The Dynamics of Riemannian Geometries

本研究将在两个不同的领域探讨弯曲黎曼几何的动力学。第一个是在广义相对论中，带电黑洞的形成和演化将被研究，以便更好地理解引力与电磁场的相互作用。第二种是植物学，在植物学中，植物生长的动力学将采用基于利玛窦流方程和描述植物中物质运输的方程的数学模型。******静态带电黑洞在其内部具有无限蓝移零面（柯西视界）。已知柯西视界对扰动是不稳定的，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缩形成的。该过程将由时间依赖的爱因斯坦-麦克斯韦方程控制，使用解析和计算方法。这将通过两种方式实现，首先，将允许带电流体落入不带电的黑洞，并监测其内部结构的变化。第二个步骤将从没有黑洞存在的带电流体开始，并采用计算方法跟踪流体坍缩形成黑洞的过程。因此，当外部事件视界和内部柯西视界形成时，我们可以同时监测它们的形成。最初的研究将以球对称进行，但将修改轴对称代码，以考虑坍缩期间发射的引力和电磁辐射，以确定此类事件的电磁和引力观测之间的对应关系。这项研究将提供关于极强引力场动力学的进一步知识。******第二个项目将继续研究叶子和花瓣的生长，使用一种新开发的几何模型，该模型可以得出一组方程，这些方程可以控制叶子或花瓣在生长过程中曲率的模式。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化的宇宙时空有很多共同之处。该模型已成功地应用于植物根部的生长，如玉米、豆类、豌豆等，以及藻类髋臼藻的生长，其中曲率经历了符号的整体变化。******该项目的下一步将是在许多不同的方向上进一步发展这一理论。首先是将各向异性添加到当前的模型中，以了解在叶子（如莲花）生长过程中褶皱图案是如何产生的。进一步的研究将在常春藤叶子上进行比较，将与UBC最近使用喷墨打印机技术开发的高精度生长测量进行比较。本文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.**