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Energy-momentum and gauge conservation laws

1 2

G.Giachetta, L.Mangiarotti, and G.Sardanashvily

1

Department of Mathematics and Physics, University of Camerino, Camerino 62032,

Italy

2

Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia

We treat energy-momentum conservation laws as particular gauge conservation laws when

generators of gauge transformations are horizontal vector elds on bre bundles. In particular,

the generators of general covariant transformations are the canonical horizontal prolongations

of vector elds on a world manifold. This is the case of the energy-momentum conserva-

tion in gravitation theories. We nd that, in main gravitational mo dels, the corresp onding

energy-momentum ows reduce to the generalized Komar sup erp otential. We show that the

sup erp otential form of a conserved ow is the common prop erty of gauge conservation laws

if generators of gauge transformations dep end on derivatives of gauge parameters. At the

same time, dep endence of a conserved ow on gauge parameters make gauge conservation laws

form-invariant under gauge transformations.

PACS numb ers: 03.50.-z, 04.20.Cv, 11.15.-q, 11.30.-g

I. INTRODUCTION

Energy-momentum in eld theory is a vast sub ject which can be studied from dif-

ferent viewp oints. The problem lies in the fact that the canonical energy-momentum

tensor fails to b e a true tensor, while the metric one is appropriate only to theories with

a background geometry. In gravitation theories, an energy-momentum ow reduces to a

sup erp otential dep ending onaworld vector eld as a gauge parameter.

Analyzing the energy-momentum problem, we follow the general pro cedure of ob-

taining di erential conservation laws in gauge theory [1-6]. Let Y ! X be a bre

 i 

bundle over a world manifold X co ordinated by(x ;y ), where x are co ordinates on X .

By gauge transformations are generally meant automorphisms of Y ! X [7]. To ob-

tain di erential conservation laws, it suces to consider one-parameter groups of gauge

transformations whose generators are pro jectable vector elds

  i  j

u = u (x )@ + u (x ;y )@ (1)

 i

on Y . If the Lie derivative L L of a Lagrangian L vanishes, wehave the weak conserva-

u

tion law



d T  0 (2)

 1

 i

of the corresp onding symmetry ow T along a vector eld u. If u = u @ is a vertical

i



vector eld, (2) is a familiar Nother conservation law, where T is a Nother current.

This is the case of internal symmetries. If u is a horizontal prolongation on Y of a

vector eld  on X , called hereafter a world vector eld, we have an energy-momentum

conservation law [5, 8, 9]. Of course, di erent horizontal prolongations of vector elds 



lead to di erent energy-momentum ows T .

Let us emphasize that the expression (2) is linear in the vector eld u. Therefore,

one can consider sup erp osition of conservation laws along di erent vector elds. In

particular, every vector eld u (1) on Y pro jected onto a world vector eld  can be

seen as the sum u = u + v of a horizontal prolongation u of  on Y and a vertical

 

vector eld v on Y . It follows that every conservation law (2) can be represented as

a sup erp osition of a Nother conservation law and an energy-momentum conservation

0

of the same world vector eld law. Conversely, two horizontal prolongations u and u





0

 di er from each other in a vertical vector eld u u . Hence, the energy-momentum





0

ows along u and u di er from each other in a Nother current. One can not single out





in a canonical way the Nother part of an energy-momentum ow. Therefore, if internal

symmetries are broken, an energy-momentum ow is not conserved in general.

In particular, a generic gravitational Lagrangian is invariant under general covariant

transformations, but not under vertical gauge transformations of the general linear group

GL(4; R). As a consequence, only the energy-momentum ow, corresp onding to the

canonical horizontal prolongation of vector elds  on X , i.e., the generators of general

covariant transformations, is generally conserved in gravitation theory. This energy-

momentum ow p ossesses the following two imp ortant p eculiarities.

(i) It reduces to a sup erp otential, i.e.,

   

T  d U ; U = U : (3)



We will see that this is a common prop erty of gauge conservation laws if a vector eld u

e

dep ends on the derivatives of gauge parameters. Indeed, the canonical prolongation  of a

 

world vector eld  =  @ dep ends on the derivatives @  of the comp onents of  which

 

play the role of gauge parameters of in nitesimal general covariant transformations.



