G en eral Rela tiv ity an d G ravi tation , Vol. 2 9, No. 1 , 1 997

Pro ject ive Invariance and Einstein’s Equations Giovan ni Giach ett a1 an d Luigi Man giar otti 1 Rece ived Ju n e 27, 1995 , Re v. ver sion J u n e 3, 1996 It is shown t hat a project ively invariant Lagran gian ® eld t heory based on linear non-sy m m et ric conn ect ions in space-t im e an d arb it rary source ® elds is equ ivalent t o E inst ein’ s st and ard t heory of grav it at ion cou pled t o a source Lagran gian dep en ding sole ly on t he original source ® elds. A key p oint is that , as in the case of Lagran gian ® eld theories based on sy m m et ric con nect ions in space-t im e, t he E uler- Lagrang e ® eld equat ions uniquely det erm ine t he project ive invariant part of t he linear con nect ion in term s of t he m et ric, their ® rst -order derivat ives, the source ® elds, an d t heir con jugat e m om ent a. KE Y W ORDS : E quivalence of t heories of gravit at ion

1. INTRODUCTION In Einst ein’ s standard theory of gravit ation the metric tensor gm u describes the geometric prop erties of t he pseudo-Riemannian space-t ime and the linear connect ion C l b a is taken to be t he Levi-Civit a connect ion, t hat is, Cl b a =

¶

1 as ( l gs b 2g

+ ¶

b gs l

±

¶ s gl b )

in a coordinat e chart . In the metric-a ne theory, ® rst discussed by Palatini, the metric tensor and the symmetric connect ion are considered as a priori indep endent dynamical ® elds. T he ® eld equat ions not only produce the Einst ein’ s equat ions in a vacuum but also det ermine the connect ion to be the Levi-Civit a connect ion of the metric tensor. A third formulat ion 1

Dep art m ent of Mat hem at ics and P hy sics, Univers ity of C am erino, I-62032 Cam erino, It aly 5 0001-7701/ 97/ 0100-0005$09.50/ 0

1997 P lenum P ublishing Corporation

6

G ia ch e t t a a n d M a n g ia rot t i

of general relat ivity is the purely a ne t heory, ® rst int roduced by J . Kijowski [1]. In this formulat ion the fundament al ® eld is the symmetric linear connect ion, while the metric tensor appears as a momentum canonically conjugat e to the connect ion. In Ref. 1 it is shown that a t heory based on a Lagrangian density depending on the derivat ives of the symmetric connect ion coe cients through the symmetric part of the Ricci tensor is equivalent to Einst ein’ s st andard t heory of gravit ation. In the last decades, much attention has been devot ed to theories which generalize general relat ivity. Torsion and non-met ricity have been called int o play to describ e the geometry of the space-t ime. In Refs. 2- 4 the aut hors consider variat ional principles based on a Lagrangian which, for the geometric part , is the simplest generalizat ion of t he scalar curvat ure, i.e., L G = ± det ( ga b ) g m u R m u (C) is constructed from t he non-symm etric connect ion and the met ric. By taking indep endent variat ions of the metric, torsion and non-m etricity, or of the metric, torsion and the symmetric part of t he connect ion, when the non-met ricity vanishes the aut hors of Refs. 2 and 3 recover the ® eld equat ions of t he U4 theory of gravit ation described by Hehl et al. [5]. In Ref. 4 no a prior i relat ion between the non-symm etric connect ion and the metric tensor is assumed. Instead, the Lagrangian is taken to be invariant under project ive transform ations of the connect ion. It is shown that the result ing theory is equivalent to Einst ein’ s standard theory of gravit ation wit h a modi® ed source ® eld Lagrangian. Other generalizat ions of the general relat ivity theory concern t he geometric Lagrangian densit ies; these are not necessarily the scalar curvat ure, but general non-linear funct ions of the curvat ure tensor. In Ref. 6 it is shown that for a general Lagrangian density L(j 2 g, j 1 C), t he condit ion that the Lagrange equat ions involve no t hird- or higher-order derivat ives of the metric tensor requires the gravit ational ® eld equat ions t o be equivalent to those of general relat ivity with modi® ed sources (here, for a section s of a ® ber bundle, j k s denot es the k-order jet of s ) . In Ref. 7 the aut hors consider a large class of a ne theories of gravit ation with non-symm etric connect ions, based on general nonlinear Lagrangians depending on the derivat ives of the connect ion param eters via some traces of the curvat ure only. T hey prove that all such theories are equivalent to Einst ein’ s standard theory of gravit ation wit h addit ional matter sources. In t his paper we consider theories of gravit ation based on a Lagrangian

