PHYSICAL REVIEW D 72, 024011 (2005)

Type III Einstein-Yang-Mills solutions

Andrea Fuster and Jan-Willem van Holten1National Institute for Nuclear and High-Energy Physics (NIKHEF), Kruislaan 409, 1098 SJ, Amsterdam, The Netherlands

(Received 7 June 2005; published 12 July 2005)

1550-7998=20

We construct two distinct classes of exact type III solutions of the D � 4 Einstein-Yang-Mills system.The solutions are embeddings of the non-Abelian plane waves in spacetimes in Kundt’s class. Reductionof the solutions to type N leads to generalized pp and Kundt waves. The geodesic equations are brieflydiscussed.

DOI: 10.1103/PhysRevD.72.024011 PACS numbers: 04.40.Nr, 04.20.Jb

1Tetrad vectors as in [1]: e1 � �e2 � @z; e3 � @u �W@z �W@ �z � �H �WW�@v; e4 � @v

I. INTRODUCTION

Exact solutions of Einstein equations have always at-tracted much attention. It is somewhat surprising to findexact solutions of such nonlinear equations. Many of themwere collected in the by now classic book by Kramer et al.which has recently been revised [1]. Among others onefinds the nondiverging spacetimes which were studied forthe first time by Kundt in [2]. Solutions which admit anonexpanding and twistfree null congruence are thereforesaid to belong to the Kundt’s class. Such metrics arealgebraically special, i.e., of Petrov type III, N, O, II orD. If one restricts the solutions to those with plane wavesurfaces they are constrained to be (at most) of type III;they can degenerate to type N and O. Conformally flatsolutions include, for example, the Bertotti-Robinson [3,4]universe and the Edgar-Ludwig metric [5]. An Einstein-Yang-Mills solution of type N was given long time ago in[6] and recovered for D � 4 supergravity in [7]. Type IIIsolutions have somehow attracted less attention. Yet, somereferences considering the type III Einstein-Maxwell sys-tem can be found [8,9]. To our knowledge, the solutionspresented in this note are the first type III solutions ofEinstein gravity coupled to a Yang-Mills field.

We also want to point out that the Kundt’s class can begeneralized to include a nonzero cosmological constant.Such solutions of type N were first considered in [10],followed by [11–14]. Type N Einstein-Maxwell solutionswere generalized to an arbitrary number of dimensions in[15] and further generalized to Lovelock-Yang-Mills solu-tions in [16]. Not very long ago the � � 0 generalizationof type III spacetimes in [1] was presented in [17].

II. YANG-MILLS IN KUNDT SPACETIMES

We consider Kundt’s class of metrics, described by theline element:

ds2 � �2du�dv�Wdz�Wd�z�Hdu�

� 2P�2dzd�z; P;v� 0 (1)

Here, P and H are real functions while W is complex; u �

�t� z�=���2

p, v � �t� z�=

���2

pare light-cone coordinates and

z � �x� iy�=���2

p, �z � �x� iy�=

���2

pcomplex conjugate co-

05=72(2)=024011(4)$23.00 024011

ordinates in the transverse plane. We consider solutions oftype III. In this case we can take P � 1 without loss ofgenerality. As a consequence the Gaussian curvature of thewave surfaces u � constant, K � 2P2�lnP�;z�z , vanishes.Type III solutions are therefore characterized by planewave surfaces. Details on specific aspects of the metrics(1) can be found in [1].

It can be seen that two distinct cases arise for vacuumsolutions of type III (for references, see [1]). In the firstcase the function W does not depend on the light-conecoordinate v. In the second case the dependence is of theform W;v� �2=�z� �z�. These two cases correspond tothe Newman-Penrose quantity � being zero or � 0,respectively.

A. Case W;v � 0

Consider the metric (1) with:

P � 1 (2a)

W � W�u; �z� (2b)

H � H0�u; z; �z� �1

2�W;�z�W;z �v (2c)

Here, W is an arbitrary complex function and H0 is a realfunction. This is a vacuum solution of Einstein equations ifthe function H0 satisfies:

1

2R33 � H;0z�z�Re�W;2�z �WW;�z �z�W;�zu � � 0 (3)

Numerical indices are tetrad indices1. All other compo-nents of the Ricci tensor vanish. This class of solutionsdegenerates to type N when �3 �

12W;�z �z� 0. In this case

one can use the remaining coordinate transformation (see[1]) to make W � 0. The metric reduces then to that offamiliar pp waves [18],

ds2 � 2dzd�z� 2dudv�H�u; z; �z�du2 (4)

where H satisfies H;z�z� 0.The function W can have an arbitrary u dependence but

must be at least quadratic in �z so that �3 � 0 and the

-1 2005 The American Physical Society

2The superscript 0 in R33 denotes the v-independent part ofR33.

3The original field in [19] was �a � �a�u�z�some function of u but the later term can be gauged awaywithout affecting gauge-invariant quantities, as pointed out byColeman.

ANDREA FUSTER AND JAN-WILLEM VAN HOLTEN PHYSICAL REVIEW D 72, 024011 (2005)

solution is truly of type III. This holds for pure radiationsolutions as well: they only enter the solution through thefunction H0 so W is as in vacuum.

