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 [164Jump To The Next Citation Point]; see also [26Jump To The Next Citation Point]). Hence the lens equation can be rewritten as
(⃗𝜃 − ⃗β ) − ⃗∇ 𝜃ψ = 0, (38 )
( 1 ) ⃗∇ 𝜃 -(⃗𝜃 − ⃗β)2 − ψ = 0. (39 ) 2
The term in brackets appears as well in the physical time delay function for gravitationally lensed images:
⃗ ⃗ 1 +-zL-DLDS-( 1-⃗ ⃗ 2 ) τ(𝜃,β ) = τgeom + τgrav = c D 2(𝜃 − β ) − ψ (𝜃) . (40 ) LS
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 (39View Equation and 40View Equation), 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 (40View Equation) depend on the value of the Hubble constant [202Jump To The Next Citation Point]; 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 [105Jump To The Next Citation Point147205Jump To The Next Citation Point] and Section 4.1).

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