In this opus, I challenge the claim that inflationary spacetimes must be past geodesi-cally incomplete. To do this, I utilize the warped product formalism of Bishop and O’Neill and build upon the venerable Friedmann Robertson Walker (FRW) space- time formalism to the Generalized Friedmann Robertson Walker (GFRW) spacetime formalism, where the achronal spacelike sections can be any geodesically complete Riemannian manifold (Σ, gΣ ). I then solve the GFRW geodesic equation in generality as a functional of the scale factor f , and derive a main theorem, which characterizes the geodesic completeness in GFRW spacetimes. After offering a definition of infla- tion which enumerates the topological requirements which permit a local foliation of a scale factor, I discuss a cohort of geodesically complete inflationary GFRWs which have averaged expansion quantity Havg > 0, proving that classical counter-examples to the theorem of Borde, Guth, and Vilenkin do exist. I conclude by introducing conjectures concerning the relationship between geodesic completeness and inflation: in particular, I speculate that if a spacetime is geodesically complete and non-trivial, it must inflate!

The double copy is a procedure that relates gravity to simpler gauge and scalar field theories. Double copy structure was first discovered in the context of scattering amplitudes, and has since been realized at the level of classical fields and curvatures. This dissertation focuses on mappings between fields (the Kerr-Schild double copy) and curvatures (the Weyl double copy). First, the connection between non-singular black holes and non-singular gauge theories is made, which illuminates a subtlety between gravitational horizons and the gauge field strength. Then, a perturbative double copy in the context of the fluid/gravity duality is presented, where the associated gauge theory quantities have surprisingly elegant interpretations in terms of certain classes of Navier-Stokes solutions. Finally, a new formula that provides a consistent treatment of external sources in the Weyl double copy is introduced. After illustrating its consistency with the Kerr-Schild double copy, the sourced Weyl double copy is applied to the most general Petrov type D electro-vac spacetime. Various limits of the general solution are analyzed, including the Kerr-Newman metric and the charged, accelerating black hole.