We can generalize the solution to include a null Yang-Mills field (�0 � �1 � 0) of the form:

A � �a�u; z; �z�Tadu; �a � �a�u; z� � ��a�u; �z� (5)

Here, �a are arbitrary complex functions. This field wasconsidered in curved spacetime for the first time in [6]. TheYang-Mills equation in flat space is just:

�;z�z� 0 (6)

This equation remains unchanged in the spacetime (1) and(2a)–(2c) because of two reasons. First, the absence ofcomponents of the YM field strength of the form Fu� inthis geometry. In the second place, the only nonvanishingChristoffel symbols of the form ���

� are precisely ��u�.

As a result only the ordinary derivative survives in the(curved) Yang-Mills equation:

@�F�� � ��u

�Fu� � �Au; Fu�� � 2�;z�z� 0 (7)

Of course Eq. (3) has to be modified to account for theenergy-momentum tensor of the Yang-Mills field:

Tuu �1

2� ab�;az �;b�z (8)

Here, ab is the invariant metric of the Lie group. In tetradindices we have T33 � Tuu. Therefore, Eq. (3) transformsto:

H;0z�z�Re�W;2�z �WW;�z �z�W;�zu � � 2 ab�;az �;b�z

It is straightforward to solve for H0:

H0�u; z; �z� � f�u; z� � �f�u; �z� � 2 ab�a ��b

� Ref�W;u�WW;�z �zg (9)

Here f is an arbitrary complex function. Eqs. (5) and (1)and (2a)–(2c) with an arbitrary (at least quadratic) W andthe givenH0 describe an exact type III solution of the EYMsystem. It is a generalized Goldberg-Kerr solution. Thetype N reduction of this solution has been known for a longtime [6].

B. Case W;v � �2=�z� �z�

We consider now the metric (1) with:

P � 1 (10a)

W � W0�u; z� �2vz� �z

(10b)

H � H0�u; z; �z� �W0 �W0

z� �zv�

v2

�z� �z�2(10c)

Here, W0 is an arbitrary complex function and H0 is a realfunction. This is a vacuum solution of Einstein equations ifthe function H0 satisfies the differential equation:

024011

1

2R033 � �z� �z�

�H0 �W0W0

z� �z

�;z�z

�W;0z W;0�z � 0 (11)

This is the only nonvanishing component of the Riccitensor2. This class of solutions degenerate to type N aswell when �3 � W;0�z =�z� �z� � 0. In this case one cantransform W0 to zero [1] and the resulting metric is knownas Kundt waves.

We consider the field (5) in this background. Features inthis geometry are the same ones as in the previous case,except for extra terms in the Yang-Mills equation:

�@z � �uzu � �vz

v�Fzv � �@�z � �u �zu � �v�z

v�F �zv � 0

(12)

After further inspection this extra term is seen to vanish as�uz

u � ��vzv, �u �z

u � ��v�zv. In this way, the Yang-Mills

equation in the spacetime (1), (10b), and (10c) is still thesame as in flat space, Eq. (6). Finally, Eq. (11) has to bemodified to include the YM energy-momentum tensorT33 � � ab=2���;az �;b�z .

We refine our ansatz in order to solve explicitly for H0.We choose the specific functionW0 � g�u�z, g�u� being anarbitrary complex function. This is the simplest form ofW0

such that �3 is nonvanishing. On the other hand we limitthe Yang-Mills field (5) to:

�a � �a�u�z (13)

Here �a�u� are bounded complex functions. This field wasconsidered in Minkowski space for the first time in [19]and is referred to as non-Abelian plane waves3. The field(13) is the only form of (5) which gives rise to a boundedenergy-momentum tensor. Under these assumptions theequation to solve reads:

�z� �z��H0 � g �gz�zz� �z

�;z �z

� g�u� �g�u� � 2 ab�a�u� ��b�u�

It is not difficult to see that:

H0 � �f�u; z� � �f�u; �z���z� �z� � g �gz�z

� $�u��z� �z�2fln�z� �z� � 1� (14)

Here f is again an arbitrary complex function and $ is thereal function $�u� � 2 ab�

a�u� ��b�u� � g�u� �g�u�.Eqs. (5), (13), (1), and (10a)–(10c) with the given W0,H0 describe another exact type III solution of the EYMsystem. The corresponding vacuum solution (�a � 0) hasto our knowledge not been considered before. The caseg�u� � 0 yields the type N reduction and can be regarded

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TYPE III EINSTEIN-YANG-MILLS SOLUTIONS PHYSICAL REVIEW D 72, 024011 (2005)

as generalized Kundt waves. A related type N solution wasvery recently given in [16], for a Yang-Mills field ofarbitrary higher (polynomial) dependence on z, �z; our(type N) solution is obtained by translating the line elementinto the Kundt canonical form and considering N � 0,D � 4.

4This choice of f is well motivated as it corresponds tohomogeneous pp waves in the (vacuum) type N reduction.

C. Geodesics

For completeness, we present the geodesic equationscorresponding to the two classes of solutions found, with-out attempting to provide detailed solutions here.

