Black holes have been a subject of investigation over years not only because they have interesting physical properties, but also because they seem to be the appropriate tool for studying gravity in quantum scale. Although a lot of effort has been made to understand the aspects of spacetime on the quantum scale, the approach which includes noncommutativity of spacetime proves to be promising among several possibilities. In this paper, we use the Hamilton-Jacobi method to study the thermodynamic properties of cylindrical black holes (black strings) in noncommutative spacetime. It is also investigated the behavior of the black string in the presence of back reaction as well as its influence on the system can be understood as an anomaly. This work aims to provide understanding about black strings thermodynamics in noncommutative spacetime since this scale of spacetime is important for the comprehension of the relation between quantum mechanics and general relativity.
From the constructions of the quantum spacetime, a four dimensional quantized spacetime can be embedded in a five dimensional continuous spacetime. Thus to observe from the five dimensional continuous spacetime where the four dimensional quantized spacetime is embedded, there exist the energy-momentum flows between the five dimensional continuous spacetime and the four dimensional quantized spacetime. This makes the energy-momentum not locally conserved generally on the four dimensional quantized spacetime. We propose that energy-momentum tensors of noncommutative field theories constructed fr…
In the last decades, noncommutative spacetimes and their deformed relativistic symmetries have usually been studied in the context of field theory, replacing the ordinary Minkowski background with an algebra of noncommutative coordinates. However, spacetime noncommutativity can also be introduced into single-particle covariant quantum mechanics, replacing the commuting operators representing the particle's spacetime coordinates with noncommuting ones. In this paper we provide a full characterization of a wide class of physically sensible single-particle noncommutative spacetime models and the …
We derive noncommutative Einstein equations for abelian twists and their solutions in consistently symmetry reduced sectors, corresponding to twisted FRW cosmology and Schwarzschild black holes. While some of these solutions must be rejected as models for physical spacetimes because they contradict observations, we find also solutions that can be made compatible with low energy phenomenology, while exhibiting strong noncommutativity at very short distances and early times.
We consider the $\mathcal{PT}$-symmetric quantum field theory on the noncommutative spacetime with angular twist and construct its pseudo-Hermitian interpretation. We explore the differences between internal and spatial parities in the context of the angular twist and for the latter we find new $\mathcal{PT}$-symmetric interactions that are nontrivial only for the noncommutative spacetime. We reproduce the same formula for the leading order $T$-matrix of the equivalent Hermitian model as the one obtained earlier for the quantum field theory on the commutative spacetime. This formula implies th…
The Sturm-Liouville eigenvalue method is employed to analytically investigate the properties of holographic superconductors in higher dimensions in the framework of Born-Infeld electrodynamics incorporating the effects of noncommutative spacetime. In the background of pure Einstein gravity in noncommutative spacetime, we obtain the relation between the critical temperature and the charge density. We also obtain the value of the condensation operator and the critical exponent. Our findings suggest that higher the value of noncommutative parameter and Born-Infeld parameter make the condensate ha…
We discuss the problem of determining the spacetime structure. We show that when we are using only topological methods the spacetime can be modelled as an R- or Q-compact space although the R-compact spaces seem to be more appropriate. Demanding the existence of a differential structure substantially narrows the choice of possible models. The determination of the differential structure may be difficult if it is not unique. By using the noncommutative geometry construction of the standard model we show that fundamental interactions determine the spacetime in the class of R-compact spaces. Fermi…
We study the effect of noncommutative spacetime geometry on one-loop corrections to the primordial curvature two-point function, arising from various forms of massless spectator matter fields interacting gravitationally with the inflaton. After deforming the algebra of functions on the inflationary background to a spatially noncommutative one, we find that this induces momentum-dependent corrections to one-loop terms which imply that the vacuum fluctuation of the energy-momentum tensor sources that of the curvature fluctuation even for distances beyond horizon scales. The one-loop corrections …
A new noncommutative spacetime of structure $ {\cal M}^4 \times Z_2 \times Z_2$ is proposed. The generalized Hilbert-Einstein action contains gravity, all known interactions and Higgs field. This theory can also provide a unified geometric framework for multigravity, which might explain the existence of dark matter and inflationary cosmology. In other words, high energy physics has laid out the crude shape of the new spacetime, while cosmology will shed light to the more details of it.
Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a loga…
Quantum group Fourier transform methods are applied to the study of processes on noncommutative Minkowski spacetime $[x^i,t]=\imathλx^i$. A natural wave equation is derived and the associated phenomena of {\it in vacuo} dispersion are discussed. Assuming the deformation scale $λ$ is of the order of the Planck length one finds that the dispersion effects are large enough to be tested in experimental investigations of astrophysical phenomena such as gamma-ray bursts. We also outline a new approach to the construction of field theories on the noncommutative spacetime, with the noncommutativity eq…
We report an investigation of the Snyder noncommutative spacetime and of some of its most natural generalizations, also looking at them as a powerful tool for comparing different notions of dimensionality of a quantum spacetime. It is known that (generalized-)Snyder noncommutativity, while having rich off-shell implications (kinematical Hilbert space), does not affect on-shell particles (physical Hilbert space), and we argue that physically meaningful notions of dimensionality should describe such spacetimes as trivially four-dimensional, without any running with scales. By studying the thermo…
Noncommutative geometry may be a starting point to a quantum gravity. We study the influence of the spacetime noncommutative parameter on the strong field gravitational lensing in the noncommutative Schwarzschild black-hole spacetime and obtain the angular position and magnification of the relativistic images. Supposing that the gravitational field of the supermassive central object of the galaxy described by this metric, we estimate the numerical values of the coefficients and observables for strong gravitational lensing. Comparing to the Reissner-Norstr\"{om} black hole, we find that the inf…
We make biframe and quaternion extensions on the noncommutative geometry, and construct the biframe spacetime for the unification of gravity and quantum field theory. The extended geometry distinguishes between the ordinary spacetime based on the frame bundle and an extra non-coordinate spacetime based on the biframe bundle constructed by our extensions. The ordinary spacetime frame is globally flat and plays the role as the spacetime frame in which the fields of the standard model are defined. The non-coordinate frame is locally flat and is the gravity spacetime frame. The field defined in bo…
We show that the cosmological constant appears as a Lagrange multiplier if nature is described by a canonical noncommutative spacetime. It is thus an arbitrary parameter unrelated to the action and thus to vacuum fluctuations. The noncommutative algebra restricts general coordinate transformations to four-volume preserving noncommutative coordinate transformations. The noncommutative gravitational action is thus an unimodular noncommutative gravity. We show that spacetime noncommutativity provides a very natural justification to an unimodular gravity solution to the cosmological problem. We ob…
It has earlier been argued that there should exist a formulation of quantum mechanics which does not refer to a background spacetime. In this paper we propose that, for a relativistic particle, such a formulation is provided by a noncommutative generalisation of the Hamilton-Jacobi equation. If a certain form for the metric in the noncommuting coordinate system is assumed, along with a correspondence rule for the commutation relations, it can be argued that this noncommutative Hamilton-Jacobi equation is equivalent to standard quantum mechanics.
We consider simple extensions of noncommutativity from flat to curved spacetime. One possibility is to have a generalization of the Moyal product with a covariantly constant noncommutative tensor $θ^{μν}$. In this case the spacetime symmetry is restricted to volume preserving diffeomorphisms which also preserve $θ^{μν}$. Another possibility is an extension of the Kontsevich product to curved spacetime. In both cases the noncommutative product is nonassociative. We find the the order $θ^2$ noncommutative correction to the Newtonian potential in the case of a covariantly constant $θ^{μν}$. It is…
Unlike noncommutative space, when space and time are noncommutative, it seems necessary to modify the usual scheme of quantum mechanics. We propose in this paper a simple generalization of the time evolution equation in quantum mechanics to incorporate the feature of a noncommutative spacetime. This equation is much more constraining than the usual Schrödinger equation in that the spatial dimension noncommuting with time is effectively reduced to a point in low energy. We thus call the new evolution equation the spacetime bootstrap equation, the dimensional reduction called for by this evoluti…
We evaluate the noncommutative Chern-Simons action induced by fermions interacting with an Abelian gauge field in a noncommutative massive Thirring model in (2+1)-dimensional spacetime. This calculation is performed in the Dirac and Majorana representations. We observe that in Majorana representation when $θ$ goes to zero we do not have induced Chern-Simons term in the dimensional regularization scheme.
