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We discuss the construction of a free scalar quantum field theory on κ-Minkowski noncommutative spacetime. We do so in terms of κ-Poincaré-invariant N-point functions, i.e., multilocal functions which respect the deformed symmetries of the spacetime. As shown in a previous paper by some of us, this is only possible for a lightlike version of the commutation relations, which allow the construction of a covariant algebra of N points that generalizes the κ-Minkowski commutation relations. We solve the main shortcoming of our previous approach, which prevented the development of a fully covariant quantum field theory: the emergence of a non-Lorentz-invariant boundary of momentum space. To solve this issue, we propose to “extend” momentum space by introducing a class of new Fourier modes and we prove that this approach leads to a consistent definition of the Pauli-Jordan function, which turns out to be undeformed with respect to the commutative case. We finally address the quantization of our scalar field and obtain a deformed, κ-Poincaré-invariant, version of the bosonic oscillator algebra.

We show that the Kantowski-Sachs model of a Schwarzschild black hole interior can be slightly generalized in order to accommodate spatial metrics of different orientations, and in this formulation the equations of motion admit a variable redefinition that makes the system regular at the singularity. This system will then traverse the singularity in a deterministic way (information will be conserved through it), and evolve into a time-reversed and orientation-flipped Schwarzschild white hole interior.

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the [Formula: see text]-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

We discuss the obstruction to the construction of a multiparticle field theory on a $\kappa$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $\kappa$-Poincar\'e group. This necessitates a braided tensor product. We study the representations of this product, and prove that $\kappa$-Poincar\'e-invariant N-point functions belong to an Abelian subalgebra, and are therefore commutative. We use this construction to define the 2-point Whightman and Pauli--Jordan functions, which turn out to be identical to the undeformed ones. We finally outline how to construct a free scalar $\kappa$-Poincar\'e-invariant quantum field theory, and identify some open problems., Comment: 27 pages, 3 figures

Recent work showed that $\kappa$-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincar\'e algebra of special-relativistic symmetries, one can toggle the curvature parameter $\Lambda$, the Planck scale quantum deformation parameter $\kappa$ and the speed of light parameter $c$ to move to the well-studied $\kappa$-Poincar\'e algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these algebras of relativistic symmetries and their associated noncommutative spacetimes, emphasizing the nontrivial effects of interplay between curvature, quantum deformation and speed of light parameters., Comment: v2 matches the version accepted for publication. We added a few references and comments, plus fixed some typos. v3 includes a few additional references

Using the methods of ordinary quantum mechanics, we study [Formula: see text]-Minkowski space as a quantum space described by noncommuting self-adjoint operators, following and enlarging T. Poulain and J.-C. Wallet, “[Formula: see text]-Poincaré invariant orientable field theories with at 1-loop: Scale-invariant couplings, preprint (2018), arXiv:1808.00350 [hep-th]. We see how the role of Fourier transforms is played in this case by Mellin transforms. We briefly discuss the role of transformations and observers.

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups $M=G/H$ equipped with an additional Poisson structure $\pi$ which is compatible with a Poisson-Lie structure $\Pi$ on $G$. Since the infinitesimal version of $\Pi$ defines a unique Lie bialgebra structure $\delta$ on the Lie algebra $\frak g=\mbox{Lie}(G)$, we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space $M^\perp=G^\ast/H^\perp$ of a given homogeneous space $M$ with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to $M^\perp$ thus showing that an even richer duality framework between $M$ and $M^\perp$ arises from them. In order to analyse physical implications of these notions, the case of $M$ being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces $M^\perp$ are explicitly constructed in the case of the well-known $\kappa$-deformation, where the cosmological constant $\Lambda$ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that $M^\perp$ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of $M$ that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces $M^\perp$ are not endowed in general with a $G^\ast$-invariant metric, we show that their geometry can be described by making use of $K$-structures., Comment: 27 pages

We present a framework which unifies a large class of non-commutative spacetimes that can be described in terms of a deformed Heisenberg algebra. The commutation relations between spacetime coordinates are up to linear order in the coordinates, with structure constants depending on the momenta plus terms depending only on the momenta. The possible implementations of the action of Lorentz transformations on these deformed phase spaces are considered, together with the consistency requirements they introduce. It is found that Lorentz transformations in general act nontrivially on tensor products of momenta. In particular the Lorentz group element which acts on the left and on the right of a composition of two momenta is different, and depends on the momenta involved in the process. We conclude with two representative examples, which illustrate the mentioned effect., 7 pages, to be published in Phys. Lett. B

