Excess Of Angle Research Articles

Hyperbolic Lunes

Summary The formula for the area of a hyperbolic triangle in terms of its angle defect is derived using hyperbolic lunes, in analogy with the argument using (elliptic) lunes to express the area of an elliptic triangle in terms of its angle excess. Several pedagogical features of this construction are then discussed.

Traversable Schwarzschild-like wormholes

In this paper we study relativistic static traversable wormhole solutions which are a slight generalization of Schwarzschild wormholes. In order to do this we assume a shape function with a linear dependence on the radial coordinate r. This linear shape function generates wormholes whose asymptotic spacetime is not flat: they are asymptotically locally flat, since in the asymptotic limit r rightarrow infty spacetimes exhibiting a solid angle deficit (or excess) are obtained. In particular, there exist wormholes which connect two asymptotically non-flat regions with a solid angle deficit. For these wormholes the size of their embeddings in a three-dimensional Euclidean space extends from the throat to infinity. A new phantom zero-tidal-force wormhole exhibiting such asymptotic is obtained. On the other hand, if a solid angle excess is present, the size of the wormhole embeddings depends on the amount of this angle excess, and the energy density is negative everywhere. We discuss the traversability conditions and study the impact of the beta -parameter on the motion of a traveler when the wormhole throat is crossed. A description of the geodesic behavior for the wormholes obtained is also presented.

Screw dislocation-induced influence of transverse modes on Hall conductivity

The Hall conductivity of an electron gas on an interface showing a topological defect called screw dislocation is investigated. This kind of defect induces a singular torsion on the medium which in turn induces transverse modes in the quantum Hall effect. It is shown that this topology decreases the plateaus’ widths and shifts the steps in the Hall conductivity to lower magnetic fields. The Hall conductivity is neither enhanced nor diminished by the presence of this kind of defect alone. We also consider the presence of two defects on a sample, a screw dislocation together with a disclination. For a specific value of deficit angle, there is a reduction in the Hall conductivity. For an excess of angle, the steps shift to higher magnetic fields and the Hall conductivity is enhanced. Our work could be tested only in common semiconductors but we think it opens a road to the investigation on how topological defects can influence other classes of Hall effect.

A Tale of Three Circles

Everyone knows that the sum of the angles of a triangle formed by three lines in the plane is 180 ◦ , but is this still true for curvilinear triangles formed by the arcs of three circles in the plane? We invite the reader to experiment enough to see that the angle sum indeed depends on the triangle, and that no general pattern is obvious. We give a complete analysis of the situation, showing along the way, we hope, what insights can be gained by approaching the problem from several points of view and at several levels of abstraction. We begin with an elementary solution using only the most basic concepts of Euclidean geometry. While it is direct and very short, this solution is not complete, since it works only in a special case. The key to another special case turns out to be a model of hyperbolic geometry, leading us to suspect that the various manifestations of the problem lie on a continuum of models of geometries with varying curvature. This larger geometric framework reveals many beautiful unifying themes and provides a single method of proof that completely solves the original problem. Finally, we describe a very simple formulation of the solution, whose proof relies on transformations of the plane, a fitting ending we think, since a transformation may be regarded as a change in one’s point of view. The background developed earlier informs our understanding of this new perspective, and allows us to give a purely geometric description of the transformations needed. For the reader who is unfamiliar with the classical noneuclidean geometries, in which the notions of line and distance are given new interpretations, we provide an overview that is almost entirely self-contained. Such a reader will be introduced to such things as angle excess, stereographic projection, and even a sphere of imaginary radius. For the reader who is familiar with the three classical geometries, we offer some new ways of looking at them, which we are confident will reveal some surprises.