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Annu. Rev. Astron. Astrophys. 1992. 30:
311-358
Copyright © 1992 by Annual Reviews. All
rights reserved |
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 (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)
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 ( (i) - )2 in Equation
11 is due to the geometrical excess path length and the
term proportional to 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() 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).