We argue that the natural framework for embedding the ideas of deformed, or doubly, special relativity (DSR) into a curved spacetime is a generalisation of Einstein-Cartan theory, considered by Stelle and West. Instead of interpreting the noncommuting "spacetime coordinates" of the Snyder algebra as endowing spacetime with a fundamentally noncommutative structure, we are led to consider a connection with torsion in this framework. This may lead to the usual ambiguities in minimal coupling. We note that observable violations of charge conservation induced by torsion should happen on a time scale of 10^3 s, which seems to rule out these modifications as a serious theory. Our considerations show, however, that the noncommutativity of translations in the Snyder algebra need not correspond to noncommutative spacetime in the usual sense.
Deformed special relativity is embedded in deformed general relativity using the methods of canonical relativity and loop quantum gravity. Phase-space dependent deformations of symmetry algebras then appear, which in some regimes can be rewritten as non-linear Poincare algebras with momentum-dependent deformations of commutators between boosts and time translations. In contrast to deformed special relativity, the deformations are derived for generators with an unambiguous physical role, following from the relationship between canonical constraints of gravity with stress-energy components. The …
By studying the notion of a fundamentally minimal length scale in asymptotically safe gravity we find that a specific version of deformed special relativity (DSR) naturally arises in this approach. We then consider two thought experiments to examine the interpretation of the scenario and discuss similarities and differences to other approaches to DSR.
We analyse the quantum geometry of 3-dimensional deformed special relativity (DSR) and the notion of spacetime points in such a context, identified with coherent states that minimize the uncertainty relations among spacetime coordinates operators. We construct this system of coherent states in both the Riemannian and Lorentzian case, and study their properties and their geometric interpretation.
Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincaré symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is t…
Deformed Special Relativity (DSR) is a generalization of Special Relativity based on a deformed Minkowski space, i.e. a four-dimensional space-time with metric coefficients depending on the energy. We show that, in the DSR framework, it is possible to derive the value of the electron mass from the space-time geometry via the experimental knowledge of the parameter of local Lorentz invariance breakdown, and of the Minkowskian threshold energy $E_{0,em}$ for the electromagnetic interaction.
Conformally deformed special relativity is mathematically consistent example of a theory with two observer independent scales. As compare with recent DSR proposals, it is formulated starting from the position space. In this work we propose interpretation of Lorentz boosts of the model as transformations among accelerated observers. We point further that the model can be considered as relativistic version of MOND program and thus may be interesting in context of dark matter problem.
We derive a scalar field theory of the deformed special relativity type, living on non-commutative kappa-Minkowski spacetime and with a kappa-deformed Poincare symmetry, from the SO(4,1) group field theory defining the transition amplitudes for topological BF-theory in 4 space-time dimensions. This is done at a non-perturbative level of the spin foam formalism working directly with the group field theory (GFT). We show that matter fields emerge from the fundamental model as perturbations around a specific phase of the GFT, corresponding to a solution of the fundamental equations of motion, and…
We explore the relation of the Holst term with the Nieh-Yan term in terms of metric variables. We show that the Holst term indeed affects the classical equations of motion in the presence of matter with spin. Therefore the correct term to add to the Einstein-Hilbert action such that the equations of motion are not affected is the Nieh-Yan term. We then calculate the torsion charge due to this term in the context of a perfect fluid sphere with torsion and show that it vanishes once a horizon is formed but not otherwise. We also show that adding on torsion to General Relativity the Einstein's eq…
We argue that a (slightly) curved space-time probed with a finite resolution, equivalently a finite minimal length, is effectively described by a flat non-commutative space-time. More precisely, a small cosmological constant (so a constant curvature) leads the kappa-deformed Poincaré flat space-time of deformed special relativity (DSR) theories. This point of view eventually helps understanding some puzzling features of DSR. It also explains how DSR can be considered as an effective flat (low energy) limit of a (true) quantum gravity theory. This point of view leads us to consider a possible g…
A couple of technicalities in the paper M. Bojowald, G. Paily, Phys. Rev. D 87, 044044 (2013) are discussed. The explicit formula given there for the function entering the modified hypersurface-deformed algebra, which presumably originates in loop quantum gravity, seems to be oversimplified, and the embedding of deformed special relativity in deformed general relativity proposed there for spherically symmetric models raises some questions as well.