Ö

L(K m u (j 1 C), Ñ

lu

i

,u

i

),

(1)

depending on a linear non-sym metric connect ion C and some source ® elds u i . T he dependence on C is assumed to be only through the symmetric

P r o j e c t iv e In v a r i a n c e a n d E in s t e in ’ s E q u a t io n s

7

part of the Ricci tensor K m u and the covariant derivat ive Ñ l u i . Moreover, since t he symmetric part of the Ricci tensor is invariant under project ive transform ations of the connect ion (i.e. transformat ions which preserve geodesics up to a reparametrizat ion) , in order to avoid algebraic constraint s on the source ® elds which can lead to inconsist encies, it is assumed that the covariant derivat ive Ñ l u i depends only on the project ive invariant part of the connect ion. We prove that any such theory is equivalent to Einst ein’ s st andard theory of gravit ation coupled t o a source Lagrangian depending on ly on t he original source ® elds. Hence, our paper extends the result s of Ref. 4 to a larger class of Lagrangian densit ies. In Ref. 7 the non-symm etric connect ion C l b a is split as follows:

~l b a + D l b a + Cl b a = C

1 a d 4 b (A l

±

~l s s ), C

~ l b a is a symmetric connect ion, D l b a is a skew-symmetric traceless where C tensor ® eld, and A l = C l s s is a connect ion on the bundle of scalar densit ies on the space-t ime. Using this decomposit ion, the aut hors treat the theory ~l b a int eracting as an a ne theory based on the symmetric connect ion C a with the addit ional `matter ® elds’ D l b and A l . In the present paper we consider a smaller class of Lagrangians than the aut hors of Ref. 7 do. However, we base our considerat ions on a diŒerent split ting of the non-symm et ric connect ion, nam ely Cl b a = * Cl b a + where * *

a

Cl b

a

= Cl b

a

±

2 d a 3 Cl b

,

2 3

Cl d

a b

,

C l = C [l b ] b .

(2)

(3)

Here C l b is the project ive invariant part of the connect ion. It turns out that for theories considered in t his paper this is the most appropriat e decomp osit ion of the connect ion. Essentially, the reason is that the ® eld equat ions deduced from the Lagrangian (1) uniquely determine t he projective invariant part * C l b a of the non-sym metric connect ion C l b a (which incorparat es “ part ” of the torsion) , and not just the symmet ric part . T he project ive invariant part of C l b a is determined in terms of the metric coe cient s (regarded as part of the momentum variables conjugat e to the connect ion param eters), their ® rst-order derivat ives, the source ® elds, and their conjugat e momenta. About the importance of results of this kind, we completely agree with t he statement contained in the Introduct ion of Ref. 7: “ T his is a mathematical result and does not dep end on t he physical (or philosophical)

8

G ia ch e t t a a n d M a n g ia rot t i

int erpretation of the variables . . .. T he relevance of such a result consist s in the fact that it enables us to analyze the dynam ical content of the theory (Cauchy problem , energy posit ivity, stability, et c.) using standard tools . . .” T he approach used in this paper is based on the theory of symplectic relat ions and their generat ing funct ions [8- 10]. T he ® eld equat ions are regarded as a subm anifold of J 1 P , the ® rst jet manifold of P ® M , where P is the phase space of the theory. T his is explained in Section 2. In Section 3 we show that the Lagrange equat ions deduced from (1) and the Einst ein equat ions coupled to a certain source Lagrangian, depending only on the original source ® elds, are just two diŒerent ways to look at the same ® eld equat ions. Finally, in t he last section, a simple example which may serve to illust rate t he theory is given. 2. THE GENERA L THEORY In this section we int roduce the general theory. It s applicat ions will be considered in the following section. Essentially, our approach is based on t he theory of sym plectic relation s and their generatin g fun ction s in the framework of the jet manifolds. T he basic concept s on the jet spaces can be found in Refs. 11 and 12. T hroughout the paper all manifolds and maps will be smoot h (C ¥ ). Let M be a manifold of dimension m ³ 1 (a space-t ime manifold, a space of param eters, etc.) , with local coordinat es (x l ) , 1 £ l £ m . We denot e by T M and T * M the tangent and cotangent spaces of M , respectively, and use the symbols Ä , Ú and Ù for tensor, symmetric and exterior product s. Let E ® M be a ® bered manifold of dimension m + l, with adapt ed coordinat es ( x l , y i ), 1 £ i £ l. T he sections s : M ® E of t his ® bered manifold represent classical ® elds and E is the con ® guration space of a given ® eld system. We denot e by V E Ì T E and V * E the vert ical subspace of T E and it s dual, respectively. T he phase space of the ® eld system is de® ned as P =

