@phdthesis{Ma2018,
  author    = {Ma, Siyuan},
  title     = {Analysis of Teukolsky equations on slowly rotating Kerr spacetimes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-414781},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vi, 89},
  year      = {2018},
  abstract  = {In this thesis, we treat the extreme Newman-Penrose components of both the Maxwell field (s=±1) and the linearized gravitational perturbations (or "linearized gravity" for short) (s=±2) in the exterior of a slowly rotating Kerr black hole. Upon different rescalings, we can obtain spin s components which satisfy the separable Teukolsky master equation (TME). For each of these spin s components defined in Kinnersley tetrad, the resulting equations by performing some first-order differential operator on it once and twice (twice only for s=±2), together with the TME, are in the form of an "inhomogeneous spin-weighted wave equation" (ISWWE) with different potentials and constitute a linear spin-weighted wave system. We then prove energy and integrated local energy decay (Morawetz) estimates for this type of ISWWE, and utilize them to achieve both a uniform bound of a positive definite energy and a Morawetz estimate for the regular extreme Newman-Penrose components defined in the regular Hawking-Hartle tetrad. We also present some brief discussions on mode stability for TME for the case of real frequencies. This says that in a fixed subextremal Kerr spacetime, there is no nontrivial separated mode solutions to TME which are purely ingoing at horizon and purely outgoing at infinity. This yields a representation formula for solutions to inhomogeneous Teukolsky equations, and will play a crucial role in generalizing the above energy and Morawetz estimates results to the full subextremal Kerr case.},
  language  = {en}
}
@phdthesis{Paganini2018,
  author    = {Paganini, Claudio Francesco},
  title     = {The role of trapping in black hole spacetimes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-414686},
  school      = {Universit{\"a}t Potsdam},
  pages     = {v, 138},
  year      = {2018},
  abstract  = {In the here presented work we discuss a series of results that are all in one way or another connected to the phenomenon of trapping in black hole spacetimes. First we present a comprehensive review of the Kerr-Newman-Taub-NUT-de-Sitter family of black hole spacetimes and their most important properties. From there we go into a detailed analysis of the bahaviour of null geodesics in the exterior region of a sub-extremal Kerr spacetime. We show that most well known fundamental properties of null geodesics can be represented in one plot. In particular, one can see immediately that the ergoregion and trapping are separated in phase space. We then consider the sets of future/past trapped null geodesics in the exterior region of a sub-extremal Kerr-Newman-Taub-NUT spacetime. We show that from the point of view of any timelike observer outside of such a black hole, trapping can be understood as two smooth sets of spacelike directions on the celestial sphere of the observer. Therefore the topological structure of the trapped set on the celestial sphere of any observer is identical to that in Schwarzschild. We discuss how this is relevant to the black hole stability problem. In a further development of these observations we introduce the notion of what it means for the shadow of two observers to be degenerate. We show that, away from the axis of symmetry, no continuous degeneration exists between the shadows of observers at any point in the exterior region of any Kerr-Newman black hole spacetime of unit mass. Therefore, except possibly for discrete changes, an observer can, by measuring the black holes shadow, determine the angular momentum and the charge of the black hole under observation, as well as the observer's radial position and angle of elevation above the equatorial plane. Furthermore, his/her relative velocity compared to a standard observer can also be measured. On the other hand, the black hole shadow does not allow for a full parameter resolution in the case of a Kerr-Newman-Taub-NUT black hole, as a continuous degeneration relating specific angular momentum, electric charge, NUT charge and elevation angle exists in this case. We then use the celestial sphere to show that trapping is a generic feature of any black hole spacetime. In the last chapter we then prove a generalization of the mode stability result of Whiting (1989) for the Teukolsky equation for the case of real frequencies. The main result of the last chapter states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation, and was recently used by Shlapentokh-Rothman (2015) for the scalar wave equation.},
  language  = {en}
}