PHYSICAL REVIEW D VOLUME 30, NUMBER 2 15 JULY 1984

Axially symmetric, static self-dual Yang-Mills and stationary Einstein-gauge field equations

Metin Giirses Department of Applied Mathematics, Tilbitak Research Institute for Basic Sciences,

P. 0. Box 74, Gebze-Kocaelt, Trtrkey (Received 21 February 1984)

It is shown that a restricted class of static, axially symmetric self-dual SU(n + 1) Yang-Mills field equa- tions are equivalent to the stationary, axially symmetric Einstein-(n - l)-Maxwell field equations.

Equivalence between the axially symmetric (AS), static by self-dual Yang-Mills and the As stationary Einstein field Gpv= ynb (Fapl, FVbu - :ga, FamB FbaB) , (1) equations was first demonstrated by witten. ' When the space-time is AS and stationary, the essential part of the Fatu," = 0 , (2)

vacuum field equations can be reduced to a complex non- Fa,,=apAP,-a,Aa, , (3) linear elliptic partial differential equation, known as the where Y,b a matrix (with positive-definite en- Ernst2 equation. Witten has shown that members of a spe- tries), a,b = 2, . , . , - ( n , O). A semicolon denotes cia1 class of the AS static self-dual SU(2) Yang-Mills equa- covarlant differentiation with respect to the Christoffel- tions, in Yaw's R gauge, also reduce to the same equation. Riemann connection. The metric y,b of the Abelian-gauge It is also known that the Ernst equation can be formulated group can be taken as the Kronecker symbol by reseal- as a o model on the symmetric space S U ( l , l ) / U ( l ) , Re- ing the gauge potentials Anp. When the space-time is sta- c e n t l ~ , ~ it was shown that this analogy exists also between tionary and axially symmetric the line element can be ex- the AS static self-dual SU(3) Yang-Mills and the AS sta- pressed as tionary Einstein-Maxwell field equations. Here the key equations are the electrovacuum Ernst4 equations, which ds2= - f (d t - od& )2+ f -1 [e2Y (dp2+dz2 )+p2d421 , (4) may be formulated as a cr model on SU(2,1)/SU(2) @ U(1).

In this work we shall show that the above analogy between the field equations of two theories can be further extended. The restricted class of the AS static self-dual S U ( n + l ) Yang-Mills and the AS stationary Ein- stein-Abelian-gauge field equations will be both formulated as a cr model on the Kahler manifold SU(n + 1 )/SU(n ) @ U(1).

The Einstein-Abelian-gauge field equations are given

where t, p, z, and d are the local coordinates and the func- tions f , w , and y depend only on p and z. We shall assume that the gauge potential one-form A" has two components,

A a ~ A R , d x P = A " , d t + A a + d 4 , (5)

where Aar and Aa+ are the components of An, in the direc- tion of time ( t ) and the azimuthal angle (4) coordinates, respectively. They also depend on p and z. Using (4) and (51, the gauge (2) and the Einstein (1) field equations reduce to

where ?, a a , and v2 are the grad, divergence, and the field equations consist of the first four equations (6a)-(6d). Laplacian operators in the flat three-dimensional cylindrical Introducing n complex scalars E and Ga these equations are coordinates, and those terms with repeated indices are written as summed up. Out of these seven equations only the first six are independent. Equations (6e) and (6f) imply the last ~ v ~ E = ( v ~ + ~ P ~ @ ~ ) . a, , equation (6g). On the other hand, Eqs. (6e) and (6f) deter-

( 7 )

mine the function y if f , w , A",, and An+ are known. Hence the essential part of the Einstein- and Abelian-gauge f v 2 G a = ( 9 E + 2 S b a a b ) . a @ " , (8)

30 486 - @ 1984 The American Physical Society

30 - BRIEF REPORTS 487

where which has unit determinant. In addition, it also satisfies

with

T B ~ = -p-lfri x ( ~ ~ A ~ + - - ~ V A ~ ~ ) , (11)

a + = - [,-lf2;i x a w + 2 1 m ( 6 ~ T ~ ~ ) J , (12)

where ri is the unit vector along the azimuthal direction and Im denotes the imaginary part. Although there is no in- teraction (by construction) between the potentials Aa, it ap- pears that the complex scalars Qa intract with one another. Equations (7) and (8) may be obtained from a variational principle where the Lagrangian density reads

with

This suggests that we should utilize the theory of harmonic mappings. The mapping F : N - IM between the Riemann manifolds M and N with metrics

is harmonic if the field equations (7) and (8) are satisfied. Here cf, denotes an ( n - 1)-dimensional complex column vector with components and a dagger denotes Hermitian conjugation. Defining a new column vector w by

and a Hermitian unimodular matrix y by

where In - 1 is the (n - 1 ) x ( n - 1 ) unit matrix, then the metric of M becomes

with

This metric describes the complex hyperbolic space5 SU(n, l ) / S U ( n ) 8 U ( 1 ) and it can be further simplified. We define a Hermitian matrix

