This article contains a local solution to the notorious Problem of Time in Quantum Gravity at the conceptual level and which is actually realizable for the relational triangle. The Problem of Time is that 'time' in GR and 'time' in ordinary quantum theory are mutually incompatible notions, which is problematic in trying to put these two theories together to form a theory of Quantum Gravity. Four frontiers to this resolution in full GR are identified, alongside three further directions not yet conquered even for the relational triangle.This article is also the definitive review on relational particle models originally due to Barbour (2003: dynamics of pure shape) and Barbour and Bertotti (1982: dynamics of shape and scale). These are exhibited as useful toy models of background independence, which I argue to be the 'other half' of GR to relativistic gravitation, as well as the originator of the Problem of Time itself. Barbour's work and my localized extension of it are shown to be the classical precursor of the background independence that then manifests itself at the quantum level as the full-blown Problem of Time. In fact 7/8ths of the Isham-Kuchař Problem of Time facets are already present in classical GR; even classical mechanics in relational particle mechanics formulation exhibits 5/8ths of these! In addition to Isham, Kuchař and Barbour, the other principal authors whose works are drawn upon in building this Problem of Time approach are Kendall (relational models only: pure-shape configuration spaces), Dirac, Teitelboim and Halliwell (Problem of Time resolving components). The recommended scheme is a combination of the Machian semiclassical approach, histories theory and records theory.This piece is far from necessarily to be read linearly, it can instead be treated as a reference book, in particular using keyword searches to find all mentions for each topic of interest to the reader. Use the Preface or the Index to identify the particular topics covered. As well as 'Quantum Gravity' and 'Semiclassical Quantum Cosmology', these in good part interdisciplinary works are of interest in Theoretical Mechanics, Shape Geometry and Shape Statistics, Molecular Physics/Theoretical Chemistry and the Physics and Philosophy of Time, Space and Relationalism. A number of conceptually interesting innovations in the Principles of Dynamics are supplied, as befit Physics cast in a relational language. As regards searching the File for which Problems of Time manifested by whichever strategy of interest, these are tagged by 'X with Y' for X the problem and Y the strategy. See Fig 86 for the evolution from the familiar names of the Problem of Time Facets to the corrected names actually used in this Article, and Figs 61 and 79 for lists of the strategies considered in the Article.

When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving 3-dimensional conformal Riemannian geometries obtained by imposing two general principles: 1) time is derived from change; 2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation but also all the equations used in York's conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom.Since Einstein created GR 4D spacetime covariance has been taken as its axiomatic basis. However, much work has been done in a dynamical approach that uses the 3+1 split into space and time of Arnowitt, Deser, and Misner (ADM) [1]. This work has been stimulated by the needs of astrophysics (especially gravitational-wave research) and by the desire to find a canonical version of GR suitable for quantization.The ADM formalism describes constrained Hamiltonian evolution of 3D spacelike hypersurfaces embedded in 4D spacetime. The intrinsic geometry of the hypersurfaces is represented by a Riemannian 3-metric g ij , which is the ADM canonical coordinate. The corresponding canonical momentum π ij is related to the extrinsic curvature κ ij of the embedding of the hypersurfaces in spacetime by π ij = − √ g(κ ij − g ij κ). The ADM dynamics, which respects full relativity of simultaneity by allowing free choice of the 3+1 split, is driven by two constraints. The linear momentum constraint π ij ;j = 0 reflects the gauge symmetry under 3D diffeomorphisms and is well understood. When it has been quotiented out, the 3 × 3 symmetric matrix g ij has three degrees of freedom. The quadratic Hamiltonian constraint gH = −π ij π ij + π 2 /2 + gR = 0 reflects the relativity of simultaneity -the time coordinate can be freely chosen at each space point. It shows that g ij has only two physical degrees of freedom. The problem is to find them. The solution, if it exists, will break 4D covariance.An important clue was obtained by York [2], who perfected Lichnerowicz's conformal technique [3] for finding initial data that satisfy the initial-value constraints of GR. In the Hamiltonian formalism, these are the ADM Hamiltonian and momentum constraints. Finding such data is far from trivial. York's is the only known effective method. He divides the 6 degrees of freedom in the 3-metric into three groups; 3 are mere coordinate freedoms, 1 is a scale part (a conformal factor), and the two remaining parts represent the conformal geometry, the 'shape of space'. York's method also introduces a distinguished foliation of spacetime -and with it a definition of simultaneity -by hypersurfaces of constant mean (extrinsic) curva...

The Problem of Time occurs because the 'time' of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the conditions of both apply, e.g. in black holes or in the very early universe. Emphasis in this Review is on the Problem of Time being multi-faceted and on the nature of each of the eight principal facets. Namely, the Frozen Formalism Problem, Configurational Relationalism Problem (formerly Sandwich Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time, Problem of Beables (alias Problem of Observables) and Spacetime Reconstruction/Replacement Problem. Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets. Particular emphasis is placed upon A) relationalism as an underpinning of the facets and as a selector of particular strategies (especially a modification of Barbour relationalism, though also with some consideration of Rovelli relationalism). B) Classifying approaches by the full ordering in which they embrace constrain, quantize, find time/history and find observables, rather than only by partial orderings such as "Dirac-quantize". C) Foliation (in)dependence and Spacetime Reconstruction for a wide range of physical theories, strategizing centred about the Problem of Beables, the Patching Approach to the Global Problem of Time, and the role of the question-types considered in physics. D) The Halliwell-and Gambini-Porto-Pullin-type combined Strategies in the context of semiclassical quantum cosmology.

