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. Reduction of the solutions to type
N leads to generalized $pp$ and Kundt waves. The geodesic equations are briefly
discussed.Comment: revtex, 4 pages, minor changes, some factors and references
  corrected, footnote adde

Andrea Fuster

H. Stephani

I. Robinson

J. van Holten

Jan-Willem van Holten

M. Cahen

M. W. Brinkmann

English

arXiv

Exact solutions of Einstein equations have always attracted much attention. It is somewhat surprising to find exact solutions of such nonlinear equations. Many of them were collected in the by now classic book by Kramer et al. which has recently been revised [1]. Among others one finds the nondiverging spacetimes which were studied for the first time by Kundt in [2]. Solutions which admit a nonexpanding and twistfree null congruence are therefore said to belong to the Kundt’s class. Such metrics are algebraically special, i.e., of Petrov type III, N, O, II or D. If one restricts the solutions to those with plane wave surfaces they are constrained to be (at most) of type III; they can degenerate to type N and O. Conformally flat solutions 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 III solutions have somehow attracted less attention. Yet, some references considering the type III Einstein-Maxwell system can be found [8,9]. To our knowledge, the solutions presented in this note are the first type III solutions of Einstein gravity coupled to a Yang-Mills field. We also want to point out that the Kundt’s class can be generalized to include a nonzero cosmological constant. Such solutions of type N were first considered in [10], followed by [11–14]. Type N Einstein-Maxwell solutions were generalized to an arbitrary number of dimensions in [15] and further generalized to Lovelock-Yang-Mills solutions in [16]. Not very long ago the ¿ ¿ 0 generalization of type III spacetimes in [1] was presented in [17]

Fuster, A.

Holten, van, J.W.