Ù

m- 1

Ä

T* M

V*E

®

E

®

M.

(4)

Obviously P ® E is a vector bundle. Induced coordinat es on P are m denot ed by (x l , y i , p i ). Let J 1 P be the ® rst-order jet manifold of the m m ® bered manifold P ® M , with coordinat es (x l , y i , p i , y li , p l i ) (y li = ¶ l y i , m m p l i = ¶ l p i ). T here is a nat ural m -form on J 1 P , nam ely

Q

Q = p li h i

: J 1P

Ù

x l,

m

® Ù

T* E , i

i

h = dy ±

ymi dx m ,

(5)

P r o j e c t iv e In v a r i a n c e a n d E in s t e in ’ s E q u a t io n s

9

where x l = ¶ l û x and x = dx 1 Ù . . . Ù dx m . T his form is obt ained by contracting the canonical vect or-valued Liouville form [14] h :P

Ù

T* E

l i p i dy

x

m+1

®

h =

Ù

Ä

T M,

Ä ¶

l

,

with the canonical inject ion l : J 1E l = dx that is

®

l

Q

Ä

T* M

Ä

(¶

l

TE , yli

+

¶ i ),

= ± lû h.

It turns out t hat t he ® eld equat ions can be regarded as a system of part ial diŒerential equat ions with sections of P ® M as unknowns or, in other words, as a submanifold of J 1 P . Now, in general, these equat ions can be written in several equivalent ways (con trol m odes ); each way consist s m m in expressing part of t he variables (y i , pi , y li , p l i ) (respon se param eters ) in terms of the remaining ones (con trol param eters ). In order to illust rate these ideas more concretely, let us consider a ® rst order Lagrangian density m

® Ù

1

L :J E

l

T * M,

i

i

L = L(x , y , y l ) x

describing a given ® eld system. T hen the ® eld equat ions are the EulerLagrange equat ions and can be written as ± dh Q

= d L,

(6)

where d is the exterior derivat ive and d h is the horizont al ext erior derivat ive on jet manifolds [11], i.e. dh Q dh Q

: J 1P

® Ù

m

= ± p l li dy i

T* M

Ù x

±

Ù

V *J 1E , p li dy li

Ù

x .

(7)

Writing eq. (6) explicit ly we ® nd the usual Euler- Lagrange equat ions in ® rst order form, i.e. p li =

¶ L , ¶ y li

p l li =

¶ L . ¶ yi

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G ia ch e t t a a n d M a n g ia rot t i

Note that the momentum variables and their derivat ives (p li , p l li ) (response paramet ers) are expressed as funct ions of the ® eld variables and their derivat ives (y i , y li ) ( control param eters). Now assume t hat the equat ions p li = ¶ L / ¶ y li can be algebraicall y solved for the yli ’ s, i.e. yli = w il (x l , y i , p li ) . (8) T hen, clearly, (6) is equivalent to the following equat ion: ± yli dp li

Ù

Ù

+ p l li dy i x

= d(L x

±

w±

Ù

p li w il )

x .

