Trying to connect a fundamentally non-commutative spacetime with the conservative perturbative approach to quantum gravity, we are led to the natural question: are non-commutative geometrical effects already present in the regime where perturbative quantum gravity provides a predictive framework? Moreover, is it necessary to introduce non-commutativity by hand, or does it arise through quantum-gravitational effects? We show that the first question can be answered in the affirmative, and the second one in the negative: perturbative quantum gravity predicts non-commutativity at the Planck scale, once one clarifies the structure of observables in the quantum theory.
In this paper we construct a non-commutative gauge theory for the deformed metric corresponding to the modified structure of a gravitational field in the case of Yukawa-Schwarzschild non-commutative space-time. The thermodynamic properties and corrections to the gravitational force on the horizon of a non-commutative Yukawa-Schwarzschild black hole are analysed.
In this paper, we analyze the thermodynamic stability of Schwarzschild Modified Gravity (MOG) black holes in a non-commutative framework. We show that, unlike a commutative MOG black hole, in the coherent state picture of non-commutativity MOG black holes are thermodynamically stable. At the final stage of evaporation, a stable remnant with zero temperature and finite entropy is left in this noncommutative framework. Also, we consider Parikh-Wilczek tunneling mechanism of massive particles from non-commutative MOG black holes and demonstrate that information leaks out of non-commutative MOG bl…
We study a non-commutative deformation of general relativity based on spectral invariants of a partial differential operator acting on sections of a vector bundle over a smooth manifold. We compute the first non-commutative corrections to Einstein equations in the weak deformation limit and analyze the spectrum of the theory. Related topics are discussed as well.
Traversable wormholes in General Relativity (GR) require exotic matter sources that violate the null energy condition (NEC), and such behavior may be avoided in modified gravity. Moreover, the concept of non-commutative geometry as a gravitational source can be leveraged both in GR and modified gravity to realize non-trivial space-time configurations. In this study, we use $f(R)$ gravity in conjunction with non-commutative geometry to analyze spherically symmetric traversable Morris-Thorne wormhole solutions from the aspect of energy condition violation, considering both constant and variable …
The field equations in modified gravity theories possess an important decoupling property with respect to certain classes of nonholonomic frames. This allows us to construct generic off--diagonal solutions depending on all spacetime coordinates via generating and integration functions containing (un--)broken symmetry parameters. Some corresponding analogous models have a nice ultraviolet behavior and seem to be (super) renormalizable in a sense of covariant modifications of Hořava--Lifshits (HL) and ghost free gravity. The apparent noncommutativity and breaking of Lorentz invariance by quantum…
Einstein-Gauss-Bonnet gravity is a generalization of the general relativity to higher dimensions in which the first and second-order terms correspond to general relativity and Einstein-Gauss-Bonnet gravity respectively. We construct a new class of five-dimensional (5D) thin-shell wormholes by the `Cut-Paste' technique from black holes in Einstein-Gauss-Bonnet gravity inspired by non-commutative geometry starting with a static spherically symmetric, Gaussian mass distribution as a source and for this structural form of the thin shell wormhole we have explored several salient features of the sol…
We study the shadow and energy emission rate of a spherically symmetric non-commutative black hole in Rastall gravity. Depending on the model parameters, the non-commutative black hole can reduce to the Schwarzschild black hole. Since the non-vanishing non-commutative parameter affects the formation of event horizon, the visibility of the resulting shadow depends on the non-commutative parameter in Rastall gravity. The obtained sectional shadows respect the unstable circular orbit condition, which is crucial for physical validity of the black hole image model.
This paper is twofold. First of all a complete unified picture of $n$-dimensional quantum gravity is proposed in the following sense: In spin foam models of quantum gravity the evaluation of spin networks play a very important role. These evaluations correspond to amplitudes which contribute in a state sum model of quantum gravity. In \cite{fk}, the evaluation of spin networks as integrals over internal spaces was described. This evaluation was restricted to evaluations of spin networks in $n$-dimensional Euclidean quantum gravity. Here we propose that a similar method can be considered to inc…
In this contribution to the proceedings of the Corfu Summer Institute 2015, I give an overview over quantum field theories on non-commutative Moyal space and renormalization. In particular, I review the new features and challenges one faces when constructing various scalar, fermionic and gauge field theories on Moyal space, and especially how the UV/IR mixing problem was solved for certain models. Finally, I outline more recent progress in constructing a renormalizable gauge field model on non-commutative space, and how one might attempt to prove renormalizability of such a model using a gener…
I review recent progress in understanding non-perturbative aspects of string theory, quantum gravity and non-commutative geometry using lattice methods.
