Gravitational waves in vacuum spacetimes

with cosmological constant.

II. Deviation of geodesics and interpretation

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

In a suitably chosen essentially unique frame tied to a given observer in a general spacetime, the equation of geodesic deviation can be decomposed into a sum of terms describing specific effects: isotropic (background) motions associated with the cosmological constant, transverse motions corresponding to the effects of gravitational waves, longitudinal motions, and Coulombtype effects. Conditions under which the frame is parallelly transported along a geodesic are discussed. Suitable coordinates are introduced and an explicit coordinate form of the frame is determined for spacetimes admitting a non-twisting null congruence. Specific properties of all nontwisting type N vacuum solutions with cosmological constant Λ (non-expanding Kundt class and expanding Robinson-Trautman class) are then analyzed. It is demonstrated that these spacetimes can be understood as exact transverse gravitational waves of two polarization modes “+” and “×”, shifted by \frac{π}{4}, which propagate “on” Minkowski, de Sitter, or anti-de Sitter backgrounds. It is also shown that the solutions with Λ > 0 may serve as exact demonstrations of the cosmic “no-hair” conjecture in radiative spacetimes with no symmetry.

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

I. INTRODUCTION AND SUMMARY

In the preceding paper [1] we classified non-twisting type N solutions of the vacuum Einstein’s equations with a non-vanishing cosmological constant Λ and analyzed their geometrical properties. Here we wish to discuss their physical properties. We shall show that these solutions can be interpreted as gravitational waves propagating in spacetimes of constant curvature — in Minkowski, de Sitter, or anti-de Sitter spaces. Our treatment focuses on the analysis of the equation of geodesic deviation.

We first discuss the equation of geodesic deviation in general spacetimes (Section II), briefly reviewing and extending [2]-[4] by using both a Newman-Penrose null tetrad and a physical frame of four independent vectors {e_{(a)}} tied to the geodesic with respect to which the relative motion is studied. In type N solutions only the Newman-Penrose scalar Ψ_{4} is non-vanishing.

Starting from Section III we study non-twisting type N solutions with Λ. As shown in [1], they comprise the non-expanding Kundt class KN(Λ) and the expanding Robinson-Trautman class RT N(Λ, ϵ). By analyzing the geodesic deviation in these spacetimes we demonstrate that they can be interpreted as exact transverse gravitational waves with two polarization modes (shifted by \frac{π}{4}) propagating “on” Minkowski, de Sitter, or anti-de Sitter space (depending on values of Λ). In the Appendix we calculate exact forms of wave amplitudes.

At the end of Section IV we discuss, for Λ > 0, special timelike geodesics explicitly. We demonstrate that observers moving along these geodesics see waves decaying exponentially fast and the spacetimes to approach locally the de Sitter space — in agreement with the cosmic no-hair conjecture (see e.g. [5], and references therein). As in our previous work [6], [7], this is an explicit demonstration of the conjecture under the presence of waves within exact theory.

II. THE RELATIVE MOTION OF FREE PARTICLES IN A GENERAL SPACE-TIME

It is natural to base the local characterization of radiative spacetimes on the equation of geodesic deviation [2]-[4]

\frac{D^{2}Z^{μ}}{dτ^{2}} = −R_{αβγ}^{μ}u^{α}Z^{β}u^{γ} ,  (1)

where u = dx/dτ, u · u = −1, is the four-velocity of a free test particle (observer), τ is the proper time, and Z(τ) is the displacement vector. In order to obtain invariant results one sets up a frame {e_{(a)}} along the geodesic. The frame components Z^{(a)}(τ), Z = Z^{(a)}e_{(a)}, are invariant quantities. Choosing e_{(0)} = u and perpendicular space-like unit vectors {e_{(1)}, e_{(2)}, e_{(3)}} in the local hypersurface orthogonal to u, we have e_{(a)} · e_{(b)} ≡ g_{αβ}e_{(a)}^{α}e_{(b)}^{β} = η_{(a)(b)} = diag(−1, 1, 1, 1). The dual basis is e^{(0)} = −u and e^{(i)} = e_{(i)} , i = 1, 2, 3. By projecting (1) onto the frame we get

Z̈^{(i)} = −R_{(0)(j)(0)}^{(i)}Z^{(j)} ,  (2)

where Z^{(j)} = e^{(j)} · Z = e_{μ}^{(j)}Z^{μ} determine directly the distance between close test particles,

Z̈^{(i)} ≡ e^{(i)} · \frac{D^{2}Z}{dτ^{2}} = e_{μ}^{(i)} \frac{D^{2}Z^{μ}}{dτ^{2}}  (3)

are physical relative accelerations, and R_{(i)(0)(j)(0)} = e_{(i)}^{α}u^{β}e_{(j)}^{γ}u^{δ}R_{αβγδ}. Eq. (1) also implies d^{2}Z^{(0)}/dτ^{2} = −u_{μ}D^{2}Z^{μ}/dτ^{2} = R_{μαβγ}u^{μ}u^{α}Z^{β}u^{γ} = 0 so that Z^{(0)} = a_{0}τ + b_{0}, a_{0}, b_{0} are constants. Setting Z^{(0)} = 0, all test particles are “synchronized” by τ (they always stay in the same local hypersurface). From the definition of the Weyl tensor we get R_{(i)(0)(j)(0)} = C_{(i)(0)(j)(0)} + \frac{1}{2}(δ_{ij}R_{(0)(0)} − R_{(i)(j)}) + \frac{1}{6}Rδ_{ij}. Using Einstein’s equations,

R_{(i)(0)(j)(0)} = C_{(i)(0)(j)(0)} − \frac{Λ}{3}δ_{ij} − \frac{κ}{2} [T_{(i)(j)} − δ_{ij}(T_{(0)(0)} + \frac{2}{3}T)] ,  (4)

T = T_{(a)}^{(a)}. Following [8] we introduce null complex tetrad {eâ} = {eˆ1, eˆ2, eˆ3, eˆ4} = {m, m̄, l, k},

m = \frac{1}{\sqrt{2}} (e_{(1)} + ie_{(2)}) , m̄ = \frac{1}{\sqrt{2}} (e_{(1)} − ie_{(2)}) ,  (5)
l = \frac{1}{\sqrt{2}} (u − e_{(3)}) , k = \frac{1}{\sqrt{2}} (u + e_{(3)}) .