(9)

Introducing the vertical exterior derivat ive on jet manifolds d v [11], this equat ion can be written as d v Q = d H, (10) where dv Q

dv Q = dp li

: J 1P

Ù

dy i

Ù

m+1

Ù

T * P,

x l ±

y li dp li

®

Ù x

+ p l li dy i

Ù

(11) x ,

and

l

H = p i dy

i

Ù

H:P

x l ±

® Ù

Hx ,

m

T* E , H = ± L ± w + p li w il ,

is the Hamiltonian form corresponding to the Lagrangian density L . T he (local) funct ion H is called the Hamilt onian density (alt hough, really, it is not a density) . Hence we see t hat now the ® eld equat ions are written in Hamilt onian form yli =

¶ H , ¶ p li

p l li = ±

¶ H , ¶ yi

(12)

with the ® eld and the momentum derivat ives (yli , p l li ) (response parameters) expressed as funct ions of the ® eld and momentum variables (y i , p li ) (control paramet ers). Of course, the relat ion (8) which led us to the equivalence of the ® eld equat ions in Lagrangian and Hamilt onian form, is just one of the possible relat ions that t he ® eld equat ions may impose on the variables (y i , p li , y li , p l li ). For example, in the applicat ion t o the t heory of gravit ation considered in the next section, the ® eld equat ions establish a relat ion between part of (y i , yli ) and part of (p l li , p li ).

P r o j e c t iv e In v a r i a n c e a n d E in s t e in ’ s E q u a t io n s

11

3. APPLICATION TO GR AVITATION THEORY Let M be a four-dimensional manifold. We denot e by C ® M the bundle of linear connect ions on M and by E ® M a vector bundle associated with t he principal bundle of linear frames on M ; hence the covariant derivat ive of sections of E ® M with respect to linear connect ions on M is de® ned. Coordinat es on J 1 C and J 1 E are denot ed by (x l , C m b a , C l , m b a ) and (x l , u i , u il ), respectively ( C l , m b a = ¶ l C m b a and u il = ¶ l u i ). We consider a ® rst-order Lagrangian density 1

L : J (C £

M

E)

® Ù

m

T* M

(13)

describing an a ne theory of gravit ation with torsion and non-met ricity. We assum e that L dep ends on the connect ion C : M ® C only through the symmet ric part of the Ricci tensor K m u (j 1 C) = (C a , b c k + C a s k C b c s ) D a b c k m u

(14)

and the covariant derivat ive

Ñ that is,

lu

i

= u

i l

+ Cl b a D a b i j u

L = L(K m u (J C), Ñ 1

lu

i

,u

i

j

,

(15)

)x .

(16)

Here D a j are a set of Kronecker d ’ s (depending on the nat ure of the ® elds u i ) and b c c D a b c k m u = d ka d ( m d u ) ± d kb d (am d u ) . bi

T he symmetric part of the Ricci tensor is invariant under projectiv e tran sform ation s of the connect ion, i.e. t ransformat ions of the following type: C l b a ® C l b a = C l b a + w l d ba (17) where w l is an arbit rary 1-form on M . Act ually, any linear connect ion C decomp oses according to ( 2) and (3). Not e that * C l b a are the param et ers of a linear connect ion, while C l is a one-form on M . T he connect ion * C is the projective in varian t part of C, that is, it is invariant under project ive transform ations ( 17) . According to (2), the Riemann- Cartan curvat ure tensor R a b c k (C) = C a , b c k ±

Cb , a c k + C a s k C b c s ±

Cb s k Ca c s

12

G ia ch e t t a a n d M a n g ia rot t i

takes the following form: 4 d k 3 C[a ,b ] c

R a b c k (C) = R a b c k ( * C) +

,

with C a , b = ¶ a C b , from which it follows that K m u = R l ( m u ) l dep end only on the project ive invariant part * C. In order to avoid const raint s on the source ® elds u i which can make the theory inconsist ent, we assume the covariant derivat ive Ñ l u i to depend on the connect ion C only via the project ive invariant part * C. It result s that the Lagrangian density L is invariant under project ive transformat ions (17) . T he phase space of the theory is given by P = C£

E£

M

Coordinat es on

Ù

3

Ù M

3

Ä

T* M

T * M Ä (T M

Ä

Ä

(TM

TM

Ä

T* M £

M

E * ).