Pure OAI Repository

 Type III Einstein-Yang-Mills solutionsCitation for published version (APA):Fuster, A., & Holten, van, J. W. (2005). Type III Einstein-Yang-Mills solutions. Physical Review D: Particles andFields, Gravitation, and Cosmology, 72(2), 1-4. Article 024011. https://doi.org/10.1103/PhysRevLett.72.024011,https://doi.org/10.1103/PhysRevD.72.024011DOI:10.1103/PhysRevLett.72.02401110.1103/PhysRevD.72.024011Document status and date:Published: 01/01/2005Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023arXiv:gr-qc/0505159v2  7 Jun 2005NIKHEF/2005-009Type III Einstein-Yang-Mills solutionsAndrea Fustera, Jan-Willem van Holtenaa National Institute for Nuclear and High-Energy Physics (NIKHEF),Kruislaan 409, 1098 SJ, Amsterdam, The Netherlands(Dated: 5th February 2008)We construct two distinct classes of exact type III solutions of the D = 4 Einstein-Yang-Millssystem. The solutions are embeddings of the non-Abelian plane waves in spacetimes in Kundt’sclass. Reduction of the solutions to type N leads to generalized pp and Kundt waves. The geodesicequations are briefly discussed.I. INTRODUCTIONExact solutions of Einstein equations have alwaysattracted much attention. It is somewhat surprisingto find exact solutions of such non-linear equations.Many of them were collected in the by now classic bookby Kramer et al, which has recently been revised [1].Among others one finds the non-diverging spacetimeswhich were studied for the first time by Kundt in [2].Solutions which admit a non-expanding and twistfreenull congruence are therefore said to belong to theKundt’s class. Such metrics are algebraically special,i.e., of Petrov type III, N, O, II or D. If one restricts thesolutions to those with plane wave surfaces they are con-strained to be (at most) of type III; they can degenerateto type N and O. Conformally flat solutions includefor example the Bertotti-Robinson [3, 4] universe andthe Edgar-Ludwig metric [5]. An Einstein-Yang-Millssolution of type N was given long time ago in [6] andrecovered for D = 4 supergravity in [7]. Type III solu-tions have somehow attracted less attention. Yet, somereferences considering the type III Einstein-Maxwellsystem can be found [8, 9]. To our knowledge, thesolutions presented in this note are the first type IIIsolutions of Einstein gravity coupled to a Yang-Millsfield.We also want to point out that the Kundt’s class can begeneralized to include a non-zero cosmological constant.Such solutions of type N were first considered in [10], fol-lowed by [11, 12, 13, 14]. Type N Einstein-Maxwell solu-tions were generalized to an arbitrary number of dimen-sions in [15] and further generalized to Lovelock-Yang-Mills solutions in [16]. Not very long ago the Λ 6= 0generalization of type III spacetimes in [1] was presentedin [17].II. YANG-MILLS IN KUNDT SPACETIMESWe consider Kundt’s class of metrics, described by theline element:ds2 = − 2du (dv + W dz + W dz̄ + H du)+ 2P−2dzdz̄, P,v = 0(1)Here, P and H are real functions while W is complex;u = (t − z)/√2, v = (t + z)/√2 are light-cone coordi-nates and z = (x + iy)/√2, z̄ = (x − iy)/√2 complexconjugate coordinates in the transverse plane. Weconsider solutions of type III. In this case we can takeP = 1 without loss of generality. As a consequence theGaussian curvature of the wave surfaces u=constant,K = 2P 2(ln P ),zz̄, vanishes. Type III solutions aretherefore characterized by plane wave surfaces. Detailson 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 ofthe form W,v = −2/(z + z̄). These two cases correspondto the Newman-Penrose quantity τ being zero or 6= 0,respectively.A. Case W,v = 0Consider the metric (1) with:P = 1 (2a)W = W (u, z̄) (2b)H = H0(u, z, z̄) +12(W,z̄ +W,z)v (2c)Here, W is an arbitrary complex function and H0 is a realfunction. This is a vacuum solution of Einstein equationsif the function H0 satisfies:12R33 = H,0zz̄ −Re(W,2z̄ +WW,z̄z̄ +W,z̄u)= 0 (3)Numerical indices are tetrad indices1. All other compo-nents of the Ricci tensor vanish. This class of solutionsdegenerates to type N when Ψ3 =12W,z̄z̄ = 0. In thiscase one can use the remaining coordinate transforma-tion (see [1]) to make W = 0. The metric reduces then1 Tetrad vectors as in [1]: e1 = ē2 = ∂z , e3 = ∂u + W ∂z +W ∂z̄ − (H + WW ) ∂v, e4 = ∂v2to that of familiar pp waves [18],ds2 = 2 dz dz̄ − 2 du dv − H(u, z, z̄) du2 (4)where H satisfies H,zz̄ = 0.The function W can have an arbitrary u dependencebut must be at least quadratic in z̄ so that Ψ3 6= 0. Thisholds for pure radiation solutions as well: they onlyenter the solution through the function H0 so W is as invacuum.We can generalize the solution to include a null Yang-Mills field (Φ0 = Φ1 = 0) of the form:A = αa(u, z, z̄)Ta du, αa = χa(u, z) + χ̄a(u, z̄) (5)Here, χa are arbitrary complex functions. This field wasconsidered in curved spacetime for the first time in [6].The Yang-Mills equation in flat space is just:α,zz̄ = 0 (6)This equation remains unchanged in the spacetime (1)-(2c) because of two reasons. First, the absence of com-ponents of the YM field strength of the form Fuµ in thisgeometry. In the second place, the only non-vanishingChristoffel symbols of the form Γ ρρν are precisely Γρρu .As a result only the ordinary derivative survives in the(curved) Yang-Mills equation:∂λFλµ + Γ ρρu Fuµ − [Au, Fuµ] = 2α,zz̄ = 0 (7)Of course Eq. (3) has to be modified to account forthe energy-momentum tensor of the Yang-Mills field:Tuu =12πγab α,az α,bz̄ (8)Here, γab is the invariant metric of the Lie group. Intetrad indices we have T33 = Tuu. Therefore, Eq. (3)transforms to:H,0zz̄ −Re(W,2z̄ +WW,z̄z̄ +W,z̄u)= 2γab α,az α,bz̄It is straightforward to solve for H0:H0(u, z, z̄) = f(u, z) + f̄(u, z̄) + 2γabχaχ̄b+ Re {(W,u +WW,z̄ ) z}(9)Here f is an arbitrary complex function. Eqs. (5) and(1)-(2c) with an arbitrary (at least quadratic) W andthe given H0 describe an exact type III solution of theEYM system. It is a generalized Goldberg-Kerr solution.The type N reduction of this solution has been knownfor a long time [6].B. Case W,v = −2/(z + z̄)We consider now the metric (1) with:P = 1 (10a)W = W 0(u, z) − 2vz + z̄(10b)H = H0(u, z, z̄) +W 0 + W0z + z̄v − v2(z + z̄)2(10c)Here, W 0 is an arbitrary complex function and H0 is areal function. This is a vacuum solution of Einstein equa-tions if the function H0 satisfies the differential equation:12R033= (z + z̄)(H0 + W0W 0z + z̄),zz̄−W,0z W,0z̄ = 0 (11)This is the only non-vanishing component of the Riccitensor2. This class of solutions degenerate to type Nas well when Ψ3 = W,0z̄ /(z + z̄) = 0. In this case onecan transform W 0 to zero [1] and the resulting metric isknown as Kundt waves.We consider the field (5) in this background. Featuresin this geometry are the same ones as in the previous case,except for an extra term in the Yang-Mills equation:(∂z + Γuuz + Γvvz )Fzv + (z ↔ z̄) = 0 (12)After further inspection this extra term is seen to vanishas Γ uuz = −Γ vvz , Γ uuz̄ = −Γ vvz̄ . In this way, theYang-Mills equation in the spacetime (1), (10b), (10c) isstill the same as in flat space, Eq. (6). Finally, Eq. (11)has to be modified to include the YM energy-momentumtensor T33 = (γab/2π) α,az α,bz̄ .We refine our ansatz in order to solve explicitly for H0.We choose the specific function W 0 = g(u)z, g(u) beingan arbitrary complex function. This is the simplest formof W 0 such that Ψ3 is non-vanishing. On the other handwe limit the Yang-Mills field (5) to:χa = λa(u) z (13)Here λa(u) are bounded complex functions. This fieldwas considered in Minkowski space for the first timein [19] and is referred to as non-Abelian plane waves3.The field (13) is the only form of (5) which gives rise toa bounded energy-momentum tensor. Under these as-sumptions the equation to solve reads:(z + z̄)(H0 + gḡzz̄z + z̄),zz̄− g(u)ḡ(u) = 2γabλa(u)λ̄b(u)2 The superscript 0 in R33 denotes the v-independent part of R33.3 The original field in [19] was χa = λa(u) z + some function of ubut the later term can be gauged away without affecting gauge-invariant quantities, as pointed out by Coleman.3It is not difficult to see that:H0 = (f(u, z) + f̄(u, z̄)) (z + z̄) − gḡzz̄+ σ(u)(z + z̄)2 {ln(z + z̄) − 1)(14)Here f is again an arbitrary complex function and σ isthe real function σ(u) = 2γabλa(u)λ̄b(u)+g(u)ḡ(u). Eqs.(5), (13) and (1), (10a)-(10c) with the given W 0, H0 de-scribe another exact type III solution of the EYM system.The corresponding vacuum solution (λa ≡ 0) has to ourknowledge not been considered before. The case g(u) ≡ 0yields the type N reduction and can be regarded as gener-alized Kundt waves. A related type N solution was veryrecently given in [16], for a Yang-Mills field of arbitraryhigher (polynomial) dependence on z, z̄; our (type N) so-lution is obtained by translating the line element into theKundt canonical form and considering N = 0, D = 4.C. GeodesicsFor completeness, we present the geodesic equationscorresponding to the two classes of solutions found,without attempting to provide detailed solutions here.· Case W,v = 0In order to get a feeling for the behavior of geodesics wechoose the specific W = z̄2. This is the simplest W suchthat the solution presented in subsection A is of typeIII. The geodesics are described in terms of real spatialcoordinates x, y by:ü −√2xu̇2 = 0 (15)ẍ + u̇2(H,x −x(x2 − y2))+ 2√2y u̇ẏ = 0 (16)ÿ + u̇2(H,y −2yx2)− 2√2y u̇ẋ = 0 (17)ẋ2 + ẏ2 − 2u̇v̇ − 2Hu̇2 −√2(x2 − y2)ẋu̇ − 2√2yx ẏu̇ = ǫ(18)Here, the overdot denotes a proper time derivative andH is:H = f + f̄ + 2(x4 − y4) + 2γabχaχ̄b +√2xv (19)We have considered the normalization condition forthe 4-velocity, Eq. (18), instead of the (complicated)equation for v. There, ǫ = −1, 0 for timelike and nullgeodesics, respectively.It can be advantageous to use u as affine parameterinstead of the proper time. If we assume u̇ > 0, from Eq.