We study the limits to the localizability of events and reference frames in the $\kappa$-Minkowski quantum spacetime. Our main tool will be a representation of the $\kappa$-Minkowski commutation relations between coordinates, and the operator and measurement theory borrowed from ordinary quantum mechanics. Spacetime coordinates are described by operators on a Hilbert space, and a complete set of commuting observables cannot contain the radial coordinate and time at the same time. The transformation between the complete sets turns out to be the Mellin transform, which allows us to discuss the localizability properties of states both in space and time. We then discuss the transformation rules between inertial observers, which are described by the quantum $\kappa$-Poincar\'e group. These too are subject to limitations in the localizability of states, which impose further restrictions on the ability of an observer to localize events defined in a different observer's reference frame., Comment: 35 pages, 4 figures

This chapter analyzes a non-viable theory of gravity based on a Jacobi-type action with a conformally invariant potential term based on the square of the Cotton tensor. This theory has striking similarities with Maxwell’s electromagnetism, of which it represents the generalization to spin 2, but unfortunately it is not dynamically consistent, leaving SD as the only viable spatially conformally invariant relational theory of gravity.

This chapter describes the fundamental problem at the core of Newton’s dynamics: the definition of inertia. This is provided by an absolute structure in Newtonian mechanics, but, as Leibniz and later Mach argued, it should be dynamically determined. This is the core of Newton’s famous ‘bucket experiment’. Assuming this law as a postulate, without first defining the notions of ‘rest’, ‘uniform motion’ and ‘right (or straight) line’, is inconsistent. In a universe that is, in Barbour’s words, like ‘bees swarming in nothing’, how is one to talk about rest/uniform motion/straight lines? With respect to what? The problem is that of establishing a notion of equilocality: in an everchanging universe, what does it mean for an object to be at the same place at dierent times?

Shape Dynamics is a new theory of gravity that is based on fewer and more fundamental first principles than General Relativity. The most important feature of this theory is the replacement of relativity of simultaneity with a more tractable gauge symmetry, namely invariance under spatial conformal transformations. This book contains both a quick introduction for readers curious about Shape Dynamics and a detailed walk-through of the historical and conceptual motivations for the theory, its logical development from first principles and an in-depth description of its present status. The book is sufficiently self-contained for an undergrad student with some basic background in General Relativity and Lagrangian/Hamiltonian mechanics. It is intended both as a reference text for students approaching the subject and as a review for researchers interested in the theory.

Shape Dynamics (SD) is a field theory that describes gravity in a different way than General Relativity (GR): it assumes a preferred notion of simultaneity, and the dynamical content of the theory consists of conformal 3- geometries. SD coincides with (GR) in most situations, in particular in the experimentally well-tested regimes, but it departs from it in some strong-gravity situations, for example at cosmological singularities or upon gravitational collapse. This chapter provides a quick introduction to the theory and a brief description of its present state.

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

The problem with the definition of inertia was solved, in the simple case of free point particles, by Tait, who introduced the concept of inertial frame. Tait’s solution would have satisfied Leibniz’ request that inertia be determined dynamically, however it only works in the absence of interactions between the material bodies. Later Mach posed again the question of the origin of inertia, suggesting the idea that it should be dynamical, which was later dubbed ‘Mach’s principle’. Moreover Mach criticized also Newton’s absolute time, and introduced the basic idea of temporal relationalism, i.e. that time should be a concept that is abstracted from change and has no independent existence. This idea is at the basis of SD and many other relational approaches to physics. This chapter concludes with the Barbour–Bertotti formulation of Mach’s principle, which they called ‘Mach–Poincaré Principle’. This formulation removes the vagueness of Mach’s original idea, and puts the principle into a precise mathematical form, which is one of the basic axioms of SD.

