Non-expanding impulsive gravitational waves

J. Podolský^{∗}

Department of Theoretical Physics,

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

July 31, 1998

Abstract

We investigate a class of impulsive gravitational waves which propagate either in Minkowski or in the (anti–) de Sitter background. These waves are constructed as impulsive members of the Kundt class P (Λ) of non-twisting, non-expanding type N solutions of vacuum Einstein equations with a cosmological constant Λ.

We show that the only non-trivial waves of this type in Minkowski spacetime are impulsive pp-waves. For Λ ≠ 0 we demonstrate that the canonical subclasses of P (Λ), which are invariantly different for smooth profiles, are all locally equivalent for impulsive profiles.

Also, we present coordinate system for these impulsive solutions which is explicitly continuous.

PACS class: 04.20.Jb, 04.30.Nk, 04.30.-w, 98.80.Hw

Keywords: impulsive waves, cosmological constant

∗E–mail: podolsky@mbox.troja.mff.cuni.cz

1 Introduction

In a now classic paper [1], Penrose proposed a method for constructing general impulsive pp-waves (plane-fronted gravitational waves with parallel rays) and spherical waves of the Robinson-Trautman type [2]. His “scissors-and-paste” approach is based on the removal of a null hypersurface from Minkowski space-time and re-attaching the parts by making an identification with a “warp”. As an explicit example Penrose constructed an impulsive plane wave. A family of impulsive spherical gravitational waves obtained by the above method have later been investigated by Hogan [3]-[5].

Another explicit impulsive pp-wave solution was found by Aichelburg and Sexl [6]. However, it was obtained by a different method: by boosting a Schwarzschild black hole to the speed of light while its mass is reduced to zero in an appropriate way. This interesting solution describes an impulsive wave generated by a single null particle. More general metrics have similarly been obtained by boosting other space-times of the Kerr-Newman family [7]-[11].

A similar approach has been adopted by Hotta and Tanaka [12] in boosting the Schwarzschild–(anti-)de Sitter metric to a similar limit. This approach also results in impulsive gravitational waves. However, the background in this case is not flat and the wavefronts are not simple planes as in Minkowski space. In [13] we demonstrated that the impulsive wave surface in the de Sitter background is a non-expanding sphere forming the cosmological horizon. It is generated by two null particles located at the poles. In the anti-de Sitter background the wavefronts are hyperboloidal surfaces containing a single null particle which propagates from one side of the universe to the other and then returns in an endless cycle. We also determined the global structure and conformal diagrams for these spacetimes.

In subsequent papers [14], [15] we presented a more general class of such solutions. Their global structure and the shape of the impulsive wave surfaces are exactly as summarized above but the sources are null particles which can have an arbitrary multipole structure and which are distributed arbitrarily over the wave surface. We also noticed that these space-times are related to the impulsive limit of a family of non-twisting and non-expanding type N solutions of the Kundt class studied in [16]. It is the purpose of the present paper to investigate the relation systematically.

In section 2 we summarize properties of the Kundt class P (Λ) and, in particular, we present its canonical subclasses. Impulsive waves of this type for Λ = 0 are investigated in section 3. Sections 4 and 5 are devoted to non-expanding impulsive waves in de Sitter and anti–de Sitter backgrounds. Continuous coordinate systems for all these solutions are presented in the final section 6.

2 Non-expanding gravitational waves P (Λ)

It can be shown [17] that all non-twisting type N solutions of the vacuum Einstein field equations with (possibly) non-vanishing cosmological constant Λ belong either to the Kundt class of non-expanding gravitational fields or to the expanding Robinson-Trautman class. In this paper we concentrate on the Kundt class which we denote P (Λ).

This class has been investigated by Ozsváth, Robinson and Rózga [16] and in [17]. In a suitable coordinate system where v is a parameter along null geodesics generated by the quadruple Debever-Penrose vector field, u is a retarded time with u =const. being wavefronts and ξ, ξ̄ are space-like coordinates, all the P (Λ) solutions can be written as

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

where

p = 1 + cξξ̄ ,

q = (1 − cξξ̄)α + β̄ξ + βξ̄ ,

F = κ\frac{q^{2}}{p^{2}}v^{2} − 2l\frac{q^{2}}{p^{2}}v + \frac{q}{p}H ,

c = \frac{Λ}{6} ,  κ = 2cα^{2} + 2ββ̄ ,  l = (ln|q|),_{u} .

