A new first integral for the equations corresponding to twisting type-N
vacuum gravitational fields with two non-commuting Killing vectors is
introduced. A new reduction of the problem to a complex second-order ordinary
differential equation is given. Alternatively, the mentioned first integral can
be used in order to provide a first integral of the second-order complex
equation introduced in a previous treatment of the problem.Comment: 7 pages, LaTeX, uses ioplppt.sty and iopl12.sty; to be published in
  Class. Quantum Gra

Chinea F J

Collinson C D

Debney G C

Ernst F J

F J Chinea

Finley J D III

Halford W D

Hauser I

Herlt E

Hoenselaers C

Ludwig G

McIntosh C B G

Stephani H

English

arXiv

© 1998 IOP Publishing Ltd.
Financial support by Direccion General de Enseñánza Superior, Spain (Project PB95-0371) is gratefully acknowledged.A new first integral for the equations corresponding to twisting type-N vacuum gravitational fields with two non-commuting Kilting vectors is introduced. A new reduction of the problem to a complex second-order ordinary differential equation is given. Alternatively, the mentioned first integral can be used in order to provide a first integral of the second-order complex equation introduced in a previous treatment of the problem.Direccion General de Enseñánza Superior, SpainDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Chinea Trujillo, Francisco Javier

Docta Complutense

arXiv:gr-qc/9801008v1  5 Jan 1998New first integral for twisting type-N vacuumgravitational fields with two non-commutingKilling vectorsF J ChineaDepartamento de F́ısica Teórica II, Facultad de Ciencias F́ısicas, UniversidadComplutense, 28040 Madrid, SpainAbstract. A new first integral for the equations corresponding to twisting type-Nvacuum gravitational fields with two non-commuting Killing vectors is introduced. Anew reduction of the problem to a complex second-order ordinary differential equationis given. Alternatively, the mentioned first integral can be used in order to provide afirst integral of the second-order complex equation introduced in a previous treatmentof the problem.1. IntroductionIt is known from the peeling-off property that the dominant term in an asymptoticexpansion of the gravitational field is of algebraic type N. Thus, such fields can beconsidered as containing the main features of gravitational radiation far from the sources.However, twist-free type-N fields are known to contain unavoidable singularities. Theonly explicitly known type-N vacuum field with twist [2] suffers from similar problems.In the quest for considering a purely type-N field throughout the whole spacetime asa theoretical model for realistic ones produced by compact sources, there remains thehope that twisting fields with the appropriate behaviour may eventually be found. Inspite of numerous efforts, no new explicit fields are known. The information gatheredon twisting type-N fields, especially when additional symmetries are considered, can beseen in references such as [1]-[19].The maximum number of independent isometries for such gravitational fields can beshown to be two, in which case the corresponding Lie algebra is necessarily non-Abelian[6]. Although the approach used in [2] was based on algebraic properties of some exteriorform equations, the solution found by Hauser turns out to possess one Killing vectorand one homothetic Killing vector. This naturally led to a more detailed investigation ofsymmetries of such a type. The special case of two proper Killing vectors was treated in2[10], where a reduction of the problem to a second-order complex differential equationwas shown. By further manipulation, it was also shown there that the problem canbe reduced to the study of a single third-order real equation. In what follows, we firstdiscuss in the next section the main features and results of the previous treatment.Within this framework, we then introduce in the final section a new first integral of thefield equations; it gives rise to a more direct reduction in terms of the original variablesappearing in the metric, while it can also be used as a (previously unnoticed) firstintegral of the second-order complex equation considered in [10]. Our aim in pursuingthe present approach is twofold: firstly, to cast the problem in a form more amenableto numerical treatment, and secondly, to open the way for the intriguing possibility offinal reduction to a second-order real equation.2. Twisting type-N vacuum fields with two non-commuting Killing vectorsThe reduction of the field and symmetry equations described in this section was carriedout in [10] by means of a compact version of the Cartan structure equations [20], interms of an sl(2, C)-valued 1-form connection; the empty space Einstein equations thentake a particularly simple form. The conditions for the existence of two non-commutingKilling vectors and the final expression for such