Generalized multi-plane gravitational lensing: time delays, recursive
  lens equation, and the mass-sheet transformation
  We consider several aspects of the generalized multi-plane gravitational lens
theory, in which light rays from a distant source are affected by several main
deflectors, and in addition by the tidal gravitational field of the large-scale
matter distribution in the Universe when propagating between the main
deflectors. Specifically, we derive a simple expression for the time-delay
function in this case, making use of the general formalism for treating light
propagation in inhomogeneous spacetimes which leads to the characterization of
distance matrices between main lens planes. Applying Fermat's principle, an
alternative form of the corresponding lens equation is derived, which connects
the impact vectors in three consecutive main lens planes, and we show that this
form of the lens equation is equivalent to the more standard one. For this,
some general relations for cosmological distance matrices are derived. The
generalized multi-plane lens situation admits a generalized mass-sheet
transformation, which corresponds to uniform isotropic scaling in each lens
plane, a corresponding scaling of the deflection angle, and the addition of a
tidal matrix (mass sheet plus external shear) to each main lens. We show that
the time delay for sources in all lens planes scale with the same factor under
this generalized mass-sheet transformation, thus precluding the use of
time-delay ratios to break the mass-sheet transformation.