T M Ä T * M £ E * ) are denot ed by ( pa b c k , M

p ai ); these are the momentum variables conjugat e to (C b c k , u i ). As usual we set p l , a b c k = ¶ l p a b c k and p l , ai = ¶ l p ai . T he funct ional form of the Lagrangian density suggest s considering the following morphism s: (i) the symbol of the symmetric Ricci morphism 3

Ä

s(K ) :

T* M

s(K ) : u a b c k ½

®

Ä

TM

Ú2 ®

T * M,

u mu = u ab c k D ab c k mu ,

(ii) the dual of s(K ) s(K ) * :

Ù

4

Ä Ú2

T* M

s(K ) * : p m u ½

TM

®

® Ù

4

T* M

Ä Ä

3

TM

Ä

T * M,

pa b c k = D a b c k m u pm u ,

(iii) the project ion ( the left inverse of s(K ) * ) i* :

Ù

4

T* M

Ä Ä

3

i * : pk m u a

TM ½

®

Ä

T* M

pb c =

2 9

® Ù

4

T* M

Ä Ú2

pk m u a D a ( b c ) k m u ,

(iv) the dual of i * i:

Ú2

i : ubc ½

T* M

®

® Ä

3

ukmua =

T* M 2 9

Ä

T M,

D ab c kmu u b c .

TM,

P r o j e c t iv e In v a r i a n c e a n d E in s t e in ’ s E q u a t io n s

13

By using these morphism s one can easily verify t hat the following decomp osit ions hold: C a b c k = CÃ a b c k + p

abc

k

= pÃ

abc

k

2 9

+D

D kmu ab c K mu ,

(18)

abc

(19)

kmup

mu

,

where Cà a b c k and pà a b c k are in the kernels of the symmetric Ricci morphism and i * , respectively. Now let us consider the Euler- Lagrange equat ions of the Lagrangian density (16) , which according to (6) are p a b c k dC a b c k + p b c k dC b c k + p ai d u

i a

+ pi du

i

= dL,

(20)

with t he not at ion pb c k = p a , a b c k and p i = p a , ai . Taking int o account (18) and (19) , these equat ions can be writ ten as pà a b c k d Cà a b c k + p m u dK m u ±

D a b c k m u p m u d(C a s k C b c s )

+ ( pà b c k + D a b c k m u p a , m u )dC b c k + pai d u = ¶

mu

+¶

L dK m u + ( ¶

a i L

du

i a

+ (¶

i i a + pi d u [b c] b ci j ± 23 i L d k D m m i j i L Dk j u l * u mj i , i + i L) d u j L Cl m D u

¶

u j ) dC b c k

¶

(21)

where we have de® ned ¶ m u = ¶ / ¶ K m u , ¶ i = ¶ / ¶ u i , and ¶ ia = ¶ / ¶ u ia . Grouping toget her t he terms containing dC l m u , after some algebra we ® nd that (21) can be written as pÃ

Ñ

à a b c k + p m u dK m u + ( D a b c k m u * a p m u k dC + 2 * C [ a k ] b p a c + pà b c k ) dC b c k + p ai d u ia + p i d u i

abc

= ¶

mu

+¶

L dK m u + ( ¶

a i L

du

i a

+ (¶

[b c] b ci j ± 23 i L d k D m m i j i L Dk j u l * u mj i . i + i L) d u j L Cl m D u

¶

u j ) dC b c k

¶

(22)

Hence, t he Euler- Lagrange equat ions for the Lagrangian (16) are given by pà a b c k = 0, = ¶

mu

ac

= J

bc

p ai

= ¶

p

D

abc

kmu

*

Ñ

ap

mu

*

b

+ 2 C [a k ] p

mu

pi = ¶

(23a) (23b)

L,

k , a i L, l * u mj i j L Cl m D u

(23c) (23d) + ¶ i L,

(23e)

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G ia ch e t t a a n d M a n g ia rot t i

where * Ñ denot es the covariant derivat ive with respect to the connect ion * C and the tensor J b c k is given by J b c k (u

i

b

, pai ) = pi D k c i j u

j

2 [b c ] d k Dm mi j 3 pi

±

u j.

