This research project studies the question of singularity formation in nonlinear dynamical theories of relativistic electromagnetic fields, both at the classical and the quantum level.

The classical part is concerned with two main problems: (1) Analysis of the well-known relativistic Vlasov-Maxwell equations, which had been conjectured to be globally well-posed as a Cauchy problem with suitable finite-energy classical data. This project will rigorously analyze a recently-developed scenario for a counterexample exhibiting finite-time collapse for a family of solutions. (2) Analysis of the nonlinear Maxwell-Born-Infeld equations for electromagnetic fields in the absence of point charges. This project will investigate recently-discovered spatially periodic plane wave solutions that exhibit finite-time blow-up to determine whether the corresponding set of Cauchy data is part of a generic bad set.

The quantum part of the research project concerns solutions of the Maxwell-Born-Infeld field equations with point defects that represent particles. The charged particles move according to a quantum velocity field obtained from a many body Dirac formalism coupled to generic electromagnetic fields. This project will study whether the Dirac Hamiltonian for the system can become unbounded below.

This project addresses fundamental issues in the theory of electromagnetism, which is central to modern science and engineering. The study of singularity formation has long been at the forefront of research in general relativity and in fluid dynamics. Recent discoveries suggest that singularities may also pose a major conceptual challenge in the nonlinear electromagnetic models that have been proposed as candidates for a consistent formulation of an electromagnetic theory without artificial regularizers. The principal investigator recently developed a consistent formulation of electromagnetic theory, incorporating intrinsic spin of particles, that is consistent at both classical and quantum levels. The current project investigates possible singularity formation in this theory. Another part of this work examines finite-time collapse in the relativistic Vlasov-Maxwell model. Such collapse would provide a novel mechanism for the formation of very small celestial bodies whose gravitational self-attraction is too weak to aid in their formation. This project aims to establish that possibility; the results could have a major impact on theories of planetary system formation.

Project Report

The grantee is a mathematical physicist working mostly on problems in statistical physics (which is concerned with the derivation of macroscopic phenomena from microscopic models), and on fundamental questions in the theory of electromagnetism. The equations of physics have predictive power as long as their solutions remain finite. Therefore it is important to understand whether solutions can develop some kind of infinities, known as "formation of singularities." This project has been concerned with the formation of singularities in some models of electromagnetism compatible with Einstein's relativity theory. The accomplishments that were funded by this grant include the following: * two graduate students (Brent Young and Yu Wang) successfully completed their Ph.D. thesis research under the grantee's supervision. Young, who now is a postdoc at Cologne University, worked on the formation of singularities in relativistic Vlasov-Poisson and -Maxwell equationswith spherical symmetry; his results are potentially relevant for understanding the formationof small celestial objects; two publications appeared, more are in preparation. Wang, who justdefended her thesis successfully, worked on charged particle systems on the sphere. One publication appeared, more are in preparation. * a postdoc (Shabnam Beheshti) was mentored by the grantee and his colleague Tahvildar-Zadeh, leading to the most general result so far of the explicit solvability of "harmonic maps into symmetric spaces;" Einstein's spacetime equations coupled to Maxwell's electromagnetic field equations define such harmonic maps whenever their solutions have two continuous symmetries (for instance, stationarity and axisymmetry). One manuscript went through the peer review process and (at the time of this writing) is awaiting final approval for publication. Beheshti is currently a clinical assistant professor in the Rutgers mathematics department. * The grantee proved that a critical voltage required to form singular so-called "struts" between point charges in the electrostatic solutions to the nonlinear Maxwell-Born-Infeld field equationscan never be reached. A publication of this result has appeared last year. (Max Born and Leo Infeldproposed this nonlinear field theory 80 years ago to avoid an infinite-energy problem for point charges in the linear field theory of Maxwell. In modern times a variant of the model has surfaced in string theory.) * Jared Speck (a former Ph.D. student jointly of the grantee and Tahvildar-Zadeh, and now at MIT) proved that singularities in the field solutions do not form in the absence of point charges, provided the initial fields are sufficiently small. This result already appeared in publication. (In contrast to the linear field equations of Maxwell, the nonlinear Maxwell-Born-Infeld field equations do have some solutions that form singularities in finite time; Speck's result shows that a critical threshold has to be surpassed for this to happen.) * During the grant period the Brookhaven National Laboratory announced the detection of about two dozen anti-alpha particles produced in relativistic high energy collisions of atomic nuclei. The grantee realized that anti-alpha particles could eventually lead to the creation in the lab of a fleeting form of matter, called "bosonic atoms," that was hitherto thought impossible to exist in the real world. This idea is discussed in two publications of the grantee on the quantum ground state energy of such matter.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Henry A. Warchall
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Rutgers University
New Brunswick
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