Singular sources in gravity and homotopy in the space of connections

Understand

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.