Junction conditions at spacetime singularities
  A classical model for the extension of singular spacetime geometries across
their singularities is presented. The regularization introduced by this model
is based on the following observation. Among the geometries that satisfy
Einstein's field equations there is a class of geometries, with certain
singularities, where the components of the metric density and their partial
derivatives remain finite in the limit where the singularity is approached.
Here we exploit this regular behavior of the metric density and elevate its
status to that of a fundamental variable -- from which the metric is
constructed. We express Einstein's field equations as a set of equations for
the metric density, and postulate junction conditions that the metric density
satisfies at singularities. Using this model we extend certain geometries
across their singularities. The following examples are discussed: radiation
dominated Friedmann-Robertson-Walker Universe, Schwarzschild black hole,
Reissner-Nordstr\"{o}m black hole, and certain Kasner solutions. For all of the
above mentioned examples we obtain a unique extension of the geometry beyond
the singularity.
