Energy-Momentum Conservation Laws in Affine-Metric Gravitation Theory

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series at a p oint x. It is a nite-dimensional smo oth manifold. Recall that a k -order

di erential op erator on sections of a bred manifold Y ,by de nition, is a morphism of

J Y toavector bundle over X . As a consequence, the dynamics of eld systems is played

out on nite-dimensional con guration and phase spaces.

In eld theory,we can restrict ourselves to the rst order Lagrangian formalism when

1 i 1

the con guration space is J Y . Given bred co ordinates (x ;y )ofY, the jet space J Y

i i

is endowed with the adapted co ordinates (x ;y ;y ):

i i

0 0

@y @y @x

0 j

y =( : y + )

@y @x

A rst order Lagrangian density on the con guration space J Y is represented bya

horizontal exterior density

i i 1 n

L = L(x ;y ;y )!; ! = dx ^ ::: ^ dx ; n = dim X:

The corresp onding rst order Euler-Lagrange equations for sections s of the bred jet

manifold J Y ! X read

i i

@ s = s ;

@ L(@ + s @ +@ s @ )@ L =0: (2)

i j

We consider the Lie derivatives of Lagrangian densities in order to obtain di erential

conservation laws. Let

u = u @ + u @

be a vector eld on a bred manifold Y and u its jet lift (15) onto the bred jet manifold

1 1

J Y ! X . Given a Lagrangian density L on J Y , let us computer the Lie derivative

L L. On solutions s of the rst order Euler-Lagrange equations (2), wehave the equality

i i

s L L = [ (s)(u u s )+u L(s)]!; = @ L: (3)

i i

In particular, if u is a vertical vector eld such that

L =0; L

the equality (3) takes the form of the current conservation law

[u (s)] = 0: (4)

In gauge theory, this conservation law is exempli ed by the No ether identities.

Let

= @

be a vector eld on X and

u = = (@ + @ )

its horizontal lift onto the bred manifold Y by a connection on Y . In this case, the

equality (3) takes the form

s L s)]! (5) L = [ T (

where

i i

T ( s)= (s ) L (6)

is the canonical energy-momentum tensor of a eld s with resp ect to the connection on

Y . The tensor (6) is the particular case of the stress energy-momentum tensors [1, 3 , 6].

In particular, when the bration Y ! X is trivial, one can cho ose the trivial connection

= 0. In this case, the temsor (6) is precisely the standard canonical energy-momentum

tensor, and if

L L =0

for all vector elds on X (e.g., X is the Minkowski space), the conservation law (5)

comes to the well-known conservation law

T (s)= 0

of the canonical energy-momentum tensor.

In general, the Lie derivative L L fails to b e equal to zero as a rule, and the equality

(5) is not the conservation law of a canonical energy-momentum tensor. For instance, in

gauge theory of gauge p otentials and scalar matter elds in the presence of a background

world metric g ,we get the covariant conservation law

r t =0 (7)

of the metric energy-momentum tensor.

In Einstein's General Relativity, the covariant conservation law (7) issues directly from

gravitational equations. But it is concerned only with zero-spin matter in the presence

of the gravitational eld generated by this matter itself. The total energy-momentum

conservation law for matter and gravityisintro duced by hand. It reads

N )

[(g ) (t + T ]= 0 (8)

where the energy-momentum pseudotensor T of a metric gravitational eld is de ned

to satisfy the relation

N N

@ @ [(g ) (g g g g ) (g ) (t + T )=

2 3

on solutions of the Einstein equations. The conservation law (8) is rather satisfactory

only in cases of asymptotic- at gravitational elds and a background gravitational eld.

The energy-momentum conservation law in the ane-metric gravitation theory and the

gauge gravitation theory was not discussed widely [5].

Thus, the Lagrangian formulation of eld theory do es not provide us with any univer-

sal pro cedure in order to analize the energy-momentum conservation lawingravitation

theory, otherwise the covariantmultimomentum Hamiltonian formalism. In the framer-

work of this formalism, we get the fundamental identity (32) whose restriction to the

Lagrangian constraint space can b e treated the energy-momentum transformation lawin

eld theory [10, 14].

Lagrangian densities of eld mo dels are almost always degenerate and the corresp ond-

ing Euler-Lagrange equations are underdetermined. To describ e constraint eld systems,

the multimomentum Hamiltonian formalism can b e utilized [9, 11 , 12 ]. In the framework

of this formalism, the nite-dimensional phase space of elds is the Legendre bundle

= ^T X TX V Y (9)

Y Y

over Y into which the Legendre morphism L asso ciated with a Lagrangian density L on

1 i

J Y takes its values. This phase space is provided with the bred co ordinates (x ;y ;p )

such that

i i

(x ;y ;p ) L =(x ;y ; ):

i i

The Legendre bundle (9) carries the multisymplectic form

=dp ^ dy ^ ! @ : (10)

In case of X = R, this form recovers the standard symplectic form in analytical mechanics.

Building on the multisymplectic form , one can develop the so-called multimomentum

Hamiltonian formalism of eld theory where canonical momenta corresp ond to derivatives

of elds with resp ect to all world co ordinates, not only the temp oral one. On the math-

ematical level, this is the multisymplectic generalization of the standard Hamiltonian

formalism in analytical mechanics to bred manifolds over an n-dimensional base X , not

only R.Wesay that a connection on the bred Legendre manifold ! X is a Hamil-

tonian connection if the form c is closed. Then, a Hamiltonian form H on is de ned

to b e an exterior form such that

dH = c (11)

for some Hamiltonian connection .Every Hamiltonian form admits splitting

i i i

H = p dy ^ ! p ! H ! = p dy ^ ! H!; ! = @ c!; (12)

i i i

where is a connection on Y ! X . Given the Hamiltonian form H (12), the equality

(11) comes to the Hamilton equations

i i

@ y (x)= @ H; @ p (x)=@H (13)

i 4

for sections of the bred Legendre manifold ! X .

The Hamilton equations (13) are the multimomentum generalization of the standard

Hamilton equations in mechanics. The energy-momentum transformation law (32) which

we suggest is accordingly the multimomentum generalization of the conventional energy

transformation law (1). Its application to the Hamiltonian gauge theory in the presence of

a background world metric recovers the familiar metric energy-momentum transformation

law (7) [10 , 14].

The identity (32) remains true also in the gravitation theory. The to ols are nowat

hand to examine the energy-momentum transformation for gravity. In this work, we

restrict our consideration to the ane-metric gravitation theory. The key p oint consists

in the feature of a metric gravitational eld whose canonical momenta on the constraint

space are equal to zero [13].

2 Technical preliminary

A bred manifold

: Y ! X

is provided with bred co ordinates (x ;y ) where x are co ordinates of the base X .A

lo cally trivial bred manifold is termed the bundle. We denote by VY and V Y the

vertical tangent bundle and the vertical cotangent bundle of Y resp ectively. For the

sake of simplicity, the pullbacks Y TX and Y T X are denoted by TX and T X

X X

resp ectively.

On bred manifolds, we consider the following typ es of di erential forms:

(i) exterior horizontal forms Y ! ^ T X ,

(ii) tangent-valued horizontal forms Y ! ^ T X TY and, in particular, soldering

forms Y ! T X VY,

(iii) pullback-valued forms

r r

Y ! ^ T Y TX; Y ! ^ T Y T X:

Y Y

Horizontal n-forms are called horizontal densities.

Given a bred manifold Y ! X , the rst order jet manifold J Y of Y is b oth the

1 1

bred manifold J Y ! X and the ane bundle J Y ! Y mo delled on the vector bundle

T X VY:

We identify J Y to its image under the canonical bundle monomorphism

: J Y ! T X TY;

= dx (@ + y @ ): (14)

Given a bred morphism of : Y ! Y over a di eomorphism of X , its jet prolongation

1 1 1 0

J :J Y !J Y reads

0 1 i i

: y J =(@ +@ y )

Every vector eld

u = u @ + u @

on a bred manifold Y ! X gives rise to the pro jectable vector eld

i i i i

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

i j

1 1

on the bred jet manifold J Y ! X where J TY is the jet manifold of the bred manifold

TY ! X.

