Conformal scattering and analysis of asymptotic properties of gauge theories in general relativity

Thesis

The study of scattering is a fundamental aspect of mathematical physics. In the second half of the 20th century, mathematical tools were developed by Penrose, Friedlander, Lax, Phillips, and others that have revealed an intriguing underlying geometry of scattering theories.

In the first part of this thesis we study the conformal scattering of Maxwell potentials on a class of asymptotically flat asymptotically simple spacetimes. We construct scattering operators as isomorphisms between Hilbert spaces on past and future null infinity, and (in the flat case) explore the structure of these Hilbert spaces with respect to the symmetries of the spacetime.

In the second part we study the Maxwell-scalar field system on de Sitter space. We prove small data peeling estimates at all orders, and construct bounded and invertible, but nonlinear, scattering operators. We discover that sufficiently regular solutions decay exponentially in time and disperse as linear waves, and find a curious asymptotic decoupling of the scalar field.

In the third part of this thesis we study the Yang–Mills–Higgs equations on the Einstein cylinder. By localizing Eardley & Moncrief's famous L^∞ estimates, we extend them to the Einstein cylinder, and remove a small data restriction in a classical theorem of Choquet-Bruhat and Christodoulou from 1981. By using conformal transformations, we deduce large data decay rates for Yang–Mills– Higgs fields on Minkowski and de Sitter spacetimes.

Authors: Taujanskas, G

• DOI: 10.5287/ora-9ozrr6rzo
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Mathematics