An Examination of Geometrical and Potential Time Delays in Gravitational
  Lensing
  In this paper we investigate the relation between the potential and geometric
time delays in gravitational lensing. In the original paper of Shapiro (1964),
it is stated that there is a time delay in the radar signals between Earth and
Venus that pass near a massive object (the Sun), compared to the path taken in
the absence of any mass. The reason for this delay is connected with the
influence of gravity on the coordinate velocity of a light ray in a
gravitational potential. The contribution from the change of the path length,
which happens to be of second order, is considered as negligible. Nevertheless,
in the gravitational lens theory the geometrical delay, related to the change
of path length, is routinely taken into account along with the potential term.
In this work we explain this apparent discrepancy. We address the contribution
of the geometric part of the time delay in different situations, and introduce
a unified treatment with two limiting regimes of lensing. One of these limits
corresponds to the time delay experiments near the Sun where the geometrical
delay is shown to be negligible. The second corresponds to the typical
gravitational lens scenario with multiple imaging where the geometrical delay
is shown to be significant. We introduce a compact, analytical, and
quantitative criteria based on relation between the angular position of source
and the Einstein radius. This criterion allows one to find out easily when it
is necessary to take the geometrical delay into account. In particular, it is
shown that the geometrical delay is non-negligible in the case of good
alignment between source, lens and observer, because in such a case it becomes
a first order quantity (the same order as the potential term).