(ii) A gravitational sup erp otential (3) dep ends on the comp onents  of a world vector

eld  . This is also a common prop erty of gauge conservation laws which make them

gauge-covariant, i.e., form-invariant under gauge transformations. Only in the Ab elian

case of electromagnetic theory, an electric current is free from gauge parameters.

A direct computation shows that General Relativity,Palatini formalism, metric-ane

gravitation theory, and gauge gravitation theory in the presence of fermion elds lead 2

to the same gravitational sup erp otential. This is the generalized Komar sup erp otential

!

@L @L

 

(D  + S  ); (4) U ( )=

  

@R @R

   

where R and S are the curvature and the torsion of a world connection K ,

     

and D is the corresp onding covariant derivative. In the particular case of the Hilb ert{



Einstein Lagrangian L and a Levi{Civita connection K , the sup erp otential (4) reduces

to the familiar Komar sup erp otential.

Throughout the pap er, we follow the convention where the covariant derivative reads

 

D  = @  K  :

   

II. GAUGE CONSERVATION LAWS

To obtain gauge conservation laws, we follow the rst variational formula of Lagran-

gian formalism.

Given a gauge eld system describ ed by sections  of a bre bundle Y ! X , its space

of elds and their rst order partial derivatives is the nite-dimensional rst order jet

1

manifold J Y of Y . Its elements are elds  identi ed by their values and the values of

their rst order derivatives at p oints of X (see, e.g., [9] for a detailed exp osition). This

 i i i

space is provided with the adapted co ordinates (x ;y ;y ), where y are the derivative

 

i i

co ordinates such that y   = @  . The transformation law of the derivative co ordinates





is

!

0i 0i 

@y @y @x

0i j

y = y + :

 

j  0

@y @x @x

A Lagrangian L of a eld system on a bre bundle Y ! X is de ned as a density

 i i 4

L = L(x ;y ;y )d x



1

Y . We will use the notation on the space J

 

 = @ L;

i i

4

! = d x; ! = @ c!; ! = @ c! :

    

Let u be a pro jectable vector eld (1) on a bre bundle Y , treated as the generator

1

of a one-parameter group of gauge transformations. Its prolongation on the space J Y

reads

1  i i i  

J u = u @ + u @ +(d u y @ u )@

 i  

 i 3

i

where d = @ + y @ denote the total derivatives. The rst variational formula provides

  i



the canonical decomp osition of the Lie derivative

  i i i  

L L! =[@ u L+(u @ +u @ +(d u y @ u )@ )L]! (5)

u   i  

 i

of a Lagrangian L in accordance with the variational problem. This decomp osition reads

  i i i  

@ u L +[u @ +u @ +(d u y @ u )@ ]L = (6)

  i  

 i

i i  

(u y u ) L d T ;

i 



where



 i i

 L =(@ d @ )L; d = @ + y @ + y @ ; (7)

i i    i

i

i  

are the variational derivatives and

    i i 

T = T ! ; T =  (u y u ) u L; (8)



i 

is said to b e a symmetry ow along the vector eld u.

It should b e emphasized that the ow (8) is de ned mo dulo the terms



i  i

d (c (y u u ));



i





where c are arbitrary skew-symmetric functions on Y [9]. Here, we leave aside these

i

b oundary terms which are indep endent of a Lagrangian, but they may be essential if

one examines integral conservation laws.

On the shell



 L =(@ d @ )L=0

i i 

i

where solutions of the Euler{Lagrange equations live, the rst variational formula (6)

leads to the weak identity

  i i i  

@ u L +[u @ +u @ +(d u y @ u )@ ]L  (9)

  i  

 i

  i i 

d [ (u y u ) u L]:



i 

If the Lie derivative L L (5) vanishes, i.e., the Lagrangian L is invariant under the

u

corresp onding one-parameter group of gauge transformations, the weak identity (9) is

brought into the weak conservation law

  i i 

0 d [ (u y u)u L]; (10)



i 

of the ow T (8) along the vector eld u. 4

The weak identity (10) leads to the di erential conservation law



d T ()= 0



on solutions  of the Euler{Lagrange equations

 L()= 0:

i

This di erential conservation law implies the integral conservation law

Z



T ()! =0; (11)



@M

where M is a compact 4-dimensional sub-manifold of X with a b oundary @M .

