9 August 1999

Ž .Physics Letters A 259 1999 104–107www.elsevier.nlrlocaterphysleta

Effective Levi-Civita dilaton theory from Metric Affine Dilatongravity

R. Scipioni 1

Department of Physics and Astronomy, The UniÕersity of British Columbia, 6224 Agricultural Road, VancouÕer, BC V6T 1Z1, Canada

Received 26 April 1999; received in revised form 9 June 1999; accepted 14 June 1999Communicated by P.R. Holland

Abstract

We show how a Metric Affine theory of Dilaton gravity can be reduced to an effective Riemannian Dilaton gravitymodel. A simple generalization of the Obukhov–Tucker–Wang theorem to Dilaton gravity is then presented. q 1999Published by Elsevier Science B.V. All rights reserved.

PACS: 04.20.-q; 04.40.-b; 04.50.qh; 04.62.qv

Among the four fundamental interactions, the twofeeble are characterised by dimensional coupling

Ž .y2constants, G s 300 GeV and Newton’s cou-FŽ 19 .y2pling constant G s 10 GeV .N

It is well known that interactions with dimen-sional coupling constants present many problemsamong which there is the renormalizability.

The success of the Weinberg–Salam model hastold us that the weak interaction is characterised by adimensionless coupling constant and the dimensionsof G are due to the spontaneous symmetry breakingF

mechanism, so that G ( 1rÕ2 where Õ (F W W

300 GeV is the vacuum expectation value of theHiggs field.

The weakness of the weak interaction being re-lated to the large vacuum expectation value of the

w xscalar field 1 .It is believed that similar mechanisms may occur

for gravity, which is characterised by a dimension-

1 E-mail: cipioni@physics.ubc.ca

less coupling constant j . The weakness of gravitythen would be related to the symmetry breaking at

w xvery high energies 2–4 .This is obtained starting from a Dilaton theory

which presents Weyl scale invariance. The potentialŽ .V c which appears in the action is assumed to have

its minimum at css , then when css the Dila-ton theory reduces to the Einstein–Hilbert action

Ž 2 .with gravitational constant G s1r 8pjs .N

In this Letter we investigate in the Tucker–Wangapproach to nonRiemannian gravity the action:

Ss kc 2R w1qb dcn wdc yV c w 1 1Ž . Ž . Ž .Hwhere R is the scalar curvature associated with thefull nonRiemannian connection.

In the Tucker–Wang approach to MAG we choosethe metric to be orthonormal g s h sa b a bŽ .y1,1,1,1, . . . and we vary with respect to thecoframe ea and the connection v a considered asb

independent gauge potentials.

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00413-2

( )R. ScipionirPhysics Letters A 259 1999 104–107 105

As we will see the nonRiemannian contribution tothe Einstein–Hilbert term times c 2 is equivalent inthe field equations to a kinetic term for the Dilatonand if no torsion terms are explicitly introduced inthe action, the coupling j is not arbitrary contrary to

w xwhat happens in Ref 5–7 where j is a free parame-ter.

Once b is fixed we obtain an effective jXy1

ny 1y1 Ž Ž ..given by j q4 see Eq. 22 .ny 2

It has to be observed however that since we havethe condition g sh , in general the Weyl groupab ab

Ž .for the action 1 is not defined in the usual way butwe may still introduce a Weyl rescaling of the formea ™ f ea for the coframe, with f an arbitrary func-tion of spacetime.

Ž .The variation of 1 with respect to c gives:

ybd wdcqkc R w1 yV Xc w 1s0 2Ž . Ž . Ž .