Noncommutative gravity in three dimensions with vanishing cosmological constant is examined. We find a solution which describes a spacetime in the presence of a torsional source. We estimate the phase shift for each partial wave of a scalar field in the spacetime by the Born approximation.
There is a growing evidence that due to quantum gravity effects the effective spacetime dimensionality might change in the UV. In this letter we investigate this hypothesis by using quantum fields to derive the UV behaviour of the static, two point sources potential. We mimic quantum gravity effects by using non-commutative fields associated to a Lie group momentum space with a Planck mass curvature scale. We find that the static potential becomes finite in the short-distance limit. This indicates that quantum gravity effects lead to a dimensional reduction in the UV or, alternatively, that po…
We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated *-products and *-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave …
Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter $z$ is related to a Planck scale, and the cosmological constant $Λ$ plays the role of a second deformation parameter of geometric nature, whose limit $Λ\to 0$ provides the corresponding noncommutative Minkowski spacetimes.
Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of plane-fronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and…
In this paper we have discussed geodesics and the motion of test particle in the gravitational field of noncommutative charged black hole spacetime. The motion of massive and massless particle have been discussed seperately. A comparative study of noncommutative charged black hole and usual Reissner-Nordstrom black hole has been done. The study of effective potential has also been included. Finally, we have examined the scattering of scalar waves in noncommutative charged black hole spacetime.
In the study of certain noncommutative versions of Minkowski spacetime there is still a large ambiguity concerning the characterization of their symmetries. Adopting as our case study the kappaMinkowski noncommutative space-time, on which a large literature is already available, we propose a line of analysis of noncommutative-spacetime symmetries that relies on the introduction of a Weyl map (connecting a given function in the noncommutative Minkowski with a corresponding function in commutative Minkowski) and of a compatible notion of integration in the noncommutative spacetime. We confirm (a…
We report a mathematical equivalence between certain models of universe relying on domain-walls and noncommutative geometries. It is shown that a two-brane world made of two domain-walls can be seen as a "noncommutative" two-sheeted spacetime under certain assumptions. This equivalence also implies a model-independent phenomenology which is presently studied. Matter swapping between the two branes (or sheets) is predicted through fermionic oscillations induced by magnetic vector potentials. This phenomenon, which might be experimentally studied, could reveal the existence of extra dimensions i…
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For ge…
Over the last decade a growing number of quantum-gravity researchers has been looking for opportunities for the first ever experimental evidence of a Planck-length quantum property of spacetime. These studies are usually based on the analysis of some candidate indirect implications of spacetime quantization, such as a possible curvature of momentum space. Some recent proposals have raised hope that we might also gain direct experimental access to quantum properties of spacetime, by finding evidence of limitations to the measurability of the center-of-mass coordinates of some macroscopic bodies…
We provide a self-contained introduction to the quantum group approach to noncommutative geometry as the next-to-classical effective geometry that might be expected from any successful quantum gravity theory. We focus particularly on a thorough account of the bicrossproduct model noncommutative spacetimes of the form [t,x_i]=i λx_i and the correct formulation of predictions for it including a variable speed of light. We also study global issues in the Poincaré group in the model with the 2D case as illustration. We show that any off-shell momentum can be boosted to infinite negative energy by …
We study the effects of noncommutativity of spacetime geometry on the thermodynamical properties of the de Sitter horizon. We show that noncommutativity results in modifications in temperature, entropy and vacuum energy and that these modifications are of order of the Planck scale, suggesting that the size of the noncommutative parameter should be close to that of the Planck. In an alternative way to deal with noncommutativity, we obtain a quantization rule for the entropy. Since noncommutativity in spacetime geometry modifies the Heisenberg algebra and introduces the general uncertainty princ…
The effects of noncommutativity in the phase-space of the classical and quantum cosmology of Bianchi models are investigated. Exact solutions in both commutative and noncommutative cases are presented and compared. Further, the Noether symmetries of the Bianchi class A spacetimes are studied in both cases and similarities and differences are discussed.
