Theory of Neutrino Flavor Mixing

The depth of our theoretical understanding of neutrino flavor mixing should match the importance of this phenomenon as a herald of long-awaited empirical challenges to the standard model of particle physics. After reviewing the familiar, simplified quantum mechanical model and its flaws, I sketch the deeper understanding of both vacuum and matter-enhanced flavor mixing that is found in the framework of scattering theory. While the simplified model gives the ``correct answer'' for atmospheric, solar, and accelerator/reactor neutrino phenomena, I argue that a key insight from the deeper picture will simplify the treatment of neutrino transport in astrophysical environments---supernovae, for example---in which neutrinos play a dynamically important role.Comment: 18 pages. Written contribution to the proceedings of ``Frontiers of Contemporary Physics--II,'' held March 5-10, 2001 at Vanderbilt University, Nashville, Tennesse

Conservative Formulations of General Relativistic Radiative Transfer

Accurate accounting of particle number and 4-momentum in radiative transfer may be facilitated by the use of transport equations that allow transparent conversion between volume and surface integrals in both spacetime and momentum space. Such conservative formulations of general relativistic radiative transfer in multiple spatial dimensions are presented, and their relevance to core-collapse supernova simulations described.Comment: 4 pages. Talk presented at MG10, Rio de Janeiro, Brazil, 20-26 July 2003. To be published in Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, edited by M. Novello, S. Perez-Bergliaffa, and R. Ruffini, World Scientific, Singapore, 200

Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3+1 Spacetime Perspective

A kinetic theory of classical particles serves as a unified basis for developing a geometric $3+1$ spacetime perspective on fluid dynamics capable of embracing both Minkowski and Galilei/Newton spacetimes. Parallel treatment of these cases on as common a footing as possible reveals that the particle four-momentum is better regarded as comprising momentum and \textit{inertia} rather than momentum and energy; and consequently, that the object now known as the stress-energy or energy-momentum tensor is more properly understood as a stress-\textit{inertia} or \textit{inertia}-momentum tensor. In dealing with both fiducial and comoving frames as fluid dynamics requires, tensor decompositions in terms of the four-velocities of observers associated with these frames render use of coordinate-free geometric notation not only fully viable, but conceptually simplifying. A particle number four-vector, three-momentum $(1,1)$ tensor, and kinetic energy four-vector characterize a simple fluid and satisfy balance equations involving spacetime divergences on both Minkowski and Galilei/Newton spacetimes. Reduced to a fully $3+1$ form, these equations yield the familiar conservative formulations of special relativistic and non-relativistic hydrodynamics as partial differential equations in inertial coordinates, and in geometric form will provide a useful conceptual bridge to arbitrary-Lagrange-Euler and general relativistic formulations.Comment: Belated upload of version accepted by MDPI Fluids. Additional material in the Introduction; added several tables and an additional appendi

Combining 3-momentum and kinetic energy on Galilei/Newton spacetime

Without the mass-energy equivalence available on Minkowski spacetime $\mathbb{M}$, it is not possible on 4-dimensional non-relativistic Galilei/Newton spacetime $\mathbb{G}$ to combine 3-momentum and total mass-energy in a single tensor object. However, given a fiducial frame, it is possible to combine 3-momentum and kinetic energy into a linear form (particle) or $(1,1)$ tensor (continuum) in a manner that exhibits increased unity of classical mechanics on flat relativistic and non-relativistic spacetimes $\mathbb{M}$ and $\mathbb{G}$. As on $\mathbb{M}$, for a material continuum on $\mathbb{G}$ the First Law of Thermodynamics can be considered a consequence of a unified dynamical law for energy-momentum rather than an independent postulate.Comment: Minor revisions and a change of Latex templat