P H Y S I C A L R E V I E W D V O L U M E 1 5 , N U M B E R 8 1 5 A P R I L 1 9 7 7

Gravitational Lagrangian and internal symmetry

Elhanan Leibowitz Department of Mathematics and Department of Physics, Ben Gurion University of the Negev, Beer-Sheva, Israel

(Received 9 September 1976)

The field equations of general relativity, when interpreted as a set of equations in a Riemannian manifold endowed with internal space at each point, are shown to be derived from a variational principle. The failures of other approaches to recover the full general-relativity theory are traced to the absence in their Lagrangians of direct coupling between space-time and internal structure. Examination of this observation sheds some light on the fundamental implications of the intimate relation between space-time covariance and internal symmetry.

I. INTRODUCTION

In the classical formulation of in stein's theory of gravitation the dynamical var iables a r e the components of the met r ic field gp,(x) correspond- ing to the Riemannian manifold of space-time, to- gether with the physical fields ("matter field") present . The field equations a r e derived f rom a principle of l eas t action

with a Lagrangian L . This Lagrangian decomposes into two parts : the gravitational Lagrangian, Lo, which i s equal to the Ricci s c a l a r R, and the mat- t e r Lagrangian L,, viz.,

(K i s a coupling constant). Variation with respec t to the met r ic field in this action integral leads to the Einstein field equation

It has been found, however, that f rom a theo- ret ical a s well a s a computational point of view, it i s desirable to r e c a s t the theory a s a formalism in space-time with internal s t ruc ture , posessing SL(2, C) symmetry. This i s the conceptual back- ground for the two-component spinor theory in curved space-time. ' This approach i s a l so the b a s i s fo r the te trad calculus and the Newman- Penrose spin-coefficients formalism,' and i s use- ful fo r the analysis and classification of gravita- tional fields. Fur thermore , i t has been shown by Carmeli3 that, using such a framework, the theory of gravitation can be c a s t into a Yang-Mills-type formal i sm. Yang in his recent integral formalism4 again re la tes gauge fields and gravitation when the la t t e r i s expressed a s an SL(2, C)-covariant theory. This i s one s tep in the cur ren t tendency to bring general relativity more in line with the prevalent t rends in par t i c le physics.

It i s therefore of importance to find out whether in formulations of that type the Einstein field equa- tions can be derived from an action principle under the variation of the fundamental quantities up- pearing in the partzcular formulation. More spec - ifically, the question a r i s e s whether a n action principle can yield the field equations without add- ing in the p r o c e s s ad hoc quantities. A part ia l answer to this question has been given in a few article^,^ where it i s demonstrated that a n appro- pr iately chosen Lagrangian leads to field equations which a r e related to Einstein equations. These equations, however, a r e weaker than the full Ein- s tein equations, and mus t be augmented by addi- tional conditions in o rder to recover general r e l - ativity. Fur thermore , auxiliary quantities must be artificially introduced in o rder to facilitate the derivation. Carmeli 's method6 of f i r s t -o rder fo rm suffers f rom the s a m e la t t e r deficiency. The m e r i t s of the aforementioned schemes l i e in their Lagrangians being functions of gauge fields, an approach which i s not attempted in the p resen t ar t ic le . Likewise, the equations obtained by Yang4 f rom a variational principle constitute a vast generalization of general relativity, and ad- mit solutions which a r e physically ~ n a c c e p t a b l e . ~ I t i s the purpose of this paper to show that the full Einstein field equations a r e obtained f rom a n SL(2, C) -invariant Lagrangian, which i s consid- e r e d a s a functional of the dynamical var iables alone, without any necessi ty to introduce any auxiliary quantities. This Lagrangian, in con t ras t to the Lagrangian considered by others , mixes up the space-t ime and internal s t ruc tures .

In Sec. I1 the framework i s s e t and the notation i s established. A basic decomposition of the c u r - vature i s represented in Sec. 111, and in Sec. IV the Lagrangian i s introduced and the resulting Euler-Lagrange equations a r e derived. The equa- tions a s applied to vacuum and to space-time ad- mitting a neutrino field a r e discussed in Sec. V. The l a s t section i s devoted to concluding r e m a r k s ,

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and in the Appendix a few identities used in the a r t i c le a r e stated.

11. RIEMANNIAN STRUCTURE IN TERMS OF SPINORS

In the s tandard spinor algebra in curved space- time8 a one-to-one correspondence i s se t between vec tors 5, in the Riemannian manifold of space- t ime (endowed with met r ic tensor g,,), and Herm- itian spinors <,A, viz.,

= ~ $ g 5 , , 5" = U " A B . tAB . The connecting quantities U X ~ sat isfy the relat ions . .

+U;~U$;) = gPV€, B J (2.1)

p , v = O , 1 , 2 , 3 , A , B = O , l .