We study the spacetime noncommutative effect on black hole as particle accelerators and, find that particle falling from infinity with zero velocity cannot collide with unbound energy when the noncommutative Kerr black hole is exactly extremal. Our results also show that the bigger of the spinning black hole's mass is, the higher of center of mass energy that the particles obtain. For small and medium noncommutative Schwarzschild black hole, the collision energy depends on the black holes' mass.
General Relativity in the noncommutative spacetime of ${\cal M}^4 \times Z_2 \times Z_2$ is constructed based on a new type of fermions in addition to the known chiral quark-leptons. While this theory has a finite physical spectrum, all the known interactions of Nature and Higgs fields can be derived from it as components of gravity with only four free parameters.
Noncommutative spacetimes lead to nonlocal quantum field theories (qft's) where spin-statistics theorems cannot be proved. For this reason, and also backed by detailed arguments, it has been suggested that they get corrected on such spacetimes leading to small violations of the Pauli principle. In a recent paper \cite{Pauli}, Pauli-forbidden transitions from spacetime noncommutativity were calculated and confronted with experiments. Here we give details of the computation missing from this paper. The latter was based on a spacetime $\mathcal{B}_{χ\vec{n}}$ different from the Moyal plane. We ar…
We study the influence of the spacetime noncommutative parameter on the strong field gravitational lensing in the noncommutative Reissner-Nordström black-hole spacetime. Supposing that the gravitational field of the supermassive central object of the Galaxy is described by this metric, we estimate the numerical values of the coefficients and observables for strong gravitational lensing. Our results show that with the increase of the parameter $\sqrt{\vartheta}$, the observables $θ_{\infty}$ and $r_m$ decrease, while $s$ increases. Our results also show that i) if $\sqrt{\vartheta}$ is strong, …
We establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of κ-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncomm…
The (3+1)-dimensional $κ$-(A)dS noncommutative spacetime is explicitly constructed by quantizing its semiclassical counterpart, which is the $κ$-(A)dS Poisson homogeneous space. This turns out to be the only possible generalization of the well-known $κ$-Minkowski spacetime to the case of non-vanishing cosmological constant, under the condition that the time translation generator of the corresponding quantum (A)dS algebra is primitive. Moreover, the $κ$-(A)dS noncommutative spacetime is shown to have a quadratic subalgebra of local spatial coordinates whose first-order brackets in terms of the …
We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a univers…
We derive a transformation of the noncommutative geometry inspired Schwarzschild solution into new coordinates such that the apparent unphysical singularities of the metric are removed. Moreover, we give the maximal singularity-free atlas for the manifold with the metric under consideration. This atlas reveals many new features e.g. it turns out to describe an infinite lattice of asymptotically flat universes connected by black hole tunnels.
A general formula is calculated for the connection of a central metric w.r.t.\ a noncommutative spacetime of Lie-algebraic type. This is done by using the framework of linear connections on central bi-modules. The general formula is further on used to calculate the corresponding Riemann tensor and prove the corresponding Bianchi identities and certain symmetries that are essential to obtain a symmetric and divergenceless Einstein Tensor. In particular, the obtained Einstein Tensor is not equivalent to the sum of the noncommutative Riemann tensor and scalar, as in the commutative case, but in a…
A pedagogical introduction to some of the main ideas and results of field theories on quantized spacetimes is presented, with emphasis on what such field theories may teach us about the problem of quantizing gravity. We examine to what extent noncommutative gauge theories may be regarded as gauge theories of gravity. UV/IR mixing is explained in detail and we describe its relations to renormalization, to gravitational dynamics, and to deformed dispersion relations in models of quantum spacetime of interest in string theory and in doubly special relativity. We also discuss some potential experi…
By considering particles as smeared objects, we investigate the effects of space noncommutativity on the orbits of particles in Schwarzschild spacetime. The effects of space noncommutativity on the value of the precession of the perihelion of particle orbit and deflection of light ray in Schwarzschild geometry are calculated and the stability of circular orbits is discussed.
