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 [81Jump To The Next Citation Point169Jump To The Next Citation Point], solutions are expanded in the small parameter (v∕c)2, where v is the typical dynamical speed inside the system. Because of the virial theorem, the dimensionless Newtonian gravitational potential ϕ ∕c2 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 (ρ) and velocity fields of the Newtonian system. If we define the spatial tensor Qjk, the second moment of the mass distribution, by the equation
∫ Qjk = ρxjxkd3x, (1 )
then the amplitude of the emitted gravitational wave is, at lowest order, the three-tensor
2 d2Qjk hjk = -----2--. (2 ) r dt
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 (2View Equation) 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:

( ) hab = h+ h× . (3 ) h× − h+
This is the definition of the wave amplitudes h+ and h× that are illustrated in Figure 1View Image. These amplitudes are referred to as the coordinates chosen for that plane. If the coordinate unit basis vectors in this plane are ˆex and ˆey, then we can define the basis tensors
e = ˆe ⊗ ˆe − ˆe ⊗ ˆe , (4 ) + x x y y e× = ˆex ⊗ ˆey + ˆey ⊗ ˆex. (5 )
In terms of these, the TT-gravitational wave tensor can be written as
h = h+e+ + h ×e×. (6 )

If the coordinates in the transverse plane are rotated by an angle ψ, then one obtains new amplitudes h ′+ and h ′× given by

h′ = cos 2ψh+ + sin 2ψh ×, (7 ) +′ h× = − sin2 ψh+ + cos 2ψh ×. (8 )
This shows the quadrupolar nature of the polarizations, and is consistent with our remark in association with Figure 1View Image that a rotation of π∕4 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 others1. 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 d2Qjk ∕dt2 will (from Equation (1View Equation)) have magnitude (M v2)nonsph, where (M v2)nonsph is twice the nonspherical part of the kinetic energy inside the source. So a bound on any component of Equation (2View Equation) is

2(M v2)nonsph h ≲ ------------. (9 ) r
It is interesting to observe that the ratio 𝜖 of the wave amplitude to the Newtonian potential ϕext of its source at the observer’s distance r is simply bounded by
h∕ ϕext < 2v2nonsph,

and this bound is attained if the entire mass of the source is involved in the nonspherical motions, so that (M v2) ∼ M v2 nonsph nonsph. By the virial theorem for self-gravitating bodies

2 vnonsph ≤ ϕint, (10 )
where ϕint is the maximum value of the Newtonian gravitational potential inside the system. This provides a convenient bound in practice [331]:
h ≲ 2ϕintϕext. (11 )
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 ϕint ∼ 0.2. If the star is in the Virgo cluster (r ∼ 18 Mpc) and has a mass of 1.4M ⊙, 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.


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