The canonical morphism (14) gives rise to the bundle monomorphism

1 1 i

b b b

: J Y TX 3 @ 7! @ = @ c 2 J Y TY; @ = @ + y @ :

X Y

This morphism determines the canonical horizontal splitting of the pullback

J Y TY = (TX) VY; (16)

J Y

i i i i

x_ @ +_y@ =_x(@ +y@)+(_y x_ y )@ :

i i i

In other words, over J Y ,wehave the canonical horizontal splitting of the tangent bundle

TY .

Building on the canonical splitting (16), one gets the corresp onding horizontal split-

tings of a pro jectable vector eld

i i i i

u = u @ + u @ = u + u = u (@ + y @ )+(u u y )@ (17)

i H V i i

on a bred manifold Y ! X .

Given a bred manifold Y ! X , there is the 1:1 corresp ondence b etween the connec-

tions on Y ! X and global sections

@ ) =dx (@ +

of the ane jet bundle J Y ! Y . Substitution of such a global section into the

canonical horizontal splitting (16) recovers the familiar horizontal splitting of the tangent

bundle TY with resp ect to the connection on Y . These global sections form the ane

space mo delled on the linear space of soldering forms on Y .

Every connection on Y ! X yields the rst order di erential op erator

D : J Y ! T X VY;

i i

D =(y )dx @ ;

on Y which is called the covariant di erential relative to the connection .

1 1

The rep eated jet manifold J J Y ,by de nition, is the rst order jet manifold of

1 i i i i

J Y ! X . It is provided with the adapted co ordinates (x ;y ;y ;y ;y ). Its subbundle

()

2 i i

J Y with y = y is called the sesquiholonomic jet manifold. The second order jet

()

2 2 i i

manifold J Y of Y is the subbundle of J Y with y = y :

3 Lagrangian formalism

Let Y ! X b e a bred manifold and L = L! a Lagrangian densityon J Y. With L, the

jet manifold J Y carries the uniqie asso ciated Poincare-Cartan form

i i

= dy ^ ! y ! + L! (18)

i i

and the Lagrangian multisymplectic form

j j i

=(@ dy + @ dy ) ^ dy ^ ! @ :

L j

i i

1 1

Using the pullback of these forms onto the rep eated jet manifold J J Y , one can construct

the exterior form

i i i

= d c =[y y )d +(@ @ @ )Ldy ] ^ !; (19)

L L L i

() i i

i i

b b

= dx @ ; @ = @ + y @ + y @ ;

()

1 1 2

on J J Y . Its restriction to the second order jet manifold J Y of Y repro duces the

familiar variational Euler-Lagrange op erator

i i i

E =[@ (@ +y @ +y @ )@ ]Ldy ^ !: (20)

L i i

The restriction of the form (19) to the sesquiholonomic jet manifold J Y de nes the

sesquiholonomic extension E of the Euler-Lagrange op erator (20). It is given by the

expression (20), but with nonsymmetric co ordinates y .

Let s b e a section of the bred jet manifold J Y ! X such that its rst order jet

1 0

prolongation J s takes its values into Ker E . Then, s satis es the rst order di erential

Euler-Lagrange equations (2). They are equivalent to the second order Euler-Lagrange

equations

j j

@ L(@ +@ s @ +@ @ s @ )@ L =0: (21)

i j

s = J s. for sections s of Y where

Wehave the following conservation laws on solutions of the rst order Euler-Lagrange

equations.

Let

u = u @ + u @

i 7

be a vector eld on a bred manifold Y and u its jet lift (15) onto the bred jet manifold

1 1

J Y ! X . Given a Lagrangian density L on J Y , let us compute the Lie derivative L L.

Wehave

i i i i

b b

L L=[@ ( (u u y )+u L)+(u u y )(@ @ @ )L]!; (22)

u i

i i

@ = @ + y @ + y @ :

s of the rst order Euler-Lagrange equations, the equality (22) comes to the On solutions

conservation law (3).