Commuting and noncommuting space-time coordinates in a class of deformed special relativity theories are investigated. Their momentum space representation, transformation behaviour, space-time algebra, invariants and the corresponding field theories are derived. Several coordinates require as a novel feature the introduction of deformed plane waves.
In this thesis, I investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. The specific deformation of general covariance is predicted by some studies into loop quantum cosmology. I firstly find the minimally-deformed model for a scalar-tensor theory, thereby establishing a classical reference point, and investigate the cosmological effects of a non-minimal coupled scalar field. By treating the deformation perturbatively, I derive the deformed gravitational action which includes the nearest order of curvature corrections.…
A spacetime interpretation of deformed relativity symmetry groups was recently proposed by resorting to Finslerian geometries, seen as the outcome of a continuous limit endowed with first order corrections from the quantum gravity regime. In this work we further investigate such connection between deformed algebras and Finslerian geometries by showing that the Finsler geometries associated to the generalisation of the Poincaré group (the so called $κ$-Poincaré Hopf algebra) are maximally symmetric spacetimes which are also of the Berwald type: Finslerian spacetimes for which the connections ar…
We recall some of the obstacles which arise when one tries to reconcile the general theory of relativity with quantum theory. We consider the possibility that gravitation theories which include torsion, and not only curvature, provide better insight into a quantum theory of gravity. We speculate on how the Dirac equation and Einstein gravity could be thought of as limiting cases of a gravitation theory which possesses torsion.
We consider deformed special relativity (DSR) theories on commutative space-time, perhaps as an first approximation to a noncommutative space-time formulation. The corresponding field theories in general possess derivatives of all orders. From a quantisation procedure and the deformed Dirac equation in momentum space we obtain for a specific DSR model enhanced expressions for momentum and charge and finite vacuum contributions.
We investigate how deformations of special relativity in momentum space can be extended to position space in a consistent way, such that the dimensionless contraction between wave-vector and coordinate-vector remains invariant. By using a parametrization in terms of an energy dependent speed of light, and an energy dependent Planck's constant, we are able to formulate simple requirements that completely determine the active transformations in position space. These deviate from the standard transformations for large velocities of the observed object. Some examples are discussed, and it is shown…
We investigate the transformation laws of coordinates in generalizations of special relativity with two observer-independent scales. The request of covariance leads to simple formulas if one assumes noncanonical Poisson brackets, corresponding to noncommuting spacetime coordinates.
We summarize some of the results we obtained in arXiv:1006.2126 (Physical Review Letters 106, 071301), arXiv:1102.4637 (Physics Letters B 700, 150-156) and in arXiv:1107.3334, giving complementary characterizations of the relativity of spacetime locality that affects certain Planck-scale-deformed phase-space descriptions of free particles.
In this paper, we propose a new deformed Poisson brackets which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity. We therefore derive the corresponding momentum transformation which is revealed to be different from previous results. Contrary to the earlier version of Fock's nonlinear relativity for which plane waves cannot be described, our resulting algebra keeps invariant for any coordinate and momentum transformations the four dimensional contraction $p_μ x^μ $, allowing therefore to associate plane waves for free particles. As in…
We examine the transformation of particle trajectories in models with deformations of Special Relativity that have an energy-dependent and observer-independent speed of light. These transformations necessarily imply that the notion of what constitutes the same space-time event becomes dependent on the observer's inertial frame. To preserve observer-independence, the such arising nonlocality should not be in conflict with our knowledge of particle interactions. This requirement allows us to derive strong bounds on deformations of Special Relativity and rule out a modification to first order in …
General relativity dynamics can be derived from different actions -- which depart from the Einstein-Hilbert action in boundary terms -- and for different choices of the dynamical variables. Among them, the teleparallel equivalent of general relativity is a torsion-based theory for the tetrad field. More general torsion-based theories have been built in the last years, intending to supersede general relativity. There are two current ways to formulate such theories; one includes a spin-connection and the other does not. We discuss the notion of Lorentz gauge invariance in such theories, and give…
We attempt to build systematically the low-energy effective Lagrangian for the Einstein--Cartan formulation of gravity theory that generally includes the torsion field. We list all invariant action terms in certain given order; some of the invariants are new. We show that in the leading order the fermion action with torsion possesses additional U(1)_L x U(1)_R gauge symmetry, with 4+4 components of the torsion (out of the general 24) playing the role of Abelian gauge bosons. The bosonic action quadratic in torsion gives masses to those gauge bosons. Integrating out torsion one obtains a point-…
In many different ways, Deformed Special Relativity (DSR) has been argued to provide an effective limit of quantum gravity in almost-flat regime. Unfortunately DSR is up to now plagued by many conceptual problems (in particular how it describes macroscopic objects) which forbids a definitive physical interpretation and clear predictions. Here we propose a consistent framework to interpret DSR. We extend the principle of relativity: the same way that Special Relativity showed us that the definition of a reference frame requires to specify its speed, we show that DSR implies that we must also ta…
In this article we elaborate on a recently proposed interpretation of DSR as an effective measurement theory in the presence of non-negligible (albeit small) quantum gravitational fluctuations. We provide several heuristic arguments to explain how such a new theory can emerge and discuss the possible observational consequences of this framework.
