  The basis of this work is the first full application of the
Poisson-Wiseman-Anderson method of `matched expansions' to compute the
self-force acting on a point particle moving in a curved spacetime. The method
employs two expansions for the Green function which are respectively valid in
the `quasilocal' and `distant past' regimes, and which are matched together
within the normal neighborhood.
  Building on a fundamental insight due to Avramidi, we provide a system of
transport equations for determining key fundamental bi-tensors, including the
tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These
bitensors are central to a broad range of problems from radiation reaction and
the self-force to quantum field theory in curved spacetime and quantum gravity.
Using their transport equations, we show how the quasilocal Green function may
be computed throughout the normal neighborhood both numerically and as a
covariant Taylor series expansion. These calculations are carried out for
several black hole spacetimes.
  Finally, we present a complete application of the method of matched
expansions. The calculation is performed in a static region of the spherically
symmetric Nariai spacetime (dS_2 X S^2), where the matched expansion method is
applied to compute the scalar self-force acting on a static particle. We find
that the matched expansion method provides insight into the non-local
properties of the self-force. The Green function in Schwarzschild spacetime is
expected to share certain key features with Nariai. In this way, the Nariai
spacetime provides a fertile testing ground for developing insight into the
non-local part of the self-force on black hole spacetimes.
