Mathematical methods and solutions in black hole astrophysics

Abstract

Black holes are nowadays widely accepted as the most likely explanation of a wide range of astrophysical phenomena from X-ray sources within our galaxy to powerfully radiating galactic cores billions of light-years away from us. The arguments that lead us to the conclusion that black holes of various masses indeed reside in such locations come from the observations in the electromagnetic spectrum and, since recently, also from gravitational-wave detections. In this thesis, I first take the reader on a brief guided tour of the essential physics of black holes and the accretion of matter onto them that leads to the observed electromagnetic signal. Then, I explore the possibility of deepening the understanding of these topics with analytical methods in the appended papers.

The topics of the papers go as follows. In the first paper (Witzany & Lämmerzahl, 2017), we have re-expressed exact relativistic equations of motion in general stationary space-times in a “pseudo-Newtonian” form, a form ready to be implemented in common Newtonian hydrodynamics codes while recovering all the essential properties of the evolution near a black hole. In the next paper (Markakis et al., 2017), we have explored various forms of equations of relativistic fluids and magnetized plasmas in curved space-time, deriving actions and Hamiltonians for fluid stream-lines, and corresponding conservation laws (such as the relativistic Kelvin theorem). In the third paper (Witzany, 2017), I have investigated whether one can use deeper geometrical properties of spinning black-hole fields to derive new conservation laws that could be used to constrain numerical simulations. The conclusion is that only weak conservation laws of the character of a differential constraint can be found. In the fourth paper (Witzany and Jefremov, 2018), we have found new closed solutions for idealized equilibria of fluids near black holes. These represent a considerable expansion of the set of closed-form prescriptions with which numerical simulations of accretion disks can be initialized.

Last but not least, one of the chapters of the thesis is dedicated to my research on the so-called spin-curvature coupling, which is experienced by rapidly rotating astrophysical objects in curved space-time. My work on these topics greatly improves the understanding of the Hamiltonian formalism that captures this coupling, and the results are likely to become important for gravitational-wave modelling.