(15):x =1√2d (lnu̇)du(20)Now, in the hyperplane y = 0, by inserting Eqs. (20) in(16), (18) and recalling ẋ = u̇ (dx/du), a coupled systemfor x(u), v(u) can be obtained.On the other hand, in the hyperplane x = 0 Eqs. (15)and (17) simplify toü = 0 ↔ u̇ = constant ≡ γ (21)ÿ + u̇2H,y = 0 (22)taking the same form as in the type N reduction (seefor example [20]). Analytical differences come from thequartic term of (19) which is absent in the type N. Thisdifference can be physically illustrated when we considerthe functions f = κ(u) z2, with κ(u) an arbitrary realfunction4, and χa = λa(u) z in Eq. (19). Then, themotion in the y direction is governed by:d2y/du2 − 8y3 + 2(γabλa(u)λ̄b(u) − κ(u))y = 0 (23)This equation describes a (parametric) non-linear oscil-lator in a potential V (y) = y2(γabλaλ̄b − κ − 2y2). Inthe type N reduction there is no cubic term in Eq. (23)and the motion reduces to that of an harmonic oscillator.Non-linear oscillators can have chaotic motion.Geodesic chaotic motion has been found in the re-lated case of non-homogeneous vacuum pp waves(f ∼ zn, n > 2), see [21]. However, chaotic behavior isnot always present in non-linear systems and thereforepossible chaotic motion arising from Eq. (23) wouldneed further investigation.· Case W,v = −2/(z + z̄)We take the u dependence out of W 0 respect to thefunction considered in subsection B, so now W 0 = z.Geodesics in this second case have the more complicatedform:ü − (1 − v/x2)u̇2 − 2/x u̇ẋ = 0 (24)ẍ + u̇2(H,x +(x − 2v/x)(1 − v/x2))+ 2(1 − 2v/x2)u̇ẋ− 2/x v̇u̇ = 0 (25)ÿ + u̇2(H,y +y(1 − v/x2))+ 2y/x u̇ẋ = 0 (26)ẋ2 + ẏ2 − 2u̇v̇ − 2Hu̇2 − 2(−2v/x + x) ẋu̇ + 2y ẏu̇ = ǫ(27)Again, the normalization condition of the 4-velocitytakes the place of the equation for v. The function H isH = H0 + v − v22x2(28)and H0 is given by (14) but with g(u) ≡ 1.4 This choice is well motivated as it corresponds to homogeneouspp waves in the (vacuum) type N reduction.4The geodesic equations can again be rewritten as dif-ferential equations in terms of an affine variable u insteadof the proper time. In particular, for the first equation:dduln(u̇x2)= 1 − vx2(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 fromthe geodesic equations, but not its character. Geodesicsfor a general vacuum type N or conformally flat pureradiation solutions have been analyzed in [22], whereit was concluded that x = 0 is a physical singularity.Whether this is also the case for our type III solutionsremains an open question.III. CONCLUSIONSIn [23] Tafel claimed that null twistfree solutions ofthe Yang-Mills equations were exhausted by the type N(class of) solutions given in [6]. A new type N solutionwas recently given in [16]. We have shown that moregeneral solutions exist, as the two type III (classes of)EYM solutions presented here are also null and twistfree.It has been proved [24] that all spacetimes withvanishing curvature invariants of all orders: 1) areKundt metrics of Petrov type III, N or O, and 2) are ofPlebański-Petrov (PP) type N or O (null radiation orvacuum). These results have been generalized to higherdimensions in [25]. Such spacetimes have been calledVSI spacetimes. Our solutions clearly meet requirements1) and 2) and are therefore new examples of D = 4VSI spacetimes. As such, they might have interestingphysical applications in supergravity and string theory,in the fashion of pp waves.AcknowledgmentsA.F. would like to thank N. Van den Bergh for dis-cussions. J.W.v.H. whishes to thank the DESY theorydivision, where part of this work was performed, for itshospitality. This work is part of the programme FP52 ofthe Foundation for Research of Matter (FOM).[1] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers,and E. Herlt (2003), cambridge, UK: Univ. Pr. 701 P.[2] W. Kundt, Z. Phys 163, 77 (1961).[3] B. Bertotti, Phys. Rev. 116, 1331 (1959).[4] I. Robinson, Bull. Acad. Polon. 7, 351 (1959).[5] S. B. Edgar and G. Ludwig, Class. Quant. Grav. 14, L65(1997), gr-qc/9612059.[6] R. Gueven, Phys. Rev. D19, 471 (1979).[7] M. Cariglia, G. W. Gibbons, R. Guven, and C. N. Pope,Class. Quant. Grav. 21, 2849 (2004), hep-th/0312256.[8] G. Hall and J. Carot, Class. Quantum Grav. 11, 475(1994).[9] M. Cahen and J. Spelkens, Academie Royale de Belgique(1967), vol.53.[10] A. Garcia Diaz and J. Plebanski, J. Math. Phys. 22, 2655(1981).[11] I. Ozsvath, I. Robinson, and K. Rozga, J. Math. Phys.26, 1755 (1985).[12] J. Bicak and J. Podolsky, J. Math. Phys. 40, 4495 (1999),gr-qc/9907048.[13] J. Bicak and J. Podolsky, J. Math. Phys. 40, 4506 (1999),gr-qc/9907049.[14] J. Podolsky and M. Ortaggio, Class. Quant. Grav. 20,1685 (2003), gr-qc/0212073.[15] Y. N. Obukhov, Phys. Rev. D69, 024013 (2004), gr-qc/0310121.[16] R. J. Gleiser and G. Dotti (2005), gr-qc/0505094.[17] J. B. Griffiths, P. Docherty, and J. Podolsky, Class.Quant. Grav. 21, 207 (2004), gr-qc/0310083.[18] M. W. Brinkmann, Proc. Natl. Acad. Sci. (1923), 9.[19] S. R. Coleman, Phys. Lett. B70, 59 (1977).[20] J. van Holten, Lect. Notes Phys. 541, 365 (2000), gr-qc/9906026.[21] J. Podolsky and K. Vesely, Phys. Rev. D58, 081501(1998), gr-qc/9805078.[22] J. Podolsky and M. Belan, Class. Quant. Grav. 21, 2811(2004), gr-qc/0404068.[23] J. Tafel, Lett. Math. Phys. 12, 163 (1986).[24] V. Pravda, A. Pravdova, A. Coley, and R. Milson, Class.Quant. Grav. 19, 6213 (2002), gr-qc/0209024.[25] A. Coley, R. Milson, V. Pravda, and A. Pravdova, Class.Quant. Grav. 21, 5519 (2004), gr-qc/0410070.

