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On the path towards quantum gravity we find friction between temporal relations in quantum mechanics (QM) (where they are fixed and field-independent), and in general relativity (where they are field-dependent and dynamic). In this talk, I will erase that distinction. I encode gravity, along with other types of interactions, in the timeless configuration space of spatial fields, with dynamics obtained through a path integral formulation. The framework demands that boundary conditions for this path integral be uniquely given. Such uniqueness arises if a reduced configuration space can be defined and if it has a profoundly asymmetric fundamental structure. These requirements place strong restrictions on the field and symmetry content of theories encompassed here. When these constraints are met, the emerging theory has no non-unitary measurement process; the Born rule is given merely by a particular volume element built from the path integral in (reduced) configuration space. Time, including space-time, emerges as an effective concept; valid for certain curves in configuration space but not assumed from the start. When some notion of time becomes available, conservation of (positive) probability currents ensues. I will show that, in the appropriate limits, a Schroedinger equation dictates the evolution of weakly coupled source fields on a classical gravitational background. Due to the asymmetry of reduced configuration space, these probabilities and currents avoid a known difficulty of standard WKB approximations for Wheeler DeWitt in minisuperspace: the selection of a unique Hamilton-Jacobi solution to serve as background. I illustrate these constructions with a simple example of a quantum gravitational theory for which the formalism is applicable, and give a formula for calculating gravitational semi-classical relative probabilities in it. Although this simple model gives the same likelihood for the evolution of all TT gravitational modes, there is evidence that a slightly more complicated model would favor modes with the smallest eigenvalues of the Laplacian and thus drive towards homogeneity.

I will present results on the quantization of an FRLW model that utilises a Schrodinger-type evolution equation. In contrast to standard Wheeler--DeWitt-type quantisations, the quantum model resolves the classical singularity, exhibits a quantum bounce, and displays novel early-universe phenomenology. A global scale emerges because of a scale anomaly, and suggests an interesting scenario for quantum shape dynamics. I will give the details of the quantization procedure and show how these techniques can be used more generally for anisotropic models. I will end by speculating about how these techniques might be applicable to a genuine quantum shape model of the universe.

I will with simple examples from spatially homogeneous and isotropic cosmology illustrate the importance of respecting the global features of a state space for a given model when reformulating field equations to useful dynamical systems. In particular I will use examples from f(R) gravity and GR with a minimally coupled scalar field. In this context I will also illustrate how various dynamical systems methods, such as, e.g., monotonic functions, center manifold techniques, averaging methods, can yield a global understanding of the solution spaces as well as approximations, complementing, e.g., the slow-roll approximation.

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.