9 research outputs found
Geometry of Universal Magnification Invariants
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.Comment: RevTEX4, 13 pages, submitted to J. Math. Phy
A Universal Magnification Theorem III. Caustics Beyond Codimension Five
In the final paper of this series, we extend our results on magnification
invariants to the infinite family of A, D, E caustic singularities. We prove
that for families of general mappings between planes exhibiting any caustic
singularity of the A, D, E family, and for a point in the target space lying
anywhere in the region giving rise to the maximum number of lensed images (real
pre-images), the total signed magnification of the lensed images will always
sum to zero. The proof is algebraic in nature and relies on the Euler trace
formula.Comment: 8 page
Mathematics of Gravitational Lensing: Multiple Imaging and Magnification
The mathematical theory of gravitational lensing has revealed many generic
and global properties. Beginning with multiple imaging, we review
Morse-theoretic image counting formulas and lower bound results, and
complex-algebraic 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 micro-minima. Multiple
imaging in optical geometry and a spacetime setting are treated. We review
global magnification relation results for model-dependent 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 Gravitatio