Here α(u) and β(u) are arbitrary real and complex functions of u, respectively. These functions play the role of two additional parameters, i.e. P (Λ) ≡ P (Λ)[α, β]. Finally, H = H(ξ, ξ̄, u) is a function restricted by the vacuum field equations with a cosmological constant, H_{,ξξ̄} + 2cH/p^{2} = 0. A general solution to this equation is

H(ξ, ξ̄, u) = (f_{,ξ} + f̄_{,ξ̄}) − \frac{2c}{p}(ξ̄f + ξf̄) ,  (2)

where f(ξ, u) is an arbitrary function of ξ and u, analytic in ξ. The space-time is conformally flat (and thus of a constant curvature) if and only if the function H is of the form H = H_{c} = [(1 − cξξ̄)A + B̄ξ + Bξ̄] /p with A(u) and B(u) being arbitrary real and complex functions, respectively. Since H_{c} of this form corresponds to (2) for f quadratic in ξ we infer that the P (Λ) solutions (1), (2) 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, de Sitter and anti-de Sitter space-time if Λ = 0, Λ > 0 and Λ < 0, respectively.

As was suggested in [16], one can base the invariant ‘canonical’ classification of all the P (Λ) solutions on signs of the parameters κ and Λ (see also [17]). There are nine possible cases since 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 = 2cα^{2} + 2ββ̄. The remaining six possibilities give the following subclasses:

A. Subclass κ = 0, Λ = 0. Using a coordinate freedom, a canonical representative of the subclass can be found and denoted as

P P ≡ P (Λ = 0)[α = 1, β = 0] .

We are using the symbol P P since the corresponding metric

ds^{2} = 2dudv − 2dξdξ̄ − 2(g + ḡ)du^{2} ,  (3)

with g(ξ, u) = \frac{1}{2}f_{,ξ} analytic in ξ, describes the well-known plane-fronted gravitational waves with parallel rays (pp-waves). These solutions have thoroughly been investigated by many authors; for details and references see [2].

B. Subclass κ > 0, Λ = 0. Any solution of this subclass can be put into the form

P K ≡ P (Λ = 0)[α = 0, β = 1] .

We have used the symbol P K since the metric

ds^{2} = 2(ξ + ξ̄)^{2}dudv − 2dξdξ̄ − 2(ξ + ξ̄)^{2} (v^{2} + \frac{g + ḡ}{ξ + ξ̄} ) du^{2} ,  (4)

with g(ξ, u) = \frac{1}{2}f_{,ξ} describes the wave solutions with Λ = 0 discovered by Kundt [18]. Details about this solution can also be found in [2].

C. Subclass κ > 0, Λ > 0. We can always find a transformation to the canonical representative of this subclass

P K(Λ)I ≡ P (Λ)[α = 0, β = 1] .

The corresponding metric reads

ds^{2} = 2(\frac{ξ + ξ̄}{1 + cξξ̄} )^{2}dudv − 2\frac{dξdξ̄}{(1 + cξξ̄)^{2}} − [2(\frac{ξ + ξ̄}{1 + cξξ̄} )^{2}v^{2} + \frac{ξ + ξ̄}{1 + cξξ̄} H ] du^{2} . (5)

We have introduced P K(Λ)I in order to indicate that this solution represents a generalization of the Kundt waves P K to the case Λ ≠ 0. Here ‘I’ means ‘of the first kind’ since there are two other P K(Λ) solutions. The solution P K(Λ)I was first discovered in [19].

D. Subclass κ > 0, Λ < 0. As in the previous case the representative is P K(Λ)I = P (Λ)[α = 0, β = 1]. The metric (5) has the same form for negative cosmological constant Λ = 6c < 0.

E. Subclass κ < 0, Λ < 0. The canonical form of this subclass is

P K(Λ^{−})II ≡ P (Λ < 0)[α = 1, β = 0]

and the metric has the form

ds^{2} = 2(\frac{1 − cξξ̄}{1 + cξξ̄})^{2}dudv − 2\frac{dξdξ̄}{(1 + cξξ̄)^{2}} − [2c(\frac{1 − cξξ̄}{1 + cξξ̄})^{2}v^{2} + \frac{1 − cξξ̄}{1 + cξξ̄}H ] du^{2} . (6)

The symbol P K(Λ^{−})II means generalized Kundt waves ‘of the second kind’ with a negative cosmological constant. This class was first discovered in [16].