Null tetrad components of the Weyl tensor are (see e.g. [8], [9])

Ψ_{0} = C_{αβγδ}k^{α}m^{β}k^{γ}m^{δ} , Ψ_{1} = C_{αβγδ}k^{α}l^{β}k^{γ}m^{δ} ,

Ψ_{2} = \frac{1}{2}C_{αβγδ}k^{α}l^{β}(k^{γ}l^{δ} − m^{γ}m̄^{δ}) ,  (6)
Ψ_{3} = C_{αβγδ}l^{α}k^{β}l^{γ}m̄^{δ} , Ψ_{4} = C_{αβγδ}l^{α}m̄^{β}l^{γ}m̄^{δ} .

Regarding expressions (6) and inverting relations (5) we obtain C_{(1)(0)(1)(0)} = \frac{1}{2}ReΨ_{0} + \frac{1}{2}ReΨ_{4} − ReΨ_{2} , C_{(2)(0)(2)(0)} = −\frac{1}{2}ReΨ_{0} − \frac{1}{2}ReΨ_{4} − ReΨ_{2} ,

C_{(1)(0)(2)(0)} = \frac{1}{2}ImΨ_{0} − \frac{1}{2}ImΨ_{4} , C_{(3)(0)(3)(0)} = 2 ReΨ_{2} ,  (7)
C_{(1)(0)(3)(0)} = −ReΨ_{1} + ReΨ_{3} , C_{(2)(0)(3)(0)} = −ImΨ_{1} − ImΨ_{3} .

Substituting Eqs. (4) and (7) into Eq. (2) we arrive at:

Z̈^{(1)} = \frac{Λ}{3}Z^{(1)} − \frac{κ}{2}(T_{(0)(0)} + \frac{2}{3}T)Z^{(1)} + \frac{κ}{2}T_{(1)(j)}Z^{(j)} + G_{1} ,
Z̈^{(2)} = \frac{Λ}{3}Z^{(2)} − \frac{κ}{2}(T_{(0)(0)} + \frac{2}{3}T)Z^{(2)} + \frac{κ}{2}T_{(2)(j)}Z^{(j)} + G_{2} ,  (8)
Z̈^{(3)} = \frac{Λ}{3}Z^{(3)} − \frac{κ}{2}(T_{(0)(0)} + \frac{2}{3}T)Z^{(3)} + \frac{κ}{2}T_{(3)(j)}Z^{(j)} + G_{3} ,

where

G_{1} ≡ + CZ^{(1)} − (L_{1} − M_{1})Z^{(3)} − (A_{+} + B_{+} )Z^{(1)} + (A_{×} − B_{×})Z^{(2)} ,
G_{2} ≡ + CZ^{(2)} + (L_{2} + M_{2})Z^{(3)} + (A_{+} + B_{+} )Z^{(2)} + (A_{×} − B_{×})Z^{(1)} ,
G_{3} ≡ −2CZ^{(3)} − (L_{1} − M_{1})Z^{(1)} + (L_{2} + M_{2})Z^{(2)} ,

and

C = ReΨ_{2} , L_{1} = ReΨ_{3} , L_{2} = ImΨ_{3} , M_{1} = ReΨ_{1} , M_{2} = ImΨ_{1} ,  (9)
A_{+} = \frac{1}{2}ReΨ_{4} , A_{×} = \frac{1}{2}ImΨ_{4} , B_{+} = \frac{1}{2}ReΨ_{0} , B_{×} = \frac{1}{2}ImΨ_{0} .

Equations (8) are well suited for physical interpretation. The relative motions depend on:

1. the cosmological constant Λ responsible for overall background isotropic motions; 2. the energy-momentum tensor T_{(a)(b)} terms describing interaction with matter-content; 3. the terms depending on the local free gravitational field, and consisting of Coulomb, longitudinal and transverse (outgoing/ingoing) components with amplitudes given by Ψ_{A}’s. In the following we put T_{(a)(b)} = 0. Individual terms in Eq. (8) can be interpreted as follows: Λ-term: Assuming T_{(a)(b)} = 0 = Ψ_{A}, Eq. (8) reduces to

Z̈^{(i)} = \frac{Λ}{3}Z^{(i)} .  (10)

Considering a sphere of test particles each having a position vector Z, Eq. (10) implies that the acceleration of each particle is in the direction Z and has the same magnitude. Assume the frame {e_{(i)}} to be parallelly transported so that Z̈^{(i)} = D^{2}(e^{(i)} · Z)/dτ^{2} = d^{2}Z^{(i)}/dτ^{2}. Eqs. (10) have solutions

Z^{(i)}(τ) = A_{i} τ + B_{i}  for Λ = 0 ,
Z^{(i)}(τ) = A_{i} exp(\sqrt{Λ/3} τ) + B_{i} exp(−\sqrt{Λ/3} τ) for Λ > 0 , (11)
Z^{(i)}(τ) = A_{i} cos(\sqrt{−Λ/3} τ) + B_{i} sin(\sqrt{−Λ/3} τ)  for Λ < 0 ,

where A_{i}, B_{i} are constants. As expected, conformally flat (Ψ_{A} = 0) vacuum backgrounds (Minkowski, de Sitter or anti-de Sitter) are homogeneous and isotropic, so that the relative motion of test particles is isotropic.

