4 Lensing Phenomena3 Basics of Gravitational Lensing3.6 Lens mapping

3.7 Time delay and ``Fermat's'' theorem

The deflection angle is the gradient of an effective lensing potential tex2html_wrap_inline2297 (as was first shown by [162Jump To The Next Citation Point In The Article]; see also [26Jump To The Next Citation Point In The Article]). Hence the lens equation can be rewritten as

equation376

or

  equation378

The term in brackets appears as well in the physical time delay function for gravitationally lensed images:

  equation383

This time delay surface is a function of the image geometry (tex2html_wrap_inline2319, tex2html_wrap_inline2321), the gravitational potential tex2html_wrap_inline2297, and the distances tex2html_wrap_inline2207, tex2html_wrap_inline2209, and tex2html_wrap_inline2211 . The first part - the geometrical time delay tex2html_wrap_inline2331 - reflects the extra path length compared to the direct line between observer and source. The second part - the gravitational time delay tex2html_wrap_inline2333 - is the retardation due to gravitational potential of the lensing mass (known and confirmed as Shapiro delay in the solar system). From Equations (39Popup Equation, 40Popup 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 (40Popup Equation) depend on the value of the Hubble constant [202Jump To The Next Citation Point In The Article]; 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 tex2html_wrap_inline2297 of the lens (see [102Jump To The Next Citation Point In The Article, 145, 205Jump To The Next Citation Point In The Article] and Section  4.1).



4 Lensing Phenomena3 Basics of Gravitational Lensing3.6 Lens mapping

image Gravitational Lensing in Astronomy
Joachim Wambsganss
http://www.livingreviews.org/lrr-1998-12
© Max-Planck-Gesellschaft. ISSN 1433-8351
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