  The finite part of the self-force on a static scalar test-charge outside a
Schwarzschild black hole is zero. By direct construction of Hadamard's
elementary solution, we obtain a closed-form expression for the minimally
coupled scalar field produced by a test-charge held fixed in Schwarzschild
spacetime. Using the closed-form expression, we compute the necessary external
force required to hold the charge stationary. Although the energy associated
with the scalar field contributes to the renormalized mass of the particle (and
thereby its weight), we find there is no additional self-force acting on the
charge. This result is unlike the analogous electrostatic result, where, after
a similar mass renormalization, there remains a finite repulsive self-force
acting on a static electric test-charge outside a Schwarzschild black hole. We
confirm our force calculation using Carter's mass-variation theorem for black
holes. The primary motivation for this calculation is to develop techniques and
formalism for computing all forces - dissipative and non-dissipative - acting
on charges and masses moving in a black-hole spacetime. In the Appendix we
recap the derivation of the closed-form electrostatic potential. We also show
how the closed-form expressions for the fields are related to the infinite
series solutions.
