Forks on a dusty road

Abstract

This thesis explores the classical and quantum aspects of dust + gravity systems with the dust field playing the role of time. In the classical setting we explored the linearized theory of dust + General Relativity around a Minkowski background. The resulting theory has three physical degrees of freedom at each spacetime point. At the linearized level we recovered two graviton modes and an ultralocal scalar mode. Remarkably the graviton modes remain Lorentz covariant despite the time gauge fixing. The other classical models we studied were the homogeneous and anisotropic Bianchi I and IX spacetimes. The dust time gauge analysis of Bianchi IX spacetime gives a new physical picture where dust Bianchi IX dynamics is characterized by transitions between dust-Kasner solutions rather than vacuum-Kasner solutions. We derived a generalized transition law between these solutions which includes a matter component. Sufficiently close to the singularity this law reduces to the usual Belinski-Khalatnikov-Lifshitz map. In the quantum setting we explored two homogeneous models with dust. We de-parameterized the theory using the dust time gauge before quantization. For homogeneous models this is the reduced phase space approach to quantization. The first model we studied was spatially flat Friedmann-Lemaˆıtre-Robertson-Walker model with dust and a cosmological constant (Λ). We showed that after gauge fixing and a canonical transformation the model reduces to a simple harmonic oscillator with frequency √ Λ. The (Lorentzian) quantum theory of this model is then immediate. The model provides a simple demonstration of non-perturbative singularity avoidance. The other model we investigated was the Bianchi I model with dust. We formulated the path integral for the model using the physical Hamiltonian obtained after gauge fixing the theory using dust as time. The quantum theory of this model is not solvable analytically. We studied the quantum dynamics using Markov Chain Monte Carlo techniques by considering the Euclidean path integral. Numerical semiclassical analysis shows that quantum fluctuations in the spatial volume and anisotropies are larger for smaller universes. We also evaluated the no-boundary wavefunction for this model. The no-boundary wavefunction implies a suppression of large universes while large anisotropies appear to dominate.