Limitations of the adiabatic approximation to the gravitational self-force

A small body moving in the field of a much larger black hole and subjected to its own gravity moves on an accelerated world line in the background spacetime of the large black hole. The acceleration is produced by the body’s gravitational self-force, which is constructed from the body’s retarded gravitational field. The adiabatic approximation to the gravitational self-force is obtained instead from the half-retarded minus half-advanced field. It is much easier to compute, and it is known to produce the same dissipative effects as the true self-force. We argue that the adiabatic approximation is limited, because it discards important conservative terms which lead to the secular evolution of some orbital elements. We argue further that this secular evolution has measurable consequences; in particular, it affects the phasing of the orbit and the phasing of the associated gravitational wave. Our argument rests on a simple toy model involving a point electric charge moving slowly in the weak gravitational field of a central mass; the charge is also subjected to its electromagnetic self-force. In this simple context the true self-force is known explicitly and it can cleanly be separated into conservative and radiation-reaction pieces. Its long-term effect on the particle’s orbital elements can be fully determined. In this model we observe that the conservative part of the self-force produces a secular regression of the orbit’s periapsis. We explain how the conclusions reached on the basis of the toy model can be extended to the gravitational self-force, and we attempt to extend them also to the case of rapid motions and strong fields. While the limitations of the adiabatic approximation are quite severe in a post-Newtonian context in which the motion is slow and the gravitational field weak, they may be less severe for rapid motions and strong fields.

Figure 1

Magnetic field produced by the orbiting particle, evaluated as a function of time in the wave zone, within the electric-dipole approximation. The units of the magnetic field are arbitrary, and the time is given in units of the initial orbital period. The initial orbital elements are $p_0=20M$, $e_0=0.5$, and ${\omega }_0=0$. We set $q^2/(m{p}_0)=0.002$. In the color version (online) of the figure, the blue curves represent the wave generated by an orbital evolution driven by the true self-force, and the red curves represent the wave calculated within the adiabatic approximation. In the black-and-white version the true wave appears thick and dark, while the approximate wave appears thin and light. The main plot shows the entire inspiral. The fact that the red/light curve appears to move upward relative to the blue/dark curve is a consequence of a relative phase shift driven by the secular evolution of $\omega$. The lower inset on the left shows the earliest part of the inspiral, before the phase shift had a chance to accumulate. The lower inset on the right shows a later portion of the inspiral, with the approximated wave (red/light) leading slightly in phase with respect to the true wave (blue/dark). Finally, the upper inset shows the latest portion of the inspiral, when the two waves are very much out of phase.