SMOOTH SANDWICH GRAVITATIONAL WAVES

J. Podolský ^{1}

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

Abstract

Gravitational waves which are smooth and contain two asymptotically flat regions are constructed from the homogeneous pp wave vacuum solution. Motion of free test particles is calculated explicitly and the limit to an impulsive wave is also considered.

PACS numbers: 04.30.-w, 04.20.Jb, 98.80.Hw

The widely known class of pp waves [1] is characterized by the existence of a quadruple Debever-Penrose null vector field which is covariantly constant. In vacuum the metric can be written as

ds^{2} = 2 dζdζ̄ − 2 dudv − (f + f̄) du^{2} ,  (1)

where f(u, ζ) is an arbitrary function of u, analytic in ζ. The simplest case for which the metric describes gravitational waves arises when f is of the form

f(u, ζ) = d(u)ζ^{2} ,  (2)

where d(u) is an arbitrary function of u; such solutions are called homogeneous pp waves (or “plane” gravitational waves). Performing the transformation [2]

ζ = \frac{1}{\sqrt{2}} (P x + iQy) ,  v = \frac{1}{2} (t + z + P P′x^{2} + QQ′y^{2}) ,  u = t − z , (3)

where real functions P (u) ≡ P (t − z), Q(u) ≡ Q(t − z) are solutions of differential equations

P′′ + d(u) P = 0 ,  Q′′ − d(u) Q = 0 ,  (4)

1E-mail: podolsky@mbox.troja.mff.cuni.cz

(here prime denotes the derivative with respect to u) the metric can be written in the (Rosen) form

ds^{2} = −dt^{2} + P^{2}dx^{2} + Q^{2}dy^{2} + dz^{2} ,  (5)

which is more suitable for physical interpretation. Considering privileged free test particles standing at fixed x, y and z, their relative motion in the x-direction is given by the function P (u) while it is given by Q(u) in the y-direction, namely Δl_{x} = P Δx and Δl_{y} = QΔy. The motions are unaffected in the z-direction which demonstrate transversality of the waves.

Assuming “profile” functions d(u) non-vanishing on some finite interval of u only, sandwich gravitational waves were constructed in [3]-[6] and elsewhere. Here we consider the function d(u) of the form

d(u) = \frac{a}{cosh^{2}(bu)} ,  (6)

where a and b are arbitrary real positive constants. Since all derivatives of d(u) are continuous the corresponding gravitational waves are smooth, contrary to “standard” sandwich waves explicitly presented in literature. On the other hand, the space-times (1) given by Eqs. (2) and (6) do not contain flat regions in front of the wave and behind it. They are curved everywhere, becoming flat only asymptotically as u → ±∞. Therefore, we should call them “smooth asymptotic sandwich waves”.

In order to find the form (5) of the metric exhibiting naturally the particle motions we have to solve the equations (4). Introducing a substitution

ξ = − tanh(bu) ,  (7)

the equations take the form

(1 − ξ^{2})\frac{d^{2}R}{dξ^{2}} − 2ξ \frac{dR}{dξ} ± \frac{a}{b^{2}} R = 0 ,  (8)

where the upper sign is applied for R = P and the lower sign for R = Q. Clearly, a general solution can be written as a linear combination of the Legendre functions of the first kind P_{α}(ξ) and the second kind Q_{α}(ξ) where α = (\sqrt{1 ± 4a/b^{2}} − 1)/2. It is natural to impose the conditions R(u → −∞) → 1 and R′(u → −∞) → 0 so that the metric (5) is written in explicit Minkowski form in the asymptotic region where u → −∞. These conditions are satisfied by particular solutions of the form

P (u) = P_{μ}(ξ(u)) ,  Q(u) = P_{ν}(ξ(u)) ,  (9)

where

μ = (\sqrt{1 + 4a/b^{2}} − 1)/2 ,  ν = (\sqrt{1 − 4a/b^{2}} − 1)/2 .  (10)

Typical behavior of the functions P, Q is shown in Fig. 1. It can be observed that in both asymptotic regions u → ±∞ the particles move uniformly. This can be shown analyticly using the definition P_{α}(ξ) ≡ F (−α, α + 1; 1; (1 − ξ)/2) where F is the hypergeometric function. Clearly, P → 1, Q → 1 as u → −∞. For u → +∞ (corresponding to ξ → −1) we use the identity 15.3.10. in [7] according to which

F ( − α, α + 1; 1; \frac{1 − ξ}{2} ) = −\frac{sin πα}{π} ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(n!)^{2}}  (11)

×[2ψ(n + 1) − ψ(n − α) − ψ(α + n + 1) − ln (\frac{1 + ξ}{2} )](\frac{1 + ξ}{2} )^{n} ,

where ψ is the digamma function. Using also the identity ψ(−α) = ψ(1 − α) + 1/α and the limit ln[(1 + ξ)/2] → −2bu for u → +∞ we conclude that