We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator $D$ on an $n$ dimensional compact Riemannian manifold with $n\geq 4$, $n$ even, the Wodzicki residue Res$(D^{-n+2})$ is the integral of the second coefficient of the heat kernel expansion of $D^{2}$. We use this result to derive a gravity action for commutative geometry which is the usual Einstein Hilbert action and we also apply our results to a non-commutative extension which, is given by the tensor product of the algebra of smooth functions on a mani…
In this paper we have solved the Bohmian equations of quantum gravity, perturbatively. Solutions up to second order are derived explicitly, but in principle the method can be used in any order. Some consequences of the solution are disscused.
We show that the effective dynamics of matter fields coupled to 3d quantum gravity is described after integration over the gravitational degrees of freedom by a braided non-commutative quantum field theory symmetric under a kappa-deformation of the Poincare group.
In this work we use the theory of Teleparallelism Equivalent to General Relativity based in non-commutative space-time coordinates. In this context, we write the corrections of the Schwarzschild solution. As a important result, we find the corrections of the gravitational energy in the realm of teleparallel gravity due to the non-commutativity of space-time. Then we interpret such corrections as a manifestation of quantum theory in gravitational field.
We describe the geomety of a set of scalar fields coupled to gravity. We consider the formalism of a differential Z_2-graded algebra of $2\times 2$ matrices whose elements are differential forms on space-time. The connection and the vierbeins are extended to incorporate additional scalar and vector fields. The resulting action describes two universes coupled in a non-minimal way to a set of scalar fields. This picture is slightly different from the description of general relativity in the framework of non-commutative geomety.
We present a survey of the application of Cones' Non-Commutative Geometry to gravitation. Bases of the theory and Euclidian gravity models are reviewed. Then we discuss the problem of a Lorentzian generalization of the theory and review existing attempts of solution.
This is my contribution to Proceedings of II Russian-Spanish Congress on Particle and Nuclear Physics at all Scales and Cosmology, Saint Petersburg, Russia, October 1 - 4, 2013. We give a brief overview of the ghost-free massive and bimetric gravity emphasising its non-perturbative aspects and ADM approach to calculating the precise number of degrees of freedom. To the standard material, we add some thoughts concerning existence and uniqueness problems of matrix square roots in non-perturbative metric formulations of massive gravity.
We compute the leading order of the three-point function in loop quantum gravity, using the vertex expansion of the Euclidean version of the new spin foam dynamics, in the region of gamma<1. We find results consistent with Regge calculus in the limit gamma->0 and j->infinity. We also compute the tree-level three-point function of perturbative quantum general relativity in position space, and discuss the possibility of directly comparing the two results.
In this paper we will study non-anticommutative perturbative quantum gravity on spacetime with a complex metric. After analysing the BRST symmetry of this non-anticommutative perturbative quantum gravity, we will also analyse the effect of shifting all the fields. We will construct a Lagrangian density which is invariant under the original BRST transformations and the shift transformations in the Batalin-Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this shift symmetry invariant Lagrangian density can be elegantly written down in …
The groupoid approach to noncommutative unification of general relativity with quantum mechanics is compared with the canonical gravity quantization. It is shown that by restricting the corresponding noncommutative algebra to its (commutative) subalgebra, which determines the space-time slicing, an algebraic counterpart of superspace (space of 3-metrics) can be obtained. It turns out that when this space-time slicing emerges the universe is already in its commutative regime. We explore the consequences of this result.
We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.
In this paper the Seiberg-Witten map is first analyzed for non-commutative Yang-Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg-Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space-time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, wh…
The description of the phase space of relativistic particles coupled to three-dimensional Einstein gravity requires momenta which are coordinates on a group manifold rather than on ordinary Minkowski space. The corresponding field theory turns out to be a non-commutative field theory on configuration space and a group field theory on momentum space. Using basic non-commutative Fourier transform tools we introduce the notion of non-commutative heat-kernel associated with the Laplacian on the non-commutative configuration space. We show that the spectral dimension associated to the non-commutati…
In this paper we will study the spontaneous breakdown of the Lorentz symmetry by ghost condensation in perturbative quantum gravity. Our analysis will be done in the Curci-Ferrari gauge. We will also analyse modification of the BRST and the anti-BRST transformations by the formation of this ghost condensate. It will be shown that even though the modified BRST and the modified anti-BRST transformations are not nilpotent, their nilpotency is restored on-shell.