In particular, let

u = = (@ + @ )

b e the horizontal lift of a vector eld

= @

on X onto the bred manifold Y by a connection on Y . In this case, the equality (3)

takes the form (5) where T ( s) (6) are co ecients of the T X -valued form

i i

T ( s)=(c ) s =[ (s ) L]dx ! (23)

on X . One can think on this form as b eing the canonical energy-momentum tensor of a

eld s with resp ect to the connection on Y .

4 Multimomentum Hamiltonian formalism

Let b e the Legendre bundle (9) over a bred manifold Y ! X . It is provided with the

bred co ordinates (x ;y ;p ):

j 0 0

@y @x @x

0 1

p = J p ; J = det( ):

@y @x @x

By J is meant the rst order jet manifold of ! X . It is co ordinatized by

i i

(x ;y ;p ;y ;p ):

i () i

We call by a momentum morphism any bundle morphism : ! J Y over Y .For

instance, let b e a connection on Y . Then, the comp osition = is a momentum

morphism. Conversely,every momentum morphism determines the asso ciated connec-

b b

tion =0 on Y ! X where 0 is the global zero section of ! Y . Every

connection on Y gives rise to the connection

i i j

[@ + (y)@ +(@ (y)p (x)p + K (x)p ] (24) =dx K )@

i j

on ! X where K is a linear symmetric connection on T X .

The Legendre manifold carries the multimomentum Liouville form

= p dy ^ ! @

and the multisymplectic form (10).

The Hamiltonian formalism in bred manifolds is formulated intrinsically in terms of

Hamiltonian connections which play the role similar to that of Hamiltonian vector elds

in the symplectic geometry.

Wesay that a connection on the bred Legendre manifold ! X is a Hamiltonian

connection if the exterior form c is closed. An exterior n-form H on the Legendre

manifold is called a Hamiltonian form if there exists a Hamiltonian connection satisfying

the equation (11).

f f

Let H b e a Hamiltonian form. For any exterior horizontal density H = H ! on ! X ,

the form H H is a Hamiltonian form. Conversely,if H and H are Hamiltonian forms,

their di erence H H is an exterior horizontal densityon!X.Thus, Hamiltonian

forms constitute an ane space mo delled on a linear space of the exterior horizontal

densities on ! X .

Let b e a connection on Y ! X and its lift (24) onto ! X .Wehave the equality

e b

c =d(c):

A glance at this equality shows that is a Hamiltonian connection and

i i

H = c = p dy ^ ! p !

i i

is a Hamiltonian form. It follows that every Hamiltonian form on can b e given by the

expression (12) where is some connection on Y ! X . Moreover, a Hamiltonian form

has the canonical splitting (12) as follows. Given a Hamiltonian form H , the vertical

tangent morphism VH yields the momentum morphism

1 i i

c c

H :!J Y; y H = @ H;

and the asso ciated connection = H 0onY. As a consequence, wehave the canonical

splitting

H = H H:

The Hamilton op erator E for a Hamiltonian form H is de ned to b e the rst order

di erential op erator

i i i

E = dH =[(y @ H)dp (p + @ H)dy ] ^ !; (25)

H i

() i i

where is the pullback of the multisymplectic form onto J .

For any connection on ! X ,wehave

E =dH c :

H 9

It follows that is a Hamiltonian jet eld for a Hamiltonian form H if and only if it takes

its values into Ker E , that is, satis es the algebraic Hamilton equations

i i

= @ H; = @ H: (26)

Let a Hamiltonian connection has an integral section r of ! X , that is, r = J r .

Then, the Hamilton equations (26) are broughtinto the rst order di erential Hamilton

equations (13).

Nowwe consider relations b etween Lagrangian and Hamiltonian formalisms on bred

manifolds in case of semiregular Lagrangian densities L when the preimage L (q ) of each

p ointof q2Qis the connected submanifold of J Y .

Given a Lagrangian density L, the vertical tangent morphism VL of L yields the

Legendre morphism

b b

L : J Y ! ; p L = :

i i

Wesay that a Hamiltonian form H is asso ciated with a Lagrangian density L if H

satis es the relations

b c b

L H j = Id ; Q = L(J Y ); (27a)

Q Q

H = H + L H; (27b)

or in the co ordinate form

@ L(x ;y ;@ H)=p ; p 2Q;

i i i

j i

L(x ;y ;@ H)=p @ HH:

Notete that di erent Hamiltonian forms can b e asso ciated with the same Lagrangian

density.

