A Visualization of Null Geodesics for the Bonnor Massive Dipole

In this work we simulate null geodesics for the Bonnor massive dipole metric by implementing a symbolic-numerical algorithm in Sage and Python. This program is also capable of visualizing in 3D, in principle, the geodesics for any given metric. Geodesics are launched from a common point, collectively forming a cone of light beams, simulating a solid-angle section of a point source in front of a massive object with a magnetic field. Parallel light beams also were considered, and their bending due to the curvature of the space-time was simulated.Comment: 9 pages, 5 figures, Presented at XIX Simposio Internacional de M\'etodos Matem\'aticos Aplicados a las Ciencias (19th International Symposium of Mathematical Methods Applied to Sciences, XIX-SIMMAC

Uncertainty in phylogenetic tree estimates

Estimating phylogenetic trees is an important problem in evolutionary biology, environmental policy and medicine. Although trees are estimated, their uncertainties are discarded by mathematicians working in tree space. Here we explicitly model the multivariate uncertainty of tree estimates. We consider both the cases where uncertainty information arises extrinsically (through covariate information) and intrinsically (through the tree estimates themselves). The importance of accounting for tree uncertainty in tree space is demonstrated in two case studies. In the first instance, differences between gene trees are small relative to their uncertainties, while in the second, the differences are relatively large. Our main goal is visualization of tree uncertainty, and we demonstrate advantages of our method with respect to reproducibility, speed and preservation of topological differences compared to visualization based on multidimensional scaling. The proposal highlights that phylogenetic trees are estimated in an extremely high-dimensional space, resulting in uncertainty information that cannot be discarded. Most importantly, it is a method that allows biologists to diagnose whether differences between gene trees are biologically meaningful, or due to uncertainty in estimation.Comment: Final version accepted to Journal of Computational and Graphical Statistic

Godel spacetime: elliptic-like geodesics and gyroscope precession

We study elliptic-like geodesic motion on hyperplanes orthogonal to the cylindrical symmetry axes of the Godel spacetime by using an eccentricity-semi-latus rectum parametrization which is familiar from the Newtonian description of a two-body system. We compute several quantities which summarize the main features of the motion, namely the coordinate time and proper time periods of the radial motion, the frequency of the azimuthal motion, the full variation of the azimuthal angle over a period, etc. Exact as well as approximate (i.e., Taylor-expanded in the limit of small eccentricity) analytic expressions of all these quantities are obtained. Finally, we consider their application to the gyroscope precession frequency along these orbits, generalizing the existing results for the circular case.Comment: 14 pages, 7 figures; revtex macro

Spacetime Embedding Diagrams for Black Holes

We show that the 1+1 dimensional reduction (i.e., the radial plane) of the Kruskal black hole can be embedded in 2+1 Minkowski spacetime and discuss how features of this spacetime can be seen from the embedding diagram. The purpose of this work is educational: The associated embedding diagrams may be useful for explaining aspects of black holes to students who are familiar with special relativity, but not general relativity.Comment: 22 pages, 21 figures, RevTex. To be submitted to the American Journal of Physics. Experts will wish only to skim appendix A and to look at the pictures. Suggested Maple code is now compatible with MapleV4r