We will see that, in gauge and gravitation mo dels, a conserved ow takes the sp ecial

form

  

T = W + d U (12)



where the term W is expressed in the variational derivatives (7), i.e., W  0, and T



on-shell reduces to a sup erp otential U (3). In this case, the integral conservation law

(11) b ecomes tautological, but the sup erp otential form (12) of T implies the following

integral relation

Z Z

 

T ()! = U ()! ; (13)

 

3 3

N @N

3 3

where N is a compact oriented 3-dimensional submanifold of X with the b oundary @N

and  is a solution of the Euler{Lagrange equations. One can think of this relation as

b eing a part of the Euler{Lagrange equations written in an integral form.



III. NOTHER CONSERVATIONS LAWS

Here, we touch on brie y the well-known Nother conservation laws in gauge theory of

internal symmetries in order to emphasize some common prop erties of gauge conservation

laws.

Let P ! X be a principal bundle with a structure Lie group G. The corresp onding

gauge mo del is formulated on the bundle pro duct

Y = E  C (14)

X

1

of a P -asso ciated vector bundle E of matter elds and the bundle C = J P=G ! X

whose sections A are principal connections on the principal bundle P . The bundle C

 m m m

is equipp ed with the co ordinates (x ;a ) such that a  A = A are gauge p otentials.

   5

 i m

The bundle Y (14) is co ordinated by(x ;y ;a ). Let us consider a one-parameter group



of vertical automorphisms of the principal bundle P . They yield the corresp onding

one-parameter group of gauge transformations of the pro duct (14). Its generator is the

vertical vector eld

r r q p  p i

 =(@  +c a  )@ +  I @ ;

 i

qp  r p

r i

where c are structure constants of the group G, I are generators of this group on the

qp p

p 

typical bre of Y ! X , and  (x ) are gauge parameters, transformed by the coadjoint

representation [5, 9, 10]. Let us use the compact notation

A p A p

 =(u @  + u  )@ ; (15)

 A

p p

A r  A r q  i

u @ =  @ ; u @ = c a @ + I @ :

A A i

p p r p qp  r p

If a Lagrangian L is gauge invariant, the rst variational formula (6) leads to the

strong equality

A p A p A p A p 

0=(u  +u @  ) L + d [(u  + u @  ) ]; (16)

 A  

p p p p A

where  L are the variational derivatives of L and

A

p

 i

d = @ + a @ + y @ :

  i

p 



p 

Due to the arbitrariness of gauge parameters  (x ), this equality is equivalent to the

following system of strong equalities:



A A

u  L + d (u  )=0; (17a)

A 

p p A



A A  A

u  L + d (u  )+u  =0; (17b)

A 

p p A p A



A A 

u  + u  =0: (17c)

p A p A

Substituting (17b) and (17c) in (17a), we obtain the well-known constraint conditions

A A

u  L d (u  L)=0

A  A

p p

of the variational derivatives of a gauge invariant Lagrangian.

On-shell, the rst variational formula (16) leads to the weak conservation law

A p A p 

0  d [(u  + u @  ) ] (18)

 

p p A

of the Nother current

 A p A p 

T = (u  + u @  ) : (19)



p p A 6

Accordingly, the equalities (17a) { (17c) on-shell lead to the equivalent system of Nother

identities



A

d (u  )  0; (20a)



p A



A  A

d (u  )+u   0; (20b)



p A p A



A A 

u  + u  =0 (20c)

p A p A

for a gauge invariant Lagrangian L [2]. They are equivalent to the weak equality (18)

p 

due to the arbitrariness of the gauge parameters  (x ).

