Twisted Electromagnetic Modes and Sagnac Ring-Lasers

A new approximation scheme, designed to solve the covariant Maxwell equations inside a rotating hollow slender conducting cavity (modelling a ring-laser), is constructed. It is shown that for well-defined conditions there exist TE and TM modes with respect to the longitudinal axis of the cavity. A twisted mode spectrum is found to depend on the integrated Frenet torsion of the cavity and this in turn may affect the Sagnac beat frequency induced by a non-zero rotation of the cavity. The analysis is motivated by attempts to use ring-lasers to measure terrestrial gravito-magnetism or the Lense-Thirring effect produced by the rotation of the Earth.Comment: LaTeX 31 pages, 3 Figure

The Electrodynamics of Inhomogeneous Rotating Media and the Abraham and Minkowski Tensors II: Applications

Applications of the covariant theory of drive-forms are considered for a class of perfectly insulating media. The distinction between the notions of "classical photons" in homogeneous bounded and unbounded stationary media and in stationary unbounded magneto-electric media is pointed out in the context of the Abraham, Minkowski and symmetrized Minkowski electromagnetic stress-energy-momentum tensors. Such notions have led to intense debate about the role of these (and other) tensors in describing electromagnetic interactions in moving media. In order to address some of these issues for material subject to the Minkowski constitutive relations, the propagation of harmonic waves through homogeneous and inhomogeneous, isotropic plane-faced slabs at rest is first considered. To motivate the subsequent analysis on accelerating media two classes of electromagnetic modes that solve Maxwell's equations for uniformly rotating homogeneous polarizable media are enumerated. Finally it is shown that, under the influence of an incident monochromatic, circularly polarized, plane electromagnetic wave, the Abraham and symmetrized Minkowski tensors induce different time-averaged torques on a uniformly rotating materially inhomogeneous dielectric cylinder. We suggest that this observation may offer new avenues to explore experimentally the covariant electrodynamics of more general accelerating media.Comment: 29 pages, 4 figures. Accepted for publication in Proc. Roy. Soc.

Stratified Graphs

Two imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, the collection of imbeddings of G may be regarded as a "stratified graph", denoted SG. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the "kth stratum", and one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is proved that the stratified graph is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three and that the spanning subgraph of SG corresponding to moving only one edge-end is a cartesian product of graphs whose underlying isomorphism types depend only on the valence sequence for G

Stratified graphs for imbedding systems

AbstractTwo imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, a collection of imbeddings S of G, called a ‘system’, may be represented as a ‘stratified graph’, and denoted SG; the focus here is the case in which S is the collection of all orientable imbeddings. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the ‘kth stratum’, and the cardinality of that set of imbeddings is called the ‘stratum size’; one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is known that the genus distribution is not a complete invariant, even when the category of graphs is restricted to be simplicial and 3-connected. However, it is proved herein that the link of each point — that is, the subgraph induced by its neighbors — of SG is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three. This supports the plausibility of a probabilistic approach to graph isomorphism testing by sampling higher-order imbedding distribution data. A detailed structural analysis of stratified graphs is presented

Direct constraints on the dark matter self-interaction cross-section from the merging galaxy cluster 1E0657-56

We compare new maps of the hot gas, dark matter, and galaxies for 1E0657-56, a cluster with a rare, high-velocity merger occurring nearly in the plane of the sky. The X-ray observations reveal a bullet-like gas subcluster just exiting the collision site. A prominent bow shock gives an estimate of the subcluster velocity, 4500 km/s, which lies mostly in the plane of the sky. The optical image shows that the gas lags behind the subcluster galaxies. The weak-lensing mass map reveals a dark matter clump lying ahead of the collisional gas bullet, but coincident with the effectively collisionless galaxies. From these observations, one can directly estimate the cross-section of the dark matter self-interaction. That the dark matter is not fluid-like is seen directly in the X-ray -- lensing mass overlay; more quantitative limits can be derived from three simple independent arguments. The most sensitive constraint, sigma/m<1 cm^2/g, comes from the consistency of the subcluster mass-to-light ratio with the main cluster (and universal) value, which rules out a significant mass loss due to dark matter particle collisions. This limit excludes most of the 0.5-5 cm^2/g interval proposed to explain the flat mass profiles in galaxies. Our result is only an order-of-magnitude estimate which involves a number of simplifying, but always conservative, assumptions; stronger constraints may be derived using hydrodynamic simulations of this cluster.Comment: Text clarified; some numbers changed slightly for consistency with final version of the accompanying lensing paper. 6 pages, uses emulateapj. ApJ in pres