Type III Einstein-Yang-Mills solutions

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 Type III Einstein-Yang-Mills solutionsCitation for published version (APA):Fuster, A., & Holten, van, J. W. (2005). Type III Einstein-Yang-Mills solutions. Physical Review D: Particles andFields, Gravitation, and Cosmology, 72(2), 1-4. Article 024011. https://doi.org/10.1103/PhysRevLett.72.024011,https://doi.org/10.1103/PhysRevD.72.024011DOI:10.1103/PhysRevLett.72.02401110.1103/PhysRevD.72.024011Document status and date:Published: 01/01/2005Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023arXiv:gr-qc/0505159v2  7 Jun 2005NIKHEF/2005-009Type III Einstein-Yang-Mills solutionsAndrea Fustera, Jan-Willem van Holtenaa National Institute for Nuclear and High-Energy Physics (NIKHEF),Kruislaan 409, 1098 SJ, Amsterdam, The Netherlands(Dated: 5th February 2008)We construct two distinct classes of exact type III solutions of the D = 4 Einstein-Yang-Millssystem. The solutions are embeddings of the non-Abelian plane waves in spacetimes in Kundt’sclass. Reduction of the solutions to type N leads to generalized pp and Kundt waves. The geodesicequations are briefly discussed.I. INTRODUCTIONExact solutions of Einstein equations have alwaysattracted much attention. It is somewhat surprisingto find exact solutions of such non-linear equations.Many of them were collected in the by now classic bookby Kramer et al, which has recently been revised [1].Among others one finds the non-diverging spacetimeswhich were studied for the first time by Kundt in [2].Solutions which admit a non-expanding and twistfreenull congruence are therefore said to belong to theKundt’s class. Such metrics are algebraically special,i.e., of Petrov type III, N, O, II or D. If one restricts thesolutions to those with plane wave surfaces they are con-strained to be (at most) of type III; they can degenerateto type N and O. Conformally flat solutions includefor example the Bertotti-Robinson [3, 4] universe andthe Edgar-Ludwig metric [5]. An Einstein-Yang-Millssolution of type N was given long time ago in [6] andrecovered for D = 4 supergravity in [7]. Type III solu-tions have somehow attracted less attention. Yet, somereferences considering the type III Einstein-Maxwellsystem can be found [8, 9]. To our knowledge, thesolutions presented in this note are the first type IIIsolutions of Einstein gravity coupled to a Yang-Millsfield.We also want to point out that the Kundt’s class can begeneralized to include a non-zero cosmological constant.Such solutions of type N were first considered in [10], fol-lowed by [11, 12, 13, 14]. Type N Einstein-Maxwell solu-tions were generalized to an arbitrary number of dimen-sions in [15] and further generalized to Lovelock-Yang-Mills solutions in [16]. Not very long ago the Λ 6= 0generalization of type III spacetimes in [1] was presentedin [17].II. YANG-MILLS IN KUNDT SPACETIMESWe consider Kundt’s class of metrics, described by theline element:ds2 = − 2du (dv + W dz + W dz̄ + H du)+ 2P−2dzdz̄, P,v = 0(1)Here, P and H are real functions while W is complex;u = (t − z)/√2, v = (t + z)/√2 are light-cone coordi-nates and z = (x + iy)/√2, z̄ = (x − iy)/√2 complexconjugate coordinates in the transverse plane. Weconsider solutions of type III. In this case we can takeP = 1 without loss of generality. As a consequence theGaussian curvature of the wave surfaces u=constant,K = 2P 2(ln P ),zz̄, vanishes. Type III solutions aretherefore characterized by plane wave surfaces. Detailson 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 ofthe form W,v = −2/(z + z̄). These two cases correspondto the Newman-Penrose quantity τ being zero or 6= 0,respectively.A. Case W,v = 0Consider the metric (1) with:P = 1 (2a)W = W (u, z̄) (2b)H = H0(u, z, z̄) +12(W,z̄ +W,z)v (2c)Here, W is an arbitrary complex function and H0 is a realfunction. This is a vacuum solution of Einstein equationsif the function H0 satisfies:12R33 = H,0zz̄ −Re(W,2z̄ +WW,z̄z̄ +W,z̄u)= 0 (3)Numerical indices are tetrad indices1. All other compo-nents of the Ricci tensor vanish. This class of solutionsdegenerates to type N when Ψ3 =12W,z̄z̄ = 0. In thiscase one can use the remaining coordinate transforma-tion (see [1]) to make W = 0. The metric reduces then1 Tetrad vectors as in [1]: e1 = ē2 = ∂z , e3 = ∂u + W ∂z +W ∂z̄ − (H + WW ) ∂v, e4 = ∂v2to that of familiar pp waves [18],ds2 = 2 dz dz̄ − 2 du dv − H(u, z, z̄) du2 (4)where H satisfies H,zz̄ = 0.The function W can have an arbitrary u dependencebut must be at least quadratic in z̄ so that Ψ3 6= 0. Thisholds for pure radiation solutions as well: they onlyenter the solution through the function H0 so W is as invacuum.We can generalize the solution to include a null Yang-Mills field (Φ0 = Φ1 = 0) of the form:A = αa(u, z, z̄)Ta du, αa = χa(u, z) + χ̄a(u, z̄) (5)Here, χa are arbitrary complex functions. This field wasconsidered in curved spacetime for the first time in [6].The Yang-Mills equation in flat space is just:α,zz̄ = 0 (6)This equation remains unchanged in the spacetime (1)-(2c) because of two reasons. First, the absence of