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 asIf 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^{1}.
It should also be clear from this that spherically-symmetric motions do not emit any gravitational radiation:
when the trace is removed, nothing remains.

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 byand 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}.

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