Gravitational waves in vacuum spacetimes

with cosmological constant.

I. Classification and geometrical properties

of non-twisting type N solutions.

Jiří Bičák and Jiří Podolský

Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Prague 8, Czech Republic

Electronic addresses: bicak@mbox.troja.mff.cuni.cz, podolsky@mbox.troja.mff.cuni.cz

Abstract

All non-twisting Petrov-type N solutions of vacuum Einstein field equations with cosmological constant Λ are summarized. They are shown to belong either to the non-expanding Kundt class or to the expanding Robinson-Trautman class. Invariant subclasses of each class are defined and the corresponding metrics are given explicitly in suitable canonical coordinates. Relations between the subclasses and their geometrical properties are analyzed. In the subsequent paper these solutions are interpreted as exact gravitational waves propagating in de Sitter or anti–de Sitter spacetimes.

PACS number(s): 04.30.-w, 04.20.Jb, 98.80.Hw

I. INTRODUCTION AND SUMMARY

The purpose of this and the subsequent paper is to analyze all non-twisting type N solutions of Einstein’s vacuum equations with Λ. There are several basic works on these solutions available in literature, in particular [1]-[3] (for pre-1980 works, see Ref. [4]). None of them, however, discusses the physical interpretation of the solutions. Such an interpretation, based on the study of the deviation of geodesics, will be presented in the following paper. In this part we summarize, compare, classify and generalize the mathematical results of Refs. [1]-[3].

We consider type N solutions in which the Debever-Penrose null vector field k is quadruple [4]. The vector field defines a congruence of null geodesics x^{α}(v) such that dx^{α}/dv = k^{α}, k_{α}k^{α} = 0, k_{α;β}k^{β} = 0, v being an affine parameter. In general a geodesic congruence is characterized by its expansion Θ = \frac{1}{2}k_{;α}^{α}, shear |σ| = \sqrt{\frac{1}{2}k_{(α;β)}k^{α;β} − Θ^{2}} and twist ω = \sqrt{\frac{1}{2}k_{[α;β]}k^{α;β}} [4]. The Bianchi identities and the Kundt-Thompson theorem for type N solutions (see [5], Theorem 7.5 in [4]) imply σ = 0 since C_{αβγδ}^{;δ} = R_{αβγδ}^{;δ} = 0 for solutions of R_{αβ} = Λg_{αβ}. In the following we assume ω = 0. Therefore, with (possibly) non-vanishing Λ we are left with two cases to consider: (i) the Kundt class of non-expanding gravitational waves, Θ = 0 (cf. [6], Ch.27 in [4]), and (ii) the Robinson-Trautman class of expanding gravitational waves of type N , Θ ≠ 0 (cf. [7], Ch.24 in [4]). Hereafter we denote the Kundt class by KN(Λ) and the Robinson-Trautman class by RT N(Λ).

Section II analyzes the class KN(Λ) in detail. We do not present any new solution of this type but we extend the results of Ref. [3] by giving the explicit forms of the transformations which leave the metrics of KN(Λ) invariant. These enable us to give the explicit transformations to different canonical subclasses of KN(Λ), which have not been given in literature so far. We also introduce a convenient new notation for these subclasses and demonstrate how they are interconnected. In particular, we show that one of the subclasses is identical to the ‘Lobatchewski waves’ studied by Siklos [8]. We also formulate the proposition (proven in Appendix) that all vacuum solutions with Λ which are conformal to the ‘Kundt waves with Λ = 0’ belong to one specific subclass of KN(Λ); we thus generalize the result of an analogous theorem for the pp-waves [8].

In Section III the Robinson-Trautman solutions [7] of type N with Λ are discussed. The transformation between two standard forms of the metric (those of Refs. [4] and [1]) is presented, and then transformations preserving the metric form given in [1] are generalized and used to demonstrate how the non-radiative part of the structural function of these solutions can be transformed away.

II. THE KUNDT CLASS OF SOLUTIONS KN(Λ)

This class has been investigated in detail by Ozsváth, Robinson and Rózga [3]. They have shown that in this case the vector k can be normalized such that \frac{1}{2}L_{k}g_{αβ} = \frac{1}{2}(k_{α;β} + k_{β;α}) = Lk_{α}k_{β}, where L_{k} is the Lie derivative and L is a scalar. Denoting L′ = L_{k}L, we find L_{k}L′ = 0 so that L′ is invariant under renormalization of k. A suitable coordinate system (v, ξ, ξ̄, u), where ξ, ξ̄ are space-like coordinates, v is a parameter along the null geodesics and u is a retarded time with u = const. being a wavefront, k ≡ ψ(u)∂_{v}, can be introduced [3] in which the KN(Λ) metrics have the form

ds^{2} = 2\frac{1}{p^{2}}dξdξ̄ − 2\frac{q^{2}}{p^{2}}dudv + F du^{2} ,  (1)

where

p = 1 + \frac{Λ}{6}ξξ̄ , q = (1 − \frac{Λ}{6}ξξ̄)α + β̄ξ + βξ̄ ,

F = κ\frac{q^{2}}{p^{2}}v^{2} − \frac{(q^{2}),_{u}}{p^{2}}v − \frac{q}{p}H , κ = \frac{Λ}{3}α^{2} + 2ββ̄ .

