### 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 [66] (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 [66]). A spacetime metric describing a plane gravitational wave traveling in the direction of the TT coordinate system (with coordinates , , , ), is described by the line element

where and are the two independent polarizations of the wave. We assume that the wave is weak, i.e., for any instant of time ,
We will neglect all terms of order or higher. The form of the line element (1) implies that the functions and describe the wave-induced perturbation of the flat Minkowski metric at the origin of the TT coordinate system (where ). 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),

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

where are spatial coordinates of the th particle and is its proper time. These two particles measure, in their proper reference frames, the frequency of the same photon traveling along a null geodesic , where is some affine parameter. The coordinate time, at which the photon’s frequency is measured by the th particle, is equal to (); we assume that . Let us introduce the coordinate time duration of the photon’s trip and the Euclidean coordinate distance between the particles:
Let us also introduce the 3-vector of unit Euclidean length directed along the line connecting the two particles. We arrange the components of this vector into the column matrix (thus, we distinguish here the 3-vector from its components being the elements of the matrix ; the same 3-vector can be decomposed into components in different spatial coordinate systems):
where the superscript denotes matrix transposition. If one neglects the spacetime curvature caused by the gravitational wave, then , , and are the angles between the path of the photon in the 3-space and the coordinate axis , , or , respectively (obviously, and ). Let us denote the value of the frequency registered by the th particle by () and let us finally define the relative change of the photon’s frequencies,
Then, it can be shown (see Chapter 5 of [66] for details) that the frequency ratio can be written [making use of the quantities introduced in Eqs. (3) and (5) – (6)] as follows (the dot means here matrix multiplication):

It is convenient to introduce the unit 3-vector 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 direction. Therefore, the components of the 3-vector , arranged into the column matrix , are

The positions of the particles with respect to the origin of the coordinate system we describe by the 3-vectors (), the components of which we put into the column matrices :
Making use of Eqs. (9) – (10) we rewrite the basic formula (8) in the following form

To obtain the response for all currently working and planned detectors it is enough to consider a configuration of three particles shown in Figure 1. 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 1 we have introduced the following notation: O denotes the origin of the TT coordinate system related to the passing gravitational wave, () are 3-vectors joining O and the particles, and () are, respectively, 3-vectors of unit Euclidean length along the lines joining the particles and the coordinate Euclidean distances between the particles, where is the label of the opposite particle. We still assume that the spatial coordinates of the particles do not change in time.

Let us denote by 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 at the moment towards the particle 2, which registers the photon with frequency at the moment . The photon is immediately transponded (without change of frequency) back to the particle 1, which registers the photon with frequency at the moment . We express the relative changes of the photon’s frequency and as functions of the instant of time . Making use of Eq. (11) we obtain

The total frequency shift of the photon during its round trip can be computed from the one-way frequency shifts and given above: