University Of Tasmania
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A Kinematical approach to gravitational lensing

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posted on 2023-05-27, 12:47 authored by Walters, SJ
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‚ÄövÑvp. 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 (‚ÄövÑvpmagnification map‚ÄövÑvp). 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‚ÄövÑvp). 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.


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