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A Kinematical approach to gravitational lensing


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Walters, SJ 2014 , 'A Kinematical approach to gravitational lensing', PhD thesis, University of Tasmania.

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The deflection of light by massive bodies has been of interest to mathematicians
and physicists from time to time since Newton suggested the possibility in his
1704 work, “Opticks”. This deflection was calculated in the late eighteenth and
early nineteenth centuries, treating light as a classical particle. The deflection was
again calculated by Einstein in the early twentieth century, using his new general
theory of relativity, to be twice the previous classical result. The measurement of
the deflection of light passing close by the Sun was widely publicized as a dramatic
confirmation of general relativity, in the now famous 1919 expeditions.
In the last three decades, gravitational lensing has become an important tool for
astrophysicists, especially in searching for dark matter and exoplanets. By 1991,
astrophysicists were suggesting that exoplanets could be found using microlensing,
and since 2004 at least ten planets have been found in this way. In microlensing,
light from a background star passes close to the lensing system, and is deflected
around the lens. Because of this, more light rays reach the observer, producing
magnification of the background source. This magnification changes over time, as
the source, lens and observer move into and out of alignment. The details of the
magnification over time are plotted in a ‘light curve’, which is simply intensity
versus time. A planet orbiting a lensing star can make changes in the light field at
the observer’s plane (”magnification map”). Such changes show up as variations
to the shape of the simple light curve.
This thesis presents a mathematical model for the paths taken by light rays near
a massive object such as a star or black hole. I begin the thesis by considering the
spacetime around a non-rotating, uncharged, spherically symmetric object. Such
a spacetime is described by the Schwarzschild metric equation. After considering
the deflection angle and travel-time delay in such a system, I derive an acceleration
vector for the massless particle (“photon”). This acceleration vector is used to plot
the light paths of many photons passing through a binary system using numerical
integration, resulting in a magnification map very similar to those currently in
use by astrophysicists. In the following chapter I consider a linear approximation
of the acceleration vector just mentioned, considering the light path as a small
perturbation about a straight line. Such an approach results in a linear third order
ordinary differential equation, but with non-constant coefficients. Unexpectedly,
a closed-form solution is found, resulting in path equations accurate to first order
in the relevant small parameter. This allows for very rapid computation of the
magnification map. Some examples are presented and compared against the fully
non-linear numerical results of the previous chapter, and also against a simpler
approach used by some other authors.
The final section of results of the thesis is given to consideration of the effect of
rotation of the lensing object. In considering the Kerr metric which describes
such a system, I follow an approach similar to that used in previous chapters.
Thus, an acceleration vector is derived, which is used to plot a magnification
map for a binary system containing a rotating object. Rotation causes bending
and asymmetry in the magnification map. This is illustrated for certain cases of
interest. A second order approximation is also considered, as well as application
of the equatorial special case to calculation of travel time delay. This delay is
compared to that expected for a non-rotating object.

Item Type: Thesis - PhD
Authors/Creators:Walters, SJ
Keywords: gravitation, optics, micro-lensing, exo-planets, general relativity
Copyright Holders: The Author
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Copyright 2014 the Author

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