Equation (33) is the result that
was first derived by DeWitt and Brehme [24
] and later corrected
by Hobbs [29
]. (The original
equation did not include the Ricci-tensor term.) In flat spacetime
the Ricci tensor is zero, the tail integral disappears (because the
Green’s function vanishes everywhere within the domain of
integration), and Equation (33
) reduces to Dirac’s
result of Equation (5
). In curved spacetime
the self-force does not vanish even when the electric charge is
moving freely, in the absence of an external force: It is then
given by the tail integral, which represents radiation emitted
earlier and coming back to the particle after interacting with the
spacetime curvature. This delayed action implies that, in general,
the self-force is nonlocal in time: It depends not only on the
current state of motion of the particle, but also on its past
history. Lest this behaviour should seem mysterious, it may help to
keep in mind that the physical process that leads to
Equation (33
) is simply an
interaction between the charge and a free electromagnetic field
; it is this field that carries the information about
the charge’s past.