General Relativity and Gravitation, Vol. 17, No. 7, 1985

The Uniqueness of the Einstein-Yang-Mills Equations

Rieardo J. Noriega 1

Received May 18, 1984

The most general gauge-invariant Lagrange density (concomitant of the metric tensor together with the gauge potentials of a gauge and its first derivatives) for which the associated Euler-Lagrange equations are precisely Yang-Mills equations is obtained. It is more general than the Lagrangian which is com- monly used, but it still has essentially the same energy momentum tensor.

1. I N T R O D U C T I O N

Let P(M, G, ~) be a principal fiber bundle over an n-dimensional manifold M, with r-dimensional structure group G, projection ~ and total space P, i.e., P is an (n + r) manifold on which G acts on the right and ~ is a map from P to M invariant under the action of G. A connection w in P, i.e., an LG-valued 1-form defined on P satisfying certain conditions (see [2] for definitions) will be called a gauge field. If U is an open subset of M then by a gauge in M with domain U we mean a section a: U ~ P . If (U, a) is a gauge in M, then wo := a*w is an LG-valued 1-form on U. (LG is the Lie algebra of G.) If (x, V) is a local chart of M for which Uc~ V r ~b and if el,..., er is a basis for LG, then on Uc~ V we many write w~ = (A~e~) dx i, where repeated lower case Greek indices are summed from 1 to r, and repeated lower case Latin indices are summed from 1 to n. The functions A~ are called the gauge potentials of w with respect to the gauge (U, ~), chart (x, V) and basis el,..., er.

1 Departamento de Matem~.tica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Republica Argentina.

629

0001-7701/85/0700-0629504.50/0 �9 1985 Plenum Publishing Corporation

630 Noriega

A quantity T will be called a type (E; r, s, w) gauge tensor field on M if T associates to each gauge (U, a) in M and E-valued, relative tensor field T~ of contravariant valence r, covariant valence s, and weight w. We say that Tis gauge invariant if T~ = T~, for any gauges (U, a), (U', a') for which Uc~U' #fb.

We will be interested in gauge-invariant, type (R; 0, 0, 1) gauge tensor fields on M. If L is such an object, g is a pseudo-Riemannian metric on M and w is a gauge field, then we say that L is an mth-order gauge-invariant Lagrangian density if there is a function F of real variables such that, for any gauge (U, a), chart (x, V), and basis el,..., e~ of LG,

L = F ( g o ; g~j,h,;...; gij, hv..h~; Ai~" A~,~2,'"," " A~.~z..~) (1)

where m : = m a x { r , s - 1 } . The Euler-Lagrange expressions for L are defined as

and

k = 0 Oxhl"'" Oxhk "Og (2)

E~(L)= ~ ( - 1 ) k - - (3) k=0 ~Xhl ' ' " ~Xhk \ OAi , hv..hk]

In the Einstein-Yang-Mills field theory of gravitation and Yang-Mills fields, the field equations are derivable from the gauge-invariant Lagrangian density

L = -- w/g R + ~ B ~ F } F ~~ (4)

where R is the scalar curvature associated to the pseudo-Riemannian metric, g = [det(gij)l, F CU := gihgikF~h k and

(5)

Also, C~7 are the structure constants of the Lie group G, and B~ are the components of a symmetric A d G invariant bilinear form on LG.

The Euler-Lagrange expressions associated with L are given by

Eli(L) = w/'g [ GO" - 2B~,je( F~ih F~Jh __ ! o-iJ F ~hk lz'~ ~ q 4 '5 ~ h k l 3 (6)

and

Ei~( L ) = 4,,/-g B ~ F ~ I Ij (7)

Uniqueness of the E-Y-M Equations 631

where

F~lJh := F~,,,-F~fkih-F;kF~h+~C~,,A~ (8)

When G = U(1) and B~, = 1/4, we obtain the Einstein-Maxwell field theory of gravitation and electromagnetism.

The Lagrangian density (4) has the form

L = -- x ~ R + A (g~ ; AT; A~5) (9)

The object of this paper is to find all Lagrangians of the form (9) for which (7) is valid and to see how (6) is affected. This is a natural generalization of the same problem for the Einstein-Maxwell field theory treated by Lovelock [5]. We will see that his theorem, with the hypothesis of gauge invariance added, is a special case of the theorem we are going to prove.

2. T HE U N I Q U E N E S S OF THE EINSTEIN-YANG-MILLS EQUATIONS

The requirement of gauge invariance for L implies the same requirement for A. Then, from the replacement theorem [1 ], we know that

A(g~j; AT; A ~ ) = A(g~; 0; - �89 A ~(gij; F~) (10)

where A~ is a type (R; 0, 0, 1) gauge tensor field on M. When L is given by (9), the identity (7) written out in full is

A O ; ~ k A ~ , k j . / l O ; h A ~ . O';hk _ i --~.~ ~ .h , j -- A~ ghk,y A~

= --4x/-g B ~ gih gjk[Fgk,j -- F~K~ Fhj" + F~sF~j+~ ~ ~k' C6~ Aj ~ ) ] (11 )

where

~?A •A OA 32A ij _ A '~ = A ~ A ij:hk

A :, OA 74' 6A 7' @ij ~A i,j OA h.k

Now, from [21, p. 68, we know that every LG-valued 1-form on an open set U of M determines uniquely a connection on U • G. Thus from Theorem 2.1 in [2] we deduce that the gauge potentials A7 are arbitrary. The same is true, then, for the second derivatives Ai.~. h of the gauge poten- tials, except for the symmetry in j, h. Then, differentiating (11) with respect to A~,,t we obtain

A~,~";rs +A~,~is;r'--4~/-gB~,~[Zg g ~r ~ -- g ~ g ~ ] (12)