F. Subclass κ = 0, Λ < 0. In this case the canonical representative is P K(Λ^{−})III ≡ P (Λ < 0)[α = 1, β = \sqrt{−c} e^{iω(u)}], ω(u) being an arbitrary real function of u. In particular, if the Debever-Penrose vector is a Killing vector, β =const. and the representative is

P K(Λ^{−})III_{K} ≡ P (Λ < 0)[α = 1, β = \sqrt{−c}] ,

where the suffix ‘K’ stands for ‘Killing’. The metric in this case reads

ds^{2} = 2(\frac{1 + \sqrt{−c} (ξ + ξ̄) − cξξ̄}{1 + cξξ̄} )^{2}dudv−2\frac{dξdξ̄}{(1 + cξξ̄)^{2}}−\frac{1 + \sqrt{−c} (ξ + ξ̄) − cξξ̄}{1 + cξξ̄} Hdu^{2} . (7)

It can be shown [20] that this subclass is the same as the ‘Lobatchevski waves’ studied by Siklos [21].

Finally, we summarize the invariant classification of the P (Λ) class of solutions in the following diagram:

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

There are three subclasses for Λ < 0 but only one subclass for Λ > 0. The reason for this asymmetry is a geometrical one, embodied in the condition κ = 2cα^{2} + 2ββ̄ which for Λ > 0 excludes the cases κ < 0 and κ = 0. We also observe natural relations between Λ = 0 and Λ ≠ 0 subclasses. We can set Λ = 6c = 0 in the metrics (5), (6), (7) and find that P K(Λ = 0)I = P K, P K(Λ = 0)II = P P = P K(Λ = 0)III_{K}. Thus, it is natural to consider the P K(Λ)I class as a generalization of the Kundt solution P K whereas the classes P K(Λ^{−})II and P K(Λ^{−})III_{K} as generalizations of pp-waves.

3 Non-expanding impulsive waves with Λ = 0

From the above classification of the P (Λ) class of non-expanding gravitational waves we observe that there are two subclasses for vanishing cosmological constant, P P and P K. Therefore, we can easily construct impulsive waves in Minkowski spacetime by assuming δ-function profiles of the form g(ξ, u) = g(ξ)δ(u) in metrics (3) or (4). These two types of impulsive solutions look quite different but we can demonstrate that, in fact, they are locally equivalent. Indeed, a transformation

U = (ξ + ξ̄)(1 + uv)u ,
V = (ξ + ξ̄)v ,  (8)
η = (ξ + ξ̄)uv + ξ ,

converts the impulsive P K solution (4) to

ds^{2} = 2dUdV − 2dηdη̄ − 2[g(η) + ḡ(η̄)]δ(U)dU^{2} ,  (9)

which is exactly the P P impulsive solution (3). To be more precise, we should note that an inverse transformation to (8) is

u = \frac{1}{2V} [η + η̄ ∓ \sqrt{(η + η̄)^{2} − 4UV} ] ,
v = ±V/\sqrt{(η + η̄)^{2} − 4UV} ,  (10)
ξ = \frac{1}{2}[η − η̄ ± \sqrt{(η + η̄)^{2} − 4UV} ] .

One can show that (η + η̄)^{2} − 4UV = (ξ + ξ̄)^{2} ≥ 0. The transformation (10) is not unique but the upper signs parametrize the part of spacetime given by ξ + ξ̄ > 0 whereas the lower signs parametrize the part given by ξ + ξ̄ < 0. For ξ + ξ̄ = 0 there is a coordinate singularity in (4).

To summarize, the only non-trivial impulsive gravitational waves of the form (1) in Minkowski spacetime are impulsive pp-waves (9).