In deformed or doubly special relativity (DSR) the action of the lorentz group on momentum eigenstates is deformed to preserve a maximal momenta or minimal length, supposed equal to the Planck length. The classical and quantum dynamics of a particle propagating in kappa-Minkowski spacetime is discussed in order to examine an apparent paradox of locality which arises in the classical dynamics. This is due to the fact that the Lorentz transformations of spacetime positions of particles depend on their energies, so whether or not a local event, defined by the coincidence of two or more particles,…
We investigate the properties of multi-particle states in Deformed Special Relativity (DSR). Starting from the Lagrangian formalism with an energy dependent metric, the conserved Noether current can be derived which is additive in the usual way. The integrated Noether current had previously been discarded as a conserved quantity, because it was correctly realized that it does no longer obey the DSR transformations. We identify the reason for this mismatch in the fact that DSR depends only on the extensive quantity of total four-momentum instead of the energy-momentum densities as would be appr…
Deformed relativistic kinematics is a framework which captures effects, that are expected from particles and fields propagating on a quantum spacetime, effectively. They are formulated in terms of a modified dispersion relation and a modified momentum conservation equation. In this work we use Finsler geometry to formulate deformed relativistic kinematics in terms of particle velocities. The relation between the Finsler geometric velocity dependent formulation and the original momentum dependent formulation allows us to construct deformed Lorentz transformations between arbitrary frames. Moreo…
We consider charged rotating BTZ black holes in noncommutative space by use of Chern-Simons theory formulation of $2+1$ dimensional gravity. The noncommutativity between the radial and the angular variables is introduced through the Seiberg-Witten map for gauge fields, and the deformed geometry to the first order in the noncommutative parameter is derived. It is found that the deformation also induces nontrivial torsion, and the Einstein-Cartan theory appears to be a suitable framework to investigate the equations of motion. Though the deformation is indeed nontrivial, the deformed and the ori…
We present an explicit example showing that the weak Gauss law of general relativity (with cosmological constant) fails in Einstein-Cartan's theory. We take this as an indication that torsion might replace dark matter.
A linear Lorentz connection has always two fundamental derived characteristics: curvature and torsion. The latter is assumed to vanish in general relativity. Three gravitational models involving non-vanishing torsion are examined: teleparallel gravity, Einstein-Cartan, and new general relativity. Their dependability is critically examined. Although a final answer can only be given by experience, it is argued that teleparallel gravity provides the most consistent approach.
The theory of the $κ$-deformed Poincare algebra is applied to the analysis of various phenomena in special relativity, quantum mechanics and field theory. The method relies on the development of series expansions in $κ^{-1}$ of the generalised Lorentz transformations, about the special-relativistic limit. Emphasis is placed on the underlying assumptions needed in each part of the discussion, and on in principle limits for the deformation parameter, rather than on rigorous numerical bounds. In the case of the relativistic Doppler effect, and the Michelson-Morley experiment, comparisons with rec…
We discuss a way to obtain the doubly special relativity kinematical rules (the deformed energy-momentum relation and the nonlinear Lorentz transformations of momenta) starting from a singular Lagrangian action of a particle with linearly realized SO(1,4) symmetry group. The deformed energy-momentum relation appears in a special gauge of the model. The nonlinear transformations of momenta arise from the requirement of covariance of the chosen gauge.
