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!

Computable properties of quantum states are given a dual gravitational interpretation via the AdS/CFT correspondence. For holographic states, boundary entanglement entropy is dual to the area of bulk geodesics, known as Ryu-Takayanagi surfaces. Furthermore, the viability of states to admit a holographic dual at all is constrained by their entanglement structure. Entanglement therefore defines a coarse classification of states in the Hilbert space. Similarly, how a state transforms under a group of operators also provides a classification on the Hilbert space. Certain states, e.g. stabilizer states, are invariant under large sets of operations, and consequently can be simulated on a classical computer. Cayley graphs offer a useful representation for a group of operators, where vertices represent group elements and edges represent group generators. In this representation, the orbit of a state under action of the group can also be represented as a ``reachability graph'', defined as a quotient of the group Cayley graph. Reachability graphs can be dressed to encode entanglement information, making them a useful tool for studying entanglement dynamics under quantum operations. Further quotienting a reachability graph by group elements that fix a chosen state property, e.g. entanglement entropy, builds a ``contracted graph''. Contracted graphs provide explicit bounds on state parameter evolution under quantum circuits. In this work, an upper bound on entropy vector evolution under Clifford group action is presented. Another important property of quantum systems is magic, which quantifies the difficulty of classically simulating a quantum state. Magic and entanglement are intimately related, but the two are not equivalent measures of complexity. Nonetheless, entanglement and magic play complementary roles when describing emergent gravitational phenomena in AdS/CFT. This manuscript describes the interplay between entanglement and magic, and offers a holographic interpretation for magic as cosmic brane back-reaction.

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.

In this dissertation I discuss about calculating one-loop partition function on curved spacetimes and various approaches to build symmetries of gravitational systems, and extending the analysis to the large dimensional spacetimes. I show the calculations pertaining to the contributions to the one-loop determinant for transverse trace-less gravitons in an $n + 3$-dimensional Schwarzschild black hole background in the large dimension limit, due to the $SO(n+2)$-type tensor and vector fluctuations, using the quasinormal mode method. Accordingly I find the quasinormal modes for these fluctuations as a function of a fiducial mass parameter $\Delta$. I show that the behavior of the one-loop determinant at large $\Delta$ accords with a heat kernel curvature expansion in one lower dimension, lending further evidence towards a membrane picture for black holes in the large dimension limit. I also find that the analysis of building one-loop determinants is similar to that of the AdS, thus serving as a motivation to explore this emergent symmetry in detail. For this, I first build these symmetries for Kerr-(A)dS black holes in arbitrary dimensions and then extend this analysis to the large dimensional Schwarzschild black hole. To study the former, in this dissertation, I discuss how to generalize the notion of hidden conformal symmetry in Kerr/CFT to Kerr-(A)dS black holes in arbitrary dimensions. I also discuss the results on building the $SL(2, R)$ generators directly from the Killing tower, whose Killing tensors and Killing vectors enforce the separability of the equations of motion. This construction amounts to an explicit relationship between hidden conformal symmetries and Killing tensors: I use the Killing tower to build a novel tensor equation connecting the $SL(2,R)$ Casimir with the radial Klein-Gordon operator. For asymptotically flat black holes in four and five dimensions I discuss that the previously known results that were obtained using the ``near-region'' limit and the monodromy method, were recovered. I also perform a monodromy evaluation of the Klein-Gordon scalar wave equation for all Kerr-(A)dS black holes, finding explicit forms for the zero mode symmetry generators. Lastly, I discuss the work on extending this analysis to the large-dimensional Schwarzschild black hole as a step towards building a Large-D/CFT correspondence.

Here I develop the connection between thermodynamics, entanglement, and gravity. I begin by showing that the classical null energy condition (NEC) can arise as a consequence of the second law of thermodynamics applied to local holographic screens. This is accomplished by essentially reversing the steps of Hawking's area theorem, leading to the Ricci convergence condition as an input, from which an application of Einstein's equations yields the NEC. Using the same argument, I show logarithmic quantum corrections to the Bekenstein-Hawking entropy formula do not alter the form of the Ricci convergence condition, but obscure its connection to the NEC. Then, by attributing thermodynamics to the stretched horizon of future lightcones -- a timelike hypersurface generated by a collection of radially accelerating observers with constant and uniform proper acceleration -- I derive Einstein's equations from the Clausius relation. Based on this derivation I uncover a local first law of gravity, connecting gravitational entropy to matter energy and work. I then provide an entanglement interpretation of stretched lightcone thermodynamics by extending the entanglement equilibrium proposal. Specifically I show that the condition of fixed volume can be understood as subtracting the irreversible contribution to the thermodynamic entropy. Using the AdS/CFT correspondence, I then provide a microscopic explanation of the 'thermodynamic volume' -- the conjugate variable to the pressure in extended black hole thermodynamics -- and reveal the super-entropicity of three-dimensional AdS black holes is due to the gravitational entropy overcounting the number of available dual CFT states. Finally, I conclude by providing a recent generlization of the extended first law of entanglement, and study its non-trivial 2+1- and 1+1-dimensional limits. This thesis is self-contained and pedagogical by including useful background content relevant to emergent gravity.

A problem of interest in theoretical physics is the issue of the evaporation of black holes via Hawking radiation subject to a fixed background. We approach this problem by considering an electromagnetic analogue, where we have substituted Hawking radiation with the Schwinger effect. We treat the case of massless QED in 1+1 dimensions with the path integral approach to quantum field theory, and discuss the resulting Feynman diagrams from our analysis. The results from this thesis may be useful to find a version of the Schwinger effect that can be solved exactly and perturbatively, as this version may provide insights to the gravitational problem of Hawking radiation.

Two ideas that extends on the theory of General Relativity are introduced and then the phenomenology they can offer is explored. The first idea shows how certain types of $f(R)$ gravity allows for traversable wormholes among its vacuum solutions. This is surprising to find in such simple setting as these type of solutions usually requires fairly complex constructions to satisfy the equations of motion of a gravitational theory. The second idea is the matter bounce description of the early universe where a fairly unique feature of the model is identified. Consequences of this feature could allow the paradigm to distinguish itself from other alternative descriptions, such as inflation, through late time observations. An explicit example of this claim is worked out by studying a model involving an interaction in the dark sector. Results of a more astrophysical nature, where a careful analysis of the morphology of blazar halos is performed, are also presented in the Appendix. The analysis determined that the $Q$-statistic is an appropriate tool to probe the properties of the intergalactic magnetic fields responsible for the halos formation.