Ψ_{4}-term: Assuming Λ = 0 = T_{(a)(b)} and Ψ_{0} = Ψ_{1} = Ψ_{2} = Ψ_{3} = 0, Eq. (8) reduces to

Z̈^{(1)} = −A_{+} Z^{(1)} + A_{×}Z^{(2)} ,
Z̈^{(2)} = A_{+} Z^{(2)} + A_{×}Z^{(1)} ,  (12)
Z̈^{(3)} = 0 ,

which describe the influence of “+” and “×” polarization modes of a transverse gravitational wave with amplitudes A_{+} and A_{×}. If particles, initially at rest, lie in the (e_{(1)}, e_{(2)}) plane, there is no motion in the longitudinal direction of e_{(3)}. The ring of particles is deformed into an ellipse, the axes of different polarizations are shifted one with respect to the other by \frac{π}{4} (such behavior is typical for linearized gravitational waves — cf. e.g. [10]). Making a rotation in the transverse plane by an angle ϑ,

e′_{(1)} = cosϑ e_{(1)} + sinϑ e_{(2)} , e′_{(2)} = − sinϑ e_{(1)} + cosϑ e_{(2)}  (13)

— which corresponds to m′ = e^{−iϑ}m — and using Eqs. (6) and (9), we find

A′_{+} (τ) = cos2ϑA_{+} − sin2ϑA_{×} , A′_{×}(τ) = sin2ϑA_{+} + cos2ϑA_{×} .  (14)

Taking ϑ = ϑ_{+} (τ) = −\frac{1}{2}ArgΨ_{4}, then A′_{+} = \frac{1}{2}|Ψ_{4}|, A′_{×} = 0 — the wave is purely “+” polarized for an observer using m_{+} = e^{−iϑ_{+}} m; if ϑ = ϑ_{×}(τ) = ϑ_{+} + \frac{π}{4}, then A′_{+} = 0, A′_{×} = \frac{1}{2}|Ψ_{4}| — the wave is purely “×” polarized for an observer using m_{×} = e^{−iϑ_{×}}m. The amplitude A = \frac{1}{2}|Ψ_{4}| is invariant under the rotation. A general observer sees a superposition of the two polarization modes shifted by \frac{π}{4}.

One can similarly show [4], [8] that Ψ_{3} and Ψ_{2} terms describe longitudinal modes and Coulombtype effects; Ψ_{1} and Ψ_{0} terms are equivalent to Ψ_{3} and Ψ_{4} terms (if k ↔ l).

For given principal null vector k and observer’s u we have chosen the frame vector e_{(3)} according to Eq. (5), which implies k_{(1)} = 0 = k_{(2)}, k_{(3)} ≠ 0, and makes the physical interpretation based on Eq. (8) simpler. This leads to essentially unique k and l. More precisely, we easily show the following

Proposition 1: Let u be the four-velocity (u · u = −1) and k be the null vector. Then there exists a unit space-like vector e_{(3)} which is the projection of the null direction given by k into the hypersurface orthogonal to u. Such e_{(3)} is unique (up to reflections e_{(3)} → −e_{(3)}) and is given by e_{(3)} = −u + \sqrt{2} k, where k satisfies k · u = −1/\sqrt{2}. Another null vector l in the plane (u, e_{(3)}) such that l · k = −1 is then given by l ≡ \sqrt{2} u − k. The only remaining freedom are rotations in the plane (e_{(1)}, e_{(2)}) perpendicular to e_{(3)}.

In what follows the orthonormal tetrad and the corresponding null frame (5) determined according to Proposition 1 will always be assumed.

Notice that Eqs. (8) represent possible motions seen by an observer with given u. By making the Lorentz boosts to other observers with u′, Ψ_{A} change (see e.g. [8]). Thus, the “strength of gravitational field” is strongly observer dependent (cf. [2], [12] and point 5. in the discussion following Eq. (34)).

III. THE CHOICE OF COORDINATES AND PARALLELLY PROPAGATED FRAMES

We shall now express our frames in coordinates suitable for spacetimes admitting a nontwisting null congruence and give the conditions for the frames to be parallelly transported. The field k is orthogonal to null hypersurfaces, say u =const., so that k^{μ} = g^{μν}u_{,ν}. It is convenient (cf. [9], [13]) to choose as coordinates u = x^{3}, parameter v = x^{0} along the null geodesics generated by k^{μ}, and two complex space-like coordinates ξ = x^{1} and ξ̄ = x^{2} that label the geodesics on each surface u =const. The metric then takes the form

g_{μν} = ( \begin{matrix}0 & 0 & 0 & g_{03} \\ 0 & g_{11} & g_{12} & g_{13} \\ 0 & g_{12} & g_{22} & g_{23} \\ g_{03} & g_{13} & g_{23} & g_{33} \end{matrix} ) ,  (15)

where g_{22} = \overline{g_{11}}, g_{23} = \overline{g_{13}} since x^{2} = \overline{x^{1}}; all other components are real, and

g_{12} > 0 , D = g_{12}^{2} − g_{11}g_{22} > 0 , (16)

since the subspace (ξ, ξ̄) is space-like. The vector k is simply

k^{μ} = (k^{0}, 0, 0, 0) ,  (17)

and the four-velocity u of a particle moving along a geodesic x^{μ}(τ), is given by u^{μ} = (v̇, ξ̇, ˙ξ̄, u̇), where the dot is d/dτ and u̇ ≠ 0 (otherwise the geodesic would not be time-like).