(24)

Note that we have used eq. (23d) . As was to be expected owing to the projective invariance of the Lagrangian, eqs. (23) involve only t he project ive invariant part of the connect ion C. We assum e the following regularity condit ions on the Lagrangian (16) : (i) det ( ¶ m u L) = / 0, (ii) the map (K m u , Ñ l u i ) ½ ® (p l m = ¶ l m L, p li = ¶ il L)

pl m

is invert ible (a local diŒeomorphism ). T hen the equat ions (23b) and (23d) can be algebraically solved for and p li ,

(p m u , p li ) ½ ®

(K m u = w m u (p a b , p ai , u

i

Ñ

),

i

lu

= w il (p a b , p ai , u

i

) ) . (25)

A key role is played by (23c) , nam ely, as in the case of symmetric connect ions these equat ions can be algebraically solved for the * C l m u ’ s. Indeed, after some calculat ions we ® nd *

C rs a =

1 2

pa k Ñ

k 9 r9 s 9 k rs

1 2

(p k 9 , r9 s 9 +

pk 9

,

lm

).

(26)

p b c p rs ) .

(27)

p r9 s 9 + S k 9 r9

s9

Here p l m denot es the inverse matrix of p l m ,

Ñ

k 9 r9 s 9 k rs

= ± d

k9 k

d

r9 r

s9 s

d

+d

k9 r

d

r9 s

d

s9 k

+ d

k9 s

d

r9 k

d

s9 r

and S k rs (p a b , p ai , u

i

) = (J b c k ±

1 b ac d a )( pb r p c s 3 kJ

±

1 2

Note that S k rs is a tensor. A second remark is important . Let us denot e the ® rst two terms on the right -hand side of (26) by S rs a and de® ne Trs

a

=

1 2

pa k Ñ

k 9 r9 s 9 k rs

S k 9 r9 s 9 .

T hen ( 26) takes the form *

C rs a = S rs a + Trs a .

(28)

P r o j e c t iv e In v a r i a n c e a n d E in s t e in ’ s E q u a t io n s

15

If we int roduce the contravariant tensor of second rank de® ned by g m u = f j det ( p a b ) j g

- 1/ 2 m u

(29)

p

and express S rs a in terms of g a b and t heir derivat ives, we ® nd that these coe cients are t he Christ oŒel symbols of g a b , i.e.,

S rs a =

1 ab ( gr , b s 2g

+ gs , b r ±

gb , rs ) .

Now let us go back to eq. (21) . After some simple manipulat ions it can be writ ten in t he form

D a b c k m u C a , b c k dp m u ± D a b c k m u C b c k dp a , m u ± u = d[D a b c k m u ( ± p a , m u C b c k + p m u C a s k C b c s ) ± pm u K m u ± p ai u ia + L] . ±

i a a dp i

+ pi d u

i

(30)

As one can easily verify, t his equat ion is invariant under project ive transformat ions; hence the connect ions param eters in (30) can be replaced by their project ive invariant combinat ion (3) . Using (25) and (28) , we now regard the quant ity t o be diŒerentiat ed on the right side of (30) as a funct ion of (p m u , p a , m u , p ai , u i ). Let us consider the expression

D a b c k m u (± p a , m u C b c k + pm u C a s k C b c s )

(31)

on the right side of (30) in more det ail. Subst ituting (28) in (31) , we ® nd

D a b c k m u (± p a , m u C b c k + p m u C a s k C b c s ) = D a b c k m u (± p a , m u S b c k + p m u S a s k S b c s ) ±

(Ñ

S

kp

bc

±

d

b k

Ñ

S

ap

ac

) Tb c k + D a b c k m u Ta s k Tb c s p m u ,

S

where Ñ denot es the covariant derivat ive with respect to S . B ut 0, since S is t he Levi-Civit a connect ion. T herefore

Ñ

S

kp

bc

=

D a b c k m u (± p a , m u C b c k + p m u C a s k C b c s ) = D a b c k m u (± pa , m u S b c k + p m u S a s k S b c s ) + D a b c k m u Ta s k Tb c s p m u and (30) reads ±

D a b c k m u C a , b c k dp m u ± D a b c k m u C b c k dp a , m u ± u ia dp ai + pi d u = d[D a b c k m u ( ± p a , m u S b c k + p m u S a s k S b c s ) + L],

i

(32)

16

G ia ch e t t a a n d M a n g ia rot t i

where L is de® ned by

L(p m u , p a , m u , p ai , u i ) = D a b c k m u Ta s k Tb c s p m u + p li Tl m u D u m i j u j ± pm u w m u ± p li (± S l m u D u m i j u j + w il ) + L ± w .