4 Non-expanding impulsive waves with Λ ≠ 0

In [15] we presented a general class of exact solutions of Einstein’s equations which describe impulsive gravitational waves in de Sitter or anti-de Sitter background generated by null particles with an arbitrary multipole structure. In five-dimensional notation the solutions can simply be written in the form

ds^{2} = dZ_{0}^{2} − dZ_{1}^{2} − dZ_{2}^{2} − dZ_{3}^{2} − ϵdZ_{4}^{2}
−H(Z_{2}, Z_{3}, Z_{4})δ(Z_{0} + Z_{1})(dZ_{0} + dZ_{1})^{2} .  (11)

The de Sitter or the anti-de Sitter background can be represented as a metric (11) with H = 0 on the four-dimensional hyperboloid

Z_{0}^{2} − Z_{1}^{2} − Z_{2}^{2} − Z_{3}^{2} − ϵZ_{4}^{2} = −ϵa^{2} ,  (12)

where a^{2} = 3/ϵΛ, ϵ = 1 for the de Sitter background (Λ > 0), and ϵ = −1 for the anti-de Sitter background (Λ < 0). The impulse is located on the null hypersurface given by

Z_{0} + Z_{1} = 0,  Z_{2}^{2} + Z_{3}^{2} + ϵZ_{4}^{2} = ϵa^{2},  (13)

which is a 2-sphere for ϵ = 1 or a 2-dimensional hyperboloid for ϵ = −1. The function H(Z_{2}, Z_{3}, Z_{4}) is determined on the wave surface (13), i.e., it must be a function of two parameters which span the surface. An appropriate parametrization is thus given by

Z_{2} = a\sqrt{ϵ(1 − z^{2})}cosϕ,  Z_{3} = a\sqrt{ϵ(1 − z^{2})}sinϕ,  Z_{4} = az ,  (14)

where |z| ≤ 1 when ϵ = 1 and |z| ≥ 1 when ϵ = −1. We showed in [15] that a non-trivial vacuum solution of this type is given by

H(z, ϕ) = ∑_{m=0}^{∞} b_{m}H_{m}(z, ϕ) = ∑_{m=0}^{∞} b_{m}Q_{1}^{m}(z) cos[m(ϕ − ϕ_{m})] ,  (15)

where b_{m} and ϕ_{m} are real constants representing an arbitrary amplitude and phase of each component and Q_{1}^{m}(z) are associated Legendre functions of the second kind generated by the relation

Q_{1}^{m}(z) = (−ϵ)^{m}|1 − z^{2}|^{m/2} \frac{d^{m}}{dz^{m}}Q_{1}(z) ,  (16)

where Q_{1}(z) = Q_{1}^{0}(z) = (z/2) log|(1 + z)/(1 − z)| − 1. The first term (m = 0) gives the simplest (axially symmetric) solution found by Hotta and Tanaka [12]. In general, these components describe point sources with an m-pole structure [15]. Geometry of the spacetimes was investigated in [13].

5 Equivalence of all the P (Λ ≠ 0) subclasses of impulsive waves

In this section we shall demonstrate that although the invariant canonical subclasses of the P (Λ ≠ 0) class of non-expanding gravitational waves described in section 2 are generically different for smooth profiles, they are locally equivalent for impulsive profiles. Moreover, they are just different four-dimensional parametrizations of the general impulsive solution in de Sitter or anti–de Sitter spacetimes given by Eqs. (11)–(14).

For the P K(Λ > 0)I class of impulsive solutions with H = H(ξ, ξ̄)δ(u) it was shown in [15] that performing the transformation

ξ = \sqrt{2}a e^{iϕ} tan \frac{θ}{2},  v = \frac{a}{\sqrt{2}} \frac{1}{t},  u = \frac{1}{\sqrt{2}a}(ρ − t),  (17)

where a = 1/\sqrt{2ϵc} = \sqrt{3/Λ} the line element (5) becomes

ds^{2} = \frac{a^{2}}{t^{2}} sin^{2} θ cos^{2} ϕ (dt^{2} − dρ^{2}) − a^{2}(dθ^{2} + sin^{2} θdϕ^{2})

− sinθ cosϕ H(θ, ϕ)δ(t − ρ)(dt − dρ)^{2}.  (18)

This can be seen to be exactly the solution (11) in which the de Sitter background in the five-dimensional form (12) is parametrized by

Z_{0} = −a cosθ + \frac{a^{2}}{t} sinθ cosϕ + \frac{ρ^{2} − t^{2}}{2t} sinθ cosϕ
Z_{1} =  a cosθ − \frac{a^{2}}{t} sinθ cosϕ
Z_{2} = \frac{ρ}{t} a sinθ cosϕ  (19)
Z_{3} =  a sinθ sinϕ
Z_{4} =  a cosθ  − \frac{ρ^{2} − t^{2}}{2t} sinθ cosϕ .