Proposition 2: In coordinates (v, ξ, ξ̄, u) the interpretation null tetrad introduced in Proposition 1 has the form

m^{μ} = (\frac{1}{g_{03}u̇} [(g_{12}ξ̇+ g_{22}˙ξ̄ + g_{23}u̇)g_{+} − (g_{11}ξ̇+ g_{12}˙ξ̄ + g_{13}u̇)g_{−} exp(−iArgg_{11})] ,

g_{−} exp(−iArgg_{11}), −g_{+} , 0) ,

m̄^{μ} = (\frac{1}{g_{03}u̇} [(g_{11}ξ̇+ g_{12}˙ξ̄ + g_{13}u̇)g_{+} − (g_{12}ξ̇+ g_{22}˙ξ̄ + g_{23}u̇)g_{−} exp( iArgg_{11})] ,

−g_{+} , g_{−} exp( iArgg_{11}), 0) ,  (18)

l^{μ} = (\sqrt{2}v̇ + \frac{1}{\sqrt{2}} \frac{1}{g_{03}u̇}, \sqrt{2}ξ̇, \sqrt{2}˙ξ̄, \sqrt{2}u̇) ,

k^{μ} = (−\frac{1}{\sqrt{2}} \frac{1}{g_{03}u̇}, 0, 0, 0) ,

where g_{±} = \sqrt{(g_{12} ± \sqrt{D})/(2D)}. The tetrad is unique up to trivial reflections and rotations m^{μ} → m^{μ}e^{iϑ}. The corresponding orthonormal frame obtained from Eq. (18) using Eq. (5) is

e_{(0)}^{μ} = u^{μ} = (v̇, ξ̇, ˙ξ̄, u̇) ,

e_{(1)}^{μ} = \frac{1}{\sqrt{2}} (\frac{2}{g_{03}u̇} Re \{(g_{12}ξ̇+ g_{22}˙ξ̄ + g_{23}u̇) G_{−}\} , −Ḡ_{−}, −G_{−}, 0) ,

e_{(2)}^{μ} = \frac{1}{\sqrt{2}} (\frac{2}{g_{03}u̇} Im \{(g_{12}ξ̇+ g_{22}˙ξ̄ + g_{23}u̇) G_{+} \} , −iḠ_{+} , iG_{+} , 0) ,  (19)

e_{(3)}^{μ} = − (v̇ + \frac{1}{g_{03}u̇}, ξ̇, ˙ξ̄, u̇) ,

where G_{±} = g_{+} ± g_{−} exp(iArg g_{11}). The expressions (18) and (19) simplify considerably if g_{11} = 0 since in this case g_{−} = 0 and g_{+} = 1/\sqrt{g_{12}} = G_{+} = G_{−}.

Proof: The last equation in (18) follows from Eq. (17) and k · u = −1/\sqrt{2}, the equation for l^{μ} follows from l = \sqrt{2}u − k. Vectors m and m̄ can then be determined from eâ · eb̂ = g_{μν}eμ ; âeν ; b̂ = g_{âb̂}. The conditions m · k = g_{ˆ1ˆ4} = 0 = g_{ˆ2ˆ4} = m̄ · k imply m^{3} = 0 = m̄^{3}, m · l = g_{ˆ1ˆ3} = 0 implies m^{0} = −(l_{1}m^{1} + l_{2}m^{2})/l_{0}. In given coordinates we have m̄^{1} = \overline{m^{2}} and m̄^{2} = \overline{m^{1}}. Denoting X = m^{1} and Y = m^{2} we get m^{μ} = (−(l_{1}X + l_{2}Y)/l_{0}, X, Y, 0) and m̄^{μ} = (−(l_{1}Ȳ + l_{2}X̄)/l_{0}, Ȳ , X̄, 0). Functions X, Y can be determined as solutions of equations m · m = g_{ˆ1ˆ1} = 0 = g_{ˆ2ˆ2} = m̄ · m̄ and m · m̄ = g_{ˆ1ˆ2} = 1,

g_{11}X^{2} + 2g_{12}XY + g_{22}Y^{2} = 0 ,  (20)
g_{11}XȲ + g_{12}(XX̄ + Y Ȳ) + g_{22}X̄Y = 1 .  (21)

(i) Assume g_{11} ≠ 0. Then X ≠ 0, and introducing a complex function C such that Y = CX, Eq. (20) implies C = (−g_{12} ± \sqrt{D})/g_{22}, and Eq. (21) gives XX̄ = |X|^{2} = (g_{12} ± \sqrt{D})/(2D). Since X = |X|e^{iφ}, φ being a real function, we have m^{1} = \sqrt{(g_{12} ± \sqrt{D})/(2D)} exp(i(ϑ − Arg g_{11})), m^{2} = −\sqrt{(g_{12} ∓ \sqrt{D})/(2D)} exp(iϑ), where ϑ = φ + Arg g_{11}. The change from the upper to lower signs accompanied by ϑ → −ϑ + π + Arg g_{11} results just in m ↔ m̄, corresponding to a reflection e_{(2)} ↔ −e_{(2)}. By performing rotation m^{μ} → m^{′μ} = e^{−iϑ}m^{μ} we can write the representatives of m and m̄ given by ϑ = 0 so that we arrive at Eq. (18). (ii) If g_{11} = 0, we simply find m^{1} = 0 and m^{2} = −1/\sqrt{g_{12}}. Hence, the null tetrad has the form (18), and this implies the orthonormal frame (19).

In general, the frames {e_{a}} and {eâ}, related by Eq. (5), are not parallelly transported along the geodesic with tangent u = e_{(0)}. However, they are if Dk/dτ = 0 = Dm/dτ. Starting with an arbitrary m, the second condition can always be satisfied by choosing m_{∥} = e^{iϑ_{∥}} m, where ϑ_{∥} = i ∫_{0}^{τ} m̄ · (Dm/dτ) dτ + ϑ_{0}, ϑ_{0} = const. We thus arrive at

