Arlie Petters - Research

Research

Petters is renowned for his pioneering work in the mathematical theory of gravitational lensing.

Over the ten-year period from 1991–2001, Petters systematically developed a mathematical theory of weak-deflection gravitational lensing, beginning with his 1991 MIT Ph.D. thesis on "Singularities in Gravitational Microlensing" and followed by the 12 papers - below. The papers resolved an array of theoretical problems in weak-deflection gravitational lensing covering image counting, fixed-point images, image magnification, image time delays, local geometry of caustics, global geometry of caustics, wavefronts, caustic surfaces, and caustic surfing. His work culminated with a 2001 mathematical tome that, among other things, systematically created a framework of stability and genericity for k-plane gravitational lensing. The book drew upon powerful tools from the theory of singularities and put the subject of weak-deflection k-plane gravitational lensing on a rigorous and unified mathematical foundation.

Following his 1991-2001 body of mathematical lensing work, Petters turned to more astrophysical lensing issues from 2002-2005. In collaboration with astronomers, he applied some of the mathematical theory in to help develop a practical diagnostic test for the presence of dark substructures in galaxies lensing quasars; classify the local astrometric (centroid) and photometric curves of an extended source when it crosses fold and cusp caustics due to generic lenses; predict the quantitative astrometric curve's shape produced by Galactic binary lenses. The classified local properties of the astrometric curves revealed a characteristic S-shape for fold crossings, parabolic and swallowtail features for cusp crossings, and a jump discontinuity for crossings over the fold arcs merging into a cusp. A formula for the size of the jump was also found.

During 2005-2007, Petters collaborated with astronomers and physicists to explore gravitational lensing in directions beyond its traditional confines in astronomy. In a series of three mathematical physics papers (2005–2006) with the astronomer Keeton, he utilized higher-order gravitational lensing effects by compact bodies to test different theories of gravity with Einstein's general theory of relativity among them. The first two papers computed beyond the standard weak-deflection limit the first- and second-order corrections to the image positions, magnifications, and time delays due to lensing in general relativity and alternative gravitational theories describable within the PPN formalism, and even determined lensing invariants for the PPN family of models. Their findings were applied to the Galactic black hole, binary pulsars, and gravitational microlensing scenarios to make testable predictions about lensed images and their time delays. The third paper took on the difficult issue of how to test hyperspace models like braneworld gravity that postulate an extra dimension to physical space. The paper developed a semi-classical wave theory of braneworld black hole lensing and used that theory along with braneworld cosmology to predict a testable signature of microscopic braneworld black holes on gamma-ray light. Additionally, in a 2007 paper, Petters and Werner found a system of equations that can be applied to test the Cosmic Censorship Hypothesis observationally using the realistic case of lensing by a Kerr black hole.

Petters's previous work (1991–2007) dealt with non-random gravitational lensing. His recent research program (2008–present) has been to develop a mathematical theory of random (stochastic) gravitational lensing. In two papers, he, Rider, and Teguia took first steps in creating a mathematical theory of stochastic gravitational microlensing. They characterized to several asymptotic orders the probability densities of random time delay functions, lensing maps, and shear maps in stochastic microlensing and determined a Kac-Rice type formula for the global expected number of images due to a general stochastic lens system. The work forms a concrete framework from which extensions to more general random maps can be made. In two additional papers, he and Aazami found geometric universal magnification invariants of higher-order caustics occurring in lensing and caustics produced by generic general maps up to codimension five. The invariants hold with a probability of 1 for random lenses and thereby form important consistency checks for research on random image magnifications of sources near stable caustics.

For more information, consult Petters's official Duke University CV for a very useful road map with detailed and extensive summaries of his research papers.

Selected papers from 1991–2001:

"Morse Theory and Gravitational Microlensing," A. O. Petters, J. Math. Phys., 33, 1915 (1992).

"Arnold's Singularity Theory and Gravitational Lensing," A. O. Petters, J. Math. Phys., 34, 3555 (1993).

"Multiplane Gravitational Lensing I: Morse Theory and Image Counting," A. O. Petters, J. Math. Phys., 36, 4263 (1995).

"Multiplane Gravitational Lensing II: Global Geometry of Caustics," A. O. Petters, J. Math. Phys., 36, 4276 (1995).

"Multiplane Gravitational Lensing III. Upper Bound on Number of Images," A. O. Petters, J. Math. Phys., 38, 1605 (1997).

"Caustics of the Double-Plane Two Point-Mass Gravitational Lens with Continuous Matter and Shear," A. O. Petters and F.J. Wicklin, Mon. Not. R. Astron. Soc., 277, 1399 (1995).

"Lower Bounds on Image Magnification in Gravitational Lensing," A. O. Petters, Proc. R. Soc. Lond. A, 452, 1475 (1996).

"Counting Formulas and Bounds on Number of Fixed Points due to Point-Mass Lenses," A. O. Petters and F.J. Wicklin, in Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini (World Scientific, Singapore, 1997).

"Bounds on Number of Cusps due to Point Mass Gravitational Lenses with Continuous Matter and Shear," A. O. Petters and H. Witt, J. Math. Phys., 37, 2920 (1996).

"Mathematical Aspects of Gravitational Lensing," A. O. Petters, in Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity, vol. B, eds. R. T. Jantzen and G. M. Keiser (World Scientific, Singapore, 1996).

"Fixed Points due to Gravitational Lenses," A.O. Petters and F.J. Wicklin, J. Math. Phys., 39, 1011 (1998)

"Stable Lens Systems, Lensed Image Magnification, and Magnification Cross Sections," A. O. Petters, in Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity, eds. V. Gurzadyan, R. T. Jantzen, and R. Ruffini (World Scientific, Singapore, 2001).

Singularity Theory and Gravitational Lensing, A. O. Petters, H. Levine, and J. Wambsganns (Birkhauser, Boston, 2001)