The weak identities (20a) { (20c) play the role of the necessary and sucient condi-

tions in order that the weak conservation law (18) b e gauge-covariant. This means that,

if the equality (18) takes place for gauge parameters  , it do es so for arbitrary deviations

 +  of  . Then the conservation law (18) is form-invariant under gauge transforma-

tions, when gauge parameters are transformed by the coadjoint representation.

The equalities (20a) { (20c) are not indep endent, for (20a) is a consequence of (20b)

and (20c). This re ects the fact that, in accordance with the strong equalities (17b) and

(17c), the Nother current (19) is brought into the sup erp otential form

 p A   p A 

T =  u  L + d U ; U =  u  ; (21)

A 

p p A



where the sup erp otential U do es not dep end on matter elds. It is readily observed

that the sup erp otential form of the Nother current (19) is caused by the fact that the

vector elds (15) dep end on derivatives of gauge parameters.

On solutions of the Euler{Lagrange equations, we have the corresp onding integral

relation (13), which reads

Z Z

 p 

T ()! =   ()! : (22)

 

p

3 3

N @N

Since the Nother sup erp otential (21) do es not contain matter elds, one can think of

(22) as b eing the integral relation b etween the Nother current (19) and the gauge eld

generated by this current. Due to the presence of gauge parameters, the relation (22) is

gauge covariant.

In electromagnetic theory, the similar relation between an electric current and the

electromagnetic eld generated by this current is well known, but it is free from gauge

parameters due to the p eculiarity of Ab elian gauge mo dels. Let us consider electromag-

j j

netic theory, where G = U(1) and I (y )= iy . In this case, a gauge parameter  is not

changed under gauge transformations. Therefore, one can put, e.g.,  = 1. Then the

Nother current (19) takes the form

 A 

T = u  :

A 7

Since the group G is Ab elian, this current do es not dep end on gauge p otentials and it

is invariant under gauge transformations. We have

 j 

T = iy  : (23)

j

It is easy to see that T (23), under the sign change, is the familiar electric current

of matter elds, while the Nother conservation law (18) is precisely the equation of

continuity. The corresp onding integral equation of continuity (11) reads

Z

j 

(y  )()! =0:



j

@M

Though the Nother current T (23) takes the sup erp otential form

  

T =  L + d U ;



the equation of continuity is not tautological. This equation is indep endent of an elec-

tromagnetic eld generated by the electric current (23) and it is therefore treated as

the strong conservation law of an electric charge. When  = 1, the electromagnetic

sup erp otential takes the form

1

 

U = F ;

4

where F is the electromagnetic strength. Accordingly, the integral equality (22) is the

integral form of the Maxwell equations. In particular, the well-known relation b etween

the ux of an electric eld through a closed surface and the total electric charge inside

this surface is restated.

IV. ENERGY-MOMENTUM IN GRAVITATION THEORIES

From nowonbyaworld manifold X is meant a 4-dimensional orientable noncompact

parallelizable manifold. As a consequence, it admits a pseudo-Riemannian metric and a

spin structure. Accordingly, a linear connection and a bre metric on the tangent and



cotangent bundles TX and T X of X are said to be a world connection and a world

metric, resp ectively.

Gravitation theories are formulated on natural bundles Y ! X , e.g., tensor bundles

which admit the canonical horizontal prolongations of any vector eld  on X . These

prolongations are the generators of general covariant transformations, where the com-

p onents of a vector eld  play the role of gauge parameters. By the reason we have

explained ab ove, we will investigate the energy-momentum conservation laws asso ciated

with these prolongations. 8

A. Tensor elds

We start from tensor elds which clearly illustrate the main p eculiarities of energy-

momentum conservation laws on natural bundles. Let

m k



Y =( TX) ( T X) (24)

X

 m

1

be a tensor bundle equipp ed with the holonomic co ordinates (x ; x_ ).