For Λ < 0 the metric (5) of impulsive P K(Λ)I subclass with H = H(ξ, ξ̄)δ(u) can be expressed, using a similar transformation

ξ = \sqrt{2}a e^{iϕ_{1}} tanh \frac{R_{1}}{2},  v = \frac{a}{\sqrt{2}} \frac{1}{t_{1}},  u = \frac{1}{\sqrt{2}a}(ρ_{1} − t_{1}),  (20)

where a = 1/\sqrt{−2ϵc} = \sqrt{−3/Λ}, in the more convenient form

ds^{2} = \frac{a^{2}}{t_{1}^{2}} sinh^{2} R_{1} cos^{2} ϕ_{1} (dt_{1}^{2} − dρ_{1}^{2}) − a^{2}(dR_{1}^{2} + sinh^{2} R_{1}dϕ_{1}^{2})

− sinhR_{1} cosϕ_{1} H(R_{1}, ϕ_{1})δ(t_{1} − ρ_{1})(dt_{1} − dρ_{1})^{2} .  (21)

Again, it can be shown by direct calculations that (21) is just the solution (11) in coordinates introduced by the parametrization

Z_{0} = −a coshR_{1} + \frac{a^{2}}{t_{1}} sinhR_{1} cosϕ_{1}
Z_{1} =  a coshR_{1} − \frac{a^{2}}{t_{1}} sinhR_{1} cosϕ_{1} + \frac{ρ_{1}^{2} − t_{1}^{2}}{2t_{1}} sinhR_{1} cosϕ_{1}
Z_{2} = \frac{ρ_{1}}{t_{1}} a sinhR_{1} cosϕ_{1}  (22)
Z_{3} =  a sinhR_{1} sinϕ_{1}
Z_{4} =  a coshR_{1}  + \frac{ρ_{1}^{2} − t_{1}^{2}}{2t_{1}} sinhR_{1} cosϕ_{1} .

For the impulsive P K(Λ^{−})II subclass, the transformation

ξ = \sqrt{2}a e^{iϕ_{2}} tanh \frac{R_{2}}{2},  v = −\frac{a^{2}}{ρ_{2}},  u = t_{2} − ρ_{2},  (23)

can be used to express the metric (6) in the form

ds^{2} = \frac{a^{2}}{ρ_{2}^{2}} cosh^{2} R_{2} (dt_{2}^{2} − dρ_{2}^{2}) − a^{2}(dR_{2}^{2} + sinh^{2} R_{2}dϕ_{2}^{2})

− coshR_{2} H(R_{2}, ϕ_{2})δ(t_{2} − ρ_{2})(dt_{2} − dρ_{2})^{2} .  (24)

However, this simple form also follows from (11) if we consider a parametrization

Z_{0} = \frac{coshR_{2}}{2ρ_{2}}(t_{2}^{2} − ρ_{2}^{2} − a^{2})
Z_{1} = \frac{coshR_{2}}{2ρ_{2}}(t_{2}^{2} − ρ_{2}^{2} + a^{2})
Z_{2} = a sinhR_{2} cosϕ_{2}  (25)
Z_{3} = a sinhR_{2} sinϕ_{2}
Z_{4} = \frac{t_{2}}{ρ_{2}} a coshR_{2} .

Finally, impulsive metric (7) of the P K(Λ^{−})III_{K} subclass for H = \sqrt{2}H(ξ, ξ̄)δ(u) using a transformation

ξ = \sqrt{2}a e^{iϕ_{3}} tanh \frac{R_{3}}{2},  v = \frac{a}{\sqrt{2}}(η − χ),  u = \frac{a}{\sqrt{2}}(η + χ),  (26)

reads

ds^{2} = a^{2}(coshR_{3} + sinhR_{3} cosϕ_{3})^{2} (dη^{2} − dχ^{2}) − a^{2}(dR_{3}^{2} + sinh^{2} R_{3}dϕ_{3}^{2})
−a(coshR_{3} + sinhR_{3} cosϕ_{3}) H(R_{3}, ϕ_{3})δ(η + χ)(dη + dχ)^{2} ,  (27)

which is exactly the metric of Eq. (21) in [15] representing the solution (11) in global coordinate system given by

Z_{0} = a(coshR_{3} + sinhR_{3} cosϕ_{3})η
Z_{1} = a(coshR_{3} + sinhR_{3} cosϕ_{3})χ
Z_{2} = a sinhR_{3} cosϕ_{3} − \frac{1}{2}a(coshR_{3} + sinhR_{3} cosϕ_{3})(χ^{2} − η^{2})  (28)
Z_{3} = a sinhR_{3} sinϕ_{3}
Z_{4} = a coshR_{3}  + \frac{1}{2}a(coshR_{3} + sinhR_{3} cosϕ_{3})(χ^{2} − η^{2}) .