Let a section r of ! X b e a solution of the Hamilton equations (13) for a Ha-

miltonian form H asso ciated with a semiregular Lagrangian density L.Ifrlives on the

constraint space Q, the section s = H r of J Y ! X satis es the rst order Euler-

Lagrange equations (2). Conversely, given a semiregular Lagrangian density L, let s be

a solution of the rst order Euler-Lagrange equations (2). Let H b e a Hamiltonian form

asso ciated with L so that

c b

H L s = s: (28)

Then, the section r = L s of ! X is a solution of the Hamilton equations (13) for H .

For sections s and r ,wehave the relations

s = J s; s = r

where s is a solution of the second order Euler-Lagrange equations (21).

We shall say that a family of Hamiltonian forms H asso ciated with a semiregular

Lagrangian density L is complete if, for each solution s of the rst order Euler-Lagrange 10

equations (2), there exists a solution r of the Hamilton equations (13) for some Hamilto-

nian form H from this family so that

b c

r = L s; s = H r; s = J ( r ): (29)

Such a complete family exists i , for each solution s of the Euler-Lagrange equations for

L, there exists a Hamiltonian form H from this family so that the condition (28) holds.

The most of eld mo dels p ossesses ane and quadratic Lagrangian densities. Com-

plete families of Hamiltonian forms asso ciated with such Lagrangian densities always exist

[10, 11 ].

5 Energy-momentum conservation laws

In the framerwork of the multimomentum Hamiltonian formalism, we get the fundamental

identity whose restriction to the Lagrangian constraint space recovers the familiar energy-

momentum conservation law [10, 14 ].

Let H b e a Hamiltonian form on the Legendre bundle (9) over a bred manifold

Y ! X . Let r b e a section of of the bred Legendre manifold ! X and (y (x);p (x)) its

lo cal comp onents. Given a connection on Y ! X ,we consider the following T X -valued

(n 1)-form on X :

T (r )=(cH ) r;

i i i i

T (r )=[p (y ) (p (y )H )]dx ! ; (30)

i i

where H is the Hamiltonian density in the splitting (12) of H with resp ect to the con-

nection .

Let

= @

be a vector eld on X . Given a connection on Y ! X ,itgives rise to the vector eld

i j i

@ +p @ )@ p @ @ +( p = @ +

i i

i j

on the Legendre bundle . Let us calculate the Lie derivative L H on a section r .We

have

+ d[ T (r )! ] ( cE ) r (31) R (L H ) r = p

V H

i e

where

R = R dx ^ dx @ =

i i i j i

(@ @ + @ @ )dx ^ dx @ ;

j j i

2 11

is the curvature of the connection , E is the Hamilton op erator (25) and

i i j i

= ( y )@ +( p @ p @ +p @ p )@

V i i

i i

j i

is the vertical part of the canonical horizontal splitting (17) of the vector eld on

over J . If r is a solution of the Hamilton equations, the equality (31) comes to the

identity

i j i i

(@ + @ @ p @ )H = T (r )+p R : (32)

i i

j i

On solutions of the Hamilton equations, the form (30) reads

i i

f f f

T (r )=[p @ H (p @ H H )]dx ! : (33)

i i

One can verify that the identity (32) do es not dep end up on choice of the connection .

For instance, if X = R and is the trivial connection, then

T (r )=Hdt

where H is a Hamiltonian function and the identity (32) consists with the familiar energy

transformation law (1).

To clarify the physical meaning of (32) when n>1, we turn to the Lagrangian

formalism. Let a multimomentum Hamiltonian form H b e asso ciated with a semiregular

Lagrangian density L. Let r b e a solution of the Hamilton equations for H which lives

on the Lagrangian constraint space Q and s the asso ciated solution of the rst order

Euler-Lagrange equations for L so that r and s satisfy the conditions (29). Then, wehave

T (r)=T (s)

where T (s) is the Lagrangian canonical energy-momentum tensor (23). It follows that

the form (33) may b e treated as a Hamiltonian canonical energy-momentum tensor with

resp ect to a background connection on the bred manifold Y ! X (or a Hamiltonian

stress energy-momentum tensor). At the same time, the identity (32) in gauge theory

turns out to b e precisely the covariant energy-momentum conservation law for the metric

energy-momentum tensor, not the canonical one [14].

