2.1 Doppler shift between freely falling particles

We start by describing the change of a photon’s frequency caused by a passing gravitational wave and registered by particles (representing different parts of a gravitational-wave detector) freely falling in the field of the gravitational wave. The detailed derivation of the formulae we show here can be found in Chapter 5 of [66Jump To The Next Citation Point] (see also [47, 15, 120]). An equivalent derivation of the response of test masses to gravitational waves in the local Lorentz gauge (without making use of the long-wavelength approximation) is given in [113].

We employ here the transverse traceless (TT) coordinate system (more about the TT gauge can be found, e.g., in Section 35.4 of [91] or in Section 1.3 of [66Jump To The Next Citation Point]). A spacetime metric describing a plane gravitational wave traveling in the +z direction of the TT coordinate system (with coordinates x0 ≡ ct, x1 ≡ x, x2 ≡ y, x3 ≡ z), is described by the line element

( ( )) ( ( )) ( ) ds2 = − c2dt2 + 1 + h+ t − z dx2 + 1 − h+ t − z dy2 + 2h× t − z- dxdy + dz2, (1 ) c c c
where h + and h × are the two independent polarizations of the wave. We assume that the wave is weak, i.e., for any instant of time t,
|h+(t)| ≪ 1, |h× (t)| ≪ 1. (2 )
We will neglect all terms of order h2 or higher. The form of the line element (1View Equation) implies that the functions h+ (t) and h ×(t) describe the wave-induced perturbation of the flat Minkowski metric at the origin of the TT coordinate system (where x = y = z = 0). It is convenient to introduce the three-dimensional matrix of the spatial metric perturbation produced by the gravitational wave (at the coordinate system’s origin),
( ) h+ (t) h×(t) 0 | h (t) − h (t) 0| H (t) := |( × + |) . (3 ) 0 0 0

Let two particles freely fall in the field (1View Equation) of the gravitational wave, and let their spatial coordinates remain constant, so the particles’ world lines are described by equations

t(τa) = τa, x (τa) = xa, y(τa) = ya, z(τa) = za, a = 1,2, (4 )
where (xa,ya,za) are spatial coordinates of the ath particle and τa is its proper time. These two particles measure, in their proper reference frames, the frequency of the same photon traveling along a null geodesic α α x = x (λ), where λ is some affine parameter. The coordinate time, at which the photon’s frequency is measured by the ath particle, is equal to ta (a = 1,2); we assume that t2 > t1. Let us introduce the coordinate time duration t12 of the photon’s trip and the Euclidean coordinate distance L12 between the particles:
∘ ---------2-----------2-----------2- t12 := t2 − t1, L12 := (x2 − x1) + (y2 − y1) + (z2 − z1) . (5 )
Let us also introduce the 3-vector n of unit Euclidean length directed along the line connecting the two particles. We arrange the components of this vector into the column 3 × 1 matrix n (thus, we distinguish here the 3-vector n from its components being the elements of the matrix n; the same 3-vector can be decomposed into components in different spatial coordinate systems):
( cosα) T ( ) n := (cosα, cosβ,cos γ) = cosβ , (6 ) cosγ
where the superscript T denotes matrix transposition. If one neglects the spacetime curvature caused by the gravitational wave, then α, β, and γ are the angles between the path of the photon in the 3-space and the coordinate axis x, y, or z, respectively (obviously, α,β, γ ∈ ⟨0;π⟩ and cos2 α + cos2β + cos2γ = 1). Let us denote the value of the frequency registered by the ath particle by νa (a = 1,2) and let us finally define the relative change of the photon’s frequencies,
y := ν2-− 1. (7 ) 12 ν1
Then, it can be shown (see Chapter 5 of [66Jump To The Next Citation Point] for details) that the frequency ratio y12 can be written [making use of the quantities introduced in Eqs. (3View Equation) and (5View Equation) – (6View Equation)] as follows (the dot means here matrix multiplication):
( ( ) ( ) ) ( ) y12 = ------1-----nT ⋅ H t1 − z1 − H t1 − z1 + (1 − cosγ )L12- ⋅ n + 𝒪 h2 . (8 ) 2(1 − cos γ) c c c

