General Relativity & Quantum Cosmology

I will show how the intrinsic definition of observables in relativity through dynamical similiarity (known as Shape Dynamics) leads to the continuation of Einstein's equations classically through the big bang singularity in simple cosmological scenarios. By appealing to general principles I argue that this is a generic feature, and that the singularity can be viewed as an artifact of the redundant description imposed by absolute length scales. I will then lay out some other welcome features of intrinsic relational systems, and discuss the broader questions raised by a theory of physics that is independent of physical dimensions such as mass and length.

I will discuss the use of (functional) Renormalization Group in models of quantum gravity. I will highlight the challenges that occur in continuum approaches to quantum gravity, such as asymptotically safe gravity, as well as challenges in discrete approaches, such as tensor models.

I review motivations for a frame-dependent dark energy action proportional to $\int d^4x (-g)^{1/2}/g_{00}^2$, and discuss implications for the black hole horizon and for perturbations on the Robertson-Walker line element.

The Weyl Theorem states that the conformal structure and the projective structure jointly suffice to fix the metric up to a global constant. This is a powerful interpretive tool in general relativity: it says in effect that if I know the paths of light rays in vacuo and I know the images of the paths of freely falling particles (i.e., the spacetime curves they follow with no preferred parametrization), then I know the metric. It has particular relevance to Shape Dynamics, where the conformal structure is taken as the object of primitive geometrical interest, and one does not generally want a preferred parametrization of timelike geodesics. The spacetimes of geometrized Newtonian gravity share many important features with the way that Shape Dynamics approaches the construction of relativistic spacetimes, in particular the fixing of a preferred foliation of spacetime by spacelike slices. Studying the way that geometrized Newtonian gravity does and does not allow one to recover a Weyl-type Theorem may, therefore, shed light on the ways that Shape Dynamics may allow one to recover the structure of relativistic spacetimes. I show that, in geometrized Newtonian gravity, the conformal structure of the spatial and temporal metrics, in conjunction with the projective structure of timelike curves, allows one to recover the full metrical structure of a geometrized Newtonian gravity only if one also fixes an affine parametreization of at least one timelike geodesic. This suggests that, in far as the analogy between geometrized Newtonian gravity and Shape Dynamics is a physically significant one, Shape Dynamics may be right not to demand preferred parametrizations of timelike geodesics from the start. The tools I develop to prove the theorem may also be applicable to problems in Shape Dynamics itself. But I make no promises.

Shape Dynamics possesses a large set of solutions in common with General Relativity. Upon close inspection, these solutions behave in surprising ways, so in order to probe the fitness of Shape Dynamics as a viable alternative to General Relativity one must understand increasingly complex solutions, on which to base perturbative studies and numerical analyses. We show that a class of time-dependent exact solutions of Shape Dynamics exists from first principles, representing a central inhomogeneity in an evolving cosmological environment. By assuming only a perfect fluid source in a spherically symmetric geometry, we show that this solution satisfies in all generality the Hamiltonian structure of Shape Dynamics. The solutions are characterized by shear-free flow of the fluid and admit an interpretation as cosmological black holes.

I will review the current status of our understanding of spherically symmetric compact solutions of Shape Dynamics, which have nontrivial degrees of freedom when matter is present. I will show some new solutions of GR in a CMC foliation: a single thin spherical shell of matter in equilibrium in a compact foliation of de Sitter, and the simplest possible model of a black hole or compact star. This is provided by a universe with the topology of a 3-sphere with two thin spherical shells of dust. One of the shells models the `fixed stars’, or the `rest of the universe’, while the other shell models collapsing matter. Both are needed for a truly relational description of gravitational collapse. It turns out that such a solution of GR cannot be evolved past a point at which the foliationceases to be admissible, but it still makes sense past that point as a solution of Shape Dynamics, because the shape degrees of freedom seem to be unaffected. My conjecture is that we have found another example of departure between GR and SD, and this departure happens whenever ordinary matter undergoes gravitational collapse.

Most practical studies in Shape Dynamics involve an N-body Newtonian interaction which is described by a homogeneous potential. This property allows one to proof several interesting features like the emergence of an arrow of time. However, more generic interactions are not described by these kind of potentials and introduce additional dimensionful coupling constants. Thus, it is an open question whether more generic interactions can be written in a fully relational manner. By studying the concrete example of the gravitational Weber interaction which is, in a sense, a more realistic theory of gravity, we show that it is possible translate non-Newtonian interactions, which have inhomogeneous potentials and additional coupling constants, into a relational language. This opens the door to study other interactions and may shed light into the relationalization of gravity as described by general relativity.

Inflation is the leading paradigm of the early Universe, according to which the tiny temperature fluctuations observed in the cosmic microwave background (CMB) originate from quantum vacuum fluctuations at very early times. Recent observations show that the Starobinsky potential is favored among the single field inflationary models. However, the calculations that match the data exclude the Planck era. I will explain why this era is important and how using techniques from loop quantum gravity, the effects of this period can be studied. In particular, we find that for a large part of the initial data surface predictions for the power spectrum of the CMB are indistinguishable from predictions neglecting the pre-inflationary era. However, initial conditions exist for which the quantum gravitational corrections to the power spectrum are potentially observable.

Relationalism is the strict disentanglement of physical law from the definition of physical object. This can be formalized in the shape dynamcis postulate that the objective evolution of the universe is described by an "equation of state of a curve in relational configuration space." The application of this postulate to General Relativity implies that gravity is described by an equation of state of a curve on conformal superspace. It turns out that the naive quantization of these equations of state introduces an undesired preferred time parametrization. However, it turns out that one can still describe the quantum evolution of the system as an equation of state of the Bohmian trajectory which remains manifestly parametrization independent. These quantum systems generically develop quasi-isolated bound states (atoms) that can be used as reference systems. It turns out that the system as a whole expands if described in units defined by these atoms. This produces phenomenological effects that are usually ascribed to the presence of a cosmological constant. This "effective cosmological constant" is however unaffected by vacuum energy. I pesent the formal argument for this statement and show this explicitly by remormalizing a scalar field coupled to shape dynamics.