Proposition 3: Consider a geodesic x^{μ}(τ) = (v, ξ, ξ̄, u) in spacetime with metric (15). Then the orthonormal frame {e_{a}} given by Eq. (19) and the null tetrad {eâ} given by Eq. (18) are parallelly transported along the geodesic if

g_{12,0} ξ̇+ g_{22,0} ˙ξ̄ + (g_{23,0} − g_{03,2})u̇ = 0 ,  (22)

and

ϑ̇_{∥}(τ) = \frac{i}{2D}[(G_{1}Ḡ_{1} − G_{2}Ḡ_{2}) E  (23)
+G_{1}(2Dṁ^{1} + m^{1}(g_{12}ġ_{12} − g_{22}ġ_{11}) + m^{2}(g_{12}ġ_{22} − g_{22}ġ_{12}))
+G_{2}(2Dṁ^{2} + m^{1}(g_{12}ġ_{11} − g_{11}ġ_{12}) + m^{2}(g_{12}ġ_{12} − g_{11}ġ_{12}))] ,

where G_{1} = g_{12}g_{−} exp(iArgg_{11})−g_{11}g_{+} , G_{2} = g_{22}g_{−} exp(iArgg_{11})−g_{11}g_{+} , E = (g_{12,1}−g_{11,2})ξ̇+ (g_{22,1} −g_{12,2})˙ξ̄+(g_{23,1} −g_{13,2})u̇ = −Ē, m^{1} = g_{−} exp(−iArgg_{11}), and m^{2} = −g_{+} . If, in addition, g_{11} = 0 then G_{1} = 0, G_{2} = −\sqrt{g_{12}} and Eqs. (22), (23) reduce to

g_{12,0} ξ̇+ (g_{23,0} − g_{03,2})u̇ = 0 ,  (24)

and

ϑ̇_{∥}(τ) = −\frac{i}{2} \frac{1}{g_{12}} [g_{12,1} ξ̇− g_{12,2} ˙ξ̄ + (g_{23,1} − g_{13,2})u̇] .  (25)

Proof: Using k_{μ}k^{μ} = 0 and k_{μ}u^{μ} = −1/\sqrt{2}, it can be shown that k is parallelly transported if Γ_{0α}^{1}u^{α} = 0. Calculating the Christoffel symbols for the metric (15) we find (22). For proving Eq. (23) we use m^{3} = 0 and m̄_{0} = 0, again the condition Γ_{0α}^{1}u^{α} = 0 and other Christoffel symbols for the metric (15).

IV. DEVIATION OF GEODESICS IN THE VACUUM NON-TWISTING TYPE N SPACETIMES WITH COSMOLOGICAL CONSTANT

In this section we apply results given above to the non-twisting type N vacuum spacetimes with non-vanishing Λ. In the preceding paper [1] we showed that all such solutions belong either to the Kundt class of non-expanding gravitational waves which we denoted by symbol KN(Λ), or to the Robinson-Trautman class of expanding gravitational waves RT N(Λ, ϵ).

The class KN(Λ) can be divided into six invariant canonical subclasses KN(Λ)[α, β], and the class RT N(Λ, ϵ) into nine invariant canonical subclasses, as analyzed in detail in [1]. All KN(Λ) metrics can be written in the form of Eq. (1, I), all RT N(Λ, ϵ) are described by Eq. (19, I). Both classes of metrics are of the form (15) in coordinates x^{μ} = (v, ξ, ξ̄, u). In the KN(Λ) class we have

g_{12} = \frac{1}{p^{2}} , g_{03} = −\frac{q^{2}}{p^{2}} , g_{33} = F ,  (26)

where p = 1 + \frac{Λ}{6}ξξ̄, q = (1 − \frac{Λ}{6}ξξ̄)α + β̄ξ + βξ̄, F = κ(q^{2}/p^{2})v^{2} − ((q^{2})_{,u}/p^{2})v − (q/p)H, κ = \frac{Λ}{3}α^{2} + 2ββ̄, H = (f_{,ξ} + f̄_{,ξ̄}) − (Λ/3p)(ξ̄f + ξf̄); and in the RT N(Λ, ϵ) class

g_{12} = v^{2} , g_{13} = vĀ , g_{23} = vA , g_{03} = ψ , g_{33} = 2(AĀ + ψB) ,  (27)

where A = ϵξ − vf, B = −ϵ + \frac{v}{2}(f_{,ξ} + f̄_{,ξ̄}) + \frac{Λ}{6}v^{2}ψ, ψ = 1 + ϵξξ̄, ϵ = −1, 0, +1, respectively.

Hence, for the KN(Λ) solutions the orthonormal frame (19) is given by

e_{(0)}^{μ} = (v̇, ξ̇, ˙ξ̄, u̇) ,
e_{(1)}^{μ} = −\frac{p}{\sqrt{2}} (\frac{2}{q^{2}} \frac{Reξ̇}{u̇}, 1, 1, 0) ,  (28)
e_{(2)}^{μ} = −\frac{p}{\sqrt{2}} (\frac{2}{q^{2}} \frac{Imξ̇}{u̇}, i, −i, 0) ,
e_{(3)}^{μ} = − (v̇ − \frac{p^{2}}{u̇q^{2}}, ξ̇, ˙ξ̄, u̇) ,

and for the RT N(Λ, ϵ) solutions we have

e_{(0)}^{μ} = (v̇, ξ̇, ˙ξ̄, u̇) ,
e_{(1)}^{μ} = \frac{1}{\sqrt{2}} \frac{1}{v} (\frac{2v}{ψu̇}Re\{vξ̇+ Au̇\} , −1, −1, 0) ,  (29)
e_{(2)}^{μ} = \frac{1}{\sqrt{2}} \frac{1}{v} (\frac{2v}{ψu̇}Im\{vξ̇+ Au̇\} , −i, i, 0) ,
e_{(3)}^{μ} = − (v̇ + \frac{1}{ψu̇}, ξ̇, ˙ξ̄, u̇) .