com-ponents of the YM field strength of the form Fuµ in thisgeometry. In the second place, the only non-vanishingChristoffel symbols of the form Γ ρρν are precisely Γρρu .As a result only the ordinary derivative survives in the(curved) Yang-Mills equation:∂λFλµ + Γ ρρu Fuµ − [Au, Fuµ] = 2α,zz̄ = 0 (7)Of course Eq. (3) has to be modified to account forthe energy-momentum tensor of the Yang-Mills field:Tuu =12πγab α,az α,bz̄ (8)Here, γab is the invariant metric of the Lie group. Intetrad indices we have T33 = Tuu. Therefore, Eq. (3)transforms to:H,0zz̄ −Re(W,2z̄ +WW,z̄z̄ +W,z̄u)= 2γab α,az α,bz̄It is straightforward to solve for H0:H0(u, z, z̄) = f(u, z) + f̄(u, z̄) + 2γabχaχ̄b+ Re {(W,u +WW,z̄ ) z}(9)Here f is an arbitrary complex function. Eqs. (5) and(1)-(2c) with an arbitrary (at least quadratic) W andthe given H0 describe an exact type III solution of theEYM system. It is a generalized Goldberg-Kerr solution.The type N reduction of this solution has been knownfor a long time [6].B. Case W,v = −2/(z + z̄)We consider now the metric (1) with:P = 1 (10a)W = W 0(u, z) − 2vz + z̄(10b)H = H0(u, z, z̄) +W 0 + W0z + z̄v − v2(z + z̄)2(10c)Here, W 0 is an arbitrary complex function and H0 is areal function. This is a vacuum solution of Einstein equa-tions if the function H0 satisfies the differential equation:12R033= (z + z̄)(H0 + W0W 0z + z̄),zz̄−W,0z W,0z̄ = 0 (11)This is the only non-vanishing component of the Riccitensor2. This class of solutions degenerate to type Nas well when Ψ3 = W,0z̄ /(z + z̄) = 0. In this case onecan transform W 0 to zero [1] and the resulting metric isknown as Kundt waves.We consider the field (5) in this background. Featuresin this geometry are the same ones as in the previous case,except for an extra term in the Yang-Mills equation:(∂z + Γuuz + Γvvz )Fzv + (z ↔ z̄) = 0 (12)After further inspection this extra term is seen to vanishas Γ uuz = −Γ vvz , Γ uuz̄ = −Γ vvz̄ . In this way, theYang-Mills equation in the spacetime (1), (10b), (10c) isstill the same as in flat space, Eq. (6). Finally, Eq. (11)has to be modified to include the YM energy-momentumtensor T33 = (γab/2π) α,az α,bz̄ .We refine our ansatz in order to solve explicitly for H0.We choose the specific function W 0 = g(u)z, g(u) beingan arbitrary complex function. This is the simplest formof W 0 such that Ψ3 is non-vanishing. On the other handwe limit the Yang-Mills field (5) to:χa = λa(u) z (13)Here λa(u) are bounded complex functions. This fieldwas considered in Minkowski space for the first timein [19] and is referred to as non-Abelian plane waves3.The field (13) is the only form of (5) which gives rise toa bounded energy-momentum tensor. Under these as-sumptions the equation to solve reads:(z + z̄)(H0 + gḡzz̄z + z̄),zz̄− g(u)ḡ(u) = 2γabλa(u)λ̄b(u)2 The superscript 0 in R33 denotes the v-independent part of R33.3 The original field in [19] was χa = λa(u) z + some function of ubut the later term can be gauged away without affecting gauge-invariant quantities, as pointed out by Coleman.3It is not difficult to see that:H0 = (f(u, z) + f̄(u, z̄)) (z + z̄) − gḡzz̄+ σ(u)(z + z̄)2 {ln(z + z̄) − 1)(14)Here f is again an arbitrary complex function and σ isthe real function σ(u) = 2γabλa(u)λ̄b(u)+g(u)ḡ(u). Eqs.(5), (13) and (1), (10a)-(10c) with the given W 0, H0 de-scribe another exact type III solution of the EYM system.The corresponding vacuum solution (λa ≡ 0) has to ourknowledge not been considered before. The case g(u) ≡ 0yields the type N reduction and can be regarded as gener-alized Kundt waves. A related type N solution was veryrecently given in [16], for a Yang-Mills field of arbitraryhigher (polynomial) dependence on z, z̄; our (type N) so-lution is obtained by translating the line element into theKundt canonical form and considering N = 0, D = 4.C. GeodesicsFor completeness, we present the geodesic equationscorresponding to the two classes of solutions found,without attempting to provide detailed solutions here.· Case W,v = 0In order to get a feeling for the behavior of geodesics wechoose the specific W = z̄2. This is the simplest W suchthat the solution presented in subsection A is of typeIII. The geodesics are described in terms of real spatialcoordinates x, y by:ü −√2xu̇2 = 0 (15)ẍ + u̇2(H,x −x(x2 − y2))+ 2√2y u̇ẏ = 0 (16)ÿ + u̇2(H,y −2yx2)− 2√2y u̇ẋ = 0 (17)ẋ2 + ẏ2 − 2u̇v̇ − 2Hu̇2 −√2(x2 − y2)ẋu̇ − 2√2yx ẏu̇ = ǫ(18)Here, the overdot denotes a proper time derivative andH is:H = f + f̄ + 2(x4 − y4) + 2γabχaχ̄b +√2xv (19)We have considered the normalization condition forthe 4-velocity, Eq. (18), instead of the (complicated)equation for v. There, ǫ = −1, 0 for timelike and nullgeodesics, respectively.It can be advantageous to use u as affine parameterinstead of the proper time. If we assume u̇ > 0, from Eq.