We conclude this section by explicit transformations relating the three different forms of impulsive P (Λ) solutions for Λ < 0 given above. The transformation between the metric (21) of the P K(Λ)I subclass for Λ < 0 and the metric (24) of the P K(Λ^{−})II subclass is

t_{1}^{2} = t_{2}^{2} − ρ_{2}^{2}(1 − tanh^{2} R_{2} cos^{2} ϕ_{2}) ,
ρ_{1} = ρ_{2} tanhR_{2} cosϕ_{2} ,
coshR_{1} = \frac{t_{2}}{ρ_{2}} coshR_{2} ,  (29)
sinΦ_{1} = −\frac{sinhR_{2} sinϕ_{2}}{\sqrt{t_{2}^{2}ρ_{2}^{−2} cosh^{2} R_{2} − 1}} .

(Note that t_{1}^{2} − ρ_{1}^{2} = t_{2}^{2} − ρ_{2}^{2} and sinhR_{1} sinϕ_{1} = − sinhR_{2} sinϕ_{2}). Performing the transformation

η̃ = D\frac{coshR_{2}}{2ρ_{2}}(t_{2}^{2} − ρ_{2}^{2} − a^{2}) ,
χ̃ = D\frac{coshR_{2}}{2ρ_{2}}(t_{2}^{2} − ρ_{2}^{2} + a^{2}) ,
x = aD ,  (30)
y = aD sinhR_{2} sinϕ_{2} ,

where D = (t_{2}ρ_{2}^{−1} coshR_{2} + sinhR_{2} cosϕ_{2})^{−1}, the metric (24) of the P K(Λ^{−})II subclass goes over to impulsive solution of the Siklos type

ds^{2} = \frac{a^{2}}{x^{2}} (dη̃^{2} − dχ̃^{2} − dx^{2} − dy^{2} − S(x, y)δ(η̃ + χ̃)(dη̃ + dχ̃)^{2}) .  (31)

This was shown in [15] to be equivalent to the metric (27) of the P K(Λ^{−})III_{K} class through the transformation

η̃ = aη, χ̃ = aχ, x = \frac{a}{coshR_{3} + sinhR_{3} cosϕ_{3}} , y = \frac{a sinhR_{3} sinϕ_{3}}{coshR_{3} + sinhR_{3} cosϕ_{3}} , (32)

with an identification S(x, y) = xH(x, y)/a.

6 Continuous coordinates for non-expanding impulsive gravitational waves

We have shown that all non-expanding impulsive waves of the P (Λ) class can be written either in the form (3) with g = g(ξ)δ(u), i.e. as pp-waves in Minkowski spacetime, or in the form (11) in de Sitter and anti-de Sitter backgrounds. Interestingly, both metrics can be expressed in a unified form

ds^{2} = dudr − dζdζ̄ − ϵdZ_{4}^{2} − H(ζ, ζ̄)δ(u)du^{2} .  (33)

Indeed, the impulsive metric (3) arises by setting ϵ = 0 and performing a transformation

r = 2v,  ζ = \sqrt{2}ξ ,  ζ̄ = \sqrt{2} ξ̄ .  (34)

The relation to (11) follows from ϵ = ±1 and a transformation

u = Z_{0} + Z_{1},  r = Z_{0} − Z_{1},  ζ = Z_{2} + iZ_{3},  ζ̄ = Z_{2} − iZ_{3} ;  (35)

note that in this case we consider the dependence of H(Z_{2}, Z_{3}, Z_{4}) on the argument Z_{4} given by Z_{4} = ±\sqrt{a^{2} − ϵζζ̄} which follows from (13).

