c Pleiades Publishing, Ltd., 2008. ISSN 0202-2893, Gravitation and Cosmology, 2008, Vol. 14, No. 3, pp. 226–229.


Gravitational Lensing by Gravitational Waves G. S. Bisnovatyi-Kogan1, 2, 3* and O. Yu. Tsupko1, 3** 1

Space Research Institute of Russian Academy of Science, Profsoyuznaya 84/32, Moscow 117997, Russia 2 Joint Institute for Nuclear Research, Dubna, Russia 3 Moscow Engineering Physics Institute, Moscow, Russia Received January 18, 2008

Abstract—Gravitational lensing by a gravitational wave is considered. We notice that, although the final and initial directions of photons coincide, there occurs a displacement between the final and initial trajectories. This displacement is calculated analytically for a plane gravitational wave pulse. Observational estimates are discussed. PACS numbers: 04., 95.30.-k, 95.30.Sf DOI: 10.1134/S0202289308030031

1. INTRODUCTION According to general relativity, any gravitational field can change trajectories of photons or, in other words, deflect light rays. Hence the gravitational field may act as a gravitational lens. Gravitational lensing by gravitational waves was considered in different cases by many authors (see [1– 3] and references therein). It was found that the deflection angle vanishes for any localized gravitational wave packet because of the transversality of gravitational waves [3]. Thus if the photon passes through a finite gravitational wave pulse, its deflection due to this wave is zero. Nevertheless we notice that a displacement between the trajectories of the photon before and after passing the wave may occur. In this work, we confirm analytically the vanishing of a deflection angle for plane wave pulses. However, we find that a gravitational wave (GW) changes the photon propagation in another way, simply shifting its whole trajectory after passing through the GW (see Figs. 1, 2). This displacement is found analytically for a photon passing through a plane GW. On the basis of this result, we obtain an approximate formula for an estimate of observational effects. 2. THE DEFLECTION ANGLE The photon wave vector ki (tangent to the trajectory) is, by definition, ki = dxi /dλ, where λ is a parameter changing along the trajectory [4]. The geodesic equation is written as Dki = 0 or Dki = 0, * **

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where D denotes a covariant derivative. It is more convenient to use the second expression. After some transformation, 
∂ki k − Γil kk dxl = 0, ∂xl dki dxl ∂ki dxl = Γkil kk , = Γk,il kk kl , l ∂x dλ dλ dλ
 1 ∂gki ∂gkl ∂gil dki = + − k kk kl , dλ 2 ∂xl ∂xi ∂x

(1)

we obtain the photon equation of motion 1 1 ∂gkl ∂gkl dki = kk kl i , or x ¨i = x˙ k x˙ l i , (2) dλ 2 ∂x 2 ∂x where the dot denotes a derivative with respect to the parameter λ. Consider a gravitational wave in flat space with the metric gik = ηik + hik , hik
1, where ηik is the flat metric (−1, 1, 1, 1) and hik is a small perturbation (gravitational wave). In this approximation, one can integrate Eq. (2) by calculating its right-hand side with using the unperturbed trajectory of the photon. Performing integration, we obtain the following expression for the deflection angle (compare with [3]): +∞  ki (+∞) − ki (−∞) 1 k l ∂hkl = k k dλ, (3) α ˆi = k 2 ∂xi −∞

where hik is calculated along the straight path and ki = const along this unperturbed trajectory. Consider a photon moving along the z axis. Its unperturbed trajectory is z = z0 + ct, and we can use 226

GRAVITATIONAL LENSING BY GRAVITATIONAL WAVES

the coordinate z as the parameter λ. Then the wave vector is ki = (1, 0, 0, 1), k = 1. When the photon passes through a finite wave packet, we denote the z coordinate of the photon entrance into the wave front as z1 and the z coordinate of its exit from the wave front as z2 (z1 < z2 ). Hence we have an expression for the deflection angle in the form (compare with [2]) α ˆi =

