General Relativity on closed spatial topologies can be derived, using a technique called "best-matching", as an evolving 3-geometry subject to constraints. These constraints can be thought of as a way of imposing temporal and spatial relationalism. The same type of constraints can be used in non-relativistic particle models to produce relational theories that suffer from the same Problem of Time as that encountered in General Relativity. As a result, these simple toy models are well suited for studying the Problem of Time in quantum gravity. In this paper, a version of these particle models is studied where we "best-match" the time translational invariance of the theory. Using insights gained from this procedure, we can move back and forth between absolute and relational time by changing the way in which the relational fields are varied. We then proceed to quantize this theory using Dirac and path integral quantizations. We discover that one of the constraints of the theory, which we call the Mach constraint, is responsible for removing the dependence of the theory on a background structure. It is this Mach constraint that is responsible for making the theory temporally relational. Because of the deep relationship between these models and General Relativity, this work may shed new light on the Problem of Time in quantum gravity and how one might expect time to emerge on quantum subsystems of the universe.
A new method involving the effective wave function is used to define the mass of a particle in a standard five-dimensional extension of general relativity. The mass is inversely proportional to the magnitude of the scalar field of the extra dimension. Since the scalar field is global and depends on the coordinates, this definition for the particle mass agrees with Machs Principle.
We discuss the emergence of relativistic effects in the Machian universe with a global preferred frame and use thermodynamic considerations to clarify the origin of gravity as an entropic force and the origin of dark energy/cosmic acceleration as related to the Hawking-Unruh temperature at the universe's horizon.
Starting from the origin of Einstein general relativity (GR) the request of Mach on the theory's structure has been the core of the foundational debate. That problem is strictly connected with the nature of the mass-energy equivalence. It is well known that this is exactly the key point that Einstein used to realize a metric theory of gravitation having an unequalled beauty and elegance. On the other hand, the current requirements of particle physics and the open questions within extended gravity theories request a better understanding of Equivalence Principle (EP). The MOND theory by Milgrom …
Classical and semiclassical schemes are presented that are timeless at the primary level and recover time from Mach's `time is to be abstracted from change' principle at the emergent secondary level. The semiclassical scheme is a Machian variant of the Semiclassical Approach to the Problem of Time in Quantum Gravity. The classical scheme is Barbour's, cast here explicitly as the classical precursor of the Semiclassical Approach. Thus the two schemes have been married up, as equally-Machian and necessarily distinct, since quantum change is part of whence the latter's timestandard is abstracted.
Gravity does not provide any scale for matter properties. We argue that this is also the implication of Mach's hypothesis of the relativity of inertia. The most general spacetime compatible with this property of gravity is that admitting three, independent spatial homothetic Killing vectors generating an arbitrary function of each one of the three spatial coordinates. The matter properties for such a spacetime are (spatially) arbitrary and the matter generating the spacetime admits {\it any} equation of state. This is also the most general spacetime containing the weak gravity physics in its e…
"It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive through the changes of things." Ernst Mach [1]. What change? Three answers to this are `any change' (Rovelli), 'all change' (Barbour) and my argument here for the middle ground of a `sufficient totality of locally relevant change' (STLRC) giving a generalization of the astronomers' ephemeris time. I then use STLRC as a selection principle on existing and new approaches to the Problem of Time in Quantum Gravity. Emergent Jacobi-Barbour-Bertotti time can be inte…
The standard model of cosmology is based on the general theory of relativity and demands more than 95\% of the universe to consist of dark matter and dark energy that has no direct observational evidence till date. The foundation of the concept these dark components are based on a fixed relation between the strength of the gravitational field and the matter density. Alternate models are put forward in past to explain the observations without dark components in the universe. Though they have their own merits and draw backs. In this paper we propose a new cosmological model based on Mach's princ…
The Problem of Time is that `time' in each of ordinary quantum theory and general relativity are mutually incompatible notions. This causes difficulties in trying to put these two theories together to form a theory of Quantum Gravity. The Problem of Time has 8 facets in canonical approaches. I clarify that all but one of these facets already occur at the classical level, and reconceptualize and re-name some of these facets as follows.The Frozen Formalism Problem becomes Temporal Relationalism, the Thin Sandwich Problem becomes Configurational Relationalism, via the notion of Best Matching. The…
We adopt a 'thermodynamical' formulation of Mach's principle that the rest mass of a particle in the Universe is a measure of its long-range collective interactions with all other particles inside the horizon. We consider all particles in the Universe as a 'gravitationally entangled' statistical ensemble and apply the approach of classical statistical mechanics to it. It is shown that both the Schrodinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to t…
We explore cosmological solutions to Lorentz breaking gravity using the gravitational sector of the Standard Model Extension (SME). By using a simple toy model for Lorentz violation and under the assumption that the so-called $\mathfrak{T}$-tensor is covariantly constant, we show that the gravity sector SME influences basic cosmology. If the approach used in this toy model is valid these results should point the way for more sofisticated studies.
