Collisionless matter in general relativity

广义相对论中的无碰撞物质

• 批准号: EP/W005956/1
• 负责人: Martin Taylor
• 金额: $ 10.27万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW005956%2F1
• 关键词: Collisionless; matter; general; relativity

General relativity is a theory of gravity first proposed by Einstein in 1915. Despite being over 100 years old, many great mathematical challenges remain in understanding the theory and its governing equations -- the Einstein equations -- which, appropriately viewed, are a nonlinear system of hyperbolic partial differential equations. One particularly striking aspect of the theory, which is in sharp contrast to the preceding Newtonian theory of gravity, is that it is nontrivial in the vacuum, i.e. in the absence of any matter. Tremendous progress has been made on the Einstein equations in vacuum recent years. Progress in the presence of matter, however, has been considerably slower. The proposed research attempts to bridge this gap.Much of the mathematical study of general relativity is driven by the so called cosmic censorship conjectures, first formulated by Penrose, which posit that any singularities arising in the theory must have certain desirable properties. In the language of partial differential equations the conjectures can be viewed as statements of global existence and global uniqueness. A fruitful direction of study, which indeed constitutes much of the progress to date on the conjectures, involves exhibiting regimes, close to certain explicit stationary solutions, in which the conjectures hold.The slow progress in the presence of matter is due to the fact that one has to not only understand difficulties arising in the vacuum and difficulties arising from the matter equations alone, but also the new difficulties which arise due to the coupling of these difficulties. This proposal concerns a particularly simple type of matter, which nonetheless is regarded as physical and exhibits a wide range of new phenomena in general relativity, described by the Vlasov equation.

广义相对论是爱因斯坦于1915年首次提出的引力理论。尽管已经有100多年的历史，但在理解这个理论及其控制方程--爱因斯坦方程--方面仍然存在许多巨大的数学挑战，适当地看，爱因斯坦方程是一个非线性双曲型偏微分方程系统。这个理论的一个特别引人注目的方面，与先前的牛顿引力理论形成鲜明对比，是它在真空中是非平凡的，即在没有任何物质的情况下。近年来，真空中的爱因斯坦方程研究取得了很大进展。然而，在物质存在的情况下，进展要慢得多。广义相对论的数学研究大多是由彭罗斯首先提出的所谓的宇宙审查理论所驱动的，该理论认为理论中出现的任何奇点都必须具有某些理想的性质。在偏微分方程的语言中，这些解可以被看作是全局存在性和全局唯一性的陈述。一个富有成效的研究方向，这确实构成了迄今为止在该理论上取得的大部分进展，涉及到展示接近于某些显式定态解的区域，其中该理论成立。在物质存在下的缓慢进展是由于这样一个事实，即人们不仅要理解真空中产生的困难和物质方程本身产生的困难，而且还包括由于这些困难的耦合而产生的新困难。这一提议涉及一种特别简单的物质，但它仍然被认为是物理的，并在广义相对论中表现出广泛的新现象，由弗拉索夫方程描述。