The Dynamics of Riemannian Geometries

黎曼几何的动力学

• 批准号: RGPIN-2017-04123
• 负责人: Hobill, David
• 金额: $ 1.53万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710895
• 关键词: Dynamics; Riemannian; Geometries

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.

本研究将探讨两个不同领域的弯曲黎曼几何的动力学。第一个是在广义相对论中，将研究带电黑洞的形成和演化，以更好地理解引力与电磁场的相互作用。第二种是在植物学中，植物生长的动力学将采用基于Ricci流方程的数学模型，再加上描述植物中物质运输的方程。 一个静态带电黑洞在其内部有一个无限蓝移的零面（柯西视界）。 已知柯西视界对扰动不稳定，该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。 该过程将由时间相关的爱因斯坦-麦克斯韦方程组使用分析和计算方法。这将通过两种方式来实现，首先，带电流体将被允许落入不带电的黑洞，内部结构的变化将被监测。第二个过程将开始与带电流体没有黑洞的存在和计算方法将被用来跟踪流体，因为它崩溃，形成一个黑洞。因此，我们可以在柯西视界形成时，同时监测外视界和内视界的形成。最初的研究将在球对称中进行，但轴对称代码将被修改，以考虑到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。这项研究将为极强引力场的动力学提供进一步的知识。 第二个项目将继续研究叶片和花瓣的生长，使用新开发的几何模型，导致一组方程，控制叶片或花瓣的曲率模式如何随着生长而发展。虽然这些方程与广义相对论中出现的方程不同，但它们确实与演化中的宇宙时空有很多共同之处。该模型已成功地应用于根的生长在植物，如玉米，豌豆等，并在生长的藻类伞藻的曲率经历了一个整体的符号变化。 该项目的下一步将是在许多不同的方向上进一步发展该理论。 第一个是在现有模型中加入各向异性，以了解荷叶等植物的叶子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上进行，其中将与最近在UBC使用喷墨打印机技术开发的高精度生长测量进行比较。还将探讨在工程应用中采用该模型的可能性。