Then the metric in (19), when expressed in terms of P takes the form

The field equations (7) and (8) may also be expressed in terms of P,. They become

Here we changed the manifold N to Ma with the metric

The exterior derivative d and the Hodge dual operation * in (24) are defined on Mo. The function a is a harmonic func- tion in Mo, i.e.,

Replacing the manifold N with M o is in a sense replacing p 2 appearing as a coefficient of d 4 2 in the space-time metric (4) with a2 . Harmonicity of the function a follows from the field equations. Hence, in summary, the AS stationary Einstein-Abelian-gauge field equations (6a)- (6d) may be reduced to a single differential equation for the matrix P, subject to some constraints. These equations read

where P, is an (n + 1) x ( n + 1 ) matrix. In the R gauge the self-dual S U ( N ) Yang-Mills field

equations are given by6

where

P = P ' , d e t P = l ,

and

Here, y = ( l / f i ) ( x l + cx2), z = ( 1 / 3 ) (x3- ,x4), and the bar operation denotes complex conjugation. Axially sym- metric static fields do not depend on the azimuthal angle ( 4 ) and time ( t ) , where y = p e l h n d z = T = t. In this case (29) becomes

where d and * are defined on M o with metric (25) as before. We notice that the AS static self-dual SU(N) Yang-Mills equations (32) are more general than the AS sta- tionary- Abelian-gauge [ ( n - 1 ) x Maxwell] field equations. They become identical when N = 12 + 1 and P = P,, which implies ( y P ) = I. This constraint is, of course, compatible with the field equations (32).

The group SU(n, 1 ) is the isometry group of the metric (23) of M. Hence the transformation

488 BRIEF REPQRTS - 30

with

i.e., S E SU(n, 11, leaves the metric (23) and the field equa- tion (24) invariant. This may be utilized to generate new solutions of the field equations. Such global transforma- tions (S = constant) produce gague-equivalent Yang-Mills field^.^ On the other hand, the SU(n, 1 ) transformation (33) leads to distinct solutions in the Einstein case. For in- stance, starting from a vacuum solution ( n = I ) , say the Kerr metric, it is possible to obtain a solution of the Einstein- ( t z - 1 )-Maxwell field equations, say an ( n - 1 ) Abelian-gauge charged-Kerr metric.

It is known that the vacuum (n = 1) and the electrovacu- um ( n = 2 ) field equations can be integrated via the ~akharov-shabat7 and the Relinski-zakharov8 techniques. These methods can be extended to integrate the field equa- tions [ ( 2 7 ) , (28)l. Such an extension bas been given by several author^.^

Quite recently,10 it has been shown that the Kerr- Newman metric ( n = 2 ) is the unique stationary black-hole solution of the Einstein-Maxwell field equations. If a black hole carries (n - 1 ) Abelian-gauge charges, the above theorem can be paraphrased. The ( n - 1 ) Abelian-gauge charged-Kerr metric is the unique stationary black-hole solution of the Einstein-Abelian-gauge field equations.

'L. Witten, Phys. Rev. D 19, 718 (1978). 8V. A. Belinski and V. E. Zakharov, Zh. Eksp. Teor. Fiz. 77, 3 2F. J. Emst, Phys. Rev. 167, 1175 (1968). (1979) [Sov. Phys. JETP. 2, 1 (197911. 3M. Gilrses and B. C. Xanthopoulos, Phys. Rev. D 3, 1912 (19821. 9A. Eris, M . Gorses, and A. Karasu; J. Math. Phys. (to be pub- 4F. J. Ernst, Phys. Rev. 168. 1415 (1968). lished); see also M, GBrses, paper presented at the Workshop on 5S. Kobayashi and K. Nomizu, Foundations of'D$ferential Geometry the Exact Solutions of Einstein's field equations, Retzbach,

(Interscience, New York, 1969). Vol. 11. Federal Republic of Germany, 1983 (unpublished). 6Y. Brihaye, D. B. Fairlie, J. Nuyts, and R. G. Yates, J. Math. Phys. "JP. Mazur, J. Phys. A u, 3173 (1982).

19, 2528 (1978). 7 ~ - ~ . Zakharov and A. B. Shabat, Funct. Anal. Appl. 13, 166

(1979).