1 2

z2

z' K '

t=0 z0 θ

y' x, x '

y

∂ (h00 + 2h03 + h33 )dz. ∂xi

227

(4)

z1

Fig. 1. Lensing of a photon by a plane GW pulse. In the initial state, the photon moves along the z axis in the reference frame K. The gravitational wave packet moves along the z  axis in the reference frame K  . The reference frame K transforms into the frame K  via rotation by the angle ϕ = −(π − θ) around the axis x (a positive rotation is anticlockwise).

z0

z' K'

K

e

wav

ξ  = 0...π, kg = ω/c,

(5)

where ω and kg are the frequency and the wave vector of the GW in K  , respectively. The pulse width δ (in space) is δ = cπ/ω. It is convenient to use dimensionless variables for ˜ = x/x0 , time, t˜ = t/t0 , t0 = 1/ω and distances, x x0 = c/ω. Hereafter we omit the tildes for simplicity. In dimensionless variables, the equation of motion (2) looks the same, and the GW form is written as sin(t − z  ). The right-hand side of (4) includes the components h00 , h03 and h33 of the gravitational perturbation in the reference frame K. The GW moves along the axis Oz  in the reference frame K  , therefore the GW has non-zero components h11 , h12 , h21 , h22 only. The reference frame K transforms into K  by rotation by the angle ϕ = −(π − θ) around the x axis (see Fig. 1). Hence we have h00 = h03 = 0,

K

e

ξ  = ωt − kg z  ,

z

wav

hik ∝ sin ξ  ,

e wav

Let us calculate the photon deflection by a plane GW pulse. Consider a light ray propagating at the angle ϕ = −(π − θ) relative to the direction of the plane GW packet propagation (see Fig. 1). Let us define, for convenience, two reference frames K and K  . The first one is connected with the direction of the light ray: the photon moves along the z axis in a positive direction in the frame K. The second one is connected with the propagation direction of the GW pulse. The GW packet moves along the z  axis in the positive direction in the reference frame K  (see Fig. 1). The reference frames are at rest relative to each other, and their origins coincide. At the initial time t = 0, the photon is situated at z0 < 0 (x0 = y0 = 0), the wave vector of the photon is ki = (k0 , 0, 0, kz ) = (1, 0, 0, 1), k = kz . The shape of the wave pulse is sinusoidal (the upper part of the sinusoid, with the phase changing from 0 to π, and with zero perturbations at the boundaries):

h33 = sin2 ϕh22 .

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y

∆y

y'

x, x ' z

Fig. 2. Passing of a photon through a GW. The positions of the wave packet at the time of the photon entrance and exit are shown by a solid line and a dashed line, respectively.

Writing h22 as h22 = h sin(t − z  ), where h is the wave amplitude, we obtain h33 = h sin2 θ sin(t + z cos θ − y sin θ).

(7)

Taking into account that the straight line ray has the trajectory z = z0 + t, one can find the intersection points of the photon and the wave front. The entrance

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point is z1 , the exit point is z2 , the point of the perturbation maximum is zm (in the frame K, the GW moves in the negative direction of the z axis, therefore we have z1 < zm < z2 ): z0 π + z0 , z2 = , z1 = 1 + cos θ 1 + cos θ π/2 + z0 . (8) zm = 1 + cos θ Due to the symmetry, the deflection may only occur in the (zy) plane: αy . Let us define Fy (z) as  1 ∂ 2 (sin(z − z0 Fy (z) = h sin θ 2 ∂y   ∂ϕy , + z cos θ − y sin θ))  = ∂y y=0 1 h sin2 θ sin(z − z0 + z cos θ − y sin θ), (9) 2 and ϕy = 0 outside the GW pulse. Then the deflection angle in the first part of the way inside the wave is zm 1 sin3 θ α1 = Fy (z)dz = − h 2 1 + cos θ ϕy =

z1

1 (10) = − h(1 − cos θ) sin θ. 2 The deflection angle in the second part of the way inside the wave is z2 1 sin3 θ α2 = Fy (z)dz = h 2 1 + cos θ zm

1 = h(1 − cos θ) sin θ. 2 And the total deflection angle is: z2 α ˆ = Fy (z)dz = 0.

