Here, these amplitudes ${\mathcal{M}}_{3,\mathrm{cl}}^{(0)}(p,k)$ and ${\mathcal{M}}_{4,\mathrm{cl}}^{(0)}(p,k_1,k_2)$ correspond to a massive particle emitting one or two gravitons, respectively.

To compare with Kovacs and Thorne and the recent work [9], we need to redefine the waveform imposing that it vanishes in the far (retarded) past $\tilde{h}(x):-h(x)-{\lim }_{u\to -\infty }h(x)$. Indeed, by exchanging the order of integration, we ignored the fact that the amplitude develops a $\frac{1}{\omega }$ pole as $\omega \to 0$ (from Weinberg’s soft theorem). The correct prescription for this pole is $1/(\omega +iϵ)$. Moreover, we also ignored terms proportional to $\delta (\omega )$ to the waveform in frequency space, which give a time-independent contribution and set ${\lim }_{u\to -\infty }h(x)=0$.

It is worth emphasizing that these three structures are not independent of each other, but they are related (linearly) by the Bianchi identity $f_{1,2}(T_1{\overline{w}}_1-T_2{\overline{w}}_2)+\sqrt{{\gamma }^2-1}(f_2{\overline{w}}_1-f_1{\overline{w}}_2)=0$ and (quadratically) by the four-dimensional identity $(f_2{\overline{w}}_1-f_1{\overline{w}}_2{)}^2=\frac{(f_1^2-2\gamma f_1{f}_2+f_2^2)(T_1{\overline{w}}_1-T_2{\overline{w}}_2{)}^2}{{\gamma }^2-1}$.

The comparison is performed by picking the frame $v_1^{\mu }=(1,0,0,0)$, $v_2^{\mu }=(\gamma ,\sqrt{{\gamma }^2-1},0,0)$, ${\tilde{b}}^{\mu }=(0,0,-1,0)$, $n^{\mu }=(1,\cos \theta ,\sin \theta \cos \varphi ,\sin \theta \sin \varphi )$ and using the following polarization vectors: ${\epsilon }^{\mu }=\frac{1}{\sqrt{2}}({\partial }_{\theta }{n}^{\mu }+i\frac{{\partial }_{\varphi }{n}^{\mu }}{\sin \theta })$, ${\epsilon }_{+}^{\mu \nu }=\Re ({\epsilon }^{\mu }{\epsilon }^{\nu })$, ${\epsilon }_{\times }^{\mu \nu }=\Im ({\epsilon }^{\mu }{\epsilon }^{\nu })$.