Annu. Rev. Astron. Astrophys. 1992. 30: 311-358
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3.4 Time Delay

The propagation time from the source to the observer varies from one image to another, and this difference can be measured when the source is variable. For a single lens in a homogeneous universe, the excess time delay associated with the image at theta (i), relative to the direct ray in the absence of the lens, is given by (Refsdal 1964b, Cooke & Kantowski 1975, Kayser & Refsdal 1983, Borgeest 1983, Schneider 1985)

Equation 11 11.

The quantity t (i) itself cannot be measured, but the relative time delay between two images, t (ij) = t (i) - t (j), can be. Since D (cf Equation 4) is inversely proportional to H0, a measurement of t (ij) provides vital information for cosmography (cf Section 4.1). The term proportional to (theta (i) - beta)2 in Equation 11 is due to the geometrical excess path length and the term proportional to psi is the gravitational time delay, known as the ``Shapiro effect'' in the solar system (Shapiro 1964).

In one formulation of gravitational lens theory, the time delay t(theta) is taken to be the fundamental quantity and the lens Equation 2 is obtained by invoking Fermat's principle, which states that images are formed at stationary points of t (Weyl 1922, Schneider 1985, Blandford & Narayan 1986). Equation 11 can be generalized to multiple lenses (Blandford & Narayan 1986, Kovner 1987b), as well as to nonstationary gravitational fields (Kovner 1990, Nityananda & Samuel 1992).