P (u → +∞) = C_{1} + C_{2}u ,  Q(u → +∞) = D_{1} + D_{2}u ,  (12)

where

C_{1} = \frac{sin πμ}{π}[\frac{1}{μ} + ψ(1 + μ) + ψ(1 − μ) − 2ψ(1)] , C_{2} = −\frac{2b}{π} sin πμ ,

D_{1} = \frac{sin πν}{π} [\frac{1}{ν} + ψ(1 + ν) + ψ(1 − ν) − 2ψ(1)] , D_{2} = −\frac{2b}{π} sin πν .(13)

For particular values of μ corresponding to μ = m = 0, 1, 2, ⋯, the constant C_{2} vanishes so that P (u → +∞) = C_{1}. These cases coincide with solutions P (u) given by Legendre polynomials P_{m}(ξ) and we easily get C_{1} = P_{m}(ξ = −1) = (−1)^{m}. Explicit particular solutions of this type arise when a/b^{2} = m(m + 1) so that P (u) = − tanh(\sqrt{a/2} u) for m = 1, P (u) = [3 tanh^{2}(\sqrt{a/6} u) − 1]/2 for m = 2, etc. Note also that there are no analogous solutions for Q(u) since (10) admits ν ∈ ⟨−1, 0⟩ only. Therefore, the only non-negative integer (for which the constant D_{2} vanishes) is ν = 0 representing a trivial case d(u) = 0. For all other values of ν the constant D_{2} is positive so that Q(u) → +∞ as u → +∞.

We can also use the above results to construct impulsive gravitational waves. The sequence of smooth functions (6) approach the Dirac δ-function (in a distributional sense) as a → ∞ if the second parameter is b = 2a (so that the normalization condition ∫_{−∞}^{+∞} d(u) du = 1 holds for arbitrary a). Considering this limit, indicated also in Fig. 1, α = (\sqrt{1 ± 1/a} − 1)/2 → 0 and using Eqs. (13) we get

P (u) = 1 − u Θ(u) ,  Q(u) = 1 + u Θ(u) ,  (14)

where Θ is the Heaviside step function (Θ = 0 for u < 0, Θ = 1 for u > 0). Therefore, particle motion in the impulsive gravitational wave is exactly the same as in the limit of “standard” sandwich waves [2]. This is just an explicit demonstration of recent general result [8] according to which the distributional limit of a geodesic motion in impulsive pp waves is independent of the specific form of (smooth) regularization. In fact, the sequence d_{ε}(u) of smooth asymptotic sandwich waves given by Eq. (6) with b = 2a = 1/ε is a simple explicit example of a regularization sequence within the largest “reasonble” class of smooth regularizations of the Dirac δ-distribution introduced in [9].

In general, sandwich pp waves may serve as (local) exact toy models of gravitational radiation far away from a finite source. However, the class of explicit solutions discussed in the paper is more interesting from purely theoretical point of view due to its specific feature that the corresponding waves have smooth profiles.

Acknowledgments

I acknowledge the support of grants No. GACR-202/96/0206 and No. GAUK-230/1996 from the Czech Republic and Charles University.

References

[1] D. Kramer, H. Stephani, M.A.H. MacCallum, and H. Herlt: Exact Solutions of the Einstein’s Field Equations, Cambridge University Press, Cambridge, 1980.

[2] R. Penrose: in General Relativity (Ed. L. O’Raifeartaigh), Clarendon Press, Oxford, 1972.

[3] H. Bondi, F.A.E. Pirani, and I. Robinson: Proc. Roy. Soc. A 251 (1959) 519.

[4] H. Bondi and F.A.E. Pirani: Proc. Roy. Soc. A 421 (1989) 395.

[5] W. Rindler: Essential Relativity, Springer, New York, 1977.

[6] J. Podolský and K. Veselý: Czech. J. Phys. 48 (1998) 871.

[7] M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions, Dover Publications, New York, 1964.

[8] R. Steinbauer: J. Math. Phys. 39 (1998) 2201.

[9] M. Kunzinger and R. Steinbauer: preprint gr-qc/9806009 (1998).

Figure Caption

Fig. 1. Typical behavior of the functions P (u) and Q(u) determining relative motion of free test particles (initially at rest) in x and y-directions, respectively, caused by smooth sandwich gravitational waves for different values of a = \frac{1}{48}, \frac{2}{48}, ⋯, \frac{12}{48}, with b = 2a. The values a = \frac{1}{8}, \frac{1}{24}, \frac{1}{48} correspond to μ = m = 1, 2, 3, respectively, for which P → (−1)^{m} as u → +∞.