Time delay and “Fermat’s” theorem

3.7 Time delay and “Fermat’s” theorem

The deflection angle is the gradient of an effective lensing potential $ψ$ (as was first shown by [164

]; see also [26

]). Hence the lens equation can be rewritten as

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 ( $⃗𝜃$ , $β⃗$ ), the gravitational potential $ψ$ , and the distances $DL$ , $DS$ , and $DLS$ . The first part – the geometrical time delay $τgeom$ – reflects the extra path length compared to the direct line between observer and source. The second part – the gravitational time delay $τ grav$ – 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 $ψ$ of the lens (see [105 , 147 , 205 ] and Section 4.1).

Going further. This section followed heavily the elegant presentation of the basics of lensing in Narayan and Bartelmann [126 ]. Many more details can be found there. More complete derivations of the lensing properties are also provided in all the introductory texts mentioned in Section 1, in particular in [167 ].
More on the formulation of gravitational lens theory in terms of time-delay and Fermat’s principle can be found in Blandford and Narayan [26 ] and Schneider [164 ]. Discussions of the concept of “distance” in relation to cosmology/curved space can be found in Section 3.5 of [167 ] or Section 14.4 of [202 ].