The Dynamics of Riemannian Geometries

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This research will explore the dynamics of curved Riemannian geometries in two different fields. The first is in general relativity where the formation and evolution of charged black holes will be studied in order to better understand the interaction of gravity with the electromagnetic field. The second is in botany where the dynamics of plant growth will employ a mathematical model based upon the equations of Ricci flow coupled with those describing material transport in the plant.******A static charged black hole has an infinite blue shifted null surface (Cauchy horizon) in its interior. The Cauchy horizon is known to be unstable to perturbations and the goal of this project is to understand the development of the structure of the interior of a charged black hole as it forms from the gravitational collapse of a charged fluid. The process will be governed by the time dependent Einstein-Maxwell equations using both analytic and computational methods. This will be approached in two ways, first the charged fluid will be allowed to fall into an uncharged black hole and the change in internal structure will be monitored. The second procedure will begin with a charged fluid with no black hole present and computational methods will be employed to follow the fluid as it collapses to form a black hole. Thus one can monitor formation of both the outer event horizon and the inner Cauchy horizon as it forms. The initial studies will take place in spherical symmetry but axisymmetric codes will be modified to account for both gravitational and electromagnetic radiation emitted during the collapse to determine the correspondence between the electromagnetic and gravitational observations of such events. This study will provide further knowledge of the dynamics of extremely strong gravitational fields.******The second project will continue to study the growth of leaves and petals using a newly developed geometric model that leads to a set of equations that govern how patterns in the curvature of a leaf or petal develop as it grows. Although the equations differ from those that appear in general relativity, they do have a great deal in common with evolving cosmological spacetimes. The model has been successfully applied to the growth of roots in plants such as corn, beans peas, etc. and in the growth of the algae Acetabularia where the curvature undergoes an overall change in sign.******The next step in this project will be to develop the theory further in a number of different directions. The first will be to add anisotropies to the the current model to understand how the ruffling patterns arise during the growth of leaves such as those of the lotus plant. Further studies will be made on ivy leaves where comparisons will be made with high precision growth measurements have been developed recently at UBC using ink jet printer technology. The possibility of employing the model in engineering applications will also be explored.**