The Mathematical Theory of Black Holes with Matter

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.

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