Self-force via Green functions and worldline integration
A compact object moving in curved spacetime interacts with its own gravitational field. This leads to 
both dissipative and conservative corrections to the motion, which can be interpreted as a self-force acting 
on the object. The original formalism describing this self-force relied heavily on the Green function 
of the linear differential operator that governs gravitational perturbations. However, because the global 
calculation of Green functions in nontrivial black-hole spacetimes has been an open problem until recently, 
alternative methods were established to calculate self-force effects using sophisticated regularization 
techniques that avoid the computation of the global Green function.We present a method for calculating the 
self-force that employs the global Green function and is therefore closely modeled after the original self-force 
expressions. Our quantitative method involves two stages: (i) numerical approximation of the retarded 
Green function in the background spacetime; (ii) evaluation of convolution integrals along the worldline 
of the object. This novel approach can be used along arbitrary worldlines, including those currently 
inaccessible to more established computational techniques. Furthermore, it yields geometrical insight into 
the contributions to self-interaction from curved geometry (backscattering) and trapping of null geodesics. 
We demonstrate the method on the motion of a scalar charge in Schwarzschild spacetime. This toy model 
retains the physical history dependence of the self-force but avoids gauge issues and allows us to focus on 
basic principles. We compute the self-field and self-force for many worldlines including accelerated 
circular orbits, eccentric orbits at the separatrix, and radial infall. This method, closely modeled after the 
original formalism, provides a promising complementary approach to the self-force problem.