Here α(u) and β(u) are arbitrary real and complex functions of u, respectively. These functions play the role of two arbitrary ‘parameters’, i.e., we can denote the Kundt class by KN(Λ) ≡ KN(Λ)[α, β]. The parameter κ is related to the invariant L′ by

L′ = κ\frac{p^{2}}{q^{2}} .  (2)

The function H = H(ξ, ξ̄, u) entering F is restricted by Einstein’s equations, H_{,ξξ̄} + (Λ/3p^{2}) H = 0. There exists a general solution to this equation

H(ξ, ξ̄, u) = (f_{,ξ} + f̄_{,ξ̄}) − \frac{Λ}{3p}(ξ̄f + ξf̄) ,  (3)

where f(ξ, u) is an arbitrary function of ξ and u, analytic in ξ. The space-time is conformally flat if and only if the structural function H is of the form

H = H_{c} = \frac{1}{p}[(1 − \frac{Λ}{6}ξξ̄)A + B̄ξ + Bξ̄] ,  (4)

with A(u) and B(u) being arbitrary real and complex functions, respectively. Since H_{c} of this form corresponds to (3) for f quadratic in ξ we easily infer the following

Lemma 1: The KN(Λ) solutions (1), (3) with f = f_{c} = c_{0}(u) + c_{1}(u)ξ + c_{2}(u)ξ^{2}, where c_{i}(u) are arbitrary complex functions of u, are isometric to Minkowski (if Λ = 0), de Sitter (Λ > 0) and anti-de Sitter spacetime (Λ < 0).

It can be proven by straightforward but tedious calculations that the following lemma (which is not formulated in [3] but is a consequence of results therein) is true:

Lemma 2: The metric of the KN(Λ)[α, β] class preserves its form (1) under the transformations (v, ξ, ξ̄, u) → (w, η, η̄, t) given by

v = a(t)[w + \frac{(1 − \frac{Λ}{6}ηη̄) γ + δ η̄ + δ̄ η}{(1 − \frac{Λ}{6}ηη̄)α′ + β′η̄ + β̄′η} ] + Δ(t) ,

ξ = \frac{B̄(t) + A(t)η}{Ā(t) − \frac{Λ}{6}B(t)η} ,  u = u(t) ,  (5)

where A(t), B(t) are arbitrary complex and a(t), u(t) are real functions of t, respectively.

In the new coordinates (w, η, η̄, t) the resulting metric KN(Λ)[α′, β′] has

α′ = \frac{\sqrt{au̇}}{Φ} [(AĀ − \frac{Λ}{6}BB̄)α + ĀBβ + AB̄β̄ ] ,
β′ = \frac{\sqrt{au̇}}{Φ} [−\frac{Λ}{3} ĀB̄α + Ā^{2}β − \frac{Λ}{6} B̄^{2}β̄ ] ,  (6)

with Φ = AĀ + \frac{Λ}{6}BB̄ and the dot denotes d/dt. The remaining real functions γ(t), Δ(t) and a complex function δ(t) in (5) must satisfy the equations

α′δ − β′γ = \frac{1}{Φ}(Ā ˙B̄ − ˙ ĀB̄) ≡ C ,
β̄′δ − β′δ̄ = \frac{1}{Φ}(ȦĀ − A ˙Ā + \frac{Λ}{6} ḂB̄ − \frac{Λ}{6}B ˙B̄) ≡ D ,  (7)
\frac{κ′}{a} Δ = \frac{1}{2}(\frac{ȧ}{a} − \frac{ü}{u̇}) − (\frac{Λ}{3}α′γ + β̄′δ + β′δ̄) .

The structural function H′(η, η̄, t) then takes the form

p′H′ = (1 − \frac{Λ}{6}ηη̄)E + F̄η + F η̄ + \frac{u̇}{Φ}\sqrt{\frac{u̇}{a}}(A − \frac{Λ}{6} B̄η̄)(Ā − \frac{Λ}{6}Bη)pH ,  (8)

where

E(t) ≡ 2γ̇ + \frac{2}{a}(α′Δ)^{·} − α′κ′ \frac{Δ^{2}}{a^{2}} − 2\frac{Δ}{a} [\frac{1}{2}α′(\frac{ȧ}{a} + \frac{ü}{u̇}) + Cβ̄′ + C̄β′ ]

−α′(2δδ̄ − \frac{Λ}{3}γ^{2}) + 2γ(β′δ̄ + β̄′δ) ,

F (t) ≡ 2δ̇+ \frac{2}{a}(β′Δ)^{·} − β′κ′ \frac{Δ^{2}}{a^{2}} − 2\frac{Δ}{a} [\frac{1}{2}β′(\frac{ȧ}{a} + \frac{ü}{u̇}) − \frac{Λ}{3}Cα′ − Dβ′ ]

−\frac{Λ}{3}γ(β′γ − 2α′δ) + 2β̄′δ^{2} .