The entropy-area relation of black holes is one of the important results of theoretical physics. It is one of the few relations that is used to test theories of quantum gravity in the absence of any experimental evidence. It states that $4 \times \ell_P^2$ is the fundamental area that holds \textit{one} bit of information. Consequently, a question arises: why $4 \times \ell_P^2$ and not $1 \times \ell_P^2$ is the fundamental holder of \textit{one} bit of information? In any case it seems the latter choice is more natural. We show that this question can be answered with a more explicit counting…
It is argued that substantial portions of both Newtonian particle mechanics and general relativity can be viewed as relational (rather than absolute) theories. I furthermore use the relational particle models as toy models to investigate the problem of time in closed-universe canonical quantum general relativity. I consider thus in particular the internal time, semiclassical and records tentative resolutions of the problem of time.
We study the propagator on a single tetrahedron in a three dimensional toy model of quantum gravity with positive cosmological constant. The cosmological constant is included in the model via q-deformation of the spatial symmetry algebra, that is, we use the Tuarev-Viro amplitude. The expected repulsive effect of dark energy is recovered in numerical and analytic calculations of the propagator at large scales comparable to the infrared cutoff. However, due to the simplicity of the model we do not obtain the exact Newton limit of the propagator. This is a first step toward the similar calculati…
Normalizing the Einstein-Hilbert action by the volume functional makes the theory invariant under constant shifts in the Lagrangian. The associated field equations then resemble unimodular gravity whose otherwise arbitrary cosmological constant is now determined as a Machian universal average. We prove that an empty space-time is necessarily Ricci tensor flat, and demonstrate the vanishing of the cosmological constant within the scalar field paradigm. The cosmological analysis, carried out at the mini-superspace level, reveals a vanishing cosmological constant for a Universe which cannot be cl…
The periodic standing wave approach to binary inspiral assumes rigid rotation of gravitational fields and hence helically symmetric solutions. To exploit the symmetry, numerical computations must solve for ``helical scalars,'' fields that are functions only of corotating coordinates, the labels on the helical Killing trajectories. Here we present the formalism for describing linearized general relativity in terms of helical scalars and we present solutions to the mixed partial differential equations of the linearized gravity problem (and to a toy nonlinear problem) using the adapted coordinate…
The quantum-to-classical transition through decoherence is a major facet of the semi-classical analysis of quantum models that are supposed to admit a classical regime, as quantum gravity should be. A particular problem of interest is the decoherence of black hole horizons and holographic screens induced by the bulk-boundary coupling with interior degrees of freedom. Here in this paper we present a first toy-model, in the context of loop quantum gravity, for the dynamics of a surface geometry as an open quantum system at fixed total area. We discuss the resulting decoherence and recoherence an…
Polarization of light rotates in a gravitational field. The accrued phase is operationally meaningful only with respect to a local polarization basis. In stationary space-times, we construct local reference frames that allow us to isolate the Machian gravimagnetic effect from the geodetic (mass) contribution to the rotation. The Machian effect is supplemented by the geometric term that arises from the choice of standard polarizations. The phase accrued along a close trajectory is gauge-independent and is zero in the Schwarzschild space-time. The geometric term may give a dominant contribution …
The two body problem in a scalar theory of gravity is investigated. We focus on the closest theory to General Relativity (GR), namely Nordström's theory of gravity (1913). The gravitational field can be exactly solved for any configuration of point-particles. We then derive the exact equations of motion of two inspiraling bodies including the exact self-forces terms. We prove that there is no innermost circular orbit (ICO) in the exact theory whereas we find (order-dependent) ICOs if post-Newtonian (PN) truncations are used. We construct a solution of the two body problem in an iterative (non-…
