## 58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15)

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#### Keywords

- Cauchy horizon (1)
- Cauchyhorizont (1)
- Gravitationswelle (1)
- Moden Stabilität (1)
- Schatten eines Schwarzen Lochs (1)
- Schwarzes Loch (1)
- Teukolsky Gleichung (1)
- Teukolsky master equation (1)
- Wellengleichung (1)
- black hole (1)

#### Institute

The Cauchy problem for the linearised Einstein equation and the Goursat problem for wave equations
(2017)

In this thesis, we study two initial value problems arising in general relativity. The first is the Cauchy problem for the linearised Einstein equation on general globally hyperbolic spacetimes, with smooth and distributional initial data. We extend well-known results by showing that given a solution to the linearised constraint equations of arbitrary real Sobolev regularity, there is a globally defined solution, which is unique up to addition of gauge solutions. Two solutions are considered equivalent if they differ by a gauge solution. Our main result is that the equivalence class of solutions depends continuously on the corre- sponding equivalence class of initial data. We also solve the linearised constraint equations in certain cases and show that there exist arbitrarily irregular (non-gauge) solutions to the linearised Einstein equation on Minkowski spacetime and Kasner spacetime.
In the second part, we study the Goursat problem (the characteristic Cauchy problem) for wave equations. We specify initial data on a smooth compact Cauchy horizon, which is a lightlike hypersurface. This problem has not been studied much, since it is an initial value problem on a non-globally hyperbolic spacetime. Our main result is that given a smooth function on a non-empty, smooth, compact, totally geodesic and non-degenerate Cauchy horizon and a so called admissible linear wave equation, there exists a unique solution that is defined on the globally hyperbolic region and restricts to the given function on the Cauchy horizon. Moreover, the solution depends continuously on the initial data. A linear wave equation is called admissible if the first order part satisfies a certain condition on the Cauchy horizon, for example if it vanishes. Interestingly, both existence of solution and uniqueness are false for general wave equations, as examples show. If we drop the non-degeneracy assumption, examples show that existence of solution fails even for the simplest wave equation. The proof requires precise energy estimates for the wave equation close to the Cauchy horizon. In case the Ricci curvature vanishes on the Cauchy horizon, we show that the energy estimates are strong enough to prove local existence and uniqueness for a class of non-linear wave equations. Our results apply in particular to the Taub-NUT spacetime and the Misner spacetime. It has recently been shown that compact Cauchy horizons in spacetimes satisfying the null energy condition are necessarily smooth and totally geodesic. Our results therefore apply if the spacetime satisfies the null energy condition and the Cauchy horizon is compact and non-degenerate.

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.