We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scaleinvariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t scaling developed in the parallel particle dynamics paper by one of the authors. In spatially-compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented.Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.

The emergent semiclassical time approach to resolving the problem of time in quantum gravity is considered in the arena of relational particle toy models. In situations with 'heavy' and 'light' degrees of freedom, two notions of emergent semiclassical WKB time emerge; these are furthermore equivalent to two notions of emergent classical 'Leibniz-Mach-Barbour' time. I study the semiclassical approach, in a geometric phase formalism, extended to include linear constraints, and with particular care to make explicit the approximations and assumptions used, which are an important part of the semiclassical approach. I propose a new iterative scheme for the semiclassical approach in the cosmologically-motivated case with one heavy degree of freedom. I find that the usual semiclassical quantum cosmology emergence of time comes hand in hand with the emergence of other qualitatively significant terms, including back-reactions on the heavy subsystem and second time derivatives. I take my analysis further for relational particle models with linearly-coupled harmonic oscillator potentials, which, being exactly soluble by means outside the semiclassical approach to quantum cosmology, are additionally useful for testing the justifiability of some of the approximations and assumptions habitually made therein. Finally, I contrast emergent semiclassical time with its hidden dilational Euler time counterpart.PACS numbers 04.60-m, 04.60.Ds * ea212@cam.ac.uk 1 I use ( ) for function dependence, [ ] for functional dependence, ( ; ] for a mixture of function dependence before the semi-colon and functional dependence after it, and ⌊ ⌋ to enclose those functions on which a derivative acts. h αβ (x) is the spatial 3-metric, with determinant h = h(h αβ (x)), covariant derivative ∇ β , and Ricci scalar R(x; h αβ (x)].˙denotes ∂ ∂λ . G || || is the norm with respect to the array G, with the G ordered to the left; this array is the DeWitt supermetric, G αβγδ = 1 √ h˘h αγ h βδ − 1 2 h αβ h γδ¯. G −1 is the inverse array G αβγδ = √ h{h αγ h βδ − h αβ h γδ }. This ordering to the left is a relatively simple choice used e.g in [2], although that does furthermore argue for the more complicated super-coordinate invariant Laplacian ordering. This is considered in my present semiclassical program in [27]. Th = G −1 ||•Bh|| 2 is the gravitational kinetic term up to factors of c and 16πG. The •B symbol is explained in Appendix A. The underline on the G −1 denotes de-densitization by division by √ h. δ h is shorthand for the functional derivative with components δ δh αβ (x) . Λ is the cosmological constant. Ψ is the wavefunction of the universe.2 I continue to use summation convention for space indices here, but sum explicitly over interparticle (cluster) separations. The R γi are the i relative Jacobi coordinates [62] with corresponding (cluster) relative masses µ i . V is the potential, which has the relational dependence V = V(||R i ||, (R i , R j ) alone). U ≡ −V. µ|| || is the norm with respect to the array µ ijαβ = 2µ i δ αβ δ ij (the mass metric)...

Relational particle models are employed as toy models for the study of the Problem of Time in quantum geometrodynamics. These models' analogue of the thin sandwich is resolved. It is argued that the relative configuration space and shape space of these models are close analogues from various perspectives of superspace and conformal superspace respectively. The geometry of these spaces and quantization thereupon is presented. A quantity that is frozen in the scale invariant relational particle model is demonstrated to be an internal time in a certain portion of the relational particle reformulation of Newtonian mechanics. The semiclassical approach for these models is studied as an emergent time resolution for these models, as are consistent records approaches. PACS numbers 04.60-m, 04.60.Ds 1 Lower and upper-case Greek letters are used as 3-d space and 4-d spacetime indices respectively. 2 Riem = {space of γ αβ 's on a fixed 3-space manifold } and Diff are the 3-space diffeomorphisms.

Relational particle models are of value in the absolute versus relative motion debate. They are also analogous to the dynamical formulation of general relativity, and as such are useful for investigating conceptual strategies proposed for resolving the problem of time in quantum general relativity. Moreover, to date there are few explicit examples of these at the quantum level. In this paper I exploit recent geometrical and classical dynamics work to provide such a study based on reduced quantization in the case of pure shape (no scale) in 2-d for 3 particles (triangleland) with multiple harmonic oscillator type potentials. I explore solutions for these making use of exact, asymptotic, perturbative and numerical methods. An analogy to the mathematics of the linear rigid rotor in a background electric field is useful throughout. I argue that further relational models are accessible by the methods used in this paper, and for specific uses of the models covered by this paper in the investigation of the problem of time (and other conceptual and technical issues) in quantum general relativity.

Relational particle dynamics include the dynamics of pure shape and cases in which absolute scale or absolute rotation are additionally meaningful. These are interesting as regards the absolute versus relative motion debate as well as discussion of conceptual issues connected with the problem of time in quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces of shapes are n-spheres and complex projective spaces, from which knowledge I construct natural mechanics on these spaces. I also show that these coincide with Barbour's indirectly-constructed relational dynamics by performing a full reduction on the latter. Then the identification of the configuration spaces as n-spheres and complex projective spaces, for which spaces much mathematics is available, significantly advances the understanding of Barbour's relational theory in spatial dimensions 1 and 2. I also provide the parallel study of a new theory for which position and scale are purely relative but orientation is absolute. The configuration space for this is an n-sphere regardless of the spatial dimension, which renders this theory a more tractable arena for investigation of implications of scale invariance than Barbour's theory itself.