The transformation (5) is also important in connection with Lemma 1. Comparing Eq. (4) with Eq. (8), we see that we can ‘generate’ H = H_{c} from H = 0 by a coordinate transformation. Therefore, the conformally flat part, H_{c}, of H generated by f_{c} cannot represent a radiative field. Only if the function f is at least cubic in ξ, the resulting spacetime is of type N and can be interpreted as radiative (see the subsequent paper).

Another important application of the coordinate freedom (5) was suggested in [3]: it is possible to use the transformation to get ‘canonical’ subclasses of KN(Λ)[α, β] corresponding to special values of parameters α and β. Without loss of generality we can assume α ≥ 0. In addition, the transformation (5) of a special form

ξ = η , v = a(t)w , u = ∫ a(t) dt ,  (9)

for a(t) ≠ 0 results in scaling α(u) = α′(t)/|a(t)|, β(u) = β′(t)/|a(t)| so that we can always assume either α = 1 or α = 0. Since the parameter L′ (2) is invariant, the sign of κ is also an invariant. We can thus base the invariant canonical classification of all KN(Λ) solutions on the sign of κ and the sign of Λ. There are nine possible cases (κ and Λ can both be positive, zero or negative). However, subclasses κ < 0, Λ > 0 and κ = 0, Λ > 0 and κ < 0, Λ = 0 are forbidden since they violate the relation κ = \frac{Λ}{3}α^{2}+2ββ̄. The remaining possibilities are six subclasses which we shall now discuss. A. Subclass κ = 0, Λ = 0. The equation κ = 0 implies β = 0 so that α = 1. A canonical representative of this subclass can be denoted as P P ≡ KN(Λ = 0)[α = 1, β = 0]. We are now using the notation P P since the corresponding metric

P P :  ds^{2} = 2dξdξ̄ − 2dudv − (g + ḡ)du^{2} ,  (10)

with arbitrary g(ξ, u) = f_{,ξ} analytic in ξ, describes well-known pp waves investigated by many authors (for details and references see section 21.5 in [4]). B. Subclass κ > 0, Λ = 0. Since 0 < κ = 2ββ̄, we have β ≠ 0. Then Eqs. (5) with A = \sqrt{β}, B = (−\frac{1}{2} \frac{α}{β} + i\frac{J}{β}) A, a = \frac{1}{ββ̄}, γ = \frac{1}{2}(\frac{α}{\sqrt{ββ̄}} )^{·} + \frac{2J}{α} (\frac{J}{\sqrt{ββ̄}} )^{·}, δ = Ȧ/A, Δ = \frac{1}{2} ȧ, u = t, where J is a real function satisfying (\frac{J}{\sqrt{ββ̄}} )^{·} = \frac{i}{4} \frac{α}{\sqrt{ββ̄}} \frac{β̄}{β} (\frac{β}{β̄} )^{·}, transform any solution of this subclass to KN ≡ KN(Λ = 0)[α = 0, β = 1]. The metric

KN :  ds^{2} = 2dξdξ̄ + 2(ξ + ξ̄)^{2} [−dv + (v^{2} − \frac{g + ḡ}{ξ + ξ̄} ) du] du ,  (11)

with arbitrary g(ξ, u) = \frac{1}{2}f_{,ξ}, describes solutions discovered by Kundt [6]. This is a special ‘Kundt solution’ in the ‘Kundt class’ of non-expanding waves. For details see [4], Chapter 27; the transformation between ṽ used there and v above is v = ṽ/(ξ + ξ̄)^{2} . C. Subclass κ > 0, Λ > 0. If α = 1 we can use (5) with A = exp(−\frac{i}{2}χ), B = cA, a = (\frac{Λ}{6} + ββ̄)^{−1}, γ = −[Ẇ + \frac{1}{2}(V̇ exp(−iχ) + ˙ V̄ exp(iχ))]/(1 + \frac{Λ}{6}cc̄), δ = Ȧ/A + \frac{Λ}{6} ċc̄/(1 + \frac{Λ}{6}cc̄), Δ = \frac{1}{4} ȧ − \frac{Λ}{12}(cc̄)^{·}(1 + \frac{Λ}{6}cc̄)^{−1}a, u = t, where c ≡ V + W Ā/A, V ≡ \frac{6}{Λ} β̄, W ≡ −\frac{6}{Λ} \sqrt{\frac{Λ}{6} + ββ̄} , χ is a real function satisfying χ̇ = \frac{i}{2}(V̇ exp(−iχ)− ˙V̄ exp(iχ))/W to obtain α′ = 0, β′ = 1. If α = 0 the scaling (9) makes |β| = 1 and then we put ξ = \frac{6}{Λ}β/η, v = w + \frac{1}{2}(β̇β̄η + β ˙β̄η̄)/(η + η̄), u = t. In both cases the representative of this subclass is KN(Λ)I ≡ KN(Λ)[α = 0, β = 1]. The metric reads

KN(Λ)I:

ds^{2} = 2\frac{dξdξ̄}{(1 + \frac{Λ}{6}ξξ̄)^{2}} − 2(\frac{ξ + ξ̄}{1 + \frac{Λ}{6}ξξ̄} )^{2}dudv + [2(\frac{ξ + ξ̄}{1 + \frac{Λ}{6}ξξ̄} )^{2}v^{2} − \frac{ξ + ξ̄}{1 + \frac{Λ}{6}ξξ̄} H] du^{2} . (12)