Barbour and Bertotti’s Mach–Poincaré Principle can be realized in classical mechanics with a mathematical procedure which was beyond the grasp of Leibniz or Newton, and turns out to be equivalent to modern gauge theory. This is the formulation of a variational principle based on ‘best matching’: one transforms subsequent configurations of the system with the Euclidean group, and by minimizing a certain functional a notion of ‘equilocality’ is established: now it makes sense to say that a particle comes back to the same point at different times.

The Hamiltonian formulation of relational particle dynamics unveils its equivalence with modern gauge theory, which admits exactly the same canonical formulation. Both are constrained Hamiltonian systems with nonhonolomic constraints, for which Dirac’s analysis, made popular by his lectures, is necessary. Dirac’s analysis is briefly summarized in this chapter for readers unfamiliar with it. The Hamiltonian formulation of the kind of systems we’re interested in is nontrivial. In fact the standard formulation fails to be predictive, precisely because of the relational nature of our dynamics.

This chapter describes the program, dubbed ‘Relativity without Relativity’, of deriving all the fundamental accepted facts at the basis of modern field theory from relational principles. A best-matching action based on Jacobi’s principle is in fact sufficient to derive the universality of the light cone (Special Relativity), the correct form of Maxwell’s action and its gauge invariance, as well as the Yang–Mills theory. Faraday is credited with the introduction of the concept of field in physics. He found it extremely useful, in particular for the description of magnetic phenomena, to use the concept of lines of force (1830s).

The best matching procedure described in Chapter 4 is equivalent to the introduction of a principal fibre bundle in configuration space. Essentially one introduces a one-dimensional gauge connection on the time axis, which is a representation of the Euclidean group of rotations and translations (or, possibly, the similarity group which includes dilatations). To accommodate temporal relationalism, the variational principle needs to be invariant under reparametrizations. The simplest way to realize this in point–particle mechanics is to use Jacobi’s reformulation of Mapertuis’ principle. The chapter concludes with the relational reformulation of the Newtonian N-body problem (and its scale-invariant variant).

This chapter recounts the main steps that led to the present formulation of SD. In particular, the attempts by Barbour, Ó Murchadha, Anderson, Kelleher and Foster of implementing conformal symmetry through best matching, which culminated in the realization that York’s method (in particular, the assumption of not transforming the trace of the extrinsic curvature when scanning the conformal equivalence class in a CMC slice) can be understood as a volume-preserving conformal transformation.

In this chapter I briefly review York’s method (or the conformal method) for solving the initial value problem of (GR). This method, developed initially by Lichnerowicz and then generalized by Choquet-Bruhat and York, allows to find solutions of the constraints of (GR) (in particular the Hamiltonian, or refoliation constraint) by scanning the conformal equivalence class of spatial metrics for a solution of the Hamiltonian constraint, exploiting the fact that, in a particular foliation (CMC), the transverse nature of the momentum field is preserved under conformal transformations. This method allows to transform the initial value problem into an elliptic problem for the solution for which good existence and uniqueness theorems are available. Moreover this method allows to identify the reduced phase space of (GR) with the cotangent bundle to conformal superspace (the space of conformal 3-geometries), when the CMC foliation is valid. SD essentially amounts to taking this phase space as fundamental and renouncing the spacetime description when the CMC foliation is not available.

This chapter deals with the most important results in SD, namely, the classical solutions of the theory in which the equivalence with (GR) breaks down. Firstly, I study the case of homogeneous but not isotropic cosmologies, known as ‘Bianchi IX’ universes in detail. In this case, each solution that reaches the big bang singularity can be continued uniquely through it, just by requiring continuity of the conformally- and scale-invariant degrees of freedom. The result is a couple of cosmological solutions with opposite orientation glued at the big bang. This result is more general than the homogeneous case, and can be extended to a large class of solutions if the BKL conjecture is valid. In the case of spherically symmetric solutions one has to couple gravity to some form of matter in order to have dynamically non-trivial degrees of freedom. The simplest case is a series of concentric infinitely thin shells of dust in a universe with the topology of a three-sphere. In this case too a departure from the dynamics of (GR) is seen, that manifests itself in a failure of the CMC slicing when one of the shells collapses (no spacetime corresponding to that solution of SD exists). The conformally invariant degrees of freedom, again, seem to still be regular when this happens. In the last part of the chapter I will discuss the sense in which one can talk about asymptotically flat solutions of SD, and past results in this regime.