1

k



Given avector eld  =  @ ,we have its canonical prolongation



@



 

m m

2 1

1

e

 =  @ +[@  x_ (25) + ::: @  x_ :::]

 

1



m

1

1 2

k k

@x_

1

k

on the tensor bundle (24) and, in particular, its prolongations

@

 

e

(26)  =  @ + @  x_

 

@ x_

on the tangent bundle TX and

@

 

e

 =  @ @  x_

 

@ x_



on the cotangent bundle T X .

Of course, one can consider the horizontal prolongation

@



) (27)  =  (@ + K x_

K  

@ x_

of a world vector eld  on TX and tensor bundles by means of any world connection

K . This is the generator of a 1-parameter group of nonholonomic automorphisms of

these bundles. These automorphisms are met with in gauge theory of the general linear

group GL(4; R) [3], but a generic gravitational Lagrangian is not invariant under these

transformations. Note that the prolongations (26) and (27) were treated as generators

of the gauge group of translations in the pioneer gauge gravitation mo dels (see [11, 12]

for a survey).

A m

1

Let us use the compact notation y = x_ such that the canonical prolongation

1

k

e

 (25) reads

 A

e

 =  @ + u @  @ : (28)

 A

This expression is the general form of the canonical prolongation of a world vector eld

 on a natural bundle Y , when this prolongation dep ends only on the rst order partial 9

derivatives of the comp onents of  . Therefore, the results obtained b elow for tensor

elds are also true for every such natural bundle Y .

Let a Lagrangian L of tensor elds b e invariant under general covariant transforma-

tions, i.e., its Lie derivative (5) vanishes for any world vector eld  . Then we have the

strong equality



A A A

@ ( L)+u @  @ L + d (u @  ) y @   =0: (29)

A 

A A

The corresp onding weak identity (10) takes the form

 A A 

0 d [ (y  u @  )  L]: (30)



A

Due to the arbitrariness of the gauge parameters  , the equality (29) is equivalent to

the system of strong equalities

@ L =0; (31a)





A A A

 L + u  L + d (u  )= y  ; (31b)

A 

A A



A A

u  + u  =0; (31c)

A A

where  L are the variational derivatives.

A

Substituting the relations (31b) and (31c) in (30), we obtain the energy-momentum

conservation law



A A

0 d [u  L + d (u   )]: (32)

 A 

A

A glance at this expression shows that, on-shell, the corresp onding energy-momentum

ow reduces to a sup erp otential, i.e.,



 A   A

T =  u  L + d U ; U = u   : (33)

A 

A

It is readily seen that the energy-momentum sup erp otential (33) emerges from the de-

e

p endence of the canonical prolongation  (28) on the derivatives of the comp onents of

the vector eld  . This dep endence guarantees that the energy-momentum conservation

law (32) is maintained under general covariant transformations.

B. General Relativity

A pseudo-Riemannian metric g onaworld manifold X is represented by a section of

the quotient

=LX=O (1; 3) ! X; (34)

where LX is the bundle of linear frames in the tangent spaces to X . It is called the

metric bundle. The linear frame bundle LX is a principal bundle with the structure 10

group GL(4; R). For the sake of simplicity, we will identify the metric bundle with a

sub-bundle of the tensor bundle

2

  _ TX; (35)

 

co ordinated by (x ;g ). In General Relativity, it is more convenient to consider the

metric bundle as a sub-bundle of the tensor bundle

2



  _ T X (36)



equipp ed with the co ordinates (x ;g ).



The second order Hilb ert{Enstein Lagrangian L of General Relativity is de ned

HE

2 

on the second order jet manifold J  of  co ordinated by(x ;g ; g ; g ): It reads

 

p

1

 

L = g g R g!; g = det (g ); (37)

HE  

2

where

1

R = [ (g + g g g )+

        

2

" "

g (f gf gf gf g)];

"    

1

g (g + g g ): f g =

      

2

Let  be avector eld on X and

  

e

 =  @ (g @  + g @  )@ (38)

  

its canonical prolongation (25) on the metric bundle  (36). Since the Lagrangian

L (37) is invariant under general covariant transformations, its Lie derivative along

HE

e

the vector eld  (38) vanishes. Then the rst variational formula for second order

Lagrangians [9, 13] leads to the energy-momentum conservation law

p

1

     

