Publications (9)30.31 Total impact
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ABSTRACT: The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morsetheoretic image counting formulas and lower bound results, and complexalgebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of microminima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for modeldependent scenarios and cover recent developments on universal local magnification relations for higher order caustics. Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitation  [Show abstract] [Hide abstract]
ABSTRACT: Recent work in gravitational lensing and catastrophe theory has shown that the sum of the signed magnifications of images near folds, cusps and also higher catastrophes is zero. Here, it is discussed how Lefschetz fixed point theory can be used to interpret this result geometrically. It is shown for the generic case as well as for elliptic and hyperbolic umbilics in gravitational lensing.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) PainleveGullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergosurface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples. Comment: 37 pages, 6 figures  [Show abstract] [Hide abstract]
ABSTRACT: We interpret the well known fact that the equations for light rays in the Kottler or Schwarzschildde Sitter metric are independent of the cosmological constant in terms of the projective equivalence of the optical metric for any value of \Lambda. We explain why this does not imply that lensing phenomena are independent of \Lambda. Motivated by this example, we find a large collection of oneparameter families of projectively equivalent metrics including both the Kottler optical geometry and the constant curvature metrics as special cases. Using standard constructions for geodesically equivalent metrics we find classical and quantum conserved quantities and relate these to known quantities.  [Show abstract] [Hide abstract]
ABSTRACT: In this geometrical approach to gravitational lensing theory, we apply the GaussBonnet theorem to the optical metric of a lens, modelled as a static, spherically symmetric, perfect nonrelativistic fluid, in the weak deflection limit. We find that the focusing of the light rays emerges here as a topological effect, and we introduce a new method to calculate the deflection angle from the Gaussian curvature of the optical metric. As examples, the Schwarzschild lens, the Plummer sphere and the singular isothermal sphere are discussed within this framework. 
Article: On Multiple Einstein Rings
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ABSTRACT: A number of recent surveys for gravitational lenses have found examples of double Einstein rings. Here, we analytically investigate the occurrence of multiple Einstein rings. We prove, under very general assumptions, that at the most one Einstein ring can arise from a mass distribution in a single plane lensing a single background source. Two or more Einstein rings can therefore only occur in multiplane lensing. Surprisingly, we show that it is possible for a single source to produce more than one Einstein ring. If two point masses, or two isothermal spheres, in different planes are aligned with observer and source on the optical axis, we show that there are up to three Einstein rings. We also discuss the image morphologies for these two models if axisymmetry is broken, and give the first instances of magnification invariants in the case of twolens planes.  [Show abstract] [Hide abstract]
ABSTRACT: A Kerr black hole with mass parameter m and angular momentum parameter a acting as a gravitational lens gives rise to two images in the weak field limit. We study the corresponding magnification relations, namely the signed and absolute magnification sums and the centroid up to postNewtonian order. We show that there are postNewtonian corrections to the total absolute magnification and centroid proportional to a/m, which is in contrast to the spherically symmetric case where such corrections vanish. Hence we also propose a new set of lensing observables for the two images involving these corrections, which should allow measuring a/m with gravitational lensing. In fact, the resolution capabilities needed to observe this for the Galactic black hole should in principle be accessible to current and nearfuture instrumentation. Since a/m >1 indicates a naked singularity, a most interesting application would be a test of the Cosmic Censorship conjecture. The technique used to derive the image properties is based on the degeneracy of the Kerr lens and a suitably displaced Schwarzschild lens at postNewtonian order. A simple physical explanation for this degeneracy is also given.  [Show abstract] [Hide abstract]
ABSTRACT: Topological invariants play an important r\^{o}le in the theory of gravitational lensing by constraining the image number. Furthermore, it is known that, for certain lens models, the image magnifications $\mu_i$ obey invariants of the form $\sum_i \mu_i=1$. In this paper, we show that this magnification invariant is the holomorphic Lefschetz number of a suitably defined complexified lensing map, and hence a topological invariant. We also provide a heat kernel proof of the holomorphic Lefschetz fixed point formula which is central to this argument, based on Kotake's proof of the more general AtiyahBott theorem. Finally, we present a new astronomically motivated lens model for which this invariant holds.  [Show abstract] [Hide abstract]
ABSTRACT: The lensing properties of the Plummer model with a central point mass and external shear are derived, including the image multiplicities, critical curves and caustics. This provides a simple model for a flattened galaxy with a central supermassive black hole. For the Plummer model with black hole, the maximum number of images is four, provided the black hole mass is less than an upper bound which is calculated analytically. This introduces a method to constrain black hole masses by counting images, thus applicable at cosmological distance. With shear, the maximum number of images is six and we illustrate the occurrence of an astroid caustic and two metamorphoses.
Publication Stats
142  Citations  
30.31  Total Impact Points  
Top Journals
Institutions

2009

Duke University
 Department of Mathematics
Durham, North Carolina, United States


20062009

University of Cambridge
 Institute of Astronomy
Cambridge, England, United Kingdom
