The Dynamics of Riemannian Geometries

这项研究将探索曲线黎曼几何在两个不同领域的动力学。第一个是广义相对论，它将研究带电黑洞的形成和演化，以便更好地了解引力与电磁场的相互作用。第二个是植物学，植物生长的动力学将采用一个数学模型，该模型基于描述植物中物质运输的Ricci流动方程。 静态带电黑洞的内部有一个无限的蓝移零面(柯西视界)。众所周知，柯西视界对微扰是不稳定的，这个项目的目标是了解带电黑洞内部结构的发展，因为它是由带电流体的引力坍塌形成的。这个过程将由时间相关的爱因斯坦-麦克斯韦方程控制，使用分析和计算方法。这将通过两种方式实现，第一，带电液体将被允许落入不带电的黑洞，并将监测内部结构的变化。第二个过程将从不存在黑洞的带电流体开始，当流体坍塌形成黑洞时，将使用计算方法来跟踪它。因此，当外事件视界和内柯西视界形成时，都可以监测它的形成。最初的研究将在球对称的情况下进行，但轴对称代码将被修改，以考虑在坍塌期间发射的引力和电磁辐射，以确定对此类事件的电磁观测和引力观测之间的对应关系。这项研究将提供有关极强引力场动力学的进一步知识。 第二个项目将继续使用新开发的几何模型来研究叶子和花瓣的生长，该模型导致了一系列方程，这些方程控制了叶子或花瓣的曲率模式如何随着生长而发展。尽管这些方程与广义相对论中出现的方程不同，但它们确实与不断演变的宇宙时空有许多共同之处。该模型已成功地应用于玉米、豆类、豌豆等植物的根的生长以及曲率发生整体变化的藻类的生长。 这个项目的下一步将是在几个不同的方向上进一步发展这一理论。第一种是将各向异性添加到当前的模型中，以了解在荷叶等植物的叶子生长过程中如何产生褶皱图案。将对常春藤叶子进行进一步的研究，并与UBC最近开发的使用喷墨打印机技术的高精度生长测量进行比较。文中还将探讨该模型在工程应用中的可能性。

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.