In this paper we have studied the nature of kinematical and dynamical laws in $κ$-Minkowski spacetime from a new perspective: the canonical phase space approach. We discuss a particular form of $κ$-Minkowski phase space algebra that yields the $κ$-extended finite Lorentz transformations derived in \cite{kim}. This is a particular form of a Deformed Special Relativity model that admits a modified energy-momentum dispersion law as well as noncommutative $κ$-Minkowski phase space. We show that this system can be completely mapped to a set of phase space variables that obey canonical (and {\it{not…
Whenever the condition of anomaly freedom is imposed within the framework of effective approaches to loop quantum cosmology, one seems to conclude that a deformation of general covariance is required. Here, starting from a general deformation we regain an effective gravitational Lagrangian including terms up to fourth order in extrinsic curvature. We subsequently constrain the form of the corrections, and then investigate the conditions for the occurrence of a big bounce and the realisation of an inflationary era, in the presence of a perfect fluid or scalar field.
We calculate the most general action for a scalar-tensor model up to quadratic order in derivatives with deformed general covariance and non-minimal coupling. We demonstrate how different choices of the free functions recover specific well known scalar-tensor models. We look at the cosmological dynamics and find the general conditions for either inflation or a big bounce. Using this we present a novel non-minimally coupled scalar model which produces a bounce, and describe how to find similar models.
Doubly special relativity has been studied for the last twenty years as a way to go beyond the special relativistic kinematics, trying to capture residual effects of a quantum gravity theory. In particular, in doubly special relativity the Einstenian relativity principle is generalized, adding to the speed of light another relativistic invariant, the Planck energy. There are several papers in the literature showing a connection between this deformed kinematics and a curved momentum space. Here we review how such kinematics can be derived from geometrical ingredients in a rigorous way, and how …
General Relativity with nonvanishing torsion has been investigated in the first order formalism of Poincare gauge field theory. In the presence of torsion, either side of the Einstein equation has the nonvanishing covariant divergence. This fact turned out to be self-consistent in the framework under consideration. By using Noether's procedure with the definition of the Lie derivative where the general coordinate transformation and the local Lorentz rotation are combined, the revised covariant divergence of the energy momentum is consistently obtained. Subsequently we have definitely derived t…
This paper reviews the Einstein Cartan theory (ECT), the famous extension of general relativity (GR) in presence of spacetime torsion. The vacuum equations are derived step by step. Vielbein formulation is discussed for determining the field equations in presence of matter. This review would be easily comprehensible for any student familiar with general relativity. Further, ECT is used to describe superstrings with intrinsic torsion, assuming a $D_p$-brane in presence of a curved background of the NS-NS Kalb-Ramond field. D-brane worldvolume is a flat spacetime governed by the Dirac-Born-Infel…
It is known that the space of momenta of DSR can be identified with de Sitter space. In this paper, we discuss the relation of the noncanonical phase space of the Magueijo-Smolin model of DSR with the canonical 5-dimensional phase space in which the de Sitter space of momenta is embedded. We suggest that in analogy with the momentum variables, also the position variables should be constrained to lie on a null hypersurface of the five-dimensional space
It was previously shown that models with deformations of special relativity that have an energy-dependent yet observer-independent speed of light suffer from nonlocal effects that are in conflict with observation to very high precision. In a recent paper it has been proposed that these paradoxa arise only in the classical limit and can be prevented by an ad-hoc introduction of a quantum uncertainty that would serve to hide the nonlocality. We will show here that the proposed ansatz for this resolution is inconsistent with observer-independence and, when corrected, is in agreement with the earl…
We investigate here all the possible invariant metric functions under the action of various kinds of semi-direct product Poincaré subgroups and their deformed partners. The investigation exhausts the possible theoretical frameworks for the spacetime realization of Cohen-Glashow's very special relativity and the deformation very special relativity approach by Gibbons-Gomis-Pope. Within Finsler-Minkowski type of spacetime, we find that the spacetime emerge a Finsler type of geometry in most cases both for undermed Poincaré subgroup and for deformed one. We give an explanation that the rotation o…