All measurements are comparisons. The only physically accessible degrees of freedom (DOFs) are dimensionless ratios. The objective description of the universe as a whole thus predicts only how these ratios change collectively as one of them is changed. Here we develop a description for classical Bianchi IX cosmology implementing these relational principles. The objective evolution decouples from the volume and its expansion degree of freedom. We use the relational description to investigate both vacuum dominated and quiescent Bianchi IX cosmologies. In the vacuum dominated case the relational dynamical system predicts an infinite amount of change of the relational DOFs, in accordance with the well known chaotic behaviour of Bianchi IX. In the quiescent case the relational dynamical system evolves uniquely though the point where the decoupled scale DOFs predict the big bang/crunch. This is a non-trivial prediction of the relational description; the big bang/crunch is not the end of physics – it is instead a regular point of the relational evolution. Describing our solutions as spacetimes that satisfy Einstein's equations, we find that the relational dynamical system predicts two singular solutions of GR that are connected at the hypersurface of the singularity such that relational DOFs are continuous and the orientation of the spatial frame is inverted.

We study a complex free scalar field theory on a noncommutative background spacetime called $\kappa$-Minkowski. In particular we address the problem of second quantization. We obtain the algebra of creation and annihilation operators in an explicitly covariant way. Our procedure does not use canonical/Hamiltonian formulations, which turn out to be ill-defined in our context. Instead we work in a spacetime covariant way by introducing a noncommutative Pauli-Jordan function. This function is obtained as a generalization of the ordinary, commutative, one by taking into account the constraints imposed by the symmetries of our noncommutative spacetime. The Pauli-Jordan function is later employed to study the structure of the light cone in $\kappa$-Minkowski spacetime, and to draw conclusions on the superluminal propagation of signals., Comment: 28 pages (+7 of appendices), 10 figures; added appendix on the derivation of finite kappa-Poincar\'e transformations of momentum space from the quantum group's commutation relations, updated references

We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincar\'e and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincar\'e and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore all of them admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous Minkowski spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed by introducing quantum extended Poincar\'e symmetries. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincar\'e algebra of symmetries., Comment: 18 pages

In this work we study the deformations into Lie bialgebras of the three relativistic Lie algebras: de Sitter, Anti-de Sitter and Poincar\'e, which describe the symmetries of the three maximally symmetric spacetimes. These algebras represent the centrepiece of the kinematics of special relativity (and its analogue in (Anti-)de Sitter spacetime), and provide the simplest framework to build physical models in which inertial observers are equivalent. Such a property can be expected to be preserved by Quantum Gravity, a theory which should build a length/energy scale into the microscopic structure of spacetime. Quantum groups, and their infinitesimal version `Lie bialgebras', allow to encode such a scale into a noncommutativity of the algebra of functions over the group (and over spacetime, when the group acts on a homogeneous space). In 2+1 dimensions we have evidence that the vacuum state of Quantum Gravity is one such `noncommutative spacetime' whose symmetries are described by a Lie bialgebra. It is then of great interest to study the possible Lie bialgebra deformations of the relativistic Lie algebras. In this paper, we develop a classification of such deformations in 2, 3 and 4 spacetime dimensions, based on physical requirements based on dimensional analysis, on various degrees of `manifest isotropy' (which implies that certain symmetries, i.e. Lorentz transformations or rotations, are `more classical'), and on discrete symmetries like P and T. On top of a series of new results in 3 and 4 dimensions, we find a no-go theorem for the Lie bialgebras in 4 dimensions, which singles out the well-known `$\kappa$-deformation' as the only one that depends on the first power of the Planck length, or, alternatively, that possesses `manifest' spatial isotropy., Comment: 13 pages, 1 table, matches version published on Nuclear Physics B

In a recent paper, one of us studied spherically symmetric, asymptotically flat solutions of Shape Dynamics, finding that the spatial metric has characteristics of a wormhole - two asymptotically flat ends and a minimal-area sphere, or `throat', in between. In this paper we investigate whether that solution can emerge as a result of gravitational collapse of matter. With this goal, we study the simplest kind of spherically-symmetric matter: an infinitely-thin shell of dust. Our system can be understood as a model of a star accreting a thin layer of matter. We solve the dynamics of the shell exactly and find that, indeed, as it collapses, the shell leaves in its wake the wormhole metric. In the maximal-slicing time we use for asymptotically flat solutions, the shell only approaches the throat asymptotically and does not cross it in a finite amount of time (as measured by a clock `at infinity'). This leaves open the possibility that a more realistic cosmological solution of Shape Dynamics might see this crossing happening in a finite amount of time (as measured by the change of relational/shape degrees of freedom)., Comment: 14 pages, 1 figure. Significantly improved and condensed version over v1

