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Abstract: Suppose a Lagrangian is constructed from its fields and their derivatives.When the field configuration is a distribution, it is unambiguously defined asthe limit of a sequence of smooth fields. The Lagrangian may or may not be adistribution, depending on whether there is some undefined product ofdistributions. Supposing that the Lagrangian is a distribution, it isunambiguously defined as the limit of a sequence of Lagrangians. But therestill remains the question: Is the distributional Lagrangian uniquely definedby the limiting process for the fields themselves? In this paper a generalgeometrical construction is advanced to address this question. We describecertain types of singularities, not by distribution valued tensors, but byshowing that the action functional for the singular fields is formallyequivalent to another action built out of \emph{smooth} fields. Thus we manageto make the problem of the lack of a derivative disappear from a system whichgives differential equations. Certain ideas from homotopy and homology theoryturn out to be of central importance in analyzing the problem and clarifyingfiner aspects of it.The method is applied to general relativity in first order formalism, whichgives some interesting insights into distributional geometries in that theory.Then more general gravitational Lagrangians in first order formalism areconsidered such as Lovelock terms for which the action principle admitsspace-times more singular than other higher curvature theories.



Author: E. Gravanis, S. Willison

Source: https://arxiv.org/







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