(as was first shown by [162
]; see also [26]). Hence the lens equation can be rewritten as
or
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 (
,
), the gravitational potential
, and the distances
,
, and
. The first part - the geometrical time delay
- reflects the extra path length compared to the direct line
between observer and source. The second part - the gravitational
time delay
- is the retardation due to gravitational potential of the
lensing mass (known and confirmed as Shapiro delay in the solar
system). From Equations (39
,
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
of the lens (see [102,
145,
205] and Section
4.1).
]. 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 Chapter
1, in particular in [166].
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 [162]. Discussions of the concept of ``distance'' in relation to
cosmology/curved space can be found in Section 3.5 of [166] or Section 14.4 of [202].
|
Gravitational Lensing in Astronomy
Joachim Wambsganss http://www.livingreviews.org/lrr-1998-12 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
