Annu. Rev. Astron. Astrophys. 1992. 30: 311-358
Copyright © 1992 by . All rights reserved


Next Contents Previous

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).