Orbifolds, the \(A, D, E\) family of caustic singularities, and gravitational lensing. (English) Zbl 1314.57022
Summary: We provide a geometric explanation for the existence of magnification relations for the \(A_n\; (n \geq 2),\; D_n\; (n \geq 4),\; E_6,\; E_7,\; E_8\) family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any noncaustic target point, the total signed magnification of the corresponding preimages vanishes. As an application to gravitational lensing, it was also shown that, independent of the choice of a lens model, the total signed magnification vanishes for a light source anywhere in the four-image region close to elliptic and hyperbolic umbilic caustics. This is a more global and higher order analog of the well-known fold and cusp magnification relations. We now extend each of these mappings to weighted projective space, which is a compact orbifold, and show that magnification relations translate into a statement about the behavior of these extended mappings at infinity. This generalizes multidimensional residue techniques developed in previous work, and introduces weighted projective space as a new tool in the theory of caustic singularities and gravitational lensing.{
©2011 American Institute of Physics}
©2011 American Institute of Physics}
MSC:
| 57R18 | Topology and geometry of orbifolds |
| 58K05 | Critical points of functions and mappings on manifolds |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 51F15 | Reflection groups, reflection geometries |
| 78A05 | Geometric optics |
| 83C50 | Electromagnetic fields in general relativity and gravitational theory |
| 83C57 | Black holes |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
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