MSc defense by Benjamin T. Søgaard
Carroll Geometry and Ultra-Relativistic Gravity
The Carroll group, which emerges as the ultra-relativistic limit of the Poincaré group, and its local realization in terms of Carroll geometry have recently received renewed interest due to their connection to flat space holography. The study of Carroll geometry, Carroll gravity and Carrollian field theories is thus currently emerging as a new direction of research. It is the intention of this thesis to further develop the subject of Carroll geometry and its application to the study of gravity.
The main goal of this thesis is to derive and analyze a theory of Carrollian gravity through a novel small c expansion of the Einstein-Hilbert action. To attain this objective we review and further develop basic notions of Carroll geometry. In particular, we will review the construction of Carroll geometry through a gauging procedure and its appearance as the natural geometry on null hypersurfaces in Lorentzian geometry. We shall develop basic aspects of Carrollian field theories by considering the construction of energy-momentum tensors. Furthermore, the degenerate metric structure of Carroll geometry does not naturally single out a distinguished connection like the Levi-Civita connection of Lorentzian geometry. Thus, we will explore what obstructions to such a natural connection exist and what choices need to be made to single out a Carrollian analog of the Levi-Civita connection.
These preliminary considerations allow us to perform an expansion of the Einstein-Hilbert action in powers of c^2. We will consider this expansion up to next-to-leading order. As the leading-order theory resembles the 3+1 decomposition of general relativity, we review this framework in order to adapt the methods thereof to the Carrollian theory. We then examine the leading-order theory in detail and develop new methods for obtaining solutions and computing boundary charges. Finally, we present several examples of solutions to the leading-order theory using the methods developed in this thesis.