fulltext

Metric-affine gravitation theory and superpotentials (*)

Dipartimento di Matematica e Fisica, Università di Camerino 62032 Camerino (MC), Italy

(ricevuto il 17 Luglio 1996; approvato il 30 Agosto 1996)

Summary. — We consider a metric-affine theory of gravity in which the dynamical fields are the Lorentzian metrics and the non-symmetric linear connections on the world manifold X. Working with a Lagrangian density which is invariant under general covariant transformations and using standard tools of the calculus of variations, we study the corresponding currents. We find that the superpotential takes a nice form involving the torsion of the linear connection in a simple way and generalizing the well-known Komar superpotential. A feature of our approach is the use of the Poincaré-Cartan form in relation to the first variational formula of the calculus of variations.

PACS 02.40 – Geometry, differential geometry and topology.

PACS 04.20.Fy – Canonical formalism, Lagrangians, and variational principles.

1. – Introduction

As is well known, generally covariant field theories lead to strong conservation laws. This means that when the field equations are satisfied the current Jl associated with the energy-momentum tensor can be written as

Jl

4 dmUlm, (1)

where dm is the formal (total) derivative and the skew-symmetric tensor density Ulm is called the superpotential. For example, in the purely metric Einstein’s gravitation theory, the superpotential corresponding to the Einstein-Hilbert Lagrangian density LEH4k2g R takes the form

Ulm₄k_{2g (˜}aumgla2 ˜aulgma) . (2)

This is the well-known Komar superpotential [1] associated with the vector field ul_on

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: mangiarottiHcamvax.cineca.it

the world manifold X. Here ˜a denotes the covariant derivative with respect to the Levi-Civita connection.

In this paper we are concerned with the metric-affine theory of gravity. Its configuration space is coordinatized by (xl_{, g}lm_{, k}a

bl), where xl are local coordinates on the world manifold X, while glm _{and k}a

blare the coefficients of a Lorentzian metric and a non-symmetric linear connection on X, respectively. We consider a Lagrangian density L(xl, glm, kabl, kabl , m), where as usual kabl , m4 ¯mkabl, which is invariant under general covariant transformations. Then, using the Lagrangian formalism to study the associated currents [2-4], we find that the corresponding superpotential takes the nice form

Ulm

4 pabml(˜bua1 Tasbus) . (3)

Now ˜b is the covariant derivative with respect to the linear connection kabl, Tasb4 ka

sb2 kabs are the corresponding torsion parameters, and pablm4 ¯LO¯kabl , m (4)

are the momenta corresponding to the connection parameters. The quantity (3) generalizes the Komar superpotential (2). Indeed, in the case of the Einstein-Hilbert Lagrangian density, using the Palatini variables, we have

pablm4k2g (gbmdla2 gbldma) , (5)

so that (3) produces the Komar superpotential (2).

There are different methods to discover differential conservation laws in Lagrangian field theories. In this work we are concerned with the so-called symmetry method. To illustrate the generality of the situation, in the following section we first consider some basic concepts of the Lagrangian formalism. Then we derive our main tool, the variational formula. Section 3 contains the characterization of generally covariant Lagrangians (in the metric-affine gravitation theory) which is used, in the final section, to derive our results concerning currents and superpotentials.

2. – Lagrangian formalism

In this section we consider some basic tools of the Lagrangian formalism [5]. In particular, we use the Poincaré-Cartan form to derive, in a new way, both the Euler-Lagrange operator and the (canonical) energy-momentum tensor. These objects will be the ingredients of our basic formula, namely the variational formula.

2.1. General considerations. – Let us consider a (first-order) field theory over a world manifold X locally parametrized by the coordinates xl

, 1 GlGm4dim X, whose classical fields are denoted by yi

, 1 GiGl. As usual, we will denote by ¯l4 ¯O¯xl, ¯i4 ¯O¯yi, yli4 ¯lyi (the field derivatives) and ¯il4 ¯O¯yli.

A basic object is the canonical Liouville form [4] U 4piluiR vl, (6)

where pil are the field momenta associated with the fields yi and ui_{4 dy}i_{2 y}lidxl, vl4 ¯lFv , v 4dx1R R R dxm. (7)

Here the symbols R and F stand for the wedge product and the inner product, respectively.

Later we will call configuration space, velocity phase space and momentum phase space of the field system the manifolds locally coordinatized by (xl_{, y}i_{), (x}l_{, y}i_{, y}

li), and (xl_{, y}i_{, p}

il), respectively.

