Radial fall: the gravitational waveform up to the second-and-half Post-Newtonian order

Abstract

We consider an application of the Multipolar Post Minkowskian formalism to the case of a two-body system in radial fall. We compute, within the post-Newtonian approximation, the associated gravitational waveform reaching the 2.5 Post-Newtonian accuracy level. At this level the presence of a radiation-reaction force manifests, modifying the fall with a corresponding bremsstrahlung radiation. We evaluate then all emissions: energy, angular momentum (vanishing identically) and linear momentum. We also evaluate the (nonlocal) inertial forces contributions appearing (at the next PN order, 4.5PN) in the center-of-mass due to the losses paving the way for future more accurate computations.

Figures (2)

• Figure 1: We show the comparison between the conservative trajectory ($\epsilon=0$ continuous blue line) with the trajectory corrected with the radiation reaction effect ($\epsilon=1$ dashed orange line) having chosen $\nu=0.1$. [Note that the expression of the radiation-reaction force (magnitude of the radial component) along the motion reduces to $|{\bf a}_{\rm 2.5PN}^r|=\frac{128 M^2\nu}{81 t^3}$]. Superposed to the other curves we show (black online) the position of the horizon ($r=2$ in units of $M$) when $\nu=0$, as a reference line.
• Figure 2: The behaviour of $\frac{dE}{d\omega}$ as a function of $\omega$, for $M=1$ and $\nu= [0\, \text{(blue online)},\frac{1}{8}\, \text{(yellow online)},\frac{1}{4}\, \text{(green online)}]$.