In the Lagrangian formalism, the metric energy-momentum tensor is de ned to b e

: gt =2

In case of a background world metric g , this ob ject is well-b ehaved. In the framework of

the multimomentum Hamiltonian formalism, one can intro duce the similar tensor

gt =2 : (34)

Recall the useful relation

@ @

: = g g

@g g

If a multimomentum Hamiltonian form H is asso ciated with a semiregular Lagrangian

density L,wehave the equalities

i i

(q )= g g t (x ;y ;@ H(q)); t

(x ;y ; (z)) = g g t (z ) t

where q 2 Q, z 2 J Y and

c b

H L (z )= z:

In view of these equalities, we can think of the tensor (34) restricted to the Lagrangian

constraint space Q as b eing the Hamiltonian metric energy-momentum tensor. On Q,

the tensor (34) do es not dep end up on choice of a Hamiltonian form H asso ciated with L.

Therefore, we shall denote it by the common symbol t. Set

t = g t :

In the presence of a background world metric g , the identity (32) takes the form

i j i i

t f g g +( @ @ p @ )H = T + p R (35)

i i

j i

where

i i

= @ + @ y @ + @ p @

and by f g are meant the Cristo el symb ols of the world metric g .

When applied to the Hamiltonian gauge theory in the presence of a background world

metric, the identity (35) recovers the familiar metric energy-momentum transformation

law (7) [10 , 14].

6 Energy-momentum conservation laws

in ane-metric gravitation theory

The contemp orary concept of gravitation interaction is based on the gauge gravitation

theory with twotyp es of gravitational elds. These are tetrad garvitational elds and

Lorentz gauge p otentials. At present, all Lagrangian densities of classical and quantum

gravitation theories are expressed in these variables. They are of the rst order with

resp ect to these elds. Only General Relativity without fermion matter sources utilizes

traditionally the Hilb ert-Einstein Lagrangian density L which is of the second order

with resp ect to a pseudo-Riemannian metric. One can reduce its order by means of the

Palatini variables when the Levi-Civita connection is regarded on the same fo oting as a

pseudo-Riemannian metric. 13

Here we consider the ane-metric gravitation theory when there is no fermion matter

and gravitational variables are b oth a pseudo-Riemannian metric g on a world manifold X

and linear connections K on the tangent bundle of X .We call them a world metric and

aworld connection resp ectively. Given a world metric, every world connection meets the

well-known decomp osition in the Cristo el symb ols, contorsion and the nonmetricity term.

We here are not concerned with the matter interecting with a general linear connection,

for it is non-Lagrangian and hyp othetical as a rule.

In the rest of the article, X is an oriented 4-dimensional world manifold which ob eys

the well-known top ological conditions in order that a gravitational eld exists on X .

4 4

Let LX ! X b e the principal bundle of linear frames in the tangent spaces to X .

The structure group of LX is the group

GL = GL (4;R)

of general linear transfromations of the standard bre R of the tangent bundle TX. The

world connections are asso ciated with the principal connections on the principal bundle

LX ! X . Hence, there is the 1:1 corresp ondence b etween the world connections and

the global sections of the principal connection bundle

C = J LX=GL : (36)

Therefore, we can apply the standard pro cedure of the Hamiltonian gauge theory in order

to describ e the con guration and phase spaces of world connections [11 , 12 , 13 ].

There is the 1:1 corresp ondence b etween the world metrics g on X and the global

sections of the bundle of pseudo-Riemannian bilinear forms in tangent spaces to X .

This bundle is asso ciated with the GL -principal bundle LX . The 2-fold covering of the

bundle is the quotient bundle

=LX=S O (3; 1) (37)

where by SO(3; 1) is meant the connected Lorentz group.