As is well known, the starting point of the Lagrangian formalism is a Lagrangian density L 4L(xl, yi, yli) v. As in analytical mechanics, we can use the Legendre map

pil4 ¯O¯ilL

to construct the Liouville form associated with L, that is U_{L4 ¯}ilLuiR vl. (8)

Then the Poincaré-Cartan form associated with the Lagrangian density L can be defined as [4]

HL4 L 1 UL, HL4 ¯ilL dyiR vl2 HLv , HL4 yli¯ilL 2L . (9)

Taking the vertical exterior differential [4] of U_L, we get

.

/

´

2 _yj_{. Then the Euler-Lagrange operator associated with the} Lagrangian density L can be defined as [4]

. / ´ EL4 d HL2 VL, EL4 (¯iL 2dl¯ilL) v 7 dyi, dl4 ¯l1 ylj¯j1 ylmj ¯jm, (11)

where d is the exterior differential.

2.2. The variational formula. – A slight modification of the Poincaré-Cartan form HLas defined in (9) leads to the (canonical) energy-momentum tensor. First of all, note

that the Liouville form U_L(associated with _{L) given by (8) can be written in the} following way:

UL4 ¯ilLvl7 ui. (12)

Then using the canonical injection

i : vO vl7 dxl, (13)

we define the (canonical) energy-momentum tensor associated with the Lagrangian density L as . / ´ T 4ii L 1UL, T 4vl7 [ (dlmL 2ymi¯ilL) dxm1 ¯ilL dyi] . (14)

Of course, T is a true tensor globally defined if L is so. To understand the meaning of (14) let

u 4ul¯l1 ui¯i, (15)

be a vector field on the configuration space of the field system, where ul _{are local} functions on X while ui can also depend on the field variables. Then the corresponding vector field on the velocity phase space reads [2]

. / ´ u 4ul¯l1 ui¯i1 uli¯il, uli4 dlui2 ymi¯lum4 ¯lui1 yl j ¯jui2 ymi¯lum. (16)

Note that u is vertical if ul4 0, In general, the vertical part uVof u can be defined as

follows:

uV4 (ui2 yliul) ¯i. (17)

Note that u 4uV iff ul4 0.

Given a vector field u, the current J associated with it is defined by the contraction of u with the energy-momentum tensor T given by (14), i.e.

. / ´ J 4uFT , J 4Jlvl, Jl4 ulL 2umymi¯ilL 1ui¯ilL . (18)

Equivalently, using the Poincaré-Cartan form HLgiven by (9) and the well-known

contact form

l 4dxl

7 (¯l1 yli¯i) , (19)

we see that the current J can also be written as J 4l*(uF HL) ,

(20)

where the projection l *, given by l * dxl

4 dxl, l * dyi

4 ylidxl, (21)

is the transpose of the injection l in (19). The current J can be integrated over appropriate (m 21)-dimensional hypersurfaces of the world manifold X to produce global conserved quantities [6].

Lemma 2.1 (The variational formula). Let L be a Lagrangian density and u be a vector field on the configuration space as in (15). Then the following identity holds:

Lu L 4uVFEL1 dHJ .

On the left side of this identity we have the Lie derivative of L with respect to u. On the right side we have two terms: the first is the contraction of the Euler-Lagrange operator ELwith the vertical part uVof u, while the second is the horizontal (formal)

derivative of the current J, i.e.

dHJ 4dlJlv . (23)

Proof. It is a direct check which follows from (16), (11), (17) and (20). _r Remark 2.2 (Noether’s theorem). – Suppose that the vector field u is a symmetry of the Lagrangian density L, i.e. Lu L 40. Then dHJ 40 on the solutions of the field

equations.

3. – Metric-affine gravitation theory

The total configuration space of the metric-affine gravitation theory is parametrized by (xl_{, g}lm_{, k}a

bl), where glmand kablare the coefficients of a Lorentzian metric and the connection parameters of a non-symmetric linear connection on the world manifold X, respectively (of course, X is assumed to satisfy the well-known topological conditions [7] to be provided with a Lorentzian metric). In this section, first we characterize the generally covariant Lagrangian densities related to the configuration space of the metric-affine gravitation theory, then, in particular, we consider those whose dependence on the linear connections on X is through their curvature tensor. Moreover, some interesting properties of the curvature tensor are discussed.

3.1. Generally covariant Lagrangians. – It is easily seen that a vector field u 4ul

¯l

on the world manifold X (ul_{are local functions on X) induces vector fields u}

Mand uCon

the configuration spaces of the Lorentzian metrics [3] and the linear connections. The field uM is given by

uM4 ul¯l1 (gla¯lub1 glb¯lua) ¯ab (24)

(¯ab4 ¯O¯gab). Now from the well-known affine transformation rule of the connection parameters under a change of coordinates, i.e.

