Travel Grant for the International Conference on "Finsler Extension of Relativity Theory"

“芬斯勒相对论扩展”国际会议旅费补助

• 批准号: 0646826
• 负责人: Pit-Mann Wong
• 金额: $ 2.5万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-12-15 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0646826&HistoricalAwards=false
• 关键词: Travel; Grant; International; Conference; Finsler

Classical general relativity is built in large part upon the extension of Riemannian Geometry to Lorentzian Geometry (more generally, semi-Riemannian Geometry). In fact, time independent questions are basically Riemannian in nature. For examples the classification of black holes is based on time independent solutions of the field equations; the (first) proof of the positive mass theorem is also based on Riemannian Geometry. At present researchers are interested in issues involved in the rigorous extension and generalization of the principal geometric and physical structures associated with the spaces of classical and relativistic physics (e.g. of Riemannian geometry) to Finsler spaces. One of the problems of basing Relativity on Riemannian Geometry is the difficulty with "casualty". This is due to the fact that geodesics in Riemannian Geometry are reversible. If a space-time admits a CTC (closed time-like curve) then one could travel back in time and all kinds of (murderous) paradoxes could occur. In contrast, there exist Finsler metrics for which the geodesics are irreversible hence, provides a more natural setting for dealing with "casualty" in Relativity. There are many issues, more technical in nature, involving Finsler Geometry that are useful in Relativity. Among these are: (a) spaces whose metric functions are symmetric polynomials depending on four variables of the third order (known as Chernov spaces in Finsler Geometry), (b) spaces whose metric functions are symmetric polynomials depending on four variables of the fourth order (known as Berwald-Moore spaces in Finsler Geometry), (c) multi-linear symmetric forms as Finsler extension of scalar product, (d) connection between Finsler spaces and hypercomplex numbers, (e) possibility of extending, in the Finsler setting, the Lorentzian splitting theorem, originally conjectured by S. T. Yau as the natural extension of the Riemannian splitting theorem of Cheeger and Gromov.

经典广义相对论在很大程度上建立在黎曼几何到洛伦兹几何（更一般地说，半黎曼几何）的扩展之上。事实上，时间无关问题基本上是黎曼性质的。例如，黑洞的分类是基于场方程的与时间无关的解;正质量定理的（第一个）证明也是基于黎曼几何。目前，研究人员感兴趣的问题涉及严格的扩展和推广的主要几何和物理结构与空间的经典和相对论物理（如黎曼几何）芬斯勒空间。把相对论建立在黎曼几何基础上的问题之一是难以处理“偶然性”。这是因为黎曼几何中的测地线是可逆的。如果一个时空允许一个封闭的类时曲线（CTC），那么人们可以回到过去，各种各样的（致命的）悖论都可能发生。相比之下，存在芬斯勒度量，其中测地线是不可逆的，因此，提供了一个更自然的设置来处理相对论中的“伤亡”。有许多问题，更多的技术性质，涉及芬斯勒几何是有用的相对论。其中包括：（a）度量函数是依赖于四个三阶变量的对称多项式的空间（在Finsler几何中被称为Besov空间），（B）空间，其度量函数是依赖于四阶变量的对称多项式（在Finsler几何中称为Berwald-Moore空间），（c）作为标量积的Finsler扩张的多线性对称形式，（d）Finsler空间与超复数之间的联系;（e）在Finsler背景下推广最初由S. T. Yau是Cheeger和Gromov的黎曼分裂定理的自然推广。