) A(vx ,r x ) (V (vx ,r x ) ) = ( , R ) R R n gln (R ).

Obviously, A obeys the usual axioms of connections.

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From the smoothness of , the pull-back (A ) is a gan -valued 1-form on F M n :

(A ) = F ,

where F is a c

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