Nondiffracting gravitational waves
aa r X i v : . [ g r- q c ] A ug Nondiffracting gravitational waves
Felipe A. Asenjo ∗ and Sergio A. Hojman
2, 3, 4, † Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile. Departamento de Ciencias, Facultad de Artes Liberales,Universidad Adolfo Ibáñez, Santiago 7491169, Chile. Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago 7800003, Chile. Centro de Recursos Educativos Avanzados, CREA, Santiago 7500018, Chile.
It is proved that accelerating nondiffracting gravitational Airy wavepackets are solutions of lin-earized gravity. It is also showed that Airy functions are exact solutions of Einstein equations fornon–accelerating nondiffracting gravitational wavepackets.
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Nondiffracting wavepackets are known solutions toSchrödinger equation [1, 2] and Maxwell equations [3, 4],and several experimental studies have been dedicated tothem [3–11]. Nevertheless, studying the implications oftheir existence on gravitational wave propagation seemsto have been neglected so far. To understand its im-portance on gravitational wave behavior, research in thisarea is needed. Furthermore, it is now well established,both theoretically and experimentally, that there existnondiffracting free wavepackets with acceleration fea-tures [1], and it is interesting to consider its existencein gravitational waves propagation.This work is devoted to investigate accelerating andnon–accelerating nondiffracting gravitational waves. Weshow below how accelerating wave packets are solutionsof linearized gravity. Additionally, we explicitly obtainan exact non–accelerating, nondiffracting gravitationalwave solutions that, as far as we know, have not beenconsidered before. This result consists on a new solutionfor the Ehlers-Kundt waves [12, 13].First, let us consider accelerating gravitational wavesthat are solutions to linearized gravity. As usual, letus consider a small perturbation h αβ of a backgroundflat spacetime metric η αβ = ( − , , , , such that thetotal metric is g αβ = η αβ + h αβ . It is straightforward toshow that the trace–reversed metric perturbation h αβ = h αβ − (1 / η αβ h µµ [13], satisfies the free wave equation ∂ µ ∂ µ h αβ = 0 , (1)under the Lorenz gauge ∂ β h βα = 0 .In order to exemplify the accelerating solutions forgravitational waves, let us consider the evolution of po-larization h zz = h zz ( t, x, y ) , i.e., ∂ z h zz = 0 , satisfyingthe Lorenz gauge. Let us assume the following form forsuch polarization h zz ( t, x, y ) = G ( ζ, y ) exp ( ik η ) , (2) ∗ Electronic address: [email protected] † Electronic address: [email protected] where k is an arbitrary constant, ζ = x − t and η = x + t [14]. Thus, as h zz evolves according to Eq. (1), we areable to find that G evolves as ik ∂ G ∂ζ + ∂ G ∂y = 0 , (3)which is the Schrödinger equation for a free particle.From all the possible solutions, it is very well–knownthat Schrödinger equation allows to have an accelerat-ing wavepacket solution [1], which in this case becomes G ( ζ, y ) = Ai (cid:0) k y − k ζ (cid:1) exp (cid:18) ik ζ y − ik ζ (cid:19) , (4)where Ai is an Airy function. Thus, the nondiffracting ac-celerating gravitational wavepacket is the real part of fullsolution (2). This wavepacket has an acceleration equalto k/ , deflecting its trajectory in a parabolic path in the y − ζ plane [3, 15]. Beyond the theoretical importanceof the accelerating wavepackets, which have encompassedseveral decades of research, the most remarkable fact isthat they have been observed [9, 11, 16]. Also, even whenAiry wavepackets have infinite energy (as plane wavesdo), it has been shown how to construct finite energy Airybeams using Gaussian beams [15, 17, 18]. In addition tothe above solution (4), other possible polarizations canpropagate independently, for example, as plane waves as h xy ( t, x ) = exp ( iη ) .Structured wavepackets are not only possible as a lin-earized solution of Einstein equations, but rather can beexact solutions. Nondiffracting gravitational waves canbe obtained in an exact manner from a modification ofEhlers–Kundt solution [12, 13]. Consider the exact met-ric in cartesian coordinates given by the interval ds = − dt + L ( u ) dx + W ( u ) dy + dz , (5)where L and W are arbitrary functions of u = z − t .Einstein equations are satisfied when L d Ldu + 1 W d Wdu = 0 . (6)Solutions of this equation describe exact nondiffractingand non–accelerating gravitational waves.Several exact solutions of Eq. (6) can be found, as forexample, L = cos u , and W = cosh u . However, this doesnot describe appropriately a wave. The simplest non–trivial solution for a structured gravitational wave canbe obtained in terms of Airy functions L ( u ) = L Ai ( u ) ,W ( u ) = W Ai ( − u ) , (7)with arbitrary constants L and W . These nondiffract-ing wavepackets have the property that the intensity ofthe gravitational wave is concentrated in the caustic.Nondiffracting wavepackets have been a fruitful field of study in quantum mechanics [1, 2, 19], electromag-netism and optics [3–11]. The solutions (4) and (7) pre-sented along this work are in the same spirit, and com-plement others structured gravitational beams, as Besselor Laguerre-Gauss, found in linearized general relativ-ity [20, 21]. 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