Now, performing a coordinate transformation

ζ = η + uΘ(u)H_{,η̄} ,
r = w + Θ(u)H + uΘ(u)H_{,η}H_{,η̄} ,  (36)

where Θ(u) is the Heaviside step function (Θ = 0 for u < 0, Θ = 1 for u > 0) and H(η, η̄) is the same function as in (33) but expressed in terms of the new coordinate η (note that η = ζ at u = 0) we get

ds^{2} = dudw − ϵdZ_{4}^{2} − |dη̄ + uΘ(u)(H_{,ηη}dη + H_{,ηη̄}dη̄)| ^{2} .  (37)

This form of non-expanding impulsive waves in Minkowski, de Sitter and anti-de Sitter space-times is explicitly continuous for all values of u (delta functions appear only in the components of the curvature tensor).

In a Minkowski background, the cosmological constant Λ vanishes, ϵ = 0 and H(ζ, ζ̄) = 2[g(ζ/\sqrt{2}) + ḡ(ζ̄\sqrt{2})] so that H_{,ηη} = g′′, H_{,ηη̄} = 0; then the metric (37) reduces to

ds^{2} = dudw − |dη̄ + uΘ(u)g′′(η)dη| ^{2} .  (38)

This continuous system for a general impulsive pp-wave was found in [22] and independently in [23]. In particular, when g(ζ) = Cζ^{n} where C = |C| exp(inφ_{n}) is an arbitrary complex constant and n is an integer, it is convenient to introduce two real coordinates ρ, φ by η = ρ exp[i(φ − φ_{n})]. In these coordinates the metric (38) takes the form

ds^{2} = dudw − [1 − uΘ(u)Dρ^{n−2}]^{2}(dρ^{2} + ρ^{2}dφ^{2})
−4uΘ(u)Dρ^{n−2}[cos(\frac{n}{2}φ)dρ − sin(\frac{n}{2}φ)ρdφ]^{2} ,  (39)

where D = n(n − 1)|C|. We may also assume coordinates x = ρ cosφ, y = ρ sinφ,

ds^{2} = dudw + 4DB_{n}uΘ(u) dxdy
−[1 + 2DA_{n}uΘ(u) + D^{2}(x^{2} + y^{2})^{n−2}u^{2}Θ(u)] dx^{2}  (40)
−[1 − 2DA_{n}uΘ(u) + D^{2}(x^{2} + y^{2})^{n−2}u^{2}Θ(u)] dy^{2}

where A_{n}(x, y) = Re {(x+iy)^{n−2}}, B_{n}(x, y) = Im {(x+iy)^{n−2}}. In particular, the metric (39) with n = 0, D ≠ 0 reduces to the continuous from of the Aichelburg and Sexl solution [6] presented in [24], [25]. For n = 2 the metric (40) gives the well-known form of the homogeneous impulsive pp-wave [1].

Continuous coordinate system for non-expanding impulsive waves in de Sitter and anti-de Sitter backgrounds is given by (37) with ϵ = +1 for Λ > 0 and ϵ = −1 for Λ < 0. The function H(η, η̄) for first few multipole terms H_{m} given by (15) and (16) is

H_{0}(η, η̄) = ±\frac{1}{2}\sqrt{1 − ϵa^{−2}ηη̄} log |\frac{1 ± \sqrt{1 − ϵa^{−2}ηη̄}}{1 ∓ \sqrt{1 − ϵa^{−2}ηη̄}}| − 1,
H_{1}(η, η̄) = (−\frac{ϵηη̄}{4a} log |\frac{1 ± \sqrt{1 − ϵa^{−2}ηη̄}}{1 ∓ \sqrt{1 − ϵa^{−2}ηη̄}}| ∓ \frac{a}{2}\sqrt{1 − ϵa^{−2}ηη̄}) (\frac{e^{iϕ_{1}}}{η} + \frac{e^{−iϕ_{1}}}{η̄} ) ,
H_{2}(η, η̄) = \frac{a^{2}}{η^{2}}e^{i2ϕ_{2}} + \frac{a^{2}}{η̄^{2}}e^{−i2ϕ_{2}},  (41)
H_{3}(η, η̄) = ∓4\sqrt{1 − ϵa^{−2}ηη̄}(\frac{a^{3}}{η^{3}}e^{i3ϕ_{3}} + \frac{a^{3}}{η̄^{3}}e^{−i3ϕ_{3}}) .

These functions satisfy the vacuum field equation (η^{2}H_{,η})_{,η} +(η̄^{2}H_{,η̄})_{,η̄} +2(ηη̄−2ϵa^{2})H_{,ηη̄} − 2H = 0 everywhere except for the point sources at η = 0. Substituting these particular forms into (37) we get continuous coordinates for the first multipole terms. Note that the simplest form of the metric arises for the quadruple term H_{2},

ds^{2} = dudw − ϵdZ_{4}^{2} − |dη̄ + CuΘ(u)\frac{dη}{η^{4}} | ^{2} ,  (42)

where C = 6a^{2}e^{i2ϕ_{1}}.

The continuous form (37) of a general impulsive solution for non-vanishing Λ is explicit and simple. However, it is written in the five-dimensional notation so that we must always restrict the metric on the hyperboloid (12) which in our coordinates is given by uw − ηη̄ − ϵZ_{4}^{2} = −ϵa^{2} for u < 0. It may thus be useful to find a continuous metric also in standard four-dimensional form. There are many suitable parametrizations of the de Sitter hyperboloid (for ϵ = 1), cf. [13]. We can consider, for example, global coordinates (u, x, y, z) such that

w = \frac{x^{2} + y^{2} − a^{2}}{u} + \frac{uz^{2}}{a^{2}},  Z_{4} = \frac{uz}{a} ,  η = x + iy,  (43)

in which the metric (37) takes the form

ds^{2} = \frac{a^{2}}{u^{2}} du^{2} − \frac{u^{2}}{a^{2}} dz^{2} − (dx − \frac{x}{u}du)^{2} − (dy − \frac{y}{u}du)^{2}
−uΘ(u) [(C + A) dx^{2} + (C − A) dy^{2} + 2B dxdy] ,  (44)

where A = (1 + uH_{,ηη̄}) (H_{,ηη} + H_{,η̄η̄}), B = (1 + uH_{,ηη̄})i(H_{,ηη} − H_{,η̄η̄}) and C = 2H_{,ηη̄} + u(H_{,ηη}H_{,η̄η̄} − H_{,ηη̄}^{2}). The function H and all its derivatives are now expressed in terms of the coordinates x and y so that the coefficients A, B and C are independent of z. A simple transformation ψ = a^{2}/u, x̃ = ax/u and ỹ = ay/u reveals for u < 0 a standard conformally flat metric of the de Sitter universe, ds^{2} = a^{2}ψ^{−2} (dψ^{2} − dx̃^{2} − dỹ^{2} − dz^{2}). Note that the singularity at ψ = ∞ is only a coordinate one.

In order to find a suitable four-dimensional system for non-expanding impulsive waves in the anti-de Sitter background (Λ < 0, ϵ = −1) it is more convenient to start from the Siklos form of the metric (31) which is explicitly conformal to pp-waves. By a simple reparametrization

u = η̃ + χ̃,  r = η̃ − χ̃,  ζ = x + iy,  ζ̄ = x − iy,  (45)

the metric (31) goes over to

ds^{2} = \frac{4a^{2}}{(ζ + ζ̄)^{2}} (dudr − dζdζ̄ − S(ζ, ζ̄)δ(u)du^{2}) .  (46)

We may use the transformation (36) with H = S and we get a continuous form

ds^{2} = \frac{4a^{2}}{[η + η̄ + uΘ(u)(S_{,η} + S_{,η̄})] ^{2}} (dudw − |dη̄ + uΘ(u)(S_{,ηη}dη + S_{,ηη̄}dη̄)| ^{2}) . (47)

Note that in his paper [21], Siklos has shown that, for the vacuum case (except for some possible point sources), the general solution for S is of the form S(η, η̄) = \frac{1}{2}(η + η̄)(f_{,η} + f̄_{,η̄})−(f +f̄) where f is an arbitrary function of η. In particular, the Kaigorodov solution (which is the simplest representative of the Siklos class, see [20]) is given by f = \frac{1}{4}η^{3} so that the continuous system for an impulsive Kaigorodov spacetime is

ds^{2} = \frac{a^{2}}{X^{2}}(1 + \frac{3}{2}uΘ(u)X)^{−2} [dudw − (1 + 3uΘ(u)X)^{2}dX^{2} − dY^{2}] ,  (48)

where we introduced η = X + iY.

Acknowledgments

This work was supported by the grant GACR-202/96/0206 of the Czech Republic and the grant GAUK-230/96 of the Charles University.

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