The approach to be followed h e r e i s to take the spinor veco t r s u i i a s the fundamental quantities of the theory, and to construct the Riemannian met r ic out of the spin s t ructure. Consequently, Eq. (2.1) i s the definition of the met r ic g,, .

Covariant der ivat ives of spinors a r e defineds by

where the spin connections rAB, a r e determined by the usual requirements f o r differentiation (Leibnitz rule , etc.) together with

The spin curvature R,,,, i s defined by the com- mutator of the covariant derivative

v A I F u - v A J v P = R A B p u ? 7 B , and i s expressed in t e r m s of the connections

~ ~ ~ ~ v = r ~ B ~ , u - r ~ B u , ~ + r A c u r C B p - r A c p r C B u

The spin curvature i s equivalent to the F,, ma- t r i ces in Carmeli 's gauge f ~ r m u l a t i o n , ~ and i s r e - lated to the Riemann curvature tensor Rx,,, by

where

111. DECOMPOSITION OF THE SPIN CURVATURE

The three bivectors S,,,OO,S,,O'=S,V'O, S,," and their complex conjugates a r e s i x independent com- plex bivectors ( a s can be shown with the aid of the "completeness relations" in the Appendix), and hence s e r v e as a bas i s in the space of complex bi-

vectors . The spin curvature R,,,, , fo r fixed val- ue of (A, B ) , i s a complex bivector and therefore can be uniquely expanded in t e r m s of the basis

where the coefficients a r e given by

Contracting (3.1) with S p V and using the "com- pleteness relations" of the Appendix and (2.4), one finds

Fur thermore , a s imi la r but longer calculation shows that the coefficients RABbb a r e related to the t race- f ree Ricci tensor:

where $pUA,;; i s the totally symmetr ic spinor ten- s o r defined in the Appendix.

Infact , RA+b i sequa l tothe spinor - QAB6b in the Newman-Penrose formalism.*

F o r fur ther reference i t i s mentioned here that the symmetr ies of RABcD (RABcD= RAmc = R,,,) entail

N. THE GRAVITATIONAL LAGRANGIAN AND ITS VARIATIONAL DERIVATIVE

If the theory i s confined to quantities which a r e of, a t most, second differential o rder in the upAb, and l inear in the second derivatives, then the only concomitant of the fundamental quantities u p A i which i s s c a l a r under coordinate t ransformations in space-t ime and invariant under the action of the group SL(2, C) i s (up to a constant, which contri- butes only a cosmological term)

Variational pr inciples of this type have been con- s ideredg f rom other points of view. Our calcula- tion, which a g r e e s with the aforementioned r e - sul ts , will be based on the decomposition scheme of Sec. 111, thus gaining in simplicity and brevity. Thus l e t u s consider the change of

under the variation

F o r this purpose we have to express the variat ions of the quantities appearing under the integral sign

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in t e rms of 6 a 5 , g o r 6apAB. Using (2.1) one finds The last relation, which i s coordinate- and SL(2, C)-invariant, can be most easily verified in

A B A B 6 g l l u = a p ~ i 6 a u + a v ~ i b a p , a "geodetic" coordinate system and spin frame, A i 6 J - g = 3 J - g e p U 6 ~ p u = q C T ~ ~ ~ ~ U ~ , adapted to the point under consideration. In such a

. . coordinate system and spin frame the fundamental 6 a p A B = -apCDuVAB6aUcb , quantities apA; a r e constant (at the point) and r e - ~ S P U A B = a [ ~ ~ d d o v ~ ~ ~ + a [ ~ ~ . b a v l ~ d duce to the Pauli matrices, and r = O (at the

C point). s ~ ~ ~ ~ ~ R ~ ~ ~ , = v ~ ~ ~ , with TIP = 2 ~ ~ ~ ~ ~ 6 ~ ~ ~ . Taking these relations into account one finds

Finally, using (Al), (3.4), and substituting (3.2), we conclude

Now a s usual we consider variations 6 0 , ~ ' such that they and their f i r s t derivatives vanish on the boundary of integration. Under these assunlptions the vector Vp too vanishes on the boundary and by Gauss's theorem the divergence c g vP 1, = (Ggvp),, does not contribute to the Euler- Lagrange equations. Thus (4.1) leads to the field equations

2RAc %aPCD + noP,& = - K T ' A ~ , (4.2)

where

That (4.2) i s indeed equivalent to the Einstein equations i s confirmed with the aid of (3.3), where- by we find that the left-hand side of (4.2) i s equal to

V. APPLICATION TO VACUUM AND NEUTRINO

In the case of a vacuum, the field equations (4.2) reduce to

~ R , , ; ~ U C' + 510uA~ = 0 .

Contracting with a," we find th is equation to be equivalent to the pair

W = O ,

RAcib = 0

which in turn a r e equivalent to

R = O ,

and these a r e the Einstein vacuum field equations R,, = 0.