(15):x =1√2d (lnu̇)du(20)Now, in the hyperplane y = 0, by inserting Eqs. (20) in(16), (18) and recalling ẋ = u̇ (dx/du), a coupled systemfor x(u), v(u) can be obtained.On the other hand, in the hyperplane x = 0 Eqs. (15)and (17) simplify toü = 0 ↔ u̇ = constant ≡ γ (21)ÿ + u̇2H,y = 0 (22)taking the same form as in the type N reduction (seefor example [20]). Analytical differences come from thequartic term of (19) which is absent in the type N. Thisdifference can be physically illustrated when we considerthe functions f = κ(u) z2, with κ(u) an arbitrary realfunction4, and χa = λa(u) z in Eq. (19). Then, themotion in the y direction is governed by:d2y/du2 − 8y3 + 2(γabλa(u)λ̄b(u) − κ(u))y = 0 (23)This equation describes a (parametric) non-linear oscil-lator in a potential V (y) = y2(γabλaλ̄b − κ − 2y2). Inthe type N reduction there is no cubic term in Eq. (23)and the motion reduces to that of an harmonic oscillator.Non-linear oscillators can have chaotic motion.Geodesic chaotic motion has been found in the re-lated case of non-homogeneous vacuum pp waves(f ∼ zn, n > 2), see [21]. However, chaotic behavior isnot always present in non-linear systems and thereforepossible chaotic motion arising from Eq. (23) wouldneed further investigation.· Case W,v = −2/(z + z̄)We take the u dependence out of W 0 respect to thefunction considered in subsection B, so now W 0 = z.Geodesics in this second case have the more complicatedform:ü − (1 − v/x2)u̇2 − 2/x u̇ẋ = 0 (24)ẍ + u̇2(H,x +(x − 2v/x)(1 − v/x2))+ 2(1 − 2v/x2)u̇ẋ− 2/x v̇u̇ = 0 (25)ÿ + u̇2(H,y +y(1 − v/x2))+ 2y/x u̇ẋ = 0 (26)ẋ2 + ẏ2 − 2u̇v̇ − 2Hu̇2 − 2(−2v/x + x) ẋu̇ + 2y ẏu̇ = ǫ(27)Again, the normalization condition of the 4-velocitytakes the place of the equation for v. The function H isH = H0 + v − v22x2(28)and H0 is given by (14) but with g(u) ≡ 1.4 This choice is well motivated as it corresponds to homogeneouspp waves in the (vacuum) type N reduction.4The geodesic equations can again be rewritten as dif-ferential equations in terms of an affine variable u insteadof the proper time. In particular, for the first equation:dduln(u̇x2)= 1 − vx2(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 fromthe geodesic equations, but not its character. Geodesicsfor a general vacuum type N or conformally flat pureradiation solutions have been analyzed in [22], whereit was concluded that x = 0 is a physical singularity.Whether this is also the case for our type III solutionsremains an open question.III. CONCLUSIONSIn [23] Tafel claimed that null twistfree solutions ofthe Yang-Mills equations were exhausted by the type N(class of) solutions given in [6]. A new type N solutionwas recently given in [16]. We have shown that moregeneral solutions exist, as the two type III (classes of)EYM solutions presented here are also null and twistfree.It has been proved [24] that all spacetimes withvanishing curvature invariants of all orders: 1) areKundt metrics of Petrov type III, N or O, and 2) are ofPlebański-Petrov (PP) type N or O (null radiation orvacuum). These results have been generalized to higherdimensions in [25]. Such spacetimes have been calledVSI spacetimes. Our solutions clearly meet requirements1) and 2) and are therefore new examples of D = 4VSI spacetimes. As such, they might have interestingphysical applications in supergravity and string theory,in the fashion of pp waves.AcknowledgmentsA.F. would like to thank N. Van den Bergh for dis-cussions. J.W.v.H. whishes to thank the DESY theorydivision, where part of this work was performed, for itshospitality. This work is part of the programme FP52 ofthe Foundation for Research of Matter (FOM).[1] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers,and E. Herlt (2003), cambridge, UK: Univ. Pr. 701 P.[2] W. Kundt, Z. Phys 163, 77 (1961).[3] B. Bertotti, Phys. Rev. 116, 1331 (1959).[4] I. Robinson, Bull. Acad. Polon. 7, 351 (1959).[5] S. B. Edgar and G. Ludwig, Class. Quant. Grav. 14, L65(1997), gr-qc/9612059.[6] R. Gueven, Phys. Rev. D19, 471 (1979).[7] M. Cariglia, G. W. Gibbons, R. Guven, and C. N. Pope,Class. Quant. Grav. 21, 2849 (2004), hep-th/0312256.[8] G. Hall and J. Carot, Class. Quantum Grav. 11, 475(1994).[9] M. Cahen and J. Spelkens, Academie Royale de Belgique(1967), vol.53.[10] A. Garcia Diaz and J. Plebanski, J. Math. Phys. 22, 2655(1981).[11] I. Ozsvath, I. Robinson, and K. Rozga, J. Math. Phys.26, 1755 (1985).[12] J. Bicak and J. Podolsky, J. Math. Phys. 40, 4495 (1999),gr-qc/9907048.[13] J. Bicak and J. Podolsky, J. Math. Phys. 40, 4506 (1999),gr-qc/9907049.[14] J. Podolsky and M. Ortaggio, Class. Quant. Grav. 20,1685 (2003), gr-qc/0212073.[15] Y. N. Obukhov, Phys. Rev. D69, 024013 (2004), gr-qc/0310121.[16] R. J. Gleiser and G. Dotti (2005), gr-qc/0505094.[17] J. B. Griffiths, P. Docherty, and J. Podolsky, Class.Quant. Grav. 21, 207 (2004), gr-qc/0310083.[18] M. W. Brinkmann, Proc. Natl. Acad. Sci. (1923), 9.[19] S. R. Coleman, Phys. Lett. B70, 59 (1977).[20] J. van Holten, Lect. Notes Phys. 541, 365 (2000), gr-qc/9906026.[21] J. Podolsky and K. Vesely, Phys. Rev. D58, 081501(1998), gr-qc/9805078.[22] J. Podolsky and M. Belan, Class. Quant. Grav. 21, 2811(2004), gr-qc/0404068.[23] J. Tafel, Lett. Math. Phys. 12, 163 (1986).[24] V. Pravda, A. Pravdova, A. Coley, and R. Milson, Class.Quant. Grav. 19, 6213 (2002), gr-qc/0209024.[25] A. Coley, R. Milson, V. Pravda, and A. Pravdova, Class.Quant. Grav. 21, 5519 (2004), gr-qc/0410070.