(11)

(12)

z1

The upper half of the sinusoid is symmetric relative to the vertical axis. We also considered a nonsymmetric plane waveform and obtained numerically that the deflection angle vanishes in this case as well. We also checked the vanishing for different photon velocities (in a medium where the photon has a velocity smaller than c). 3. THE DISPLACEMENT Let us find the displacement analytically for a plane GW pulse. To find the displacement we need to

further integrate the equation of motion (4), which gives: z y(z) = z1

dz 

z 

Fy (z  )dz  ,

y(z1 ) = 0.

(13)

z1

We obtain for the trajectory y(z) of the photon inside the wave (z1 < z < z2 ): 1 (−1 + cos(z − z0 + z cos θ)) sin3 θ , y(z) = h 2 (1 + cos θ)2 (14) y(z1 ) = 0. We see that the photon trajectory has a sinusoidal form inside the wave. Using (14), we see that the total displacement along the y axis does not vanish and amounts to ∆y = y(z2 ) − y(z1 ) = −h = −h

sin3 θ (1 + cos θ)2

1 − cos θ sin θ. 1 + cos θ

(15)

Thus, although the initial and final directions of the photon coincide and the deflection angle vanishes, there occurs a displacement of the trajectory. This displacement is absent in the case θ = 0 (the photon and GW directions are parallel) and reaches its maximum when θ = π/2 (the photon and GW directions are orthogonal). It is clear that this displacement will be zero if we consider the whole sinusoid with the upper and lower parts since the displacement due to the upper half of the sinusoid will be cancelled by that due to the lower part. Therefore this displacement takes place mainly in the case of isolated wave pulses which have shapes similar to the upper half of the sinusoid, or when it has non-symmetric upper and lower parts of wave profile, and it may, in principle, vanish for a periodic wave of long duration. The wave pulses may be produced, for example, during stellar collapse (see [5], the paper contains many figures with waveforms) or in the course of formation of largescale structure of the Universe (see [6]). We have calculated the displacement in the dimensionless variables. In dimensional variables we have ∆y = −h

sin3 θ δ . π (1 + cos θ)2

(16)

Let us estimate the observational effects caused by this displacement. GRAVITATION AND COSMOLOGY Vol. 14

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GRAVITATIONAL LENSING BY GRAVITATIONAL WAVES

change due to passing of the light ray through a GW pulse, ∆αd , is (see Fig. 3)

Source

∆αd = ∆y

∆αd

229

∆y hδ  , Ds Ds

(17)

where h is the amplitude of the GW pulse, δ is its thickness, and Ds is the distance between the source and the observer. Let us estimate the angular position change for GW pulses produced during formation of the largescale structure of the Universe in dark matter (see [6]). For these estimates we put h = 10−11 , δ = 1 Mpc, Ds = 100 Mpc, then we obtain

Ds

∆αd  2 × 10−8 arcsec.

(18)

ACKNOWLEDGMENTS The authors thank M. Barkov for useful discussions. This work was partly supported by RFBR grants 08-02-00491 and 08-02-90106, the RAN Program “Formation and evolution of stars and galaxies” and Grant for Leading Scientific Schools NSh-2977.2008.2. The work of O.Yu. Tsupko was also partly supported by the Dynasty Foundation.

Observer Fig. 3. The observational effect of the displacement in the photon trajectory.

REFERENCES

4. OBSERVATIONAL EFFECTS OF THE DISPLACEMENT The directions of photons passing through a GW packet does not change, therefore there is no focusing of rays in this case. Thus the displacement of trajectories does not lead to any magnification effect. But the displacement leads to a change in the angular position of an object for a distant observer. This

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