2.3 Solar-system-based detectors
Real gravitational-wave detectors do not stay at rest with respect to the TT coordinate system related
to the passing gravitational wave, because they also move in the gravitational field of the solar system
bodies, as in the case of the LISA spacecraft, or are fixed to the surface of the Earth, as in the case of
Earth-based laser interferometers or resonant bar detectors. Let us choose the origin O of the TT coordinate
system employed in Section 2.1 to coincide with the solar system barycenter (SSB). The motion of the
detector with respect to the SSB will modulate the gravitational-wave signal registered by the detector. One
can show that as far as the velocities of the particles (modeling the detector’s parts) with respect to the
SSB are non-relativistic, which is the case for all existing or planned detectors, Eqs. (12) can
still be used, provided the 3-vectors
and
(
) will be interpreted as made
of the time-dependent components describing the motion of the particles with respect to the
SSB.
It is often convenient to introduce the proper reference frame of the detector with coordinates
.
Because the motion of this frame with respect to the SSB is non-relativistic, we can assume that the
transformation between the SSB-related coordinates
and the detector’s proper reference frame
coordinates
has the form
where the functions
describe the motion of the origin
of the proper reference frame with
respect to the SSB, and the functions
account for the different orientations of the spatial axes of the
two reference frames. One can compute some of the quantities entering Eqs. (12) in the detector’s
coordinates rather than in the TT coordinates. For instance, the matrix
of the wave-induced
spatial metric perturbation in the detector’s coordinates is related to the matrix
of the
spatial metric perturbation produced by the wave in the TT coordinate system through equation
where the matrix
has elements
. If the transformation matrix
is orthogonal, then
,
and Eq. (19) simplifies to
See [31, 54, 68
, 78
] for more details.
For a ground-based laser-interferometric detector, the long-wavelength approximation can be employed
(however, see [27, 115, 114] for a discussion of importance of high-frequency corrections, which modify
the interferometer response function computed within the long-wavelength approximation).
In the case of an interferometer in a standard Michelson and equal-arm configuration (such
configurations can be represented by Figure 1 with the particle 1 corresponding to the corner station of
the interferometer and with
), the observed relative frequency shift
is equal to the difference of the round-trip frequency shifts in the two detector’s arms [136]:
Let
be the components (with respect to the TT coordinate system) of the 3-vector
connecting the origin of the TT coordinate system with the corner station of the interferometer. Then
,
, and, making use of Eq. (17), the relative frequency shift (21) can be
written as
The difference
of the phase fluctuations measured, say, by a photo detector, is related to the
corresponding relative frequency fluctuations
by
One can integrate Eq. (22) to write the phase change
as
where the dimensionless function
,
is the response function of the interferometric detector to a plane gravitational wave in the long-wavelength
approximation. To get Eqs. (24) – (25) directly from Eqs. (22) – (23) one should assume that
the quantities
,
, and
[entering Eq. (22)] do not depend on time
. But the
formulae (24) – (25) can also be used in the case when those quantities are time dependent, provided
the velocities
of the detector’s parts with respect to the SSB are non-relativistic. The
error we make in such cases is on the order of
, where
is a typical value of the
velocities
. Thus, the response function of the Earth-based interferometric detector equals
where all quantities here are computed in the SSB-related TT coordinate system. The same response
function can be written in terms of a detector’s proper-reference-frame quantities as follows
where the matrices
and
are related to each other by means of formula (20). In Eq. (27) the
proper-reference-frame components
and
of the unit vectors directed along the interferometer arms
can be treated as constant (i.e., time independent) quantities.
From Eqs. (26) and (3) it follows that the response function
is a linear combination of the two wave
polarizations
and
, so it can be written as
The functions
and
are the interferometric beam-pattern functions. They depend on the location
of the detector on Earth, the position of the gravitational-wave source in the sky, and the polarization angle
of the wave (this angle describes the orientation, with respect to the detector, of the axes relative to which
the plus and cross polarizations of the wave are defined, see, e.g., Figure 9.2 in [135
]). Derivation of the
explicit formulae for the interferometric beam patterns
and
can be found, e.g., in Appendix C
of [66
].
In the long-wavelength approximation, the response function of the interferometric detector can be
derived directly from the equation of geodesic deviation [126]. Then the response is defined as the
relative difference between the wave-induced changes of the proper lengths of the two arms, i.e.,
, where
and
are the instantaneous values of
the proper lengths of the interferometer’s arms and
is the unperturbed proper length of these
arms.
In the case of an Earth-based resonant-bar detector the long-wavelength approximation is very accurate
and the dimensionless detector’s response function can be defined as
, where
is
the wave-induced change of the proper length
of the bar. The response function
can be computed
in terms of the detector’s proper-reference-frame quantities from the formula (see, e.g., Section 9.5.2
in [135
])
where the column matrix
consists of the components (computed in the proper reference
frame of the detector) of the unit vector
directed along the symmetry axis of the bar. The
response function (29) can be written as a linear combination of the wave polarizations
and
, i.e., the formula (28) is also valid for the resonant-bar response function but with
some bar-pattern functions
and
different from the interferometric beam-pattern
functions. Derivation of the explicit form of the bar patterns can be found, e.g., in Appendix C
of [66
].