In Loop Quantum Gravity, tremendous progress has been made using the Ashtekar-Barbero variables. These variables, defined in a gauge-fixing of the theory, correspond to a parametrization of the solutions of the so-called simplicity constraints. Their geometrical interpretation is however unsatisfactory as they do not constitute a space-time connection. It would be possible to resolve this point by using a full Lorentz connection or, equivalently, by using the self-dual Ashtekar variables. This leads however to simplicity constraints or reality conditions which are notoriously difficult to impl…
We review some approaches to the Hamiltonian dynamics of (loop) quantum gravity, the main issues being the regularization of the Hamiltonian and the continuum limit. First, Thiemann's definition of the quantum Hamiltonian is presented, and then more recent approaches. They are based on toy models which provide new insights into the difficulties and ambiguities faced in Thiemann's construction. The models we use are parametrized field theories, the topological BF model of which a special case is three-dimensional gravity which describes quantum flat space, and Regge lattice gravity.
We tackle the question of motion in Quantum Gravity: what does motion mean at the Planck scale? Although we are still far from a complete answer we consider here a toy model in which the problem can be formulated and resolved precisely. The setting of the toy model is three dimensional Euclidean gravity. Before studying the model in detail, we argue that Loop Quantum Gravity may provide a very useful approach when discussing the question of motion in Quantum Gravity.
It is often argued that gravity has to be a quantum theory simply because a fundamentally semiclassical approach would necessarily be inconsistent. Here I review recent Newtonian toy models of (stochastic) semiclassical gravity. They provide one option to implement a force semiclassically without getting into the known problems associated with mean-field. These models are not complete theories and should not be considered too seriously, but their consistency shows that semiclassical gravity is hard to dismiss on purely theoretical grounds.
We propose a toy model of quantum gravity in two dimensions with Euclidean signature. The model is given by a kind of discretization which is different from the dynamical triangulation. We show that there exists a continuum limit and we can calculate some physical quantities such as the expectation value of the area, that is, the volume of the two dimensional euclidean space-time. We also consider the extensions of the model to higher dimensions.
Why are "analogue spacetimes'' interesting? For the purposes of this workshop the answer is simple: Analogue spacetimes provide one with physically well-defined and physically well-understood concrete models of many of the phenomena that seem to be part of the yet incomplete theory of "quantum gravity'', or more accessibly, "quantum gravity phenomenology''. Indeed "analogue spacetimes'' provide one with concrete models of "emergence'' (whereby the effective low-energy theory can be radically different from the high-energy microphysics). They also provide many concrete and controlled models of …
We present a dynamical toy model for an expanding universe inside a black hole. The model is built by matching a spherically symmetric collapsing matter cloud to an expanding Friedmann-Robertson-Walker universe through a phase transition that occurs in the quantum-gravity dominated region, here modeled with semi-classical corrections at high density. The matching is performed on a space-like hyper-surface identified by the co-moving time at which quantum-gravity induced effects halt collapse. The purpose of the model is to suggest a possible reconciliation between the observation that black ho…
Some relations of the quantum potential to Weyl geometry are indicated with applications to the Friedmann equations for a toy quantum cosmology. Osmotic velocity and pressure are briefly discussed in terms of quantum mechanics and superfluids with connections to gravity.