.

/

´

(¯lmn4 ¯O¯klmn). As one can easily check directly, (24) and (26) preserve the Lie bracket of vector fields, i.e. they are representations of R-Lie algebras.

Using (16), we can write the corresponding vector field on the velocity phase space of the linear connections. Then taking the direct sum of this representation with (24),

we get the following representation of R-Lie algebras:

.

`

/

`

´

Now let us consider a Lagrangian L 4L(xl_{, g}lm_{, k}a

bl, kabl , m) v. Then the condi-tion for L to be generally covariant is

Lu L 40 (28)

for each vector field u on the world manifold X.

Using the explicit expression given in (27), we see that (28) is equivalent to the following four conditions:

(29) pabml1 palbm1 pamlb1 pambl1 palmb1 pablm4 0 , (30) kg bmpabml2 kbampbgml2 kbmapbmgl1 ¯aglL 1 1klbmpabmg2 kbampblmg2 kbmapbmlg1 ¯algL 40 , (31) 2 gbl ¯abL 1Ldla2 kbnm , apbnml1 klbm¯abmL 1 1klbn , mpabnm2 kbam¯blmL 2kban , mpblnm2 kbma¯bmlL 2kbna , mpbnlm4 0 , ¯lL 40 , (32) where pabml4 ¯L ¯ka bm , l (33)

are the momenta corresponding to the connection parameters ka bm.

Remark 3.1. As in gauge theory [3], let us consider the curvature 2-form

.

/

´

The 2-form R can also be seen as a 2-form V in the following way: V 4 [ dkabmR dxm1 1 2(k g blkagm2 kgbmkagl) dxl7 dxm] R eab, (35)

which looks like a generalized symplectic structure on the configuration space of the linear connections.

There is a canonical linear connection D whose connection parameters are [8] ¯gelFDeab4 0 ,

(36)

¯lFDeab4 (kbeldga2 kgaldeb) ege. (37)

One can easily check that V is closed with respect to the exterior covariant derivative induced by D, i.e. DV 40. Of course, this is just the Bianchi identity.

Let u 4ul

¯l be a vector field on X. Then, denoting by ˜ the covariant derivative associated with a linear connection ka

bl, we have ˜_{u 4 (¯}lua2 kablub) eal. (38)

Using the torsion

T 4 1 2 (k

a

bl2 kalb) dxbR dxl, (39)

we can modify ˜ by adding T, thereby getting the new connection ˜

×

u 4˜u1uFT , ˜×_{u 4 (¯}lua2 kalbub) eal. (40)

Now applying the canonical covariant derivatives (36) and (37) to ˜×u, we find D(˜×_{u) 4 ]2u}m_dka

bm1 [umba1 (kgblkagm2 kgbmkagl) ul] dxm( 7 eab, (41)

where umba is given by (27).

It follows immediately that the vector field (26) is uniquely determined by the condition

uCFV 4D(˜

× u) , (42)

where V is given by (35). Thus the vector field uC is the Hamiltonian vector field

associated with u.

3.2. Dependence through the curvature. – Let us consider a Lagrangian density whose dependence on the linear connections on X is only through their curvatures, i.e.

L 4 LiR ,

(43)

where R is given by (34). It is easily seen that the condition (43) is equivalent to the two following ones [3]:

pablm1 pabml4 0 , (44)

¯aglL 1kgbmpabml2 kbampbgml4 0 . (45)

Of course, the conditions (44) and (45) imply the conditions (29) and (30), respectively. The other two conditions (31) and (32) are just equivalent to L being invariant under the infinitesimal transformations of the world manifold X.

4. – Currents and superpotentials

Let us consider a generally covariant Lagrangian density L as in (43). Recalling (11), we see that the field operator for the metric is simply

Eab4 ¯abL , (46)

while that for the connection it is

Eabl4 ¯ablL 2dmpablm. (47)

From (18) we get the current (48) Jl

4 ulL 2um_ka

bn , mpabnl1 (kgbn¯gua2 kagn¯bug2 kabm¯num1 ¯nb2 ua) pabnl, where we have used (26).

We have the following lemma.

