### 3.7 Time delay and “Fermat’s” theorem

The deflection angle is the gradient of an effective lensing potential (as was first shown by [164]; see
also [26]). Hence the lens equation can be rewritten as
or
The term in brackets appears as well in the physical time delay function for gravitationally lensed images:
This time delay surface is a function of the image geometry (, ), the gravitational potential , and
the distances , , and . The first part – the geometrical time delay – reflects the
extra path length compared to the direct line between observer and source. The second part –
the gravitational time delay – is the retardation due to gravitational potential of the
lensing mass (known and confirmed as Shapiro delay in the solar system). From Equations (39
and 40), it follows that the gravitationally lensed images appear at locations that correspond
to extrema in the light travel time, which reflects Fermat’s principle in gravitational-lensing
optics.
The (angular-diameter) distances that appear in Equation (40) depend on the value of the Hubble
constant [202]; therefore, it is possible to determine the latter by measuring the time delay between
different images and using a good model for the effective gravitational potential of the lens
(see [105, 147, 205] and Section 4.1).