What can we learn about quantum gravity from a simple toy model, without actually quantizing it? The toy model consists of a finite number of point particles, coupled to three dimensional Einstein gravity. It has finitely many physical degrees of freedom. These are basically the relative positions of the particles in spacetime and the conjugate momenta. The resulting reduced phase space is derived from Einstein gravity as a topological field theory. The crucial point is thereby that we do not make any a priori assumptions about this phase space, except that the dynamics of the gravitational fi…
In the present paper we analyze a toy model for an $f(ϕ,R)$ gravity which has the form of a power-law modified gravity in which the exponent is space-time dependent. Namely, we investigate the effects of adding to the Hilbert-Einstein action an $R^ϕ$-term. We present possible equivalences of the model with known models of modified gravity theories and examine the problem of matter stability in this model. Like $f(R)$-gravity toy models, the present one offers the possibility of unifying the early and the late-time evolution of the Universe. We show that the behavior of the scalar field depends…
The idea that quantum gravity effects might leak outside the horizon of a black hole has recently been intensively considered. In this study, we calculate the quasinormal modes as a function of the location and amplitude of a generic metric perturbation distorting to the Schwarzschild spacetime. We conclude on the possible observability of quantum metric corrections by current and future gravitational wave experiments.
We compare the recent loop quantum cosmology approach of Bojowald and co-workers with earlier quantum cosmological schemes. Because the weak-energy condition can now be violated at short distances, and not necessarily with a high energy density, a number of possible instabilities are suggested: flat space unstable to expansion or baby universe production. Or else a Machian type principle is required to prevent such behaviour. Allowing a bounce to prevent an approaching singularity seems incompatible with other standard notions concerning the arrow of time and unitarity. Preventing rapid oscill…
We present a one loop calculation in the context of Horava-Lifshitz gravity. Due to the complexity of the calculation in the full theory we focus here on the study of a toy model, namely the conformal reduction of the z=2 projectable theory in 2+1 dimensions. For this value of the dimension there are no gravitons, hence the conformal mode is the only physical degree of freedom, and thus we expect our toy model to lead to qualitatively correct answers regarding the perturbative renormalization of the full theory. We find that Newton's constant (dimensionless in Horava-Lifshitz gravity) is asymp…
We find a class of electrically charged exact solutions for a toy model of metric-affine gravity. Their metric is of the Plebański-Demiański type and their nonmetricity and torsion are represented by a triplet of covectors with dilation, shear, and spin charges.
In recent years several approaches to quantum gravity have found evidence for a scale dependent spectral dimension of space-time varying from four at large scales to two at small scales of order of the Planck length. The first evidence came from numerical results on four-dimensional causal dynamical triangulations (CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then little progress has been made in analytically understanding the numerical results coming from the CDT approach and showing that they remain valid when taking the continuum limit. Here we argue that the spectral dimens…
The standard formulation of general relativity fails to describe some recent interests in the universe. It impels us to go beyond the standard formulation of gravity. The $f(Q)$ gravity theory is an interesting modified theory of gravity, where the gravitational interaction is driven by the nonmetricity $Q$. This study aims to examine the cosmological models with the presence of bulk viscosity effect in the cosmological fluid within the framework of $f(Q)$ gravity. We construct three bulk viscous fluid models, i.e. (i) for the first model, we assuming the Lagrangian $f(Q)$ as linear dependence…
I investigate the relation between an operative definition of the area of a surface specified by matter fields and the area operators recently introduced in the canonical/loop approach to Quantum Gravity and in Rovelli's variant of the Husain-Kuchar Quantum-Gravity toy model. The results suggest that the discreteness of the spectra of the area operators might not be observable.