For a space admitting a neutrino field qA, the matter Lagrangian L, i s taken to be

- qAq'lA&) - Variation of qA and 77A in the action integral leads to the Euler-Lagrange equations

q A I A i = O (and c.c.) , i.e., the Weyl equations for the neutrino field. This equation can now be used to find the "energy- momentum spinor vector" TLLAg, to be substituted in the right-hand side of (4.2), namely in order to calcylate the variational derivative 6 (G L,)/ 6oLLAB. The variational variables oLLAB appear in

(the contribution of this t e rm i s known from Sec. IV), and in the spin connections rAB, in L,. Thus

6LM =i(VAI .Vi - 71A7?BILL)60LLAB

+ iqA77B(aLLcir61?cALL - auA;6 rci,). (5.1)

Firs t observe that the indentity

AB A& o =a,, I,,= u ~ ~ ~ , ~ - rALLv~X + rACv~LL CB

+ rBEva, AE

yields the following expressions for the variation of the spin connections:

6 rABW = - $ U " ~ ~ [ ( ~ C T , ~ ~ ) ~ ~ - oXAE6rXWV]

=+.+ovAC[(60VBE)lLL + 0ABE6rYLLAI.

(Again this relation i s most easily verified in a "geodetic" coordinate system and spin frame.) Substituting in (5.1), the t e r m s containing 6rX,, cancel, and we have

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After some manipulations the following expression is obtained:

with

= - i(SpYAC77i)77C - quuBE77ATE)6U,,ai.

Again, the divergence t e rm does not contribute to the variational equations.

The contribution of the variation of 6, i.e., L , 6 e , vanishes also due to the neutrino equa- tion. Hence we conclude from (5.2)

+gcl~i77d - q c r l ; l ~ ~ ) u " ~ * . (5.3)

From the field equations thus obtained, viz.,

2 ~ ~ ~ ~ ~ o ~ ~ ~ + f b U A g = - K T ~ ~ ~ , , (5.4)

with TpA;, given by (5.3), the familiar form can be recovered. Contracting (5.4) with a p A B yields

and therefore

VI. CONCLUSION

It has been demonstrated that general relativity can be viewed a s a theory involving the fundamen- tal quatities upA* alone (along with external phys- ical fields). The field equations a r e derived from a variational principle, the gravitational Lagran- gian being

It will be instructive to compare this Lagrangian to other Lagrangians considered r e ~ e n t l y , ~ ' ~ " j which have not led directly to the full general- relativistic field equations, but have nevertheless been adopted by proponents of the gauge-field ap- proach to gravitation. A typical Lagrangian pro- posed by those authors i s (in our notation)

In such a Lagrangian the space-time indices ( p , v) a r e contracted with space-time in- dices, and internal-group indices (A, B) a r e con- tracted with internal-group indices, and no mixing occurs. The only way to couple directly indices of one kind with indices of the other kind i s to recas t

RABbv into a form where all the indices appear on the same footing. This i s accomplished by the usual way of changing tensorial indices into spin- orial indices, viz.,

R A B ~ ~ - R A B P ~ x ; = ~ ~ ~ P ~ ~ ~ x ~ R A B ~ ~ . Now the coupling emerges naturally: . ,

E E B X ~ Q Y ~ A B P 6 X P = S p u A B ~ A B p u , and this i s precisely the Lagrangian of the present work, which indeed leads to the full general-rela- tivity field equations. (We mention in passing that the program outlined here can be car r ied out, mutatis mutandis, within the framework of spinor calculus in the five-dimensional unified theory of relativity," thus tying up the electromagnetic field with internal degrees of freedom.)

The las t observation may provide more insight into the centrality in general relativity of the in- separability of space-time aspects and internal group, and into the fundamental role played by the intimate relation between general covariance and internal symmetry.

APPENDIX: ALGEBRAIC PROPERTIES OF THE S r v A B

Several properties of the basic quantities sPuAB, which a r e used in this work, a r e listed here. All of them can be verified from the definition (2.6) and the relations (2.1).

"Completeness relations" a r e

(e,.,, i s the alternating tensor e p v X 7 = c g t J J U 1234 = '1' Single-index contraction yields

+ S P U a ( c ~ D ) ~

+ E A ( ~ E D ~ B ~ " ' ) , S, XAB sU = -$fi A B ~ ; ,

where +pVAB&b i s the totally symmetric spinor tensor

+ p U A B d b = ~ ( p a ( b ~ ' " ~ b )

( $ p U A B ; ~ = @ v p a e ~ b = $ b V s ~ ~ b = $ " ~ ~ , gpv+'" 'a~tb =O). Another useful identity re la tes products of a 's with S's:

upA;uVc~- o u A i ~ p c i = S p u A C ~ & + S " ' & E ~ ~ . (A l l

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