  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.