0 d f2g   L + d [ r  g r  )]g; g(g

  HE   

2

  

r  = @  f g :

  

A glance at this conservation law shows that, on-shell, the energy-momentum ow re-

duces to the well-known Komar sup erp otential [14]:

q

1

    

U = j g j(g r  g r  ): (39)

 

2 11

C. Metric-ane gravitation theory

In metric-ane gravitation theory, gravity is describ ed by a pseudo-Riemannian met-

ric g and a world connection K on X . Since world connections are asso ciated with prin-

cipal connections on the linear frame bundle LX , there is one-to-one corresp ondence

between the world connections and the sections of the quotient

1

C = J LX=GL(4; R) ! X: (40)

K

 

This bundle is provided with the co ordinates (x ;k ) so that, for any section K of



C ! X ,

K

 

k  K = K

 

are the co ecients of the world connection K . The bundle C (40) admits the canonical

K

horizontal prolongation

@

   

e

 =  @ +[@  k @  k @  k +@  ] (41)

K        

@k



of vector elds  on X . We will use the compact notation

 A" A"

e

 =  @ +(u @  + u @  )@ ;

K  " " A

where

A

y = k ; (42)



" " " "

u = k  k  k  ;

  



" " 

u =    :





Metric-ane gravitation theory is formulated on the bundle pro duct

Y =C ; (43)

K

X



co ordinated by(x ;g ;k ), where  is the sub-bundle (35). The corresp onding space



1

J Y is equipp ed with the co ordinates



(x ;g ;k ;g ;k ):

  

We will assume that a metric-ane Lagrangian L factorizes through the curvature

MA

" "

R = k k + k k k k ;

    "   "  12

and do es not dep end on the derivative co ordinates g of a world metric. Then the



following relations take place:

 

 =  ; (44)

@L

MA

   

=  k  k : (45)

   

@k



We also have the equalities

  A   " A " "

 u =  ;  u = @ L  k :

MA 

A A

Given a vector eld  on a world manifold X , its canonical prolongation on the

pro duct (43) reads

  

e

 =  @ +(g @  + g @  )@ +

  

A A 

(u @  + u @  )@ :

 A

Let a metric-ane Lagrangian L be invariant under general covariant transforma-

MA

tions, i.e.,

L L =0 (46)

MA

e

for anyworld vector eld  . Then, on-shell, the rst variational formula (6) leads to the

weak conservation law

 A A A" 

0 d [ (y  u @  u @  )  L ] (47)

 " MA

A

where

  A A A" 

T =  (y  u @  u @  )  L (48)

" MA

A

is the energy-momentum ow of the metric-ane gravity.

It is readily observed that, in the lo cal gauge where the vector eld  is constant,

the energy-momentum ow (48) leads to the canonical energy-momentum tensor

   

T =( k  L ) :

  MA

This tensor was suggested in order to describ e the energy-momentum complex in the

Palatini mo del [15].



Due to the arbitrariness of the gauge parameters  , the equality (46) is equivalent

to the system of strong equalities

@ L =0;





 A A A

 L +2g  L + u  L + d ( u ) y  =0; (49)

MA  MA A MA 

A A

A" A" 

(u @ + u @ )L @  =0; (50)

A MA "

A

("  )

 =0; (51)

13

where  L and  L are the corresp onding variational derivatives. It is readily

 MA A MA

observed that the equality (50) holds owing to the relation (45), while the equality (51)

do es due to the relation (44).

A

Substituting the term y  from the expression (49) in the energy-momentum con-

A

servation law (47), we bring this conservation law into the form

 A  A

0 d [2g   L + u   L  u @  + (52)

  MA A MA

A



 A 

d ( )@  + d ( u ) d ( @  )]:

  

A

After separating the variational derivatives, the energy-momentum conservation law (52)

of the metric-ane gravity takes the sup erp otential form

       

0 d [2g   L +(k  L k  L k  L ) +

  MA  MA  MA MA  

      

 L @  d ( L ) + d ( (@  k  ))];

MA   MA    

where the energy-momentum ow on-shell reduces to the generalized Komar sup erp o-

tential

@L

MA

 

U = (@  k  ) (53)

  

k

 

that we have written in the form (4) [16].