Moeller's energy-momentum complex is employed in order to determine the energy and momentum distributions for a spacetime described by a "generalized Schwarzschild" geometry in (3+1)-dimensions on a noncommutative curved D3-brane in an effective, open bosonic string theory. The geometry considered is obtained by an effective theory of gravity coupled with a nonlinear electromagnetic field and depends only on the generalized (effective) mass and charge which incorporate corrections of first order in the noncommutativity parameter.
We propose a variation of spacetime noncommutative field theory to realize the stringy spacetime uncertainty relation without breaking any of the global symmetries of the homogeneous isotropic universe. We study the spectrum of metric perturbations in this model for a wide class of accelerating background cosmologies. Spacetime noncommutativity leads to a coupling between the fluctuation modes and the background cosmology which is nonlocal in time. For each mode, there is a critical time at which the spacetime uncertainty relation is saturated. This is the time when the mode is generated. Thes…
In a recent paper we have suggested that a formulation of quantum mechanics should exist, which does not require the concept of time, and that the appropriate mathematical language for such a formulation is noncommutative differential geometry. In the present paper we discuss this formulation for the free point particle, by introducing a commutation relation for a set of noncommuting coordinates. The sought for background independent quantum mechanics is derived from this commutation relation for the coordinates. We propose that the basic equations are invariant under automorphisms which map o…
We showed before that self-dual electromagnetism in noncommutative (NC) spacetime is equivalent to self-dual Einstein gravity. This result implies a striking picture about gravity: Gravity can emerge from electromagnetism in NC spacetime. Gravity is then a collective phenomenon emerging from gauge fields living in fuzzy spacetime. We elucidate in some detail why electromagnetism in NC spacetime should be a theory of gravity. In particular, we show that NC electromagnetism is realized through the Darboux theorem as a diffeomorphism symmetry G which is spontaneously broken to symplectomorphism H…
We study the effects of spacetime noncommutativity on the nonequilibrium dynamics of particles in a thermal bath. We show that the noncommutative thermal bath does not suffer from any further IR/UV mixing problem in the sense that all the finite-temperature non-planar quantities are free from infrared singularities. We also point out that the combined effect of finite temperature and noncommutative geometry has a distinct effect on the nonequilibrium dynamics of particles propagating in a thermal bath: depending on the momentum of the mode of concern, noncommutative geometry may switch on or s…
The focus of this PhD thesis is on applications, new developments and extensions of the noncommutative gravity theory proposed by Julius Wess and his group. In part one we propose an extension of the usual symmetry reduction procedure to noncommutative gravity. We classify in the case of abelian Drinfel'd twists all consistent deformations of spatially flat Friedmann-Robertson-Walker cosmologies and of the Schwarzschild black hole. The deformed symmetry structure allows us to obtain exact solutions of the noncommutative Einstein equations in many of our models. In part two we develop a new for…
We propose a "Copenhagen interpretation" for spacetime noncommutativity. The goal is to be able to predict results of simple experiments involving signal propagation directly from commutation relations. A model predicting an energy dependence of the speed of photons of the order E/E_Planck is discussed in detail. Such effects can be detectable by the GLAST telescope, to be launched in 2006.
We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $κ$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $κ$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $κ$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corres…
Motivated by the doubly special relativity theories and noncommutative spacetime structures, thermodynamical properties of the photon gas in a phase space with compact spatial momentum space is studied. At the high temperature limit, the upper bounds for the internal energy and entropy are obtained which are determined by the size of the compact spatial momentum space. The maximum internal energy turns out to be of the order of the Planck energy and the entropy bound is then determined by the factor $\big(V/l_{_{\rm Pl}}^3\big)$ through the relevant identification of the size of the momentum s…
In this paper we investigated the causality problem present in the recent work about the Doplicher-Fredenhagen-Roberts-Amorim (DFRA) noncommutative framework which analyzed the complex scalar field. To accomplish this task we provided a brief review of the main ingredients of the problem and we demonstrated precisely that the DFRA algebra obeys the rules of the Canonical Commutation Relations algebra. This fact permitted us to prove the form of the DFRA operators previously constructed in the usual way. After that, we introduced the solution of its Klein-Gordon equation with a source term. Its…
In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel'd twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green's operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation…