KN(Λ)I indicates that this solution is a generalization of the KN waves to the case Λ ≠ 0, ‘I’ means ‘of the first kind’. The KN(Λ)I solutions were first discovered by García Díaz and Plebański [1],[9], transformation between their coordinates and those used here is ξ = \sqrt{\frac{6}{Λ}} tanh\sqrt{\frac{Λ}{6}} ξ̃ , v = r/sinh\sqrt{\frac{Λ}{6}}(ξ̃ + ¯ ξ̃) , u = −\sqrt{\frac{Λ}{6}}t. However, in [1],[9] not all KN(Λ ≠ 0) solutions were found since (invariantly different) subclasses E and F mentioned below were omitted.

D. Subclass κ > 0, Λ < 0. The same transformation as in the previous case leads to the metric (12) which has thus the same form for Λ < 0. The transformation to the coordinates used in [1] is ξ = \sqrt{−\frac{6}{Λ}} tan \sqrt{−\frac{Λ}{6}} ξ̃ , v = r/sin \sqrt{−\frac{Λ}{6}}(ξ̃ + ¯ ξ̃) , u = −\sqrt{−\frac{Λ}{6}}t. E. Subclass κ < 0, Λ < 0. Now κ < 0 implies α = 1. Using (5) with A = exp(iϕ), B = cA, a = (1 + \frac{6}{Λ}ββ̄)^{−1}, γ = 0, δ = Ā^{2}˙c̄/(1 + \frac{Λ}{6}cc̄), Δ = \frac{3}{2Λ} ȧ, u = t, where c ≡ \frac{1}{β} (\sqrt{1 + \frac{6}{Λ}ββ̄} −1), ϕ is a real function satisfying ϕ̇ = i\frac{Λ}{12}(ċc̄−c˙c̄)/(1+ \frac{Λ}{6}cc̄), we find β′ = 0 and the canonical representative, denoted by KN(Λ^{−})II ≡ KN(Λ < 0)[α = 1, β = 0], is

KN(Λ^{−})II:

ds^{2} = 2\frac{dξdξ̄}{(1 + \frac{Λ}{6}ξξ̄)^{2}} − 2(\frac{1 − \frac{Λ}{6}ξξ̄}{1 + \frac{Λ}{6}ξξ̄})^{2}dudv + [\frac{Λ}{3} (\frac{1 − \frac{Λ}{6}ξξ̄}{1 + \frac{Λ}{6}ξξ̄})^{2}v^{2} − \frac{1 − \frac{Λ}{6}ξξ̄}{1 + \frac{Λ}{6}ξξ̄}H] du^{2} . (13)

Here KN(Λ^{−})II means generalized Kundt waves ‘of the second kind’ with Λ < 0. This class was first discovered by Ozsváth et.al. [3]. Observe that with κ > 0, Λ > 0, KN(Λ^{+} )II ≡ KN(Λ^{+} )I, since transformation (5) where A = \frac{1}{\sqrt{2}}, B = \sqrt{\frac{3}{Λ}}, a = 1, γ = δ = Δ = 0, u = \frac{Λ}{6}t identifies KN(Λ^{+} )[α = 0, β = 1] with KN(Λ^{+} )[α′ = 1, β′ = 0]. F. Subclass κ = 0, Λ < 0. The relation κ = 0 implies α = 1. Thus, |β| = \sqrt{−\frac{Λ}{6}} and the representative can be denoted by KN(Λ^{−})III ≡ KN(Λ < 0)[α = 1, β = \sqrt{−\frac{Λ}{6}}e^{iω(u)}], ω(u) being an arbitrary real function of u. The metric takes the form of (1) with q = (1 + \sqrt{−\frac{Λ}{6}}ξe^{−iω(u)})(1 + \sqrt{−\frac{Λ}{6}} ξ̄e^{iω(u)}). This class was also first discovered in [3]. One can distinguish two subsubclasses of KN(Λ^{−})III according to the value of L: 1. L ≠ 0 (k is not a Killing vector). 2. L = 0 (k is a Killing vector). In case 2. β =const. and the transformation ξ = \sqrt{β/β̄} η leads to KN(Λ^{−})III_{K} ≡ KN(Λ < 0)[α = 1, β = \sqrt{−\frac{Λ}{6}}], where the suffix ‘K’ stands for ‘Killing’:

KN(Λ^{−})III_{K} :  ds^{2} = 2\frac{dξdξ̄}{(1 + \frac{Λ}{6}ξξ̄)^{2}} − 2(\frac{(1 + \sqrt{−\frac{Λ}{6}}ξ)(1 + \sqrt{−\frac{Λ}{6}} ξ̄)}{1 + \frac{Λ}{6}ξξ̄} )^{2}dudv
−\frac{(1 + \sqrt{−\frac{Λ}{6}}ξ)(1 + \sqrt{−\frac{Λ}{6}} ξ̄)}{1 + \frac{Λ}{6}ξξ̄} Hdu^{2} .  (14)

