General Shells and Generalized Functions

Abstract: In this work, standard methods of the mixed thin-shell foramlism are refined using the framework of Colombeau's theory of generalized functions. To this end, systematic use is made of smooth generalized functions, in particular regularizations of the Heaviside step function and the delta distribution, instead of working directly with the corresponding Schwartz distributions. Based on this change of method, the resulting extended thin shell formalism is shown to offer a decisive advantage over traditional approaches to the subject: it avoids dealing with ill-defined products of distributions in the calculation of nonlinear curvature expressions, thereby allowing for the treatment of problems that prove intractable with the 'conventional' thin-shell formalism. This includes, in particular, the problem of matching singular spacetimes with distributional metrics (containing a delta distribution term) across a joint boundary hypersurface in spacetime, the problem of setting up the dominant energy condition for thin shells, and the problem of defining reasonably rigorously nonlinear distribution-valued curvature invariants needed in higher-derivative theories of gravity. Eventually, as a further application, close links to Penrose's cut-and-paste method are established by proving that results of said method can be re-derived using the generalized formalism presented.