We study tree-unitarity and renormalizability in Lifshitz-scaling theory, which is characterized by an anisotropic scaling between the spacial and time directions. Due to the lack of the Lorentz symmetry, the conditions for both unitarity and renormalizability are modified from those in relativistic theories. For renormalizability, the conventional discussion of the power counting conditions has to be extended. Because of the dependence of $S$-matrix elements on the reference frame, unitarity requires stronger conditions than those in relativistic cases. We show that the conditions for unitari…
Non-perturbative approaches to quantum gravity call for a deep understanding of the emergence of geometry and locality from the quantum state of the gravitational field. Without background geometry, the notion of distance should entirely emerge from the correlations between the gravity fluctuations. In the context of loop quantum gravity, quantum states of geometry are defined as spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. Here, we develop the condensed matter point of view on extracting…
In this paper, we propose a toy model of the renormalizable theory describing massive spin two field. Although the model is renormalizable, we show that the model contains ghost. The coupling of the theory with gravity can be regarded as a new kind of bimetric gravity or bigravity. We show that the massive spin two field plays the role of the cosmological constant.
We investigate the dynamics of two quantum mechanical oscillator system-bath toy models obtained by truncating to zero spatial dimensions linearized gravity coupled to a massive scalar field and scalar QED. The scalar-gravity toy model maps onto the phase damped oscillator, while the scalar QED toy model approximately maps onto an oscillator system subject to two-photon damping. The toy models provide potentially useful insights into solving for open system quantum dynamics relevant to the full scalar QED and weak gravitational field systems, in particular operational probes of the decoherence…
In this PhD thesis, we develop a new approach to classical gravity starting from Mach's principles and the idea that the local shape of spatial configurations is fundamental. This new theory, "shape dynamics", is equivalent to general relativity but differs in an important respect: shape dynamics is a theory of dynamic conformal 3-geometry, not a theory of spacetime. Equivalence is achieved by trading foliation invariance for local conformal invariance (up to a global scale). After the trading, what is left is a gauge theory invariant under 3d diffeomorphisms and conformal transformations that…
We revisit singularities of two distinct kinds in the Cauchy problem of general scalar-tensor theories of gravity (previously discussed in the literature), and of metric and Palatini f(R) gravity, in both their Jordan and Einstein frame representations. Examples and toy models are used to shed light onto the problem and it is shown that, contrary to common lore, the two conformal frames are equivalent with respect to the initial value problem.
A sharp version of the information paradox involves a seeming violation of the monogamy of entanglement during black hole evaporation. We construct an analogous paradox in empty anti-de Sitter space. In a local quantum field theory, Bell correlations between operators localized in mutually spacelike regions are monogamous. We show, through a controlled calculation, that this property can be violated by an order-1 factor in a theory of gravity. This example demonstrates that what appears to be a violation of the monogamy of entanglement may just be a subtle violation of locality in quantum grav…
We construct a toy model for isothermal spheres in Horava gravity, which includes Einstein's gravity if a parameter is appropriately chosen. The equations for the isothermal spheres are derived from the partition function of the gravitating particle system. We confirm that the Newtonian limit of the system coincides with the model of the well-known isothermal sphere. The stability of the isothermal sphere is found to be sensitive to the energy density at the center of the sphere.
In this paper, we attempt to resolve the dark matter problem in f(T) gravity. Specifically, from our model we successfully obtain the flat rotation curves of galaxies containing dark matter. Further, we obtain the density profile of dark matter in galaxies. Comparison of our analytical results shows that our torsion-based toy model for dark matter is in good agreement with empirical data-based models. It shows that we can address the dark matter as an effect of torsion of the space.