This subsubclass can be shown to be identical with the ‘Lobatchevski waves’ studied by Siklos [8]: indeed, the transformation ξ = −\sqrt{−\frac{6}{Λ}}(x + \frac{1}{2} + iy)/(x − \frac{1}{2} + iy), v = \frac{12}{Λ} r, brings the metric (14) into the form

ds^{2} = −\frac{3}{Λ} · \frac{1}{x^{2}} (dx^{2} + dy^{2} + 2dudr + H̃du^{2}) ,  (15)

where H̃ ≡ −\frac{Λ}{6}xH. This is the Siklos metric [8]. Recently we analyzed in detail the behaviour of test particles in these solutions and interpreted them as waves in the antide Sitter spacetime [10]. Impulsive waves of this type were investigated in [11]. G. Relations of the subclasses. Using the above results we can summarize the invariant canonical classification of the KN(Λ) class of solutions in the following diagram:

KN(Λ) \{ \begin{matrix}Λ = 0 \{ \begin{matrix}κ = 0 : P P \\ κ > 0 : KN \end{matrix} \\ Λ ≠ 0 \{ \begin{matrix}κ > 0 : KN(Λ)I \\ κ < 0 : KN(Λ^{−})II \\ κ = 0 : KN(Λ^{−})III → KN(Λ^{−})III_{K} . \end{matrix} \end{matrix}

There is an asymmetry with respect to the sign of Λ: there are three distinct classes of non-expanding waves for Λ < 0 whereas there is only one such class for Λ > 0. The reason is in the condition κ = \frac{Λ}{3}α^{2} + 2ββ̄ which for Λ > 0 excludes the cases κ < 0 and κ = 0. Intuitively, fewer non-expanding waves ‘fit’ into the de Sitter universe which admits closed spacelike sections than into the anti-de Sitter space.

There exist natural relations between the Λ = 0 and Λ ≠ 0 subclasses. The metrics (12), (13) and (14) do not diverge as Λ → 0, we can set Λ = 0 and thus find

KN(Λ = 0)I = KN , KN(Λ^{−} = 0)II = P P , KN(Λ^{−} = 0)III = P P .  (16)

Thus, it is natural to consider the KN(Λ)I class as a generalization of the Kundt solution KN, and the classes KN(Λ^{−})II and KN(Λ^{−})III as generalizations of P P waves. There exists no generalization of P P waves to the case of Λ > 0.

From metrics (10) and (15), with ξ = (1/\sqrt{2})(x + iy), and from (11) and (12) we find ds_{KN(Λ^{−})III_{K}}^{2} = −\frac{6}{Λ}(ξ + ξ̄)^{−2}ds_{P P}^{2} , and ds_{KN(Λ)I}^{2} = (1 + \frac{Λ}{6}ξξ̄)^{−2}ds_{KN}^{2}. Therefore, the class KN(Λ^{−})III_{K} is conformal to the P P -class, and the class KN(Λ)I is conformal to the KN-class. In fact, the solutions KN(Λ^{−})III_{K} and KN(Λ)I are the only nontrivial space-times conformal to P P and KN, respectively. The theorem proven by Siklos [8] states that the only vacuum solutions (other than P P solutions themselves) which are properly (with non-constant factor) conformal to non-flat P P metrics are KN(Λ^{−})III_{K} metrics. However, we see from (16) that P P metrics are a special case of metrics KN(Λ^{−})III_{K} for Λ = 0, and Siklos’ theorem may just be formulated as follows:

Proposition 1: The only vacuum solutions conformal to non-flat P P metrics are KN(Λ^{−})III_{K} metrics.

In addition, the following analogous proposition can be proven for the KN solutions:

Proposition 2: The only vacuum solutions conformal to KN metrics are KN(Λ)I metrics.

The proof is contained in Appendix.

The conformal, homothetic, and isometric symmetries of the KN(Λ) solutions have been systematically investigated by Salazar, García and Plebański [2] and by Siklos [8]. It is only in the subclasses P P and KN(Λ^{−})III_{K} that the vector k = ∂_{v} is a Killing vector; k_{α;β} = 0 only in the P P subclass. Let us finally summarize the classification of all KN(Λ) solutions and compare our notation with notations used in literature:

Notation in this paper Notations in literature
KN(Λ)  R(Λ, α, β) [3]
P P  pp , R [4], [3] for Refs.
KN  K [6], [4]
KN(Λ)I  K(Λ) [1], [3]
KN(Λ^{−})II  R(Λ) [3]
KN(Λ^{−})III  (IV)_{1} [3]
KN(Λ^{−})III_{K}  (IV)_{0} [8], [3]

Impulsive waves in the KN(Λ) spacetimes were recently studied in [12].