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_ _ j i We have the symmetry condition A~= A~ from (10). indices in (12) we deduce

A=~r;ti. + A~i;tr, _- _ 4 x / - ~ B=~[2gSt gi, _ g~i g~r _ g , ~g , ]

Noriega

Changing

(13)

The right-hand sides of (12) and (13) are the same from the symmetry "" s i ; t r _ _ i s ; t r _ _ i s ; r t of g,S. Also A~ ~ - - A ~ , - A s ~. We deduce

A~'; 7 =A~'~;"~ (14)

We have, from (14), the symmetry relations

Aij;hk _ Aji;hk -- __Ao;kh -- Ahk;ij (15)

Let us consider the antisymmetric gauge tensor field ~;hk given by

,~0"; h k ,4 [/j;hk ]

i.e., the alternate of A ~s:hx From (15) we have af t "

3:0;hk = A0;hk. Aik;# + A ~h;~J

It is easy then to derive, from (13)

AU.hk _ ~U;hk _ 4X/ - ~ B=~[gih gkj__ g~k ghj] (16)

By (10) we have

2~j;hk_ 2~j;hk(,, ~ s ) ct f l - - ct f l ~ e S r s ,

Also from (10) we see that

A ij;hk _ l_Aij:hk l e f t - -4 " ' cz f l

where

~A 1 A d = - - ~F~

is a type (R, 2, 0, 1) gauge tensor field on M. For n = 2, 3 we have

)~O';hk __ 0

in which case we integrate (16) to obtain

o _ _ 2 x / ~ B=~F~O+ B ~ A l e -- (17)

Uniqueness of the E-Y-M Equations 633

where

B~ = B~(ghk )

Now, by (17), B~ are the components of a type (~; 2, 0, 0) gauge ten- sor field on M. Thus, for each a, B~ are the components of a 2-con- travariant skew tensor concomitant of the metric tensor ghk. Loveloek [3] has proved that, under these circumstances, there is a number B~ for each such that

o_{-2B~e~J for n = 2 B ~ - 0 for n > 2

where e ~ is the Levi-Civitfi symbol. Since A ~ is gauge invariant as is - 2 x / ~ B~F ~U, it follows that B~ is

gauge invariant. It is easy then to prove that the B~ are the components of an AdG invariant linear form on LG.

We deduce from (17) that

A=,,fgB~F~.F~J+B~e~ for n = 2 (18)

where b is a constant and

where

A=x/-gB~F~.F~iJ+bxfg for n = 3

For n = 4 it is clear that

(19)

;v = 2(gh~; F~k)

However, for n = 4 However, for n = 4 A O:hk;~s = 0

sO that, by (16) 2~,~= 0. Then 2 = 2(gh~) and from [3] we conclude

2 = 8a (21)

where a is a constant. Integrating (16) we find

A=x/-gB~F~F~~ for n = 4 (22)

and once again the gauge invariance requirement implies that the D ~ are the components of a symmetric AdG invariant bilinear form on LG.

2~hk = 2D ~ e 0hk (20)

634 Noriega

From (13) it is clear that

A it;rs;hk ..L A is;rt; hk - - f'l

from which is easily seen that A il i2;...;i2p-~i2e is completely antisymmetric in all Ctl

its Lating indices for p >/3. Now it is standard to modify Lovelock's proof in [5] to conclude that

A=,e / -gB~BF~i jFaiJ+bx/ -g for n odd, n>~5 (23)

and

A = x//-g B ~ F } F r Oexl...C~m~,ili2""i2m-'i2mFqli2...F~2m_li2m

+ bx//-g for n=2m>~6 (24)

where the D~,...~ are the components of a symmetric A d G invariant mul- tilinear form on LG.

In D=r..~il '""i~F~.i2, , , the only term which is not a divergence is D e ir'~2 . . . . �9 �9 A ~ A ~1 �9 .. A~ ~ A~2m. Since the A~' are arbitrary, it ~ , . . .~ C~'~, C~,.~, m __,, __,~ ,~,,_~

follows from Theorem 3 in [4] that (24) will satisfy (7) if and only if

C~m = 0 D cq...um C}l171" " C Bm,;m (25)

for all ]~'s and y's, If L G 2 = [ L G , L G ] and (LG2) h =- L G 2 x L G 2 x . . . x L G 2,

then (25) means that the aforementioned multilinear form vanishes on (LG2) m (which is always true if the Lie group is Abelian). We deduce the following.

Theorem. which (7) is valid is

L = - x / g R + ~ B~t~F~F~iJ + b ~ - g

and

The only gauge-invariant scalar density of the type (9) for

for n odd

+ for . = 2 m

where D~r..~ are the components of a symmetric A d G invariant multilinear form on L G vanishing in (LG2) m. In any case, the energy momentum ten- sor is only affected by the cosmological term.

If we consider G = U(1) and B ~ = 1/4, we obtain Lovelock's theorem in [5] (with the hypothesis of gauge invariance added).

Uniqueness of the E-Y-M Eq,Jations 635

A C K N O W L E D G M E N T

I wish to t h a n k the referee for helpful sugges t ions .

R E F E R E N C E S

1. Horndeski, G. W. (1981). Utilitas Math., 19, 215-246. 2. Kobayashi, S., and Nomizu, K. (1963). Foundations of Differential Geometr,v,

(Wiley-Interscience, New York). 3. Lovelock, D. (1969). Arch, Rat. Mech. Anal,, 33, 54-70. 4. Lovelock, D. (1972). J. Austral Math. Soc., 14(4), 482495. 5. Lovelock, D. (1974). Gen. Rel. Gray., 5, 399-408.

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