Killing fields were performed at thetetrad rather than at the metric level, by using techniques modelled on similar ones ingauge theories [9]. Based on [6], it was required that the Lie algebra of isometries tookthe form[ξ1, ξ2] = ξ1. (1)The connection 1-form was chosen to be in a gauge of maximum simplicity, followingthe gauge used in [2]. Such a gauge is compatible with the requirement that the fieldbe of type N. It leads to the vanishing of all components of the Weyl spinor, except forΨ4. The metric is expressed in terms of the null tetrad {l, n, m, m̄} (where l and n arereal 1-forms, while m is a complex one, with complex conjugation being denoted by abar) by means of the usual relationg = −2l ⊗s n + 2m ⊗s m̄, (2)where ⊗s means the symmetrized tensor product. The 1-form l is the repeated principalnull eigenform, and its twist is assumed not to vanish. The four independent spacetimecoordinates are {u, σ, ζ, ζ̄}, where u and σ are real, while ζ is complex. Details of theprocedure need not concern us here; they are given in [9] and [10]. The Killing vectorscan be shown to beξ1 = i∂ζ − i∂ζ̄ (3)3andξ2 = ζ∂ζ + ζ̄∂ζ̄ . (4)The tetrad can be expressed as †l = (ζ + ζ̄)du + Ddζ + D̄dζ̄ (5)n = (ζ + ζ̄)−1dσ + (−σ + FD̄ − M)(ζ + ζ̄)−2dζ + (−σ + F̄D − M̄)(ζ + ζ̄)−2dζ̄ (6)m = D̄−1M̄du − σ(ζ + ζ̄)−1dζ + M̄(ζ + ζ̄)−1dζ̄ , (7)where D, M and F are complex functions of the real independent variable u only. Dueto the first Bianchi identities, the requirement that the resulting metric be of type N,the conditions for the existence of the Killing fields, and the vacuum field equations,these functions are subject to the following system of equations:Mu = −F, (8)Fu = −2FD̄−1, (9)Du = 1 − D−1M, (10)2M − 2M̄ − 2FD̄ + 2F̄D + FM̄ − F̄M = 0, (11)where the subscript denotes a derivative with respect to u. Note that (11) takes the formof a first integral of the previous equations (8)-(10) (in the sense that the derivative ofits left-hand side with respect to u vanishes when those first three equations are takeninto account). Two points should be stressed, however; firstly, (11) appears by itself asone of the combined field and symmetry equations, rather than being searched for; andsecondly, its right-hand side is not an arbitrary constant, but has to take the specificvalue zero.If we define the new independent variable s = ln v, withv(u) =∫F (u) ¯F (u)du, (12)and the new complex dependent variable z asz = 2v−1M, (13)† The tetrad was denoted by k, m, t, t̄ in [10]; the correspondence with the notation used here is thefollowing: k → −l, m → n, t → m.4then the set of equations (8)-(11) reduce to the single second-order autonomous ordinarydifferential equationz̈ + ż +2(z + ż)2(z̄ + ˙̄z)zz̄ + z̄ż + ż= 0, (14)where dots denote derivatives with respect to the real independent variable s. Byrewriting the set (8)-(11) as equations in the independent variable v, it can be easilyseen that M, D and F are explicitly given algebraically in terms of z, z̄ and their firstderivatives (the second and successive derivatives of z and z̄ can be expressed in termsof the first-order ones by means of (14) and its complex conjugate). Finally, the originalindependent variable u is simply given by a quadrature, corresponding to the inversionof (12). The details are given in [10].By further manipulation of (14) (involving a Legendre transformation and thecompatibility conditions for an associated linear system), it was shown in [10] thatthe problem can be reduced to a single, third-order, real ordinary differential equation.Similar techniques have been subsequently used in [12] and [13] in the more general caseof the existence of one Killing vector and one homothetic Killing vector, showing also areduction to a final, third-order equation. It is rather remarkable that the simplest suchequation was found in [12] for one specific case of the existence of a proper homotheticKilling vector (i.e. a case not reducible to the situation considered in the present paper).The reduction to third-order equations and the relations among different approacheshave been further investigated in [15], [17], and [18].For completeness, the remainder of this section will be used for recording someimportant quantities that appear in the present problem. They can be computed ina straightforward fashion from the tetrad (5)-(7) (taking into account (8)-(11) whenderivatives are involved), especially if one uses some computer algebra software. Theonly non-vanishing spin coefficients are the following:λ =FD̄(σD̄ + M̄D)(ζ + ζ̄)(15)ν =FM̄(σD̄ + M̄D)(ζ + ζ̄)2(16)ρ = −D̄(ζ + ζ̄)(σD̄ + M̄D)(17)τ = −M̄(σD̄ + M̄D). (18)5As ρ = θ + iω, where θ is the expansion and ω is the twist of the congruence defined byl, the twist is given by the following expression:ω =i2(ζ + ζ̄)MD̄2 − M̄D2(σD̄ + M̄D)(σD + MD̄). (19)All components of the Weyl spinor vanish, except for Ψ4, as corresponds to the type-Ncharacter in the gauge we are using. The