Case W;v� 0

In order to get a feeling for the behavior of geodesics wechoose the specific W � �z2. This is the simplest W suchthat the solution presented in subsection A is of type III.The geodesics are described in terms of real spatial coor-dinates x, y by:

�u����2

px _u2 � 0 (15)

�x� _u2�H;x�x�x2 � y2�� � 2

���2

py _u _y � 0 (16)

�y� _u2�H;y�2yx2� � 2���2

py _u _x � 0 (17)

_x2 � _y2 � 2 _u _v�2H _u2 ����2

p�x2 � y2� _x _u

� 2���2

pyx _y _u � & (18)

Here, the overdot denotes a proper time derivative and His:

H � f� �f� 2�x4 � y4� � 2 ab�a ��b �

���2

pxv (19)

We have considered the normalization condition for the4-velocity, Eq. (18), instead of the (complicated) equationfor v. There, & � �1; 0 for timelike and null geodesics,respectively.

It can be advantageous to use u as affine parameterinstead of the proper time. If we assume _u > 0, fromEq. (15):

x �1���2

pd�ln _u�du

(20)

Now, in the hyperplane y � 0, by inserting Eq. (20) in (16)and (18) and recalling _x � _u�dx=du�, a coupled system forx�u�, v�u� can be obtained.

On the other hand, in the hyperplane x � 0 Eqs. (15) and(17) simplify to

�u � 0 $ _u � constant � (21)

�y� _u2H;y� 0 (22)

taking the same form as in the type N reduction (see forexample [20]). Analytical differences come from the

024011

quartic term of (19) which is absent in the type N. Thisdifference can be physically illustrated when we considerthe functions f � (�u�z2, with (�u� an arbitrary real func-tion4, and �a � �a�u�z in Eq. (23). Then, the motion in they direction is governed by:

d2y=du2 � 8y3 � 2� ab�a�u� ��b�u� � (�u��y � 0 (23)

This equation describes a (parametric) nonlinear oscillatorin a potential V�y� � y2� ab�

a ��b � (� 2y2�. In the typeN reduction there is no cubic term in Eq. (23) and themotion reduces to that of an harmonic oscillator.

Nonlinear oscillators can have chaotic motion. Geodesicchaotic motion has been found in the related case of non-homogeneous vacuum pp waves (f zn; n > 2), see [21].However, chaotic behavior is not always present in non-linear systems and therefore possible chaotic motion aris-ing from Eq. (23) would need further investigation.

Case W;v� �2=�z� �z�

We take the u dependence out of W0 respect to thefunction considered in subsection B, so now W0 � z.Geodesics in this second case have the more complicatedform:

�u� �1� v=x2� _u2 � 2=x _u _x � 0 (24)

�x� _u2�H;x � �x� 2v=x��1� v=x2��

� 2�1� 2v=x2� _u _x�2=x _v _u � 0 (25)

�y� _u2�H;y�y�1� v=x2�� � 2y=x _u _x � 0 (26)

_x 2 � _y2 � 2 _u _v�2H _u2 � 2��2v=x� x� _x _u�2y _y _u � &

(27)

Again, the normalization condition of the 4-velocitytakes the place of the equation for v. The function H is

H � H0 � v�v2

2x2(28)

and H0 is given by (14) but with g�u� � 1.The geodesic equations can again be rewritten as differ-

ential equations in terms of an affine variable u instead ofthe proper time. In particular, for the first equation:

ddu

ln�_u

x2

�� 1�

v

x2(29)

Then the remaining geodesics become equations for�x; y; v� in terms of u.

The existence of a singularity at x � 0 is clear from thegeodesic equations, but not its character. Geodesics for ageneral vacuum type N or conformally flat pure radiation

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ANDREA FUSTER AND JAN-WILLEM VAN HOLTEN PHYSICAL REVIEW D 72, 024011 (2005)

solutions have been analyzed in [22], where it was con-cluded that x � 0 is a physical singularity. Whether this isalso the case for our type III solutions remains an openquestion.

III. CONCLUSIONS

In [23] Tafel claimed that null twistfree solutions of theYang-Mills equations were exhausted by the type N (classof) solutions given in [6]. A new type N solution wasrecently given in [16]. We have shown that more generalsolutions exist, as the two type III (classes of) EYMsolutions presented here are also null and twistfree.

It has been proven [24] that all spacetimes with vanish-ing curvature invariants of all orders: 1) are Kundt metricsof Petrov type III, N or O, and 2) are of Plebanski-Petrov

024011

(PP) type N or O (null radiation or vacuum). These resultshave been generalized to higher dimensions in [25]. Suchspacetimes have been called VSI spacetimes. Our solutionsclearly meet requirements 1) and 2) and are therefore newexamples of D � 4 VSI spacetimes. As such, they mighthave interesting physical applications in supergravity andstring theory, in the fashion of pp waves.

ACKNOWLEDGMENTS

A. F. would like to thank N. Van den Bergh for discus-sions. J. W. v. H. whishes to thank the DESY theory divi-sion, where part of this work was performed, for itshospitality. This work is part of the programme FP52 ofthe Foundation for Research of Matter (FOM).

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