Quantum groups and non-commutative spaces have been repeatedly utilized in approaches to quantum gravity. They provide a mathematically elegant cut-off, often interpreted as related to the Planck-scale quantum uncertainty in position. We consider here a different geometrical interpretation of this cut-off, where the relevant non-commutative space is the space of directions around any spacetime point. The limitations in angular resolution expresses the finiteness of the angular size of a Planck-scale minimal surface at a maximum distance 1/\sqrt{Lambda} related the cosmological constant Lambda.…
This article is based on the study of wormhole geometries in the context of symmetric teleparallel gravity or $f(Q)$ gravity, where $Q$ is the non-metricity scalar, and it is responsible for the gravitational interaction. To discuss the wormhole solutions, we consider spherically symmetric static spacetime metric with anisotropic matter contents under well-known non-commutative distributions known as Gaussian and Lorentzian distributions with an extra condition of permitting conformal killing vectors (CKV). This work aims to obtain wormhole solutions under these distributions, and through we f…
In this work, we construct a non-commutative (NC) gauge theory of gravity for any metric with spherical symmetries, where we use a non-diagonal tetrad field. The deformed gauge potentials (tetrad fields) and the components of deformed metric are computed to the second order in the NC parameter $Θ^{μν}$, as the application to the Schwarzschild black hole we show that the NC geometry removes the singularity at the origin of the black hole, and increase the event horizon. The non-commutativity correction to the effective potential of the Schwarzschild metric is also computed and we show how this …
Usually, General Relativity (GR) is known to be unrenormalizable perturbatively from the viewpoint of quantum field theory. But in the modern sense of renormalizability, there still remains the possibility to investigate whether GR is "nonpertubatively" renormalizable or not. Here I review the basics and results in this topic based on the "Effective Average Action" approach which was proposed by M.Reuter, and discuss its application to the balck hole geometry.
I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide against just selling my mathematical wares, and for a survey, necessarily very selective, but taking a global phenomenological approach to its subject matter. After all, non-commutative geometry does not purport yet to solve the riddle of quantum gravity; it is more of an insurance policy against the probable failure of the other approaches. The plan is as foll…
We consider the problem of (1+1)-dimensional quantum gravity coupled to particles. Working with the canonically reduced Hamiltonian, we obtain its post-Newtonian expansion for two identical particles. We quantize the (1+1)-dimensional Newtonian system first, after which explicit energy corrections to second order in 1/c are obtained. We compute the perturbed wavefunctions and show that the particles are bound less tightly together than in the Newtonian case.
Spinning black holes in dynamical Einstein-Chern-Simons gravity are constructed by directly solving the field equations, without resorting to any perturbative expansion. This model is obtained by adding to the Einstein-Hilbert action a particular higher-curvature correction: the Pontryagin density, linearly coupled to a scalar field. The spinning black holes are stationary, axi-symmetric, asymptotically flat generalisations of the Kerr solution of Einstein's gravity, but they possess a non-trivial (odd-parity) scalar field. They are regular on and outside the horizon and satisfy a generalized …
Euclidean gravity provides an interesting test system for an analysis of cosmological perturbations in an effective Hamiltonian constraint with holonomy modifications from loop quantum gravity. This paper presents a discussion of scalar modes, with a specific form of the holonomy modification function derived from a general expansion in a connection formulation. Compared with some previous models, the constraint brackets are deformed in a different and more restricted way. A general comparison of anomaly-free brackets in various effective and operator versions shows overall consistency between…
We find a gravitational wave solution to the linearized version of quadratic gravity by adding successive perturbations to the Einstein's linearized field equations. We show that only the Ricci squared quadratic invariant contributes to give a different solution of those found in Einstein's general relativity. The perturbative solution is written as a power series in the $β$ parameter, the coefficient of the Ricci squared term in the quadratic gravitational action. We also show that, for monochromatic waves of a given angular frequency $ω$, the perturbative solution can be summed out to give a…