According to Proposition 3 these frames are parallelly transported along time-like geodesics x^{μ}(τ) = (v, ξ, ξ̄, u) in the KN(Λ)[α, β] spacetimes if

(\frac{q}{p})_{,ξ} = 0 = (\frac{q}{p})_{,ξ̄} ,  ϑ̇_{∥}(τ) = i (\frac{p_{,ξ}}{p} ξ̇− \frac{p_{,ξ̄}}{p} ˙ξ̄) ,  (30)

and in the case of RT N(Λ, ϵ) solutions if

ξ̇= fu̇ ,  ˙  ξ̄ = f̄u̇ ,  ϑ̇_{∥}(τ) = \frac{i}{2} (f_{,ξ} − f̄_{,ξ̄}) u̇ .  (31)

The equation of geodesic deviation is now given by Eq. (8) with T_{(a)(b)} = 0. The amplitudes (9) for both classes of spacetimes are calculated in the Appendix. We find that the invariant form of the equation of geodesic deviation with respect to the interpretation frame along any time-like geodesic in the KN(Λ) and RT N(Λ, ϵ) spacetimes takes the form

Z̈^{(1)} = \frac{Λ}{3}Z^{(1)} − A_{+} Z^{(1)} + A_{×}Z^{(2)} ,
Z̈^{(2)} = \frac{Λ}{3}Z^{(2)} + A_{+} Z^{(2)} + A_{×}Z^{(1)} ,  (32)
Z̈^{(3)} = \frac{Λ}{3}Z^{(3)} ,

where the amplitudes of the transverse gravitational wave are given by

A_{+} (τ) = \frac{1}{2}pqu̇^{2} Re \{f_{,ξξξ}\} ,  A_{×}(τ) = \frac{1}{2}pqu̇^{2} Im \{f_{,ξξξ}\} ,  (33)

for the KN(Λ) spacetimes, and by

A_{+} (τ) = −\frac{1}{2} \frac{ψ}{v}u̇^{2} Re \{f_{,ξξξ}\} ,  A_{×}(τ) = −\frac{1}{2} \frac{ψ}{v}u̇^{2} Im \{f_{,ξξξ}\} ,  (34)

in the RT N(Λ, ϵ) spacetimes (see Eqs. (45) and (50) in Appendix). Equations (32)-(34) give relative accelerations of the free test particles in terms of their actual positions. They enable us to draw a number of simple conclusions:

1. All particles move isotropically one with respect to the other according to Eqs. (11) if no gravitational wave is present, i.e., if f_{,ξξξ} = 0. In this case both the KN(Λ) and RT N(Λ, ϵ) spacetimes are vacuum conformally flat (cf. (43), (48)), and therefore Minkowski (Λ = 0), de Sitter (Λ > 0) and anti-de Sitter (Λ < 0) (see Lemma 1 and 3 in [1]). Such spaces are maximally symmetric, homogeneous, isotropic, and they represent a natural background for other “non-trivial” KN(Λ) and RT N(Λ, ϵ) type N solutions.

2. If amplitudes A_{+} and A_{×} do not vanish (f_{,ξξξ} ≠ 0), the particles are influenced by the wave (see Eq. (12) and subsequent discussion) in a similar way as they are affected by a standard gravitational wave on Minkowski background (cf. [10]). However, if Λ ≠ 0, the influence of the wave adds with the (anti-) de Sitter isotropic expansion (contraction). This makes plausible our interpretation of the KN(Λ) and RT N(Λ, ϵ) metrics as exact gravitational waves propagating on the constant curvature backgrounds.

3. The wave propagates in the space-like direction of e_{(3)} and has a transverse character since only motions in the perpendicular directions of e_{(1)} and e_{(2)} are affected. The propagation direction given by e_{(3)} coincides with the projection of the Debever-Penrose vector k on the hypersurface orthogonal to the observer’s velocity u (cf. Proposition 1).

4. There are two polarization modes of the wave — “+” and “×”, A_{+} and A_{×} being the amplitudes. Under rotation (13) in the transverse plane they transform according to Eq. (14) so that the helicity of the wave is 2, as with linearized waves on Minkowski background. For the special choice of the frame given by ϑ(τ) = ϑ_{+} = −\frac{1}{2}pqu̇^{2}Arg\{f_{,ξξξ}\} for the KN(Λ), and by ϑ_{+} = \frac{1}{4}(ψ/v)u̇^{2}Arg\{f_{,ξξξ}\} for the RT N(Λ, ϵ) spacetimes, the observer views pure “+” polarization, and for ϑ_{×} = ϑ_{+} + \frac{π}{4} — pure “×” polarization.

5. The waves have amplitude A = \frac{1}{2}pqu̇^{2} |f_{,ξξξ}| for the KN(Λ) class and A = \frac{1}{2}(ψ/v)u̇^{2} |f_{,ξξξ}| for RT N(Λ, ϵ); this is invariant under rotations (13). However, the amplitude changes under Lorentz transformations to another observer u′ with a spatial velocity v⃗ = (v_{1}, v_{2}, v_{3}) with respect to the original observer. For type N solutions we get A′ = (1 − v_{3})^{2}/(1 − v_{1}^{2} − v_{2}^{2} − v_{3}^{2}) A. By increasing speed in the wave-propagation direction e_{(3)} (v_{1} = v_{2} = 0, v_{3} > 0), he experiences a weakening of the wave amplitude by factor (1 − v_{3})/(1 + v_{3}) (A′ → 0 as v_{3} → 1) , and by moving in the opposite direction, an increase of the amplitude (A′ → ∞ as v_{3} → −1). By increasing speed in the transverse directions e_{(1)}, e_{(2)}, (v_{1}^{2} + v_{2}^{2} ≠ 0, v_{3} = 0), she experiences an increase by the factor 1/(1 − v_{1}^{2} − v_{2}^{2}).

In general, all KN(Λ) spacetimes contain singularities except for the homogeneous ppwaves [8] given by p = 1 = q and f_{,ξξξ} = 6c_{3}(u), where c_{3}(u) is a finite function of u. All other KN(Λ) spacetimes are singular at |ξ| = ∞ where the amplitudes A_{+} , A_{×} diverge. Additional singularities in the amplitudes may occur if the coefficients c_{n}(u), n ≥ 3, of the analytic expansion of function f(ξ, u) are badly behaved at some u.