We study different aspects of the Hořava-Lifshitz inspired Bianchi-II cosmology and its relations with the mixmaster universe model. First, we present exact solutions for a toy model, where only the cubic in spatial curvature terms are present in the action; then we briefly discuss some exotic singularities, which can appear in this toy model. We study also the toy model where only the quadratic in spatial curvature terms are present in the action. We establish relations between our results and those obtained by using the Hamiltonian formalism. Finally, we apply the results obtained by studyin…
We formulate holomorphic theory of gravity and study how the holomorphy symmetry alters the two most important singular solutions of general relativity: black holes and cosmology. We show that typical observers (freely) falling into a holomorphic black hole do not encounter a curvature singularity. Likewise, typical observers do not experience Big Bang singularity. Unlike Hermitian gravity \cite{MantzHermitianGravity}, Holomorphic gravity does not respect the reciprocity symmetry and thus it is mainly a toy model for a gravity theory formulated on complex space-times. Yet it is a model that de…
General ideas of gauge/gravity duality allow for the possibility of time dependent solutions that interpolate between a perturbative gauge theory phase and a weakly curved string/gravity phase. Such a scenario applied to cosmology would exhibit a non-geometric phase before the big bang. We investigate a toy model for such a cosmology, whose endpoint is the classical limit of the two-dimensional non-critical string. We discuss the basic dynamics of this model, in particular how it evolves toward the double scaling limit required for stringy dynamics. We further comment on the physics that will …
General aspects of vielbein representation, ADM formulation and canonical quantization of gravity are reviewed using pure gravity in three dimensions as a toy model. The classical part focusses on the role of observers in general relativity, which will later be identified with quantum observers. A precise definition of gauge symmetries and a classification of inequivalent solutions of Einstein's equations in dreibein formalism is given as well. In the quantum part the construction of the physical Hilbert space is carried out explicitly for a torus and cylinder type space manifold, which has no…
Unified dynamical system formulations for $f(R,ϕ,X)$ gravity with applications to nonminimal derivative coupling and $R^2$-Higgs inflation
Institute of Research and Development, Duy Tan University（延大大学研发研究所）；Faculty of Natural Sciences, Duy Tan University（延大大学自然科学学院）；Center for Space Research, North-West University（西北大学太空研究中心）；Department of Applied Mathematics, University of Campinas（坎皮纳斯大学应用数学系）；Instituto de Física, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学物理研究所）
In [1] we initiated an approach towards quantizing the Hamiltonian constraint in Loop Quantum Gravity (LQG) by requiring that it generates an anomaly-free representation of constraint algebra off-shell. We investigated this issue in the case of a toy model of a 2+1-dimensional $U(1)^{3}$ gauge theory, which can be thought of as a weak coupling limit of Euclidean three dimensional gravity. However in [1] we only focused on the most non-trivial part of the constraint algebra that involves commutator of two Hamiltonian constraints. In this paper we continue with our analysis and obtain a represen…
The objective of the paper was to examine gravitational evolutions in the Higgs--dark matter sector toy model. The real part of the Higgs doublet was modelled by a neutral scalar. Two dark matter candidates introduced were the dark photon and a charged complex scalar. Non-minimal couplings of both scalars to gravity were included. The coupling channels between the ordinary and dark matter sectors were kinetic mixing between the electromagnetic and dark $U(1)$ fields and the Higgs portal coupling among the scalars. The structures of emerging singular spacetimes were either of Schwarzschild or R…
We propose a mathematical structure, based on a noncommutative geometry, which combines essential aspects of general relativity and quantum mechanics, and leads to correct "limiting cases" of both these theories. We quantize a groupoid constructed on space-time rather than space-time itself. Both space and time emerge in the transition process to the commutative case. Our approach clearly suggests that quantum gravitational observables should be looked for among correlations of distant phenomena rather than among local effects. A toy model is computed (based on a finite group) which predicts t…
We compute quantum corrections for the gravitational potential obtained by including a derivative self-coupling in its classical dynamics as a toy model for analysing quantum gravity in the strong field regime. In particular, we focus on quantum corrections to the classical solutions in the vacuum outside localised matter sources.
An inhomogeneous (1+1)-dimensional model of the quantum gravity is considered. It is found, that this model corresponds to a string propagating against some curved background space. The quantization scheme including the Wheeler-DeWitt equation and the "particle on a sphere" type of the gauge condition is suggested. In the quantization scheme considered, the "problem of time" is solved by building of the quasi-Heisenberg operators acting in a space of solutions of the Wheeler-DeWitt equation and the normalization of the wave function corresponds to the Klein-Gordon type. To analyze the physical…