It is increasingly widely believed that at high levels of resolution spacetime geometry may be inherently non-commutative. In this work we introduce an effective mechanism that maps matter fields onto a non-commutative background manifold. The mechanism, which is relevant at high densities in strongly gravitating systems (black holes, early universe) deforms a density field into two distinct fields, one residing dominantly on the lattice tops (hereafter, on-cell) and the other residing dominantly in the inter-lattice regions (hereafter, off-cell). The two fields have different physical and the…
This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
Investigations of possible violations of the Pauli Exclusion Principle represent critical tests of the microscopic space-time structure and properties. Space-time non-commutativity provides a class of universality for several Quantum Gravity models. In this context the VIP-2 Lead experiment sets the strongest bounds, searching for Pauli Exclusion Principle violating atomic-transitions in lead, excluding the $θ$-Poincaré Non Commutative Quantum Gravity models far above the Planck scale for non-vanishing $θ_{μν}$ ``electric-like'' components, and up to $6.9 \cdot 10^{-2}$ Planck scales if $θ_{0i…
Axisymmetric black hole in a non-commutative gauge theory: classical and quantum gravity effects
Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Universidade Federal de Campina Grande（坎皮纳格兰德联邦大学）；Khazar University（阿塞拜疆汗沙尔大学）；Damghan University（达姆甘大学）；Eastern Mediterranean University（东地中海大学）
We review the main recent progresses in non-commutative space-time phenomenology in underground experiments. A popular model of non-commutative space-time is $θ$-Poincaré model, based on the Groenewold-Moyal plane algebra. This model predicts a violation of the Spin-statistic theorem, in turn implying an energy and angular dependent violation of the Pauli Exclusion principle. Pauli Exclusion Principle Violating transitions in nuclear and atomic systems can be tested with very high accuracy in underground laboratory experiments such as DAMA/LIBRA and VIP(2). In this paper we derive that the $θ$…
Quantum features of a non-commutative Schwarzschild black hole
Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Universidade Federal de Campina Grande（坎皮纳格兰德联邦大学）；Khazar University（哈扎尔大学）；Federal University of Paraíba（帕拉伊巴联邦大学）；Universidade Federal do Piauí (UFPI)（皮奥伊联邦大学）；Universidade Estadual do Maranhão (UEMA)（马拉尼昂州立大学）
In this paper we discuss the absolutely anticommuting nilpotent symmetries for perturbative quantum gravity in general curved spacetime in linear and non-linear gauges. Further, we analyze the finite field-dependent BRST (FFBRST) transformation for perturbative quantum gravity in general curved spacetime. The FFBRST transformation changes the gauge-fixing and ghost parts of the perturbative quantum gravity within functional integration. However, the operation of such symmetry transformation on the generating functional of perturbative quantum gravity does not affect the theory on physical grou…
We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter θ. We thus express the action at order θ^n+2 in terms of noncommutative fields of order at most θ^n+1 and, iterating the procedure, in terms of noncommuta…
The Hilbert space of loop quantum gravity is usually described in terms of cylindrical functionals of the gauge connection, the electric fluxes acting as non-commuting derivation operators. It has long been believed that this non-commutativity prevents a dual flux (or triad) representation of loop quantum gravity to exist. We show here, instead, that such a representation can be explicitly defined, by means of a non-commutative Fourier transform defined on the loop gravity state space. In this dual representation, flux operators act by *-multiplication and holonomy operators act by translation…
We explore asymptotic safety of gravity-matter systems, discovering indications for a near-perturbative nature of these systems in the ultraviolet. Our results are based on the dynamical emergence of effective universality at the asymptotically safe fixed point. Our findings support the conjecture that an asymptotically safe completion of the Standard Model with gravity could be realized in a near-perturbative setting.
We consider noncommutative gravity on a space with canonical noncommutativity that is based on the commutative MacDowell-Mansouri action. Gravity is treated as gauge theory of the noncommutative $SO(1,3)_\star$ group and the Seiberg-Witten (SW) map is used to express noncommutative fields in terms of the corresponding commutative fields. In the commutative limit the noncommutative action reduces to the Einstein-Hilbert action plus the cosmological term and the topological Gauss-Bonnet term. After the SW expansion in the noncommutative parameter the first order correction to the action, as expe…