In particular, let us consider the Hilb ert{Einstein Lagrangian density

p

1



L = R g!; R = g R ;

HE  

2

in the framework of metric-ane gravitation theory. Then the generalized Komar sup er-

p otential (4) comes to the Komar sup erp otential (39) if we substitute the Levi{Civita

connection k = f g. One may generalize this example by considering the Lagran-

   

gian

p

L = f (R ) g!;

where f (R ) is a p olynomial of the scalar curvature R . In the case of a symmetric

connection, we restate the sup erp otential

p

@f

    

U = g(g D  g D  )

 

@R

of the Palatini mo del [17] as like as the sup erp otential in the recent work [18] where

Lagrangians of the Palatini mo del factorize through the pro duct R R .

D. Gauge gravitation theory 14

Turning to the energy-momentum conservation law in gauge gravitation theory, we

meet with the problem that spinor bundles over aworld manifold do not admit general

covariant transformations. This diculty can b e overcome as follows [9, 19]. The linear

frame bundle LX is the principal bundle LX ! over the metric bundle  (34) whose

structure group is the Lorentz group. Let us consider a spinor bundle S !  asso ciated

with the Lorentz bundle LX ! . For each gravitational eld g , the restriction of this

g

spinor bundle to g (X )   is isomorphic to the spinor bundle S ! X whose sections

describ e Dirac fermion elds in the presence of a gravitational eld g . Moreover, the

spinor bundle S !  can b e provided with the Dirac op erator and the Dirac Lagrangian

whose restrictions to g (X ) are the familiar Dirac op erator and Dirac Lagrangian of

fermion elds in the presence of a background gravitational eld g . It follows that

sections of the bre bundle S ! X describ e the total system of fermion and gravitational

elds on a world manifold X . At the same time, the bre bundle S ! X is not a

spinor bundle, and it inherits general covariant transformations of the frame bundle

LX . The corresp onding horizontal prolongation on S of world vector elds on X can b e

constructed.

As a consequence, gauge gravitation theory of metric gravitational elds, world con-

nections and Dirac fermion elds can b e formulated on the the bundle pro duct

Y = C  S; (54)

K



  A 

co ordinated by (x ;h ;k ; ), where h are tetrad co ordinates on , i.e.,



a a

   ab

g = h h 

a b

1

where  is the Minkowski metric. The corresp onding space J Y is provided with the

adapted co ordinates



  A A

(x ;h ;k ; ;h ;k ; ):

 

a 

a

The total Lagrangian on this space is the sum

L = L + L (55)

MA D

of the metric-ane Lagrangian L in the previous Section and the Dirac Lagrangian

MA

i 1



 + 0 q A B kb a ka b   B C

L = f h [ ( ) ( ( h  h )(h h k )L )

D B   ab C

q A    k

k

2 4

1



+ +

kb a ka b   + C 0 q A B

( ( h  h )(h h k ) L )( ) ]

  A B

  k C

A k ab

4

p

1

+ 0 A B

g; L = [ ; ]: m ( ) g

ab a b B

A

4 15

 

Note that, in fact, the Dirac Lagrangian L dep ends only on the torsion k k

D    

of a world connection, while the pseudo-Riemannian part is given by the derivative



co ordinates h .

k

Given a vector eld  on a world manifold X , its horizontal prolongation on the

pro duct (54) is

e e

 =  + v; (56)

Y

@

 

e e

 =  + @  h ;

K 



c

@h

c

1 @



d A kb a ka b  B + A + B

v = Q ( h  h )[L h + L @ + L @ ];

ab c ab B A B

k   d ab A



4 @h

c

d

e

where  is the vector eld (41), L are generators of the Lorentz group in the

K ab c



ob ey the condition Minkowski space, and the terms Q

c



   ab

(Q h + Q h ) =0

a b a

b

e

(see [9, 19] for a detailed exp osition). The horizontal part  of the vector eld (56) is

the generator of a one-parameter group of general covariant transformations of the bre

bundle (54), whereas the vertical one v is the generator of a one-parameter group of

vertical Lorentz automorphisms of the spinor bundle S ! . By construction, the total

