Generalized Darmois-Israel junction conditions
  We present a general method to derive the appropriate Darmois-Israel junction
conditions for gravitational theories with higher-order derivative terms by
integrating the bulk equations of motion across the singular hypersurface. In
higher derivative theories, the field equations can contain terms which are
more singular than the Dirac delta distribution. To handle them appropriately,
we formulate a regularization procedure based on representing the delta
function as the limit of a sequence of classical functions. This procedure
involves imposing suitable constraints on the extrinsic curvature such that the
field equations are compatible with the singular source being a delta
distribution. As explicit examples of our approach, we demonstrate in detail
how to obtain the generalized junction conditions for quadratic gravity,
$\mathcal{F}(R)$ theories, a 4D low-energy effective action in string theory
and action terms that are Euler densities. Our results are novel, and refine
the accuracy of previously claimed results in $\mathcal{F} (R)$ theories and
quadratic gravity. In particular, when the coupling constants of quadratic
gravity are those for the Gauss-Bonnet case, our junction conditions reduce to
the known ones for the latter obtained independently by boundary variation of a
surface term in the action. Finally, we briefly discuss a couple of
applications to thin-shell wormholes and stellar models.