III. THE ROBINSON-TRAUTMAN CLASS OF SOLUTIONS RT N(Λ, ϵ)

The Robinson-Trautman solutions [7] satisfying the vacuum equations with Λ can be written as (see [4])

ds^{2} = 2\frac{r^{2}}{P^{2}}dζdζ̄ − 2dudr − [Δ lnP − 2r(lnP)_{,u} − \frac{2m}{r} − \frac{Λ}{3}r^{2}]du^{2} ,  (17)

where ζ is a complex spatial coordinate, r is an affine parameter along the rays generated by the vector field k, u is a retarded time, m is a function of u which in some cases can be interpreted as mass, and Δ ≡ 2P^{2}∂^{2}/∂ζ∂ζ̄. The function P ≡ P (ζ, ζ̄, u) satisfies the equation ΔΔ(lnP) + 12m(lnP)_{,u} − 4m_{,u} = 0. Here we restrict attention to the solutions of type N and denote these as RT N(Λ). In this case m = 0 and Δ lnP = K(u). By a transformation u = g(ũ), r = r̃/ġ, where ġ = dg/dũ, we can set the Gaussian curvature K(u) of the 2-surfaces 2P^{−2}dζdζ̄ to be K = 2ϵ, where ϵ = +1, 0, −1 (since P̃ = ġP and K̃ = ġ^{2}K, the sign of K is invariant). Thus, the different subclasses can be denoted as RT N(Λ, ϵ). The corresponding metrics can be written as

ds^{2} = 2\frac{r^{2}}{P^{2}}dζdζ̄ − 2dudr − 2[ϵ − r(lnP)_{,u} − \frac{Λ}{6}r^{2}]du^{2} .  (18)

Since ϵ = +1, 0, −1 and Λ > 0, Λ = 0, Λ < 0, there are 9 invariant subclasses.

Another coordinates for the RT N(Λ, ϵ) class, suitable for physical interpretation, has been given by García Díaz and Plebański [1]. Their metric is expressed in terms of a function f(ξ, u) which is an arbitrary function of u, analytic in spatial coordinate ξ

ds^{2} = 2v^{2}dξdξ̄ + 2vĀdξdu + 2vAdξ̄du + 2ψdudv + 2(AĀ + ψB)du^{2} ,  (19)

where

A = ϵξ − vf ,  B = −ϵ + \frac{v}{2}(f_{,ξ} + f̄_{,ξ̄}) + \frac{Λ}{6}v^{2}ψ ,  ψ = 1 + ϵξξ̄ .

It can be shown that the transformation relating (18) with (19) has the form

ξ = F (ζ, u) = ∫ f(ξ(ζ, u), u) du ,
v = \frac{r}{1 + ϵF F̄} ,  (20)
u → −u ,  P = (1 + ϵF F̄)(F_{,ζ} F̄_{,ζ̄})^{− \frac{1}{2}} .

If f does not depend on ξ, we put ξ = F (ζ, u) = ζ + ∫ f(u) du.

The non-vanishing Weyl tensor components are proportional to f_{,ξξξ} so that the solutions are conformally flat if f is quadratic in ξ. Thus, we can formulate

Lemma 3: The RT N(Λ, ϵ) solutions (19) with f = f_{c} = c_{0}(u) + c_{1}(u)ξ + c_{2}(u)ξ^{2}, where c_{i}(u) are arbitrary complex functions of u, are isometric to Minkowski (if Λ = 0), de Sitter (Λ > 0) and anti-de Sitter spacetime (Λ < 0).

Transformations preserving the form of (19) were studied in [2]. However, more general transformations (Eq. (2.4) in [2] follows from Eqs. (21), (24) if we put Δ = 0 and αᾱ + ϵββ̄ = 1) are given in the following

Lemma 4: The coordinate transformations (v, ξ, ξ̄, u) → (w, η, η̄, t), which maintain invariant the form of the RT N(Λ, ϵ) metric (19), are

ξ = \frac{αη + β}{γη + δ} ,
v = \frac{(γη + δ)(γ̄η̄ + δ̄)}{\sqrt{(αδ − βγ)(ᾱδ̄ − β̄γ̄)}} w ,  (21)
u = ∫ \frac{\sqrt{(αδ − βγ)(ᾱδ̄ − β̄γ̄)}}{δδ̄ + ϵββ̄} dt ,

where α(t), β(t), γ(t), δ(t) are arbitrary complex functions of t which satisfy the conditions

γ̄δ = −ϵᾱβ ,  γγ̄ + ϵαᾱ = ϵ(δδ̄ + ϵββ̄) .  (22)

In the coordinates (w, η, η̄, t), new structural function f′(η, t) is related to f as follows:

f′ = \frac{ᾱδ̄ − β̄γ̄}{δδ̄ + ϵββ̄} \frac{(γη + δ)^{2}}{\sqrt{(αδ − βγ)(ᾱδ̄ − β̄γ̄)}} f
−\frac{1}{αδ − βγ} [(β̇δ − βδ̇) + (α̇δ − αδ̇+ β̇γ − βγ̇)η + (α̇γ − αγ̇)η^{2} ] . (23)

From this relation and Lemma 3 we see that the conformally flat part f_{c} is indeed unimportant since it can be generated from f = 0 by the coordinate transformation (21).