RT N(Λ, ϵ) spacetimes also contain singularities. The character of the singularities depends on parameter ϵ and on the form of the function f(ξ, u). As follows from Eq. (34), there is always a singularity at v = 0. Another singularity is given by ψ = ∞ which occurs only for ϵ ≠ 0 at |ξ| = ∞. There may be singularities for special forms of f, namely if f_{,ξξξ} = ∞. This occurs at |ξ| = ∞ if f contains the terms c_{n}ξ^{n}, n ≥ 4. Another type of singularities may appear if some of the coefficients c_{n}(u) diverge for some values of u. Singularities might be considered as “sources” of waves; however, it is far from certain whether non-singular sources “covering” the regions in which singularities occur can be constructed. The singularities of the RT N(Λ, ϵ) spacetimes can invariantly be characterized by the non-vanishing invariant constructed recently [14] from the second derivatives of the Riemann tensor.

Finally, we shall discuss a special class of geodesics explicitly. Since for f = f_{c} = c_{0}(u) + c_{1}(u)ξ +c_{2}(u)ξ^{2} the metrics represent Minkowski, de Sitter or anti-de Sitter space there always exists a transformation of coordinates which brings g_{μν}[f = f_{c}] to g_{μν}[f = 0] (see Lemmas 2 and 4 in [1]). It is thus sufficient to consider only the non-trivial part f_{w} ≡ f − f_{c} of function f(ξ, u). Moreover, one can always rearrange its analytic expansion so that f = ∑_{n=0}^{∞} c_{n}(u)ξ^{n} = ∑_{n=0}^{∞} c̃_{n}(u)(ξ − ξ_{0})^{n} = ∑_{n=3}^{∞} c̃_{n}(u)(ξ − ξ_{0})^{n} + f̃_{c}, ξ_{0} being an arbitrary complex constant. Therefore, it is natural to consider structural functions of the form

f_{w} = c_{3}(u)(ξ − ξ_{0})^{3} + c_{4}(u)(ξ − ξ_{0})^{4} + ... .  (35)

Consider a special class of geodesics characterized by ξ = ξ_{0} =const. For the RT N(Λ, ϵ) solutions these are geometrically privileged since the interpretation frame (29) is parallelly propagated along them (Eq. (31) is satisfied). One can also find special geodesics for some subclasses of KN(Λ): ξ = ξ_{0} for the P P subclass, ξ_{0} = ±\sqrt{\frac{6}{Λ}} for KN(Λ)I, and ξ_{0} = 0 for KN(Λ^{−})II. The geodesics ξ = ξ_{0} have the same forms as geodesics in the “background” since Christoffel symbols for f = f_{w} and ξ = ξ_{0} coincide with those for f = 0. However, the test particles feel the tidal forces proportional to A_{+} and A_{×} given by Eqs. (33), (34). The amplitudes do not vanish since f_{w,ξξξ} = 6c_{3}(u) is non-vanishing.

The timelike geodesics ξ = ξ_{0} =const. in the RT N(Λ > 0, ϵ)[f_{w}] spacetimes are given by

v = \frac{α}{1 + ϵξ_{0}ξ̄_{0}} (C_{1} cosh \frac{τ}{α} + C_{2} sinh \frac{τ}{α}) ,  u̇ = − (C_{1} sinh \frac{τ}{α} + C_{2} cosh \frac{τ}{α} + C_{3}) ^{−1} , (36)

where α = \sqrt{3/Λ}, C_{1}, C_{2}, C_{3} are real constants satisfying C_{1}^{2} − C_{2}^{2} + C_{3}^{2} = 2ϵ. The integration of Eq. (36) can be performed explicitly but we do not give it here since only u̇ enters the amplitudes. The wave amplitudes (34) are A_{+} = Re A and A_{×} = Im A where

A(τ) = −\frac{3}{α}(1 + ϵξ_{0}ξ̄_{0})^{2} (C_{1} cosh \frac{τ}{α} + C_{2} sinh \frac{τ}{α}) ^{−1} ×
× (C_{1} sinh \frac{τ}{α} + C_{2} cosh \frac{τ}{α} + C_{3}) ^{−2} c_{3}(u(τ)) .  (37)

As proper time τ along geodesics increases, τ → ∞, particles recede from v = 0 and amplitudes decay as A ∼ exp(−3\sqrt{Λ/3} τ), i.e., waves are damped exponentially. The spacetime locally approaches the de Sitter universe. This is an explicit demonstration of the cosmic no-hair conjecture (see, e.g. [5]) under the presence of waves within exact model spacetimes. (For cosmic no-hair conjecture in the Robinson-Trautman spacetimes of Petrov type II see [6], [7].)

Similarly, for the KN(Λ > 0)I[f_{w}] subclass (representing the only spacetimes of the KN(Λ) type admitting Λ > 0) the geodesics ξ = ξ_{0} = ±\sqrt{6/Λ} are given by

v = C_{1} exp (\frac{τ}{α}) ,  u = −\frac{1}{2C_{1}} exp (−\frac{τ}{α}) + C_{2} ,  (38)

and

v = C_{1} sinh(\frac{τ}{α} + 2τ_{0}) ,  u = \frac{1}{2C_{1}} tanh(\frac{τ}{2α} + τ_{0}) + C_{2} ,  (39)

with C_{1}, C_{2}, τ_{0} constants. For observers moving along these geodesics,

A(τ) = ±12\sqrt{\frac{6}{Λ}}u̇^{2}(τ) c_{3}(u(τ)) .  (40)

After substitution of the explicit dependence of u(τ) we see that as τ → +∞ the amplitudes behave like A ∼ exp(−2\sqrt{Λ/3} τ). Again, gravitational waves are damped exponentially and the cosmic no-hair conjecture is confirmed.

ACKNOWLEDGMENTS

We thank Jerry Griffiths for reading the manuscript and for very helpful comments. We also acknowledge the support of grant No. GACR-202/99/0261 from the Czech Republic.

