Quantum Aspects of Spacetime Geometry
In this dissertation, we investigate how some properties of quantum mechanics integrate in our description of the geometry of spacetime. This is done both at a theoretical and phenomenological level. In the first chapter, we give a brief introduction to the development of quantum mechanical ideas in our current understanding of spacetime and gravity. In particular, we motivate the necessity of a holographic description of gravity, as well as review some of the essential properties of the AdS/CFT correspondence.
In the second chapter, we use this holographic duality to ask how the geometry of a particular bulk system is encoded in a non-gravitational field theory. In particular, we review and use the light-cone cuts prescription, which has been proposed as a method to reconstruct the conformal metric of a holographic spacetime. With this framework, we explore how additional information about the bulk geometry gets encoded in the structure of these light-cone cuts. We pay special attention on how the hyperbolic angle related to a cusp in the light-cone cut encodes information about the matter content of the spacetime. We provide an explicit numerical example reconstructing the metric for a four-dimensional spacetime composed by the superposition of a boson star and a gas of radiation in AdS.
The full bulk path integral in a Lorentzian formulation of holography includes metrics that violate boundary causality. This leads to the following puzzle: The commutator of two field theory operators at spacelike-separated points on the boundary must vanish. However, if these points are causally related in a bulk metric, then the bulk calculation of the commutator will be nonzero. It would appear that the integral over all metrics of this commutator must vanish exactly for holography to hold. This is puzzling since it must also be true if the commutator is multiplied by any other operator. Upon a careful treatment of boundary conditions in holography, in the third chapter of this thesis we show how the bulk path integral leads to a natural resolution of this puzzle.
In chapter four, we explore some phenomenological implications stemming from quantum modifications to a classical theory of gravity. Two new observational windows have been opened to strong gravitational physics: gravitational waves, and very long baseline interferometry. This suggests observational searches for new phenomena in this regime, and in particular for those necessary to make black hole evolution consistent with quantum mechanics. We describe possible features of ``compact quantum objects" that replace classical black holes in a consistent quantum theory, and approaches to observational tests for these using gravitational waves. This is an example of a more general problem of finding consistent descriptions of deviations from general relativity, which can be tested via gravitational wave detection. Simple models for compact modifications to classical black holes are described via an effective stress tensor, possibly with an effective equation of state. A general discussion is given of possible observational signatures, and of their dependence on properties of the colliding objects. The possibility that departures from classical behavior are restricted to the near-horizon regime raises the question of whether these will be obscured in gravitational wave signals, due to their mutual interaction in a binary coalescence being deep in the mutual gravitational well. Numerical simulation with such simple models will be useful to clarify the sensitivity of gravitational wave observation to such highly compact departures from classical black holes.