If ϵ ≠ 0, Eq. (22) implies that |α| = |δ|, |β| = |γ| so that \sqrt{(αδ − βγ)(ᾱδ̄ − β̄γ̄)} = δδ̄ + ϵββ̄. Equations (21)-(23) then simplify to

ξ = \frac{αη + β}{γη + δ} , v = \frac{(γη + δ)(γ̄η̄ + δ̄)}{αᾱ + ϵββ̄} w , u = t ,  (24)

where α(t) and β(t) are arbitrary complex functions of t and γ(t) = β̄(t)e^{iΓ(t)}, δ(t) = ᾱ(t)e^{iΔ(t)} , with Γ(t), Δ(t) being arbitrary real functions of t satisfying Δ − Γ = (1 + ϵ)\frac{π}{2}. The structural function is given by f′ = {(γη + δ)^{2} f − [(β̇δ − βδ̇) + (α̇δ − αδ̇+ β̇γ − βγ̇)η + (α̇γ − αγ̇)η^{2}]}e^{−iΔ} /(αᾱ + ϵββ̄).

For ϵ = 0, equations (22) imply γ = 0 so that the transformations (21) yield

ξ = A(t)η + B(t) , v = \frac{w}{\sqrt{A(t)Ā(t)}} , u = ∫ \sqrt{A(t)Ā(t)}dt ,  (25)

where A(t) and B(t) are arbitrary complex functions of t. The relation (23) between the structural functions is now f′ = \sqrt{Ā/A} f − (Ḃ/A + Ȧ/A η), so that, in contrast to the case ϵ ≠ 0, the term quadratic in η vanishes. Hence, Eq. (25) does not enable us to transform away the complete conformally flat part f_{c}. (We tried but without success to generalize (25) so that quadratic terms could be removed.)

Symmetries of the RT N(Λ, ϵ) solutions have been investigated in [13] (see also [4], Table 33.2) and, more systematically, in [2]. The solutions which are not conformally flat allow the existence of at most two Killing vectors.

ACKNOWLEDGMENTS

We thank Jerry Griffiths for reading the manuscript and useful suggestions. We also acknowledge the support of grant GACR-202/99/0261 from the Czech Republic.

APPENDIX: PROOF OF PROPOSITION 2 For the conformally related metrics g_{αβ} and ĝ_{αβ}, ĝ_{αβ} = Ω^{−2}g_{αβ}, the trace-free Ricci tensors are related by [4]

S_{αβ} = R_{αβ} − \frac{1}{4}Rg_{αβ} ,
Ŝ_{αβ} = R̂_{αβ} − \frac{1}{4}R̂ĝ_{αβ} = S_{αβ} + \frac{2}{Ω}(Ω_{;αβ} − \frac{1}{4}g_{αβ}□Ω) ,  (26)

where □Ω = g^{μν}Ω_{;μν}, the covariant derivative is taken with respect to g_{αβ}, and the scalar curvatures is

R̂ = Ω^{2}R + 6Ω□Ω − 12g^{αβ}Ω_{;α}Ω_{;β} .  (27)

Since R̂_{αβ} = Λĝ_{αβ}, R̂ = 4Λ, we have Ŝ_{αβ} = 0. In coordinates (v, ξ, ξ̄, u),

g_{12} = 1 , g_{03} = −(ξ + ξ̄)^{2} , g_{33} = 2(ξ + ξ̄)^{2}(v^{2} − \frac{1}{2} \frac{H}{ξ + ξ̄} ) ,  (28)

R_{33} = (ξ + ξ̄) H_{,ξξ̄}, and the other Ricci tensor components vanish. Therefore, R = 0 and (26) can be written as

Ω_{,αβ} − Γ_{αβ}^{γ}Ω_{,γ} − \frac{1}{4}g_{αβ}□Ω + \frac{1}{2}ΩR_{αβ} = 0 ,  (29)

which gives

(\frac{Ω_{,v}}{ξ + ξ̄} )_{,ξ} = 0 = (\frac{Ω_{,v}}{ξ + ξ̄} )_{,ξ̄} ,  (30)
Ω_{,vv} = 0 ,  (31)
Ω_{,ξξ} = 0 = Ω_{,ξ̄ξ̄} ,  (32)
Ω_{,vu} + 2vΩ_{,v} − (ξ + ξ̄)(Ω_{,ξ} + Ω_{,ξ̄}) + \frac{1}{4}(ξ + ξ̄)^{2}□Ω = 0 ,  (33)
Ω_{,ξξ̄} = \frac{1}{4}□Ω ,  (34)
Ω_{,ξu} − \frac{Ω_{,u}}{ξ + ξ̄} − \frac{1}{2} (\frac{H}{ξ + ξ̄} )_{,ξ} Ω_{,v} = 0 = Ω_{,ξ̄u} − \frac{Ω_{,u}}{ξ + ξ̄} − \frac{1}{2} (\frac{H}{ξ + ξ̄} )_{,ξ̄} Ω_{,v} ,  (35)

Ω_{,uu} − 2vΩ_{,u} − [2v (2v^{2} − \frac{H}{ξ + ξ̄} ) + \frac{1}{2} (\frac{H}{ξ + ξ̄} )_{,u}] Ω_{,v}

−\frac{1}{2}[(ξ + ξ̄)H]_{,ξ̄}Ω_{,ξ} − \frac{1}{2}[(ξ + ξ̄)H]_{,ξ}Ω_{,ξ̄} + 2v^{2}(ξ + ξ̄)(Ω_{,ξ} + Ω_{,ξ̄})