(33)

Now we observe that the quant ity

D a b c k m u (± p a , m u S b c k + p m u S a s k S b c s ) coincides with the Einst ein- Hilbert Lagrangian density up to a divergence, namely D a b c k m u (± p a , m u S b c k + p m u S a s k S b c s ) = pm u R m u + ¶

m (p

mu

S l u l ± pl u S l u m ) ,

where R m u is t he Ricci tensor of the Levi-Civit a connect ion S . Moreover, it is easily seen that (32) is equivalent to the following equat ions:

¶ [D a b c k m u (± p a , m u S b c k + p m u S a s k S b c s ) ] ¶ ps r ¶ ± ¶ a [ D a b c k m u (± pa , m u S b c k + pm u S a s k S b c s )] ¶ pa , s r ¶ L ¶ L = ± + ¶ a , ¶ ps r ¶ pa , s r

{

(

)

¶ L

i

u a = ±

p a , ai =

¶ p ai

¶

¶ L u i

.

,

} (34a)

(34b) (34c)

Equat ion (34a) is the Einst ein equat ions in Lagrangian form for the gravitational ® elds p m u . Equat ions (34b) and (34c) are the source ® eld equat ions in Hamiltonian form. Of course, only those solut ions of these equat ions which correspond to the right signat ure (+ , ± , ± , ± ) of p m u have physical meaning. It follows that the purely a ne theory of gravit ation wit h torsion and non-m etricity, described by the Lagrangian density (16) , is equivalent to Einst ein’ s standard theory of gravit ation, wit h Hamilt onian density for the source ® elds given by (33) .

P r o j e c t iv e In v a r i a n c e a n d E in s t e in ’ s E q u a t io n s

17

4. EXA MPLE As a simple example which may help to illust rate the met hod used in this paper, let us consider the Lagrangian for the scalar linear ® eld theory L m a t (p m u , p l , m u , u , u

l)

1 mu u 2 (p

=

mu u

Ö ±

± det (p m u ) m 2 u

T he momenta correspondin g to the scalar ® eld u pm =

¶ Lm a t = pm u u ¶ u m

2

).

are given by

u

so that the Hamiltonian density (33)

L = L m at ± p m u p m p u reduces to

L(p m u , p l , m u , u , p l ) =

1 2 (±

pm u p m p u ±

Ö±

det (p m u ) m 2 u

2

),

(35)

since all coe cients D u m i j vanish. Hence, according to (28) , the ® eld equat ions determine the project ive invariant part of the connect ion to be the Levi-Civit a connect ion. T he Einst ein equat ions (34a) now read

¶ L

Rmu = ±

¶ pm u

and, using the relat ions

¶ pa u ¶ pl m

= ± pa ( l pm ) u ,

¶ ¶ pm u

Ö

± det (p a b ) =

1 2

Ö±

det ( p a b ) pm u ,

we ® nd that Rm u =

1 2

(± p a ( m p u ) b p a p b +

1 2

Ö

± det (p a b ) p m u m 2 u

T his implies pm u R m u = ±

1 m u 2 pm u p p

+

Ö

± det (p a b ) m 2 u

so that , according to (33) , L = L + pm u R m u + p m u p m p u .

2

2

).

(36)

18

G ia ch e t t a a n d M a n g ia rot t i

Subst itut ing (35) and (37) in this expression, aft er simple calculat ions we get L = 12 ± det (p a b )m 2 u 2 . Now t he term (36) , i.e.,

Ö

Ö

± det (p a b ) may be obt ained from the Einst ein equat ions

det (R m u +

1 2

pa ( m p u ) b p a p b ) = ( 12 m 2 u

) det ( pa b )

2 4

and hence L(C l a b , C m , l a b , u , p l ) = ACK NOW LEDGEMENT

2 m 2u

2

Ö

± det

(

K mu +

1 pa ( m p u ) b p a pb 2

)

.

T his work has been support ed by the Minist ero della P ubblica Istruzione, Italy (nat ional and local funds) . R EFER ENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16.

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