We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schr\"odinger equation. As with the Wheeler--DeWitt equation, time disappears, and a frozen formalism that gives a static wavefunction on the space of possible shapes of the system is obtained. However, if one follows the Dirac procedure and quantizes by imposing constraints, the potential that ensures scale invariance gives rise to a conformal anomaly, and the scale invariance is broken. A behaviour closely analogous to renormalization-group (RG) flow results. The wavefunction acquires a dependence on the scale parameter of the RG flow. We interpret this as time evolution and obtain a novel solution of the problem of time in quantum gravity. We apply the general procedure to the three-body problem, showing how to fix a natural initial value condition, introducing the notion of complexity. We recover a time-dependent Schr\"odinger equation with a repulsive cosmological force in the `late-time' physics and we analyse the role of the scale invariant Planck constant. We suggest that several mechanisms presented in this model could be exploited in more general contexts., Comment: 31 pages, 5 figures

Spherically symmetric, asymptotically flat solutions of Shape Dynamics were previously studied assuming standard falloff conditions for the metric and the momenta. These ensure that the spacetime is asymptotically Minkowski, and that the falloff conditions are Poincar\'e-invariant. These requirements however are not legitimate in Shape Dynamics, which does not make assumptions on the structure or regularity of spacetime. Analyzing the same problem in full generality, I find that the system is underdetermined, as there is one function of time that is not fixed by any condition and appears to have physical relevance. This quantity can be fixed only by studying more realistic models coupled with matter, and it turns out to be related to the dilatational momentum of the matter surrounding the region under study., Comment: 13 pages, 6 figures

Shape dynamics is a 3D conformally invariant theory of gravity that possesses a large set of solutions in common with general relativity. When looked at closely, these solutions are found to behave in surprising ways; in order to probe the fitness of shape dynamics as a viable alternative to General Relativity one must find and understand increasingly more-complex, less-symmetrical exact solutions on which to base perturbative studies and numerical analyses to compare them with data. Spherically symmetric exact solutions have been studied, but only in a static vacuum setup. In this work we construct a class of time-dependent exact solutions of Shape Dynamics 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 fully dynamic nonvacuum solution satisfies in all generality the Hamiltonian structure of shape dynamics. The simplest choice of solutions is shown to be a member of the McVittie family.

We study a family of noncommutative spacetimes constructed by one four-vector. The large set of coordinate commutation relations described in this way includes many cases that are widely studied in the literature. The Hopf-algebra symmetries of these noncommutative spacetimes, as well as the structures of star product and twist, are introduced and considered at first order in the deformation, described by four parameters. We also study the deformations to relativistic kinematics implied by this framework, and calculate the most general expression for the momentum dependence of the Lorentz transformations on momenta, which is an effect that is required by consistency. At the end of the paper we analyse the phenomenological consequences of this large family of vector-like deformations on particles propagation in spacetime. This leads to a set of characteristic phenomenological effects., Comment: 9 pages, 1 figure

The literature on quantum-gravity-inspired scenarios for the quantization of spacetime has so far focused on particle-physics-like studies. This is partly justified by the present limitations of our understanding of quantum-gravity theories, but we here argue that valuable insight can be gained through semi-heuristic analyses of the implications for gravitational phenomena of some results obtained in the quantum-spacetime literature. In particular, we show that the types of description of particle propagation that emerged in certain quantum-spacetime frameworks have striking implications for gravitational collapse and for the behaviour of gravity at large distances., Comment: This essay received honorable mention in the Gravity Research Foundation 2010 Awards for Essays on Gravitation

We introduce an area operator for the Moyal noncommutative plane. We find that the spectrum is discrete, but, contrary to the expectation formulated by other authors, not characterized by a "minimum-area principle". We show that an intuitive analysis of the uncertainty relations obtained from Moyal-plane noncommutativity is fully consistent with our results for the spectrum, and we argue that our area operator should be generalizable to several other noncommutative spaces. We also observe that the properties of distances and areas in the Moyal plane expose some weaknesses in the line of reasoning adopted in some of the heuristic analyses of the measurability of geometric spacetime observables in the quantum-gravity realm. © 2009 Elsevier B.V. All rights reserved.