Thus, the total con guration space of the ane-metric gravitational variables is rep-

resented by the pro duct of the corresp onding jet manifolds:

1 1

J C J : (38)

Given a holonomic bundle atlas of LX asso ciated with induced co ordinates of TX and

T X , this con guration space is provided with the adapted co ordinates

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

Also the total phase space of the ane-metric gravity is the pro duct of the Leg-

endre bundles over the ab ove-mentioned bundles C and . It is co ordinatized by the

corresp onding canonical co ordinates

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

On the con guration space (38), the Hilb ert-Einstein Lagrangian density of General

Relativity reads

L = g F g!; (39)

F = k k + k k k k :

" "

The corresp onding Legendre morphism is given by the expressions

L =0; p

g: (40) ( g g ) = L = p

These relations de ne the constraint space of General Relativityinmultimomentum

canonical variables.

Building on the set of connections on the bundle C , one can construct the complete

family of multimomentum Hamiltonian forms asso ciated with the Lagrangian density (39).

To minimize it, we consider the following subset of these connections.

Let K be a world connection and

" "

S = [k k k k + @ K + @ K

K " "

" " "

2K (K k )+K k + K k

" " " "

()

" "

K k K k ]

" "

the corresp onding connection on the bundle C (36). Let K b e another symmetric world

connection which induces an asso ciated connection on the bundle . On the bundle C ,

we consider the following connection

0 " 0 "

= K g K g ;

" "

= S R (41)

where

" "

R = K K + K K K K

" "

is the curvature of the connection K . The corresp onding multimomentum Hamiltonian

form is given by the expression

H =(p dg + p dk ) ^ ! H !;

HE HE

" " 0 0

g )+p H = p g + K S (K

" HE "

(p R ): (42)

It is asso ciated with the Lagrangian density L . We shall justify that the multimo-

mentum Hamiltonian forms (42) parameterized by all the world connections K and K

constitute the complete family. 15

Given the multimomentum Hamiltonian form H (42) plus that H of matter, the

HE M

corresp onding covariant Hamilton equations for General Relativity read

0 " 0 "

@ g + K g + K g =0; (43a)

" "

@ k = S R ; (43b)

" "

0 0

K K + p = p @ p

" "

1 1 @ H

(R g R) g ; (43c)

2 2 @g

"[ ] [ ] "

k k = p + p @ p

" "

" "( ) ( ) "

K K K p p + p : (43d)

" "

(" )

The Hamilton equations (43a) and (43b) are indep endent of canonical momenta and so,

reduce to the gauge-typ e condition. The gauge-typ e condition (43b) breaks into two parts

F = R ; (44)

@ (K k )+@ (K k )

" " "

2K (K k )+K k + K k

() " " " "

" "

K k K k =0: (45)

" "

It is readily observed that, for a given world metric g and a world connection k , there

always exist the world connections K and K such that the gauge-typ e conditions (43a),

(44) and (45) hold (e.g. K is the Levi-Civita connection of g and K = k ). It follows that

the multimomentum Hamiltonian forms (42) consitute really the complete family.

Being restricted to the constraint space (40), the Hamilton equations (43c) and (43d)

comes to

1 1 @ H

(R g R) g = ; (46)

2 @g

p p p

D ( gg ) D ( gg )+ g[g (k k )

+g k )+ g k )] = 0 (47) (k (k

where

g : g + k D g = @ g + k

Substituting Eq.(44) into Eq.(46), we obtain the Einstein equations

1 1

F g F = t (48)

where t is the metric energy-momentum tensor of matter. It is easy to see that Eqs.(47)

and (48) are the familiar equations of gravitation theory phrased in terms of the general-

ized Palatini variables. In particular, the former is the equation for the torsion and the 16

nonmetricity term of the connection k . In the absence of matter sources of a general

linear connection, it admits the well-known solution

k V ; = f g

D g = V g

where V is an arbitrary covector eld corresp onding to the well-known pro jective freedom.

Turn now to the identity (32). It takes the form

i j i

f f

(@ + @ + @ @ p @ )(H + H )=

i i HE M

(T R + T )+p + p R (49)

where T and T are the canonical energy-momentum tensors of ane-metric gravity and

matter resp ectively.

