Carroll physics in general relativity: understanding the Carroll limit of the singularity theorem
The Carroll limit of general relativity is an ultra-relativistic limit that is obtained by taking the speed of light (c) to zero. This is in contrast to the Galilean limit, where you can resolve it by taking the c → ∞ limit. This limit was formulated in the study of the Carroll group as an ultra-relativistic limit of the Poincaré group, as such, one can exploit the various Carroll symmetries that arise. In this thesis, we study the Carroll limit of general relativity and its applications to Cosmology and the Singularity Theorem. We begin by reviewing the representation theory of the Carroll group and its applications to non-relativistic physics. We then study the Carroll scalar field and its properties in the Carroll limit, as well as how the Friedmann equations reduce in the small c expansion in comparison to the evolution of the scalar field in our Standard Cosmology. We then study how fluids evolve in the Carroll regime, which gives us insight into the early universe and its dynamics. This study allows us to explore how non-relativistic matter, as well as the cosmological constant, would evolve in the Carroll limit. To understand radiation in the Carroll limit, we take a purely Classical route and study Maxwell's Theory of electromagnetism in a curved background. That then allows us to study the singularity theorem and how singularities are affected by taking this limit, which prompts us to look into a relatively recent theorem known as the BGV theorem, which looks at singularities from the perspective of geodesic incompleteness in contrast to the Hawking and Penrose ideas on the singularity theorem.
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Department of Mathematics and Applied Mathematics