引力中的奇异源与联络空间中的同伦
Singular sources in gravity and homotopy in the space of connections
摘要：
假设一个拉格朗日量是由它的场及其导数构成的。当场位形是分布时，它被明确地定义为光滑场序列的极限。拉格朗日量可能是也可能不是一个分布，这取决于是否存在一些未定义的分布乘积。假设拉格朗日量是一个分布，它被明确地定义为拉格朗日量序列的极限。但仍然存在一个问题：分布拉格朗日量是否由场本身的极限过程唯一定义？本文提出了一种通用的几何构造来解决这个问题。我们描述某些类型的奇点，而不是分布值张量，但通过显示的奇异领域的行动功能是（正式）相当于另一个行动建立了\emph{smooth}领域。因此，我们设法使问题的缺乏一个衍生物消失的系统，使微分方程。同伦和同调理论中的某些思想在分析问题和澄清问题的细节方面具有重要意义，这种方法被应用于广义相对论的一阶形式，从而对该理论中的分布几何给出了一些有趣的见解。然后更一般的引力拉格朗日在一阶形式主义被认为是如洛夫洛克条款（其中的作用原理承认时空比其他更高的曲率理论奇异）。
英文摘要：
Suppose a Lagrangian is constructed from its fields and their derivatives. When the field configuration is a distribution, it is unambiguously defined as the limit of a sequence of smooth fields. The Lagrangian may or may not be a distribution, depending on whether there is some undefined product of distributions. Supposing that the Lagrangian is a distribution, it is unambiguously defined as the limit of a sequence of Lagrangians. But there still remains the question: Is the distributional Lagrangian uniquely defined by the limiting process for the fields themselves? In this paper a general geometrical construction is advanced to address this question. We describe certain types of singularities, not by distribution valued tensors, but by showing that the action functional for the singular fields is (formally) equivalent to another action built out of \emph{smooth} fields. Thus we manage to make the problem of the lack of a derivative disappear from a system which gives differential equations. Certain ideas from homotopy and homology theory turn out to be of central importance in analyzing the problem and clarifying finer aspects of it. The method is applied to general relativity in first order formalism, which gives some interesting insights into distributional geometries in that theory. Then more general gravitational Lagrangians in first order formalism are considered such as Lovelock terms (for which the action principle admits space-times more singular than other higher curvature theories).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