Lemma 4.1. Using only the condition (31), the current (48) can be written as Jl_{4 2 ( 2 g}bl Eab1 klbn Eabn2 kban Ebln2 kbnaEbnl) ua2

(49)

2dm(klbnpabnm2 kbanpblnm) ua1

1dm(kbnapbnlm) ua2 kabmpabnl¯num1 dm(pbmla ¯bua) 1 1(kgbn¯gua2 kagn¯bug) pabnl2 dm(pabml) ¯bua.

Proof. Using (46) and (47), the condition (31) can be rewritten as follows: (50) 2 gbl Eab1 Ldla2 kbnm , apbnml1 klbn Eabn1 klbndmpabnm1 klbn , mpabnm2 2kban Ebln2 kbandmpblnm2 kban , mpblnm2 kbnaEbnl2 kbnadmpbnlm2 kbna , mpbnlm4 0 , or, equivalently, (51) Ldla2 kbnm , apbnml4 2 ( 2 gbl Eab1 klbn Eabn2 kban Ebln2 kbnaEbnl) 2 2 dm(klbnpabnm2 kbanpblnm2 kbnapbnlm) . Now the result (49) follows immediately by substituting (51) into the expression (48) of the current Jl_.

r

Let us consider the last three rows in (49) separately. The first row can be rewritten as follows:

(52) _{2 d}m(klbnpabnm2 kbanpblnm) ua4 dm(¯almL) ua4

where we have used (44) and (45). The second row leads to (53) dm(kbnapbnlm) ua2 kabmpabnl¯num1 dm(pabml¯bua) 4

4 2dm[pablm(¯bua2 kabgug) ] 4dm]pabml[˜bua1 (kagb2 kabg) ug]( , where we have used (44). The expression of the covariant derivative ˜buais that in (38). Finally, the third row yields

(54) (kg

bn¯gua2 kagn¯bug) pabnl2 dm(pabml) ¯bua4

4 (kgbnpabnl2 kbanpbgnl) ¯gua2 dm(pabml) ¯bua4 4 (2¯ablL 1dmpablm) ¯bua4 2 Eabl¯bua, where we have used (44) and (45).

Collecting these results together, we get the following proposition.

Proposition 4.2. Let L be a generally covariant Lagrangian density as in (43). Then, on the solutions of the field equations associated with (46) and (47), the corresponding superpotential is

Ulm_{4 p}abml[˜bua1 (kagb2 kabg) ug] 4pabml˜ ×

bua, (55)

where ˜× has been defined in (40).

Of course, the superpotential (55) is defined up to a closed (m 22)-form. Note that the covariant derivative ˜×u appearing in the superpotential (55) is just the same which determines the Hamiltonian lift uC in (42).

Remark 4.3. From (49), using (52), (53) and (54), we see that the current Jl_{can be} written in the following way:

Jl 4 dm(pabml˜ × bua) 22gbl Eab2 Eabl˜ × bua1 (56) 1(dm Ealm2 klbm Eabm1 kbam Eblm) ua.

The intrinsic character of each term of the first row in (56) is obvious. It may be interesting to see directly that also the second row in (56) is intrinsic. Indeed, we have

(57) (dm Ealm2 klbm Eabm1 kbam Eblm) ua4

4 dm( Ealmua) 2klbm Eabmua2 Ealm˜mua4 ˜m( Ealmua) 2 Ealm˜mua, where ˜m( Ealmua) denotes the exterior covariant derivative of the tangent valued (m 21)-form Ealmua. Thus the covariant character of the current (56) is manifest.

* * *

This work was supported by GNFM-CNR, by the National Research Project 40% “Geometria e Fisica» of the MURST, and by the University of Camerino.

R E F E R E N C E S

[1] KOMAR A., Phys. Rev., 113 (1959) 934.

[2] MANGIAROTTIL. and MODUGNO M., Ann. Inst. Henri Poincaré A, 39 (1983) 29. [3] GIACHETTA G. and MANGIAROTTI L., Int. J. Theor. Phys., 29 (1990) 789. [4] GIACHETTA G. and MANGIAROTTI L., Int. J. Theor. Phys., 34 (1995) 2351.

[5] MANGIAROTTIL. and MODUGNOM., in Proceedings of the International Meeting on Geometry and Physics, Florence, October 12-15, 1982, edited by M. MODUGNO (Pitagora Editrice, Bologna) 1983, p. 135.

[6] GOLDBERGN., in General Relativity and Gravitation. One Hundred Years After the Birth of Albert Einstein, edited by A. HELD, VOL. 1 (Plenum Press, New York) 1980, p. 469. [7] DODSONC. T. J., Categories, Bundles and Spacetime Topology (Kluwer Academic Publishers,

London) 1988.