latter is given byΨ4 =2F(σD̄ + M̄D)(ζ + ζ̄)2. (20)From (19) and (20), we should keep in mind for non-triviality thatnon−flat spacetime ⇐⇒ F 6= 0, (21)whilenon−vanishing twist ⇐⇒ MD̄2 − M̄D2 6= 0. (22)Note that the condition (21) is precisely what is required for the coordinate change (12)to be defined.As the twist measures the failure of the 1-form l to be Frobenius integrable, it issometimes convenient to consider the following alternative expression to (19):dl ∧ l = −1DD̄(MD̄2 − M̄D2)du ∧ dζ ∧ dζ̄ . (23)3. New first integral and reductionThe present author realized recently that the system of equations (8)-(10) admit thefollowing first integral, which went unnoticed in [10]:FD̄2 + F̄D2 − 2MM̄ = k, (24)where k is an arbitrary real constant. As an alternative to the reduction to a final second-order complex equation given in [10] (equation (14) above), in the manner discussed inthe previous section, the following new possibility presents itself: from (10) one canwriteM = D − DDu, (25)and then, upon differentiation of (25) with respect to u and substitution into (8), F canbe expressed asF = DDuu + D2u − Du. (26)By substituting M, M̄, F and F̄ as given by (25) and (26) (and their complex conjugates)into the constraint equation (11) and the first integral (24), one ends up with two real6equations where only D, D̄, and their derivatives up to the second order appear. Theycan be solved for Duu (and its complex conjugate). The result is the following:Duu =AB, (27)whereA = −D̄2D3u + DD̄D̄uD2u − 3DD̄D2u + DD̄DuD̄u−2D2Du + D̄2Du + DD̄Du + kDu + 2D2 + k, (28)andB = DD̄(DD̄u + D̄Du + D + D̄). (29)Finally, there remains equation (9), which is third order in D. But third-orderderivatives can be obtained from (27). When this is done, (9) is satisfied identically.In other words, the only equation to be solved is precisely the second-order complexequation (27). While this equation is rather complicated, it has the following property:it determines D as a function of the original independent variable u, without the needof transformations such as (12). It seems that this will allow for a more direct numericaltreatment of the twisting type-N problem, as compared with an approach based on (14)or its associated third-order real equation. On the other hand, the new integral (24)can also be used in connection with (14): By carrying out the changes discussed in theprevious section, one ends up with additional information about (14). Namely, it admitsthe following first integral:I0 =e2sz̄ + ˙̄z(zz̄ + z ˙̄z + ˙̄z)2 +e2sz + ż(zz̄ + z̄ż + ż)2 + 4e2szz̄, (30)where I0 is an arbitrary real constant. Numerical treatment of the problem with theaid of the new first integral is currently under consideration [21]. Finally, the followingremark can be made: As the second-order complex differential equation (14) was shownto be reducible to a third-order real one, there is some hope that the new first integralmay result in the reduction to a second-order real differential equation.AcknowledgmentsFinancial support by Dirección General de Enseñanza Superior, Spain (Project PB95-0371) is gratefully acknowledged.References[1] Debney G C, Kerr R P and Schild A 1969 J. Math. Phys. 10 1842[2] Hauser I 1974 Phys. Rev. Lett. 33 1112; 33 1525(E)7[3] Hauser I 1978 J. Math. Phys. 19 661[4] Ernst F J and Hauser I 1978 J. Math. Phys. 19 1816[5] Halford W D 1979 J. Math. Phys. 20 1115[6] Collinson C D 1980 J. Math. Phys. 21 2601[7] McIntosh C B G 1985 Class. Quantum Grav. 2 87[8] Stephani H and Herlt E 1985 Class. Quantum Grav. 2 L63[9] Chinea F J 1988 Class. Quantum Grav. 5 135[10] Chinea F J 1988 Phys. Rev. D 37 3080[11] Plebański J F and Przanowski M 1991 Phys. Lett. 152A 257[12] Herlt E 1991 Gen. Rel. Grav. 23 477[13] Ludwig G and Yu Y B 1992 Gen. Rel. Grav. 24 93[14] Stephani H 1993 Class. Quantum Grav. 10 2187[15] Finley J D III, Plebański J F and Przanowski M 1994 Class. Quantum Grav. 11 157[16] Hoenselaers C, Ferwagner R and Schief W 1994 Relativity Today, Proc. 4th Hungarian RelativityWorkshop ed R P Kerr and Z Perjés (Budapest: Akadémiai Kiadó) p 47[17] Ludwig G and Edgar S B 1996 Gen. Rel. Grav. 28 707[18] Edgar S B and Ludwig G 1997 Gen. Rel. Grav. 29 19[19] Finley J D III, Plebański J F and Przanowski M 1997 Class. Quantum Grav. 14 489[20] Chinea F J 1984 Phys. Rev. Lett. 52 322[21] Chinea F J and Fernández-Jambrina L 1997 in preparation

Classical and Quantum Gravity

New first integral for twisting type-N vacuum gravitation fields with two non-commuting Killing vectors

Crossref

New first integral for twisting type-N vacuum gravitational fields with two non-commuting Killing vectors

New first integral for twisting type-N vacuum gravitational fields with
  two non-commuting Killing vectors

New first integral for twisting type-N vacuum gravitational fields with two non-commuting Killing vectors

Abstract

Similar works

Full text

Available Versions

Docta Complutense

Crossref