Limitations of the adiabatic approximation to the gravitational self-force

TL;DR

The paper critiques the adiabatic approximation for gravitational self-force calculations, highlighting its neglect of conservative effects that influence orbital evolution and gravitational wave phasing, especially in strong-field, rapid-motion regimes.

Contribution

It demonstrates the limitations of the adiabatic approximation by analyzing conservative effects through a toy model and discusses implications for gravitational wave modeling.

Findings

Adiabatic approximation neglects conservative self-force effects.

Conservative effects cause secular orbital changes affecting gravitational wave phase.

Limitations are more severe in slow, weak-field regimes than in strong, rapid-motion scenarios.

Abstract

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 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. 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 to fast 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 appear to be less so for rapid motions and strong fields.