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The growing reality of gravitational wave astronomy is giving age-oldproblems a new lease of life; one such problem is that of the self-force. Acharged or massive particle moving in a curved background space-time produces afield that affects its motion, pushing it off its expected geodesic. Thisself-field gives rise to a so-called self-force acting on the particle. Inmodelling this motion, the self-force approach uses a perturbative expansion inthe mass ratio. One of the most interesting sources of gravitational waves areextreme mass ratio inspirals - systems perfectly suited to self-forcemodelling. One of the key problems within the self-force model is thedivergence of the field at the particle. To resolve this, the field is splitinto a singular component and a smooth regular field. This regular-singularsplit, introduced by Detweiler and Whiting, is used in most modern self-forcecalculations. In this thesis, we derive high-order expansions of theDetweiler-Whiting singular field, and use these to push the boundaries oncurrent precision limits of self-force calculations. Within the mode-sumscheme, we give over 14 previously unknown regularisation parameters, almostdoubling the current regularisation parameter database. We also produce smootheffective sources to high order, and propose an application of the higher termsto improve accuracy in the m-mode scheme. Finally, we investigate the status ofthe cosmic censorship conjecture and the role that the self-force plays. Tothis end, we give regularisation parameters for non-geodesic motion. We alsoshow the necessity of our results in the exciting area of second orderself-force calculations, which benefit significantly from high-order coordinateexpansions of the singular field. We calculate several parameters that theseschemes require, and highlight the further advancements possible from theresults of this thesis.

The Self-Force Problem: Local Behaviour of the Detweiler-Whiting Singular Field