−\frac{1}{2}(ξ + ξ̄)^{2} (v^{2} − \frac{1}{2} \frac{H}{ξ + ξ̄} ) □Ω + \frac{1}{2}(ξ + ξ̄)H_{,ξξ̄}Ω = 0 .  (36)

By using (34), Eq. (27) takes the form

ΩΩ_{,ξξ̄} − Ω_{,ξ}Ω_{,ξ̄} + \frac{Ω_{,v}}{(ξ + ξ̄)^{2}} [Ω_{,u} + (v^{2} − \frac{1}{2} \frac{H}{ξ + ξ̄} ) Ω_{,v}] = \frac{Λ}{6} .  (37)

Eqs. (30) imply Ω_{,v} = A(u, v)(ξ + ξ̄). Eq. (31) gives A_{,v} = 0 so that Ω = A(u)(ξ + ξ̄)v + B(ξ, ξ̄, u). By virtue of Eq. (32) it must be of the form Ω = A(u)(ξ + ξ̄)v + C(u)ξξ̄ + D_{1}(u)ξ + D_{2}(u)ξ̄ + E(u). Eq. (33) combined with Eq. (34) gives dA/du = D_{1} + D_{2} so that Ω = A(u)(ξ +ξ̄)v +C(u)ξξ̄+ \frac{1}{2}dA/du(ξ +ξ̄)+D(u)(ξ −ξ̄)+E(u) with A, C, E being real functions of u, and D(u) being pure imaginary. We distinguish the possibilities:

1) A = 0

In this case Eq. (35) implies Ω = Cξξ̄ + D(ξ − ξ̄) + E, with C, E real constants, D a pure imaginary constant. Let (i) C ≠ 0. Without loss of generality we can set D = 0 by transformation ξ → ξ′ = ξ − D/C. If E ≠ 0, we can set E = 1 by ξ′ = ξ/E which implies Ω = Cξξ̄ + 1. Eq. (37) gives C = Λ/6 so that Ω = 1 + (Λ/6)ξξ̄, and the metric ĝ_{αβ} takes the canonical form (12) of the KN(Λ)I metric. If E = 0 we can make transformation ξ′ = 1/(Cξ), after which ĝ_{αβ} takes the canonical form (11) of the KN metric. Now (ii) C = 0. If D ≠ 0 we can set E = 0 by ξ′ = ξ + E/(2D). The relation (37) gives Ω = \sqrt{Λ/6}(ξ −ξ̄), with Λ necessarily being negative. It represents just another coordinate form of the KN(Λ)I metric since by ξ = (ξ′ + i\sqrt{−6/Λ})/(iξ′ + \sqrt{−6/Λ}) we get the KN(Λ^{−})I metric in the canonical form (12). If D = 0 then Ω = E so that ĝ_{αβ} is the KN metric by transformation ξ′ = |E|ξ.

2) A ≠ 0

Without loss of generality we can assume A = 1 by using transformation w = A(u)v + \frac{1}{2}dA/du, t = ∫ A^{−1}(u) du. Therefore, Ω = (ξ + ξ̄)v + C(u)ξξ̄ + D(u)(ξ − ξ̄) + E(u). In this case Eq. (35) implies H = 2[\frac{dC}{du} ξξ̄ + α(u)(ξ + ξ̄) + \frac{dD}{du}(ξ − ξ̄) + \frac{dE}{du}], with α being an arbitrary real function of u. Eq. (37) reduces to α(u) = C(u)E(u) + D(u)^{2} − Λ/6, whereas Eq. (36) is satisfied identically. The solution given by ĝ_{αβ} is then conformally flat since g_{αβ} for H of this form is conformally flat ( C_{αβγδ} = 0). Therefore, the solution describes Minkowski, de Sitter or anti-de Sitter spacetime, according to the sign of Λ. It is interesting to notice that although ĝ_{αβ} of this form describes a conformally flat vacuum solution, the KN solution to which it is conformal (given by g_{αβ} with H of the same form) is conformally flat but not necessarily a vacuum solution. In general, it is a pure radiation solution, becoming a vacuum solution (Minkowski) only for C =const. In particular, if α = 0 and D, E =const., then the conformally flat KN pure radiation solution given by H = 2\frac{dC}{du} ξξ̄ is Wils’ solution (3.10) [14] for N = 0 (where Q + Q̄ = \frac{dC}{du}, v → v/(ξ + ξ̄)^{2} ). The solution was used by Wils as an explicit counterexample of theorem 32.17 in [4] according to which there are no other conformally flat pure radiation solutions besides the special P P wave of McLenaghan et al. [15].

We have thus shown that all vacuum solutions conformal to the KN class are 1) KN(Λ)I solutions, 2) KN solutions themselves (KN = KN(Λ = 0)I), 3) Minkowski, de Sitter and anti-de Sitter space-times (these are special cases of KN(Λ)I given by KN(Λ)I[H = H_{c}]). Therefore, all vacuum solutions conformal to the KN class belong to the KN(Λ)I class, which proves Proposition 2 in Section II.

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