Drawing with Complex Numbers

It is not commonly realized that the algebra of complex numbers can be used in an elegant way to represent the images of ordinary 3-dimensional figures, orthographically projected to the plane. We describe these ideas here, both using simple geometry and setting them in a broader context.Comment: 14 pages, 5 figure

La Cohomologie des Figures Imposibles

On explique ici le lien étroit entre certains types de figures impossibles et la notion mathématique de cohomologie en relation avec la tripoutre et avec un autre type de figures impossibles lié au cube de Necker.The close relationship between certain types of impossible figure and the mathematical idea of cohomology is explained in relation to the tribar and to another type of impossible figure related to the Necker cube.Peer Reviewe

Exploring the unification of quantum theory and general relativity with a Bose-Einstein condensate

Despite almost a century's worth of study, it is still unclear how general relativity (GR) and quantum theory (QT) should be unified into a consistent theory. The conventional approach is to retain the foundational principles of QT, such as the superposition principle, and modify GR. This is referred to as `quantizing gravity', resulting in a theory of `quantum gravity'. The opposite approach is `gravitizing QT' where we attempt to keep the principles of GR, such as the equivalence principle, and consider how this leads to modifications of QT. What we are most lacking in understanding which route to take, if either, is experimental guidance. Here we consider using a Bose-Einstein condensate (BEC) to search for clues. In particular, we study how a single BEC in a superposition of two locations could test a gravitizing QT proposal where wavefunction collapse emerges from a unified theory as an objective process, resolving the measurement problem of QT. Such a modification to QT due to general relativistic principles is testable near the Planck mass scale, which is much closer to experiments than the Planck length scale where quantum, general relativistic effects are traditionally anticipated in quantum gravity theories. Furthermore, experimental tests of this proposal should be simpler to perform than recently suggested experiments that would test the quantizing gravity approach in the Newtonian gravity limit by searching for entanglement between two massive systems that are both in a superposition of two locations.Comment: 51 pages, 10 figure

Polarization modes for strong-field gravitational waves

Strong-field gravitational plane waves are often represented in either the Rosen or Brinkmann forms. These forms are related by a coordinate transformation, so they should describe essentially the same physics, but the two forms treat polarization states quite differently. Both deal well with linear polarizations, but there is a qualitative difference in the way they deal with circular, elliptic, and more general polarization states. In this article we will describe a general algorithm for constructing arbitrary polarization states in the Rosen form.Comment: 4 pages. Prepared for the proceedings of ERE2010 (Granada, Spain

Stephen Hawking

Tributes poured in on Wednesday to, the brightest star in the firmament of science, whose insights shaped modern cosmology and inspired global audiences in the millions. He died at the age of 76 in the early hours of Wednesday morning. In a statement that confirmed his death at home in Cambridge, Hawking’s children said: “We are deeply saddened that our beloved father passed away today. He was a great scientist and an extraordinary man whose work and legacy will live on for many years. His courage and persistence with his brilliance and humour inspired people across the world

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.Comment: 38 pages, 8 figures, late