We consider some alternative scenarios for the fate of Poincaré/Lorentz symmetry at the Planck scale, and we discuss some opportunities to test these scenarios.

A lot has been written about the tragic story of Rapa Nui (Easter Island). A thriving culture, capable of building hundreds of moai, the emblematic giant stone statues, which obliterated itself due to unsustainable practices, mainly deforestation and overpopulation. The people who cut the last palm tree on the island are often mentioned. Did they understand they were committing suicide? Or were they just too concentrated on other things which they misjudged as more important than trees?

Have our fundamental theories got time right? Does size really matter? Or is physics all in the eyes of the beholder? In this essay, we question the origin of time and scale by reevaluating the nature of measurement. We then argue for a radical scenario, supported by a suggestive calculation, where the flow of time is inseparable from the measurement process. Our scenario breaks the bond of time and space and builds a new one: the marriage of time and scale.

It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behavior of any kind that defines an `arrow of time'. We show that this is not so. The simplest non-trivial time-symmetric law that can be used to model a dynamically closed universe is the Newtonian $N$-body problem with vanishing total energy and angular momentum. Because of special properties of this system (likely to be shared by any law of the Universe), its typical solutions all divide at a uniquely defined point into two halves. In each a well-defined measure of shape complexity fluctuates but grows irreversibly between rising bounds from that point. Structures that store dynamical information are created as the complexity grows and act as `records'. Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and will only be aware of the records of one branch and deduce a unique past and future direction from inspection of the available records., To appear in Physical Review Letters

We provide a bottom-up argument to derive some known results from holographic renormalization using the classical bulk-bulk equivalence of General Relativity and Shape Dynamics, a theory with spatial conformal (Weyl) invariance. The purpose of this paper is twofold: 1) to advertise the simple classical mechanism: trading of gauge symmetries, that underlies the bulk-bulk equivalence of General Relativity and Shape Dynamics to readers interested in dualities of the type of AdS/CFT; and 2) to highlight that this mechanism can be used to explain certain results of holographic renormalization, providing an alternative to the AdS/CFT conjecture for these cases. To make contact with usual the semiclassical AdS/CFT correspondence, we provide, in addition, a heuristic argument that makes it plausible why the classical equivalence between General Relativity and Shape Dynamics turns into a duality between radial evolution in gravity and the renormalization group flow of a conformal field theory. We believe that Shape Dynamics provides a new perspective on gravity by giving conformal structure a primary role within the theory. It is hoped that this work provides the first steps towards understanding what this new perspective may be able to teach us about holographic dualities., 27 pages, no figures. Version to appear in EPJC. Title changed. Minor corrections to text

We show that there are 2 equivalent first order descriptions of 2+1 gravity with non-zero cosmological constant. One is the well-known spacetime description and the other is in terms of evolving conformal geometry. The key tool that links these pictures is Cartan geometry, a generalization of Riemannian geometry that allows for geometries locally modelled off arbitrary homogeneous spaces. The two different interpretations suggest two distinct phase space reductions. The spacetime picture leads to the 2+1 formulation of General Relativity due to Arnowitt, Deser, and Misner while the conformal picture leads to Shape Dynamics. Cartan geometry thus provides an alternative to symmetry trading for explaining the equivalence of General Relativity and Shape Dynamics., 29 preprint pages, 3 figures

The absence of unique time evolution in Einstein's spacetime description of gravity leads to the hitherto unresolved `problem of time' in quantum gravity. Shape Dynamics is an objectively equivalent representation of gravity that trades spacetime refoliation invariance for three-dimensional conformal invariance. Its logical completion presented here gives a dimensionless description of gravitational dynamics. We show that in this framework the classical problem of time is completely solved. Since a comparable definitive solution is impossible within the spacetime description, we believe Shape Dynamics provides a key ingredient for the creation of quantum gravity., 14 pages, no figures