The Mathematical Theory of Black Holes with Matter

黑洞与物质的数学理论

基本信息

批准号：
2128386
负责人：
Elena Giorgi
金额：
$ 11.96万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2128386&HistoricalAwards=false
关键词：
Mathematical Theory Black Holes Matter

项目摘要

General Relativity is the fundamental physical theory of gravity and has a role of primary importance in our understanding of the universe. The key equation of General Relativity is due to Einstein, and black holes are its most surprising solutions. Black holes occupy a central stage in our understanding of gravity and tremendous progress in their research has been accomplished in the past decades. They are expected to form as a result of gravitational collapse and are surrounded by matter with which they interact. They are expected to radiate energy away in the form of gravitational waves (as detected by LIGO) and settle to a stationary state. Most mathematical models used in the study of such evolution of black holes do not consider any matter or energy field present in the spacetime: more precisely, they only assume the presence of the gravitational field. This is called the case of the vacuum Einstein equation. Even though the vacuum Einstein equation already presents many difficulties from the mathematical point of view, they hardly represent a complete picture about the physics involved. In order to obtain a realistic model for astrophysical black holes, matter fields should be added to the Einstein equation to model the surrounding of the black holes. This research is aimed at the study of black hole stability both for the vacuum Einstein equation and for the coupled equations with electromagnetic radiation. The plan of this research is to create a rigorous and systematic approach to understand the interaction of gravitational radiation with other matter fields present in astrophysical objects. We plan to consider the interaction between gravitation and electromagnetic fields, governed by the Maxwell equations, and develop a rigorous and clear understanding of their interactions. Our approach is based on the Teukolsky formalism. The Einstein-Maxwell equation has many features in common with the vacuum Einstein equation, but also presents new substantial difficulties related to the coupling of the gravitational and electromagnetic interactions. One of the difficulties is to identify gauge-invariant quantities which transport electromagnetic and gravitational radiation and derive the partial differential equations they satisfy. We then plan to be able to generalize the main ideas in dealing with those interactions to other matter systems, like Einstein-Vlasov, null dust or complex scalar, which are of fundamental importance in astrophysical systems. In general, in the case of the Einstein equation coupled with matter fields, we expect to obtain coupled hyperbolic PDEs with sources which interact one with another.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

一般相对论是重力的基本物理理论，在我们对宇宙的理解中起主要重要性。一般相对性的关键方程是由于爱因斯坦引起的，黑洞是其最令人惊讶的解决方案。在过去的几十年中，黑洞占据了我们对重力和研究的巨大进步的中心阶段。预计它们会由于重力崩溃而形成，并被与之相互作用的物质所包围。预计它们将以重力波的形式辐射能量（由Ligo检测到），并定居至固定状态。黑洞演变的研究中使用的大多数数学模型都不认为时空中存在任何问题或能量场：更确切地说，它们仅假定引力场的存在。这称为真空爱因斯坦方程的情况。即使真空爱因斯坦方程从数学的角度出现了许多困难，但它们几乎不代表有关所涉及的物理学的完整图片。为了获得天体物理黑洞的现实模型，应将物质场添加到爱因斯坦方程中，以模拟黑洞的周围。这项研究旨在研究真空爱因斯坦方程和具有电磁辐射的耦合方程的黑洞稳定性。这项研究的计划是创建一种严格而系统的方法，以了解引力辐射与天体物理物体中存在的其他物质领域的相互作用。我们计划考虑重力与电磁场之间的相互作用，受麦克斯韦方程约束，并对它们的相互作用产生严格而清晰的了解。我们的方法基于Teukolsky形式主义。 Einstein-Maxwell方程在真空爱因斯坦方程中具有许多共同的特征，但也带来了与引力和电磁相互作用偶联有关的新的实质困难。困难之一是确定传输电磁和重力辐射的量规不变量，并得出所满足的部分微分方程。然后，我们计划能够将这些相互作用与其他物质系统（例如Einstein-Vlasov，Null Dust或复杂标量）等相互作用进行概括，这在天体物理系统中至关重要。通常，如果爱因斯坦方程与物质领域相结合，我们希望获得与源头耦合的双曲线PDE，这些PDE与另一个相互作用的PDE相互作用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来支持的。