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
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
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
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:
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Living Rev. Relativity 15, (2012), 4
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