By considering the variation with respect to theconnection we get the equation:

2w b w bD e ne sy dcn e neŽ . Ž .a a

c

w bsA c dcn e ne 3Ž . Ž .Ž .a

2Ž .with A c sy .c

To prove the previous one we have to observethat the Einstein–Hilbert term which appears in the

Ž .action 1 can be written as:

R w1sRa n w e neb 4Ž .Ž .b a

Where Ra are the curvature two forms which areb

defined by Ra sdv a qv a nv c . So we get:b b c b

R w1s dv a qv a nv c n w e neb 5Ž .Ž . Ž .b c b a

Then we have to calculate the variation of:

c 2 dv a qv a nv c n w e neb 6Ž .Ž . Ž .b c b a

We have:

c 2 dv a n w e nebŽ .Ž .b a

2 a w bsd c v n e neŽ .b a

a 2 w bqv nd c e ne 7Ž .Ž .b a

Ž .so mod d :

dv a n w e neb c 2Ž .b a

a 2 w bsv nd c e neŽ .b a

sv a nc 2 d w e nebŽ .Ž .b a

q2 c v a ndcn w e neb 8Ž .Ž .Ž .b a

Considering the connection variation and using thew xdefinition of the covariant exterior derivative D 8

Ž .we obtain formula 3 .The full nonRiemannian Einstein–Hilbert term

can be written as:

ow w a c w bˆ ˆR 1sR 1yl nl n e neŽ .c b a

ˆa w byd l n e ne 9Ž .Ž .Ž .b a

ˆawhere l is the traceless part of the nonRieman-b

nian part of the connection la .b

By considering the coframe variation we get thenthe generalized Einstein equations:

o2 a w bkc R n e ne neŽ .b a c

a w bˆy2kc l ndcn e ne neŽ .b a c

w w xwyb dcn i dcq i dcn dcc c

2 a d w bˆ ˆqkc l nl n e ne neŽ .d b a c

yV c w e s0 10Ž . Ž .c

The Cartan equation can be written as:

w a w a aD e ne sA c dcn e ne sFŽ .Ž . Ž .b b b

11Ž .

To solve the previous we need the 0-forms f cab

defined by F a s f ca we .b b c

We decompose the nonmetricity and torsion as:

1 1ˆQ sQ q g Q , T sT q e nTŽ .ab ab ab a a aˆn ny1

12Ž .

where Q s Dg ,T s de q v b n e ,Q s Qa ,ab ab a a a b a

Ts i T a.a

( )R. ScipionirPhysics Letters A 259 1999 104–107106

We have the relations:

1d d aQ s g f q f eŽ .bc bc da adˆ n

y f q f y f ea , 13Ž . Ž .b ac bca abc

1a dT s e ne fŽ .c c adˆ ny1

1 b ay e ne f q f q f ,Ž . Ž .b ac bca cab2

ny1 1c c aTy Qs f q 1yn f eŽ .Ž .ac ca2n n ny2Ž .

We get:

f sA c i w dcn w e ne 14Ž . Ž . Ž .Ž .Ž .cab c a b

from which we get:

ˆabQ s0, T s0 15Ž .c

and

ny1 1ynTs Qq A c dc 16Ž . Ž .

2n ny2

the solution for the nonmetricity and torsion can thenbe written as:

1Q s g Q,ab abn

1 1a a aT s e nQ y e ndc A c 17Ž . Ž . Ž . Ž .

2n ny2

Using the expression of la as a function of T a andb

Q :ab

2l s i T y i T y i i T q i Q y i Q ecŽ .ab a b b a a b c b ac a bc

yQ 18Ž .ab

we get:

1l sy g Qab ab2n

1q A c i dc e y i dc eŽ . Ž . Ž .Ž .a b b any2

19Ž .

and the traceless part:

1l s A c i dc e y i dc e 20Ž . Ž . Ž . Ž .Ž .ab a b b any2

By using the previous expression in the generalisedEinstein equations we get after some calculations:

owX2

wkc G yb dcn i dcq i dcn dcc c c

yV c w e s0 21Ž . Ž .co o a w bŽ .where G sR n e ne ne and:c b a c

ny1X

b sbq4k 22Ž .ny2

Then if we choose:

ny1bsy4k 23Ž .

ny2

we get the generalized Einstein equations reduced to:o2 wkc G yV c e s0 24Ž . Ž .c c

which are equivalent to:o y2 wkG yV c c e s0 25Ž . Ž .c c

This are the Einstein equations we would have ob-tained from an Einstein theory with the potentialŽ . y2V c c .

The nonRiemannian contribution to theEinstein–Hilbert term is:

4 ny1w wDR 1sy dcn dc 26Ž . Ž .2 ny2c

so, the equation for c becomes:

ny1 4ow w wybd dcqkc R 1 yk dcn dcŽ .Ž .

ny2 c

yV Xc w 1s0 27Ž . Ž .

Observe the following interesting case.Ž .Suppose we start from the bs0 in the action 1

that is:

Ss kc 2R w1yV c w 1 28Ž . Ž .Hthen we get the equations:

ny1ow2

wkG c y4k dcn i dcq i dcn dcc c cny2

yV c w e s0,Ž . c

4 ny1ow wqkc R 1y dcn dcŽ .

c ny2

yV Xc w 1s0 29Ž . Ž .

( )R. ScipionirPhysics Letters A 259 1999 104–107 107

For ns4 we get:o

w2wkG c y6k dcn i dcq i dcn dcc c c

yV c w e s0, 30Ž . Ž .c

6o Xw w wqkc R 1y dcn dc yV c 1s0Ž . Ž .c

The Einstein equations in this case coincide formallywith the conformally invariant Einstein equationsobtained starting from the action:

o2 w wSs kc R 1q6k dcn dcŽ .HyV c w 1 31Ž . Ž .

We have to remember however that this equivalenceholds with the amendment that the Weyl rescaling isdefined for the coframe and not for the metric sinceg is fixed to be orthonormal.ab

What found above can be extended to more gen-eral actions like for example:

Ss kc 2R w1qb dcn wdcŽ .Ha g

w wq f c dQn dQ q f c Qn QŽ . Ž . Ž . Ž .1 22 2

yV c w 1 32Ž . Ž .where Q is the trace of the nonmetricity 1-forms

ab Ž . Ž .Qsg Q and f c , f c are two arbitrary func-ab 1 2

tions of the Dilaton field c .We get in this case the generalized Einstein equa-

tions:o

wX2wkc G yb dcn i dcq i dcn dcc c c

yV c w eŽ . c

aw

wy f c dQn i dQq i dQn dQŽ .1 c c2g

w wq f c Qn i Qy i Qn Q s0 33Ž . Ž .2 c c2

we get the equation for Q.

a d f c w dQ q f c g w Qs0 34Ž . Ž . Ž .Ž .1 2

Ž .Eq. 33 can be considered as a generalization of thew xObukhov–Tucker–Wang theorem 9–11 to the Dila-

Ž .ton Gravity action 32 .

Ž .The equation for c 27 contains other two termswhich are:a g

X Xw wf c dQn dQ q f c Qn Q 35Ž . Ž . Ž . Ž . Ž .1 22 2ny 1If bsy4k then the Einstein equations wouldny 2

reduce to:o2 wkc G yV c eŽ .c c

aw wy f c dQn i dQq i dQn dQŽ .1 c c2

gw wq f c Qn i Qy i Qn Q s0 36Ž . Ž .2 c c2

The inclusion of torsion terms in the action likeTn wT and T c n wT would complicate the analy-c

sis since in that case the traceless part of the Cartanequation and then the expression for la would beb

w xmodified 10 , the study of these more general casesas well as the study of further possible generaliza-tions of Obukhov–Tucker–Wang theorem will beconsidered in following investigations.

Acknowledgements

I wish to thank the International center for cul-Ž .tural cooperation and development NOOPOLIS

Italy for partial financial support, and C. Wang andŽ .R. Tucker Lancaster for stimulating discussions on

the topic.

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