The Mathematical Theory of Black Holes with Matter

广义相对论是引力的基本物理理论，在我们对宇宙的理解中起着至关重要的作用。广义相对论的关键方程源于爱因斯坦，而黑洞是它最令人惊讶的解决方案。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，对黑洞的研究取得了巨大的进步。它们预计会由于引力坍塌而形成，并被与之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探测到的那样)，并稳定在一个稳定的状态。大多数用于研究这种黑洞演化的数学模型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。这就是真空爱因斯坦方程的情况。尽管真空爱因斯坦方程从数学的角度已经提出了许多困难，但它们很难代表所涉及的物理的完整图景。为了获得天体物理黑洞的真实模型，应该在爱因斯坦方程中加入物质场来模拟黑洞的周围。本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和系统的方法来了解引力辐射与天体中存在的其他物质场的相互作用。我们计划考虑由麦克斯韦方程支配的引力场和电磁场之间的相互作用，并对它们的相互作用有一个严格而清晰的理解。我们的方法是基于特科尔斯基的形式主义。爱因斯坦-麦克斯韦方程与真空爱因斯坦方程有许多共同之处，但也提出了与引力和电磁相互作用耦合有关的新的实质性困难。困难之一是确定传输电磁辐射和引力辐射的规范不变量，并推导出它们所满足的偏微分方程式。然后，我们计划将处理这些相互作用的主要思想推广到其他物质系统，如爱因斯坦-弗拉索夫、零尘埃或复标量，这些物质系统在天体物理系统中具有基本的重要性。一般来说，在爱因斯坦方程与物质场相结合的情况下，我们希望获得具有相互作用的源的耦合双曲线偏微分方程组。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.