Research

• General Relativistic Fluids
• Einstein-Yang/Mills Equations
• Einstein-Dirac-Maxwell Equations
• Einstein-Dirac-Yang/Mills Equations
• Compressible Navier-Stokes Equations
• Decay Results in the Kerr Geometry
• Compressible Fluids

Einstein-Dirac-Maxwell Equations

Very recently, I have been studying the coupling of Einstein's equations to the Dirac equation (ED equations) as well as to both the Dirac and Maxwell's equations (EDM equations). This couples quantum mechanics to gravity and electromagnetism.

In [22] we derive the coupled ED equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Using numerical methods, we construct an infinite number of soliton-like solutions. For weak coupling (i.e. small rest mass of the fermions), the solutions are linearly stable, whereas for stronger coupling, we show via topological methods, that the stable solution becomes unstable . Although gravitation is not renormalizable ( which means that the problem cannot be treated in a perturbation expansion), our solutions of the ED equations are regular and well-behaved even for strong coupling. For physical interpretations, we discuss how the energy of the fermions and the (ADM) mass behave as functions of the rest mass of the fermions.

The results of [22] are extended in [23] to the coupled EDM equations. Soliton-like solutions are constructed and properties of the solutions, as well as their stability are discussed, as functions of the electromagnetic coupling constant. We obtain solutions even when the electromagnetic coupling constant is so strong that the total interaction is repulsive in the Newtonian limit. Our solutions are regular and well-behaved; this shows that combined electromagnetic and gravitational self-interaction of the Dirac particles becomes finite when gravitation is taken into account.

It is shown analytically in [24] that the Dirac equation has no normalizable, time-periodic solutions in a Reissner-Nordstrom black hole background (Reissner-Nordstrom black holes have charge, and result from the coupling of gravity to electromagnetism); in particular, there are no static solutions of the Dirac equation in such a background field. This means that Dirac particles which are attracted to a Reissner-Nordstrom black hole can "fall into" the singularity, but cannot stay on a periodic orbit around the black hole. The result can also be applied to Dirac particles of matter in gravitational collapse, and indicates that all the matter must eventually disappear into the black hole. Our result is basically a consequence of the Heisenberg Uncertainty Principle and of the particular form of the Dirac current. It is essential for our methods and results that the particles have spin, and shows that spin is an important effect to be taken into account in the study of gravitational collapse.

In the paper [25], we consider for

a spherically symmetric, static system of (2j+ 1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein-Dirac-Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the event horizon, the only black hole solutions of the EDM equations are the Reissner-Nordstrom solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a "cloud" of Spin- -particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon.

In [29] we study the coupled Einstein-Dirac-Maxwell equations for a static spherically symmetric system of two fermions in a singlet spinor state. Stable soliton-like solutions are shown to exist, and we discuss the regularizing effect of gravity from a Feynman diagram point of view. (This paper received an "Honorable Mention" by the Gravity Research Foundation Competition, 1999.)

In [44] we show that for any black hole solution of the Einstein-Dirac-Maxwell equations, the Direac spinors cannot be normalized outside of the black hole.

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