Exact solutions of Einstein equations have always attracted much attention. It is somewhat surprising to find exact solutions of such nonlinear equations. Many of them were collected in the by now classic book by Kramer et al. which has recently been revised [1]. Among others one finds the nondiverging spacetimes which were studied for the first time by Kundt in [2]. Solutions which admit a nonexpanding and twistfree null congruence are therefore said to belong to the Kundt’s class. Such metrics are algebraically special, i.e., of Petrov type III, N, O, II or D. If one restricts the solutions to those with plane wave surfaces they are constrained to be (at most) of type III; they can degenerate to type N and O. Conformally flat solutions 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 III solutions have somehow attracted less attention. Yet, some references considering the type III Einstein-Maxwell system can be found [8,9]. To our knowledge, the solutions presented in this note are the first type III solutions of Einstein gravity coupled to a Yang-Mills field.
We also want to point out that the Kundt’s class can be generalized to include a nonzero cosmological constant.
Such solutions of type N were first considered in [10], followed by [11–14]. Type N Einstein-Maxwell solutions were generalized to an arbitrary number of dimensions in [15] and further generalized to Lovelock-Yang-Mills solutions in [16]. Not very long ago the ¿ ¿ 0 generalization of type III spacetimes in [1] was presented in [17]

Fuster, A Andrea

Holten, JW Jan-Willem  van

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Type III Einstein-Yang-Mills solutions

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