It is convenient to introduce the unit 3-vector k directed from the origin of the coordinate system to the source of the gravitational wave. In the coordinate system adopted by us the wave is traveling in the +z direction. Therefore, the components of the 3-vector k, arranged into the column matrix k, are

T k = (0,0,− 1) . (9 )
The positions of the particles with respect to the origin of the coordinate system we describe by the 3-vectors xa (a = 1,2), the components of which we put into the column matrices xa:
T xa = (xa,ya,za) , a = 1,2. (10 )
Making use of Eqs. (9View Equation) – (10View Equation) we rewrite the basic formula (8View Equation) in the following form
( ) T ( kT ⋅ x1) ( L12 kT ⋅ x2) n ⋅ H t1 + ------ − H t1 + ----+ ------ ⋅ n ( ) y12 = ----------------c----------------c------c--------- + 𝒪 h2 . (11 ) 2(1 + kT ⋅ n)

To obtain the response for all currently working and planned detectors it is enough to consider a configuration of three particles shown in Figure 1View Image. Two particles model a Doppler tracking experiment, where one particle is the Earth and the other is a distant spacecraft. Three particles model a ground-based laser interferometer, where the masses are suspended from seismically-isolated supports or a space-borne interferometer, where the three test masses are shielded in satellites driven by drag-free control systems. In Figure 1View Image we have introduced the following notation: O denotes the origin of the TT coordinate system related to the passing gravitational wave, xa (a = 1, 2,3) are 3-vectors joining O and the particles, na and La (a = 1,2,3) are, respectively, 3-vectors of unit Euclidean length along the lines joining the particles and the coordinate Euclidean distances between the particles, where a is the label of the opposite particle. We still assume that the spatial coordinates of the particles do not change in time.

View Image

Figure 1: Schematic configuration of three freely-falling particles as a detector of gravitational waves. The particles are labelled 1, 2, and 3, their positions with respect to the origin O of the coordinate system are given by 3-vectors xa (a = 1,2, 3). The Euclidean separations between the particles are denoted by La, where the index a corresponds to the opposite particle. The unit 3-vectors na point between pairs of particles, with the orientation indicated.

Let us denote by ν0 the frequency of the coherent beam used in the detector (laser light in the case of an interferometer and radio waves in the case of Doppler tracking). Let the particle 1 emit the photon with frequency ν0 at the moment t0 towards the particle 2, which registers the photon with frequency ν ′ at the moment t′ = t + L ∕c + 𝒪 (h) 0 3. The photon is immediately transponded (without change of frequency) back to the particle 1, which registers the photon with frequency ν at the moment t = t0 + 2L3∕c + 𝒪 (h). We express the relative changes of the photon’s frequency ′ y12 := (ν − ν0)∕ν0 and y21 := (ν − ν′)∕ν′ as functions of the instant of time t. Making use of Eq. (11View Equation) we obtain

1 ( ( 2L3 kT ⋅ x1) ( L3 kT ⋅ x2)) ( ) y12(t) = -------T-----nT3 ⋅ H t − ----+ ------ − H t −--- + ------ ⋅ n3 + 𝒪 h2 , (12a ) 2(1 − k ⋅ n3) ( c c c )c 1 T ( L3 kT ⋅ x2) ( kT ⋅ x1) ( 2) y21(t) = 2(1-+-kT-⋅ n-)n3 ⋅ H t − c--+ ---c-- − H t + ---c-- ⋅ n3 + 𝒪 h . (12b ) 3

The total frequency shift y := (ν − ν )∕ν 121 0 0 of the photon during its round trip can be computed from the one-way frequency shifts y12 and y21 given above:

′ y = -ν-− 1 = ν-ν--− 1 = (y + 1 )(y + 1) − 1 = y + y + 𝒪 (h2). (13 ) 121 ν0 ν′ν0 21 12 12 21

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