The first term in the second line is dropped because it is a total derivative, the third line corresponds to the explicit symmetrization of the result due to the symmetry of γij . Applying the previous procedure a second time we obtain:

In addition if we impose that the perturbation γij should be traceless in the 3- dimensional sense we get the semifinal formula:

Undoing the time Fourier transform we can also write:

where, by comparison with (7.3.54), Qij (t) is recognized to be the time dependent quadrupole moment.

7.3.4 Energy Loss by Quadrupole Radiation

When we calculated the stress-energy tensor of the plane wave we expressed it in terms of the following quadratic form:

We focus on the structure of such an expression. Given a symmetric traceless tensor in three dimensions

Fig. 7.11 Integration on the solid angle. The unit vector n singles out the infinitesimal solid angle around its direction

we look for an SO(3) invariant way of rewriting the tensor combination appearing in (7.3.71), namely:

This can be done in terms of a unit vector:

whose physical meaning is that of propagation axis of the considered gravitational plane wave.

We begin with some identities:

Hence we conclude that:

Using this result in our expression for the stress-energy tensor of a plane wave traveling in the n direction we are in a position to write down the energy radiated away per unit time and per unit solid angle by gravitational emission. Indeed in a solid angle unit dΩ around the direction n we have (see Fig. 7.11):

where A is a multiplicative constant that we will fix later by comparison with our previous results.

The constant can be immediately fixed by taking the trace = k and comparing with the previous result: