A. Shadi Tahvildar-Zadeh

ON THE STATIC SPACETIME OF A SINGLE POINT CHARGE

Among all electromagnetic theories which (a) are derivable from a Lagrangian, (b) satisfy the dominant energy condition, and (c) in the weak field limit coincide with classical linear electromagnetics, we identify a certain subclass with the property that the corresponding spherically symmetric, asymptotically flat, electrostatic spacetime metric has the mildest possible singularity at its center, namely, a conical singularity on the time axis. The electric field moreover has a point defect on the time axis, its total energy is finite, and is equal to the ADM mass of the spacetime. By an appropriate scaling of the Lagrangian, one can arrange the total mass and total charge of these spacetimes to have any chosen values. For small enough mass-to-charge ratio, these spacetimes have no horizons and no trapped null geodesics. We also prove the uniqueness of these solutions in the spherically symmetric class, and we conclude by performing a qualitative study of the geodesics and test-charge trajectories of these spacetimes.

Integrability and vesture for harmonic maps into symmetric spaces

After giving the most general formulation to date of the notion of integrability for axially symmetric harmonic maps from R^3 into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular we show that the problem of finding N-solitonic harmonic maps into a noncompact Grassmann manifold SU(p,q)/S(U(p) x U(q)) is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method by explicitly computing a 1-solitonic harmonic map for the two cases (p = 1, q = 1) and (p = 2, q = 1); and we show that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr-Newman family of solutions to the Einstein-Maxwell equations.

On a zero-gravity limit of the Kerr–Newman spacetimes and their electromagnetic fields

We discuss the limit of vanishing

Dirac’s Point Electron in the Zero-Gravity Kerr–Newman World

G

Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations

(Newtons constant of universal gravitation) of the maximal analytically extended Kerr--Newman electrovacuum spacetimes {represented in Boyer--Lindquist coordinates}. We investigate the topologically nontrivial spacetime emerging in this limit and show that it consists of two copies of flat Minkowski spacetime glued at a timelike solid cylinder. As

On the regularity of spherically symmetric wave maps

G\to 0

Strichartz estimates for the wave and Schroedinger equations with potentials of critical decay

, the electromagnetic fields of the Kerr-Newman spacetimes converge to nontrivial solutions of Maxwells equations on this background spacetime. We show how to obtain these fields by solving Maxwells equations with singular sources supported only on a circle in a spacelike slice of the spacetime. These sources do not suffer from any of the pathologies that plague the alternate sources found in previous attempts to interpret the Kerr--Newman fields on the topologically simple Minkowski spacetime. We characterize the singular behavior of these sources and prove that the Kerr-Newman electrostatic potential and magnetic stream function are the unique solutions of the Maxwell equations among all functions that have the same blow-up behavior at the ring singularity.

On the cauchy problem for equivariant wave maps

The results of a study of the Dirac Hamiltonian for a point electron in the zero-gravity Kerr–Newman spacetime are reported; here, “zero-gravity” means G → 0, where G is Newton’s constant of universal gravitation, and the limit is effected in the Boyer–Lindquist coordinate chart of the maximal analytically extended, topologically nontrivial, Kerr–Newman spacetime. In a nutshell, the results are: the essential self-adjointness of the Dirac Hamiltonian; the reflection symmetry about zero of its spectrum; the location of the essential spectrum, exhibiting a gap about zero; and (under two smallness assumptions on some parameters) the existence of a point spectrum in this gap, corresponding to bound states of Dirac’s point electron in the electromagnetic field of the zero-G Kerr–Newman ring singularity. The symmetry result of the spectrum extends to the Dirac Hamiltonian for a point electron in a generalization of the zero-G Kerr–Newman spacetime with different ratio of electric-monopole to magnetic-dipole moment. The results are discussed in the context of the general-relativistic hydrogen problem. Also, some interesting projects for further inquiry are listed in the lastsection.

Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields

Motivated by integrability of the sine-Gordon equation, we investigate a technique for constructing desired solutions to Einsteins equations by combining a dressing technique with a control-theory approach. After reviewing classical integrability, we recall two well-known Killing field reductions of Einsteins equations, unify them using a harmonic map formulation, and state two results on the integrability of the equations and solvability of the dressing system. The resulting algorithm is then combined with an asymptotic analysis to produce constraints on the degrees of freedom arising in the solution-generation mechanism. The approach is carried out explicitly for the Einstein vacuum equations. Applications of the technique to other geometric field theories are also discussed.