2.4 Amplitude of gravitational waves – the quadrupole approximation
The Einstein equations are too difficult to solve analytically in the generic case of a strongly gravitating
source to compute the luminosity and amplitude of gravitational waves from an astronomical source. We
will discuss numerical solutions later; the most powerful available analytic approach is called the
post-Newtonian approximation scheme. This approximation is suited to gravitationally-bound systems,
which constitute the majority of expected sources. In this scheme [81
, 169
], solutions are expanded in the
small parameter
, where
is the typical dynamical speed inside the system. Because of the virial
theorem, the dimensionless Newtonian gravitational potential
is of the same order, so that the
expansion scheme links orders in the expanded metric with those in the expanded source terms. The
lowest-order post-Newtonian approximation for the emitted radiation is the quadrupole formula,
and it depends only on the density (
) and velocity fields of the Newtonian system. If we
define the spatial tensor
, the second moment of the mass distribution, by the equation
then the amplitude of the emitted gravitational wave is, at lowest order, the three-tensor
This is to be interpreted as a linearized gravitational wave in the distant almost-flat geometry far from the
source, in a coordinate system (gauge) called the Lorentz gauge.
2.4.1 Wave amplitudes and polarization in TT-gauge
A useful specialization of the Lorentz gauge is the TT-gauge, which is a comoving coordinate system: free
particles remain at constant coordinate locations, even as their proper separations change. To get the
TT-amplitude of a wave traveling outwards from its source, project the tensor in Equation (2)
perpendicular to its direction of travel and remove the trace of the projected tensor. The result of doing this
to a symmetric tensor is to produce, in the transverse plane, a two-dimensional matrix with only two
independent elements:
This is the definition of the wave amplitudes
and
that are illustrated in Figure 1. These
amplitudes are referred to as the coordinates chosen for that plane. If the coordinate unit basis vectors in
this plane are
and
, then we can define the basis tensors
In terms of these, the TT-gravitational wave tensor can be written as
If the coordinates in the transverse plane are rotated by an angle
, then one obtains new amplitudes
and
given by
This shows the quadrupolar nature of the polarizations, and is consistent with our remark in association
with Figure 1 that a rotation of
changes one polarization into the other.
It should be clear from the TT projection operation that the emitted
radiation is not isotropic: it will be stronger in some directions than in
others.
It should also be clear from this that spherically-symmetric motions do not emit any gravitational radiation:
when the trace is removed, nothing remains.
2.4.2 Simple estimates
A typical component of
will (from Equation (1)) have magnitude
, where
is twice the nonspherical part of the kinetic energy inside the source. So a bound on any
component of Equation (2) is
It is interesting to observe that the ratio
of the wave amplitude to the Newtonian potential
of its
source at the observer’s distance
is simply bounded by
and this bound is attained if the entire mass of the source is involved in the nonspherical motions, so that
. By the virial theorem for self-gravitating bodies
where
is the maximum value of the Newtonian gravitational potential inside the system. This
provides a convenient bound in practice [331]:
The bound is attained if the system is highly nonspherical. An equal-mass star binary system is a good
example of a system that attains this bound.
For a neutron star source, one has
. If the star is in the Virgo cluster (
) and
has a mass of
, and if it is formed in a highly-nonspherical gravitational collapse, then the upper
limit on the amplitude of the radiation from such an event is 1.5 × 10–21. This is a simple way to get the
number that has been the goal of detector development for decades, to make detectors that can observe
waves at or below an amplitude of about 10–21.