APPENDIX: GRAVITATIONAL WAVE AMPLITUDES

We calculate amplitudes A_{+} = \frac{1}{2}ReΨ_{4} and A_{×} = \frac{1}{2}ImΨ_{4} by using differential forms. Let {eâ} = {m, m̄, l, k} be a null tetrad, m = eˆ1 = m^{μ}∂_{μ}, m̄ = eˆ2 = m̄^{μ}∂_{μ}, l = eˆ3 = l^{μ}∂_{μ}, k = eˆ4 = k^{μ}∂_{μ}. The dual basis {ω ω ωâ} is given by one-forms ω ω ωˆ1 = m̄_{μ}dx^{μ}, ω ω ωˆ2 = m_{μ}dx^{μ}, ω ω ωˆ3 = −k_{μ}dx^{μ}, ω ω ωˆ4 = −l_{μ}dx^{μ}; the metric is ds^{2} = g_{âb̂} ω ω ωâω ω ωb̂ = 2 ω ω ωˆ1ω ω ωˆ2 − 2 ω ω ωˆ3ω ω ωˆ4 with g_{ˆ1ˆ2} = eˆ1 · eˆ2 = 1 and g_{ˆ3ˆ4} = eˆ3 · eˆ4 = −1. The natural choice of the null basis for the metric KN(Λ) is

ω ω ωˆ1 = \frac{dξ̄}{p} , ω ω ωˆ2 = \frac{dξ}{p} , ω ω ωˆ3 = \frac{q^{2}}{p^{2}}du , ω ω ωˆ4 = dv − \frac{1}{2} \frac{p^{2}}{q^{2}}F du ;  (41)

in coordinates x^{μ} = (v, ξ, ξ̄, u) we have

m^{μ} = (0, 0, p, 0) ,  m̄^{μ} = (0, p, 0, 0) ,  (42)

k^{μ} = (1, 0, 0, 0) , l^{μ} = (\frac{1}{2} \frac{p^{4}}{q^{4}}F, 0, 0, \frac{p^{2}}{q^{2}}) .

The non-vanishing components of the Weyl tensor in this null tetrad are

C_{ˆ3ˆ2ˆ3ˆ2} ≡ Ψ_{4} = \frac{1}{2} \frac{p^{5}}{q^{3}}f_{,ξξξ} = \overline{C_{ˆ3ˆ1ˆ3ˆ1}} .  (43)

The interpretation null tetrad for the KN(Λ) spacetimes, given by Eq. (18), reads

m^{μ} = ( − \frac{p}{q^{2}} \frac{ξ̇}{u̇}, 0, −p, 0) ,  m̄^{μ} = ( − \frac{p}{q^{2}} \frac{˙ξ̄}{u̇}, −p, 0, 0) ,  (44)

k^{μ} = (\frac{1}{\sqrt{2}u̇} \frac{p^{2}}{q^{2}}, 0, 0, 0) , l^{μ} = (\sqrt{2}v̇ − \frac{1}{\sqrt{2}u̇} \frac{p^{2}}{q^{2}}, \sqrt{2}ξ̇, \sqrt{2}˙ξ̄, \sqrt{2}u̇) .

The relation between tetrads (42) and (44) is given by the Lorentz transformation, k_{natur} = Ak_{interp}, l_{natur} = (l_{interp} + Be^{iϑ}m̄_{interp} + B̄e^{−iϑ}m_{interp} + BB̄ k_{interp})/A, m_{natur} = e^{−iϑ}m_{interp} + B k_{interp}, where A = \sqrt{2}u̇q^{2}/p^{2}, B = −\sqrt{2} ξ̇/p, ϑ = π. The coefficients Ψ_{A} transform (see [8])

Ψ_{4}^{interp} = A^{2}Ψ_{4}^{natur} = pqu̇^{2}f_{,ξξξ} ,  (45)
Ψ_{3}^{interp} = Ψ_{2}^{interp} = Ψ_{1}^{interp} = Ψ_{0}^{interp} = 0 .

Similarly, the natural choice of null basis for the RT N(Λ, ϵ) metric is

ω ω ωˆ1 = vdξ̄ + Ādu , ω ω ωˆ2 = vdξ + Adu , ω ω ωˆ3 = ψdu , ω ω ωˆ4 = −dv − Bdu ,  (46)

so that

m^{μ} = (0, 0, \frac{1}{v}, 0) ,  m̄^{μ} = (0, \frac{1}{v}, 0, 0) ,  (47)

k^{μ} = (−1, 0, 0, 0) , l^{μ} = (−\frac{B}{ψ}, −\frac{A}{vψ} , −\frac{Ā}{vψ} , \frac{1}{ψ}) .

For this tetrad we obtain non-vanishing components

C_{ˆ3ˆ2ˆ3ˆ2} ≡ Ψ_{4} = −\frac{1}{2vψ} f_{,ξξξ} = \overline{C_{ˆ3ˆ1ˆ3ˆ1}} .  (48)

The interpretation null tetrad (18) reads

m^{μ} = (\frac{1}{ψu̇}(vξ̇+ Au̇), 0, −\frac{1}{v}, 0) ,  m̄^{μ} = (\frac{1}{ψu̇}(v ˙ξ̄ + Āu̇), −\frac{1}{v}, 0, 0) ,  (49)

k^{μ} = (−\frac{1}{\sqrt{2}} \frac{1}{ψu̇}, 0, 0, 0) , l^{μ} = (\sqrt{2}v̇ + \frac{1}{\sqrt{2}} \frac{1}{ψu̇}, \sqrt{2}ξ̇, \sqrt{2}˙ξ̄, \sqrt{2}u̇) .

The relation between the tetrads (47) and (49) is again given by the Lorentz transformation with A = \sqrt{2} u̇ψ, B = −\sqrt{2}(vξ̇+ Au̇), ϑ = π. We thus get

Ψ_{4}^{interp} = A^{2}Ψ_{4}^{natur} = −\frac{ψ}{v}u̇^{2} f_{,ξξξ} ,  (50)
Ψ_{3}^{interp} = Ψ_{2}^{interp} = Ψ_{1}^{interp} = Ψ_{0}^{interp} = 0 .

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