On the initial value problem in general relativity and wave propagation in black-hole spacetimes
The first part of this thesis is concerned with the question of global uniqueness of solutions to the initial value problem in general relativity. In 1969, Choquet-Bruhat and Geroch proved, that in the class of globally hyperbolic Cauchy developments, there is a unique maximal Cauchy development. The original proof, however, has the peculiar feature that it appeals to Zorn’s lemma in order to guarantee the existence of this maximal development; in particular, the proof is not constructive. In the first part of this thesis we give a proof of the above mentioned theorem that avoids the use of Zorn’s lemma. The second part of this thesis investigates the behaviour of so-called Gaussian beam solutions of the wave equation - highly oscillatory and localised solutions which travel, for some time, along null geodesics. The main result of this part of the thesis is a characterisation of the temporal behaviour of the energy of such Gaussian beams in terms of the underlying null geodesic. We conclude by giving applications of this result to black hole spacetimes. Recalling that the wave equation can be considered a “poor man’s” linearisation of the Einstein equations, these applications are of interest for a better understanding of the black hole stability conjecture, which states that the exterior of our explicit black hole solutions is stable to small perturbations, while the interior is expected to be unstable. The last part of the thesis is concerned with the wave equation in the interior of a black hole. In particular, we show that under certain conditions on the black hole parameters, waves that are compactly supported on the event horizon, have finite energy near the Cauchy horizon. This result is again motivated by the investigation of the conjectured instability of the interior of our explicit black hole solutions.