The energy-momentum tensor (33) of ane-metric gravity on solutions of the Hamilton

equations reads

1 1

R = P g: (50) H = T =

" HE

2 2

It follows that

(@ + @ )H = T :

One can verify also that, on solutions of the Hamilton equations, the curvature of the

connection (41) vanishes. Then, the identity (49) takes the form (35). In gauge theory,

it is reduced to the familiar conservation law (7). This is however a very particular case

b ecause the Hilb ert-Einstein Hamiltonian density H is indep endent of gravitational

momenta.

Since the canonical momenta p of the world metric are equal to zero and the

Hamilton equation (43c) comes to

f f

@ (H + H )=0;

HE M

the equality (49) can b e rewritten

i j i i

)H = @ p @ @ +( @ R : (51) R + T )+p (T

M i i M

j i

In gauge theory, it takes the form

@ R p R = (T + T ): (52)

It is the form of the energy-momentum transformation law whichwe observe also in the

case of the quadratic Lagrangian densities of ane-metric gravity.

As a test case, let us consider the sum

1 1

L =( g F g g g g F g! (53) + F )

2 4"

of the Hilb ert-Einstein Lagrangian density and the Yang-Mills one. The corresp onding

Legendre morphism reads

L =0; (54a) p

()

L =0; (54b) p

[] "

+ L = p g g g g F g: (54c)

To construct the complete family of multimomentum Hamiltonian forms asso ciated

with the Lagrangian density (53), we consider the following connection

0 " 0 "

= K g K g ;

" "

= S (55)

on the bundle C where we utilize the notations of the expression (41). Then the

multimomentum Hamiltonian forms

H =(p dg + p dk ) ^ ! H!;

0 " 0 "

H = p (K g + K g )+p S

" "

[] ["] "

+ g g g g (p )(p ) (56)

are asso ciated with the Lagrangian density 53) and constitute the complete family.

The corresp onding Hamilton equations read

0 " 0 "

@ g + K g + K g =0; (57a)

" "

["] "

@ k = S + "g g g g (p ); (57b)

@ H @ H

@ p = ; (57c)

@g @g

"[ ] [ ] "

= p k + p k @ p

" "

" "( ) ( ) "

p K p + p : (57d) K K

" "

(" )

The equation (57b) breaks into the equation (54c) and the equation (45). It is readily

observed that, for a given world metric g andaworld connection k , there always exist the

world connections K and K such that the gauge-typ e conditions (57a) and (45) hold (e.g.

K is the Levi-Civita connection of g and K = k ). It follows that the multimomentum

Hamiltonian forms (56) consitute really the complete family.

Substituting Eq.(57b) into Eq.(57c) on the constraint space (54a), we get the Einstein

equations. On the constraint space (54b) - (54c), the equation (57d) is the Yang-Mills

generalization

"[ ] [ ] "

D p = @ p + p k p k =0

" " 18

of the equation (47).

Turn now to the energy-momentum conservation law. For the sake simplicity, let us

consider the splitting of the multimomentum Hamiltonian form (52) with regard to the

connection (41) where the Hamiltonian densityis

f f

R H =H+ p :

Wehave the relation

@ H

= R

on solutions of the Hamilton equations. The canonical energy-momentum tensor asso ci-

ated with the Hamiltonian density H is written

1 "

[] [] ["] "

T = p R + g g g g p (p )

2 2

["] "

g g g g (p )) = (H +

1 1 1

R R + R ( R R + R):

" 4" 2

The identity (32) takes the form

@ @

i j i

f f

(@ + @ + @ @ p @ p S )(H + H )=

i i M

@k @P

(T + T )+p R + p R

On solutions of the Hamilton equations, wehave

@ @ d

i j i i

f f

( @ @ p @ )H p S (T + T )+p R H =

i i M M

j i

@k @p dx

where the term

@ @

p S H = k ( R R k p R

( )

@k @p

do es not vanish. In case of gauge theory and gravity without non-metricity,we obtain

1 1 d

(T + T ): (58) K K R R + R p R =

( ) ( )

" 2 dx

Let us cho ose the lo cal geo detic co ordinate system at a p oint x 2 X . Relative to this

co ordinate system, the equality (58) at x comes to the conservation law

(T + t )= 0:

dx 19

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