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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：
2006741
负责人：
Elena Giorgi
金额：
$ 11.96万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-15 至 2021-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006741&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.

广义相对论是引力的基本物理理论，在我们理解宇宙中起着至关重要的作用。广义相对论的关键方程是爱因斯坦提出的，而黑洞是它最令人惊讶的解。黑洞在我们对引力的理解中占据着中心地位，在过去的几十年里，黑洞的研究取得了巨大的进展。它们被认为是引力坍缩的结果，并被与它们相互作用的物质所包围。预计它们将以引力波的形式辐射能量（如LIGO所探测到的那样），并稳定在一个稳定的状态。大多数用于研究黑洞演化的数学模型都没有考虑时空中存在的任何物质或能量场：更准确地说，它们只假设引力场的存在。这就是所谓的真空爱因斯坦方程。尽管真空爱因斯坦方程已经从数学的角度提出了许多困难，但它们很难代表所涉及的物理学的全貌。为了得到一个真实的天体物理黑洞模型，需要在爱因斯坦方程中加入物质场来模拟黑洞的周围环境。本研究的目的是研究真空爱因斯坦方程和电磁辐射耦合方程的黑洞稳定性。这项研究的计划是建立一个严格和系统的方法来理解引力辐射与天体物理对象中存在的其他物质场的相互作用。我们计划考虑引力和电磁场之间的相互作用，由麦克斯韦方程组，并制定一个严格和明确的理解他们之间的相互作用。我们的方法是基于Teukolsky形式主义。爱因斯坦-麦克斯韦方程与真空爱因斯坦方程有许多共同的特征，但也提出了新的实质性困难，涉及到引力和电磁相互作用的耦合。其中一个困难是确定规范不变的量，传输电磁和引力辐射，并推导出它们满足的偏微分方程。然后，我们计划能够将处理这些相互作用的主要思想推广到其他物质系统，如Einstein-Vlasov，空尘或复杂标量，这些在天体物理系统中具有根本重要性。在一般情况下，在爱因斯坦方程与物质场耦合的情况下，我们希望获得耦合双曲偏微分方程与源相互作用another.This奖反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。