Lagrangian L (55) ob eys the relations

L L =0: (57)

v D

L L =0; L L =0: (58)

MA D

e e

The relation (57) results in the Nother conservation law, while the equalities (58) lead

to the energy-momentum one [4, 6].

e

Using the compact notation (42), let us rewrite the horizontal part  of the vector

eld (56) in the form

@

   A A"

e

 =  @ + @  h +(u @  + u @  )@ :

  " A



a

@h

a

Due to the arbitrariness of the functions  , the conditions (58) lead to the strong

equalities



 A A A

 L +2h  L + u  L + d ( u )=y @ L ; (59)

MA  MA A MA  MA

A A

p

@L

D

A

gt + h + @ L u = (60)  L +

A D D

c

@h

c

@L @L @L

D D D

 A +

h + + ;



c A

+

A

@h @ @

c A 16

where

p

@L

D

gt = h

a

@h

a

is the metric energy-momentum tensor of fermion elds. We also have the relations (44),

(45) and

@L @L @L

D D D



= = h : (61)



c

 

@k @k @h

   

c

The corresp onding energy-momentum conservation law reads

 A A A"

0 d [@ L (y  u @  u @  ) (62)

 MA "

A

@L @L @L

D D D

 A + 

(@  h h  )+  +   L]:

c c A

+

A

@h @ @

c  A

A

Substituting the term y @ L from (59) and the term

MA

A

@L @L @L

D D D

 A +

h + +



c A

+

A

@h @ @

c A

from (60) in (62), we bring this conservation lawinto the sup erp otential form

       

0 d [2h   L +(k  L k  L k  L) + (63)

     

      

 L@  d ( L) + d ( (@  k  ))]

     

@L @L

D D

 

d [( h + h )@  ]:

 

a a

@h @h

a a

By virtue of the relations (61), the last term in the expression (63) vanishes, i.e., fermion

elds do not contribute to the sup erp otential. It follows that the energy-momentum

conservation law (62) of gauge gravitation theory takes the sup erp otential form (12),



where U is the generalized Komar sup erp otential (53).

V. CONCLUSIONS

The ab ove energy-momentum conservation laws in gravitation theories are derived

from the condition of invariance of gravitational Lagrangians under general covariant

transformations. We have shown that they p ossess the generic prop erties of gauge con-

servation laws. Since the generators of general covariant transformations dep end on

the derivatives of gauge parameters, i.e., comp onents of a world vector eld  , the



corresp onding energy-momentum ow T takes the sup erp otential form. As in electro-

magnetic theory, one can write the integral relation (13) on-shell between the energy-

 

momentum ow T and the gravitational sup erp otential U . The examples of fermion 17



elds and Pro ca elds [9] show that the sup erp otential U do es not dep end on matter

elds. Then, by analogy with electromagnetic and gauge theories, one can think of (13)



as b eing the relation b etween the energy-momentum ow T in a compact 3-dimensional



manifold N  X and the ux U of a gravitational eld generated by this ow through

the b oundary @N .

The p eculiarityofgravitation theory as likeasany non-Ab elian gauge theory lies in

the fact that the integral relation (22) dep ends on a gauge parameter. In the case of

general covariant transformations, this gauge parameter is a world vector eld  whose

canonical prolongation on a natural bundle is the generator of general covariant trans-

formations. As a consequence, the energy-momentum conservation law in gravitation

theories as likeasinmechanics [20] dep ends on a reference frame, but it is form-invariant

under general covariant transformations. In particular, with resp ect to the lo cal gauge

 =const., the energy-momentum ow in the conservation law (62) is the canonical

energy-momentum tensor.

A direct computation also shows that, for a number of gravitation mo dels, the gen-

eralized Komar sup erp otential (4) app ears to be an energy-momentum sup erp otential

asso ciated with gauge invariance under general covariant transformations.

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