We consider the conformal group of a space of dim n=p+q, with SO(p,q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to a curved 2n-dim geometry admits an action functional linear in the curvatures. Because symmetry is maintained between the translations and the special conformal transformations in the construction, these spaces are called biconformal; this same symmetry gives biconformal spaces overlapping structures with double field theories, including manifest T duality.…
We reformulate the general theory of relativity in the language of Riemann-Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan…
Canonical methods can be used to construct effective actions from deformed covariance algebras, as implied by quantum-geometry corrections of loop quantum gravity. To this end, classical constructions are extended systematically to effective constraints of canonical quantum gravity and applied to model systems as well as general metrics, with the following conclusions: (i) Dispersion relations of matter and gravitational waves are deformed in related ways, ensuring a consistent realization of causality. (ii) Inverse-triad corrections modify the classical action in a way clearly distinguishable…
Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full th…
In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyse the gravitational theories, starting from the standard Einstein's general theory of relativity and focusing on the Rastall theory. After that, we discuss the important role of the torsion in this mathematical framework. In the Appendix of the paper we analyse the importance of the nascent gravitational wave astronomy as a tool to discriminate among the general theory of relativity and alternative theories of gravity.
We revisit one of the earliest proposals for deformed dispersion relations in the light of recent results on dynamical dimensional reduction and production of cosmological fluctuations. Depending on the specification of the measure of integration and addition rule in momentum space the model may be completed so as to merely deform Lorentz invariance, or so as to introduce a preferred frame. Models which violate Lorentz invariance have a negative UV asymptotic dimension and a very red spectrum of quantum vacuum fluctuations. Instead, models which preserve frame independence can exhibit running …
In recent years, Planck-scale modifications to particles' dispersion relation have been deeply studied for the possibility to formulate some phenomenology of Planckian effects in astrophysical and cosmological frameworks. There are some indications [arXiv:gr-qc/0611024] that Finsler geometry can provide some generalization of Riemannian geometry which may allow to account for non-trivial (Planckian) structure of relativistic particles' configuration space. We investigate the possibility to formalize Planck-scale deformations to relativistic models in curved spacetime, within the framework of F…
We show that General Relativity (GR) with cosmological constant may be formulated as a rather simple constrained SO(D-1,2) (or SO(D,1))-Yang-Mills (YM) theory. Furthermore, the spin connections of the Cartan-Einstein formulation for GR appear as solutions of a genuine SO(D-1,1)-YM. We also present a theory of gravity with torsion as the most natural extension of this result. The theory comes out to be strictly an YM-theory upon relaxation of a suitable constraint. This work sets out to enforce the close connection between YM theories and GR by means of a new construction.
In general relativity the affine connection is required to be symmetric so torsion is zero while according to the Einsten- Cartan's theory torsion is connected to the spin tensor as expressed by the Cartan's equations. We consider the theory of spinors in general relativity in the light of the results of Einstein Cartan's theory.In general relativity the affine connection is required to be symmetric so torsion is zero while according to the Einsten- Cartan's theory torsion is connected to the spin tensor as expressed by the Cartan's equations. We consider the theory of spinors in general relat…
In many different ways, Deformed Special Relativity (DSR) has been argued to provide an effective limit of quantum gravity in almost-flat regime. Some experiments will soon be able to test some low energy effects of quantum gravity, and DSR is a very promising candidate to describe these latter. Unfortunately DSR is up to now plagued by many conceptual problems (in particular how it describes macroscopic objects) which forbids a definitive physical interpretation and clear predictions. Here we propose a consistent framework to interpret DSR. We extend the principle of relativity: the same way …
A hypothesis of general relativity is that spacetime torsion vanishes identically. This assumption has no empirical support; in fact, a nonvanishing torsion is compatible with all the experimental tests of general relativity. The first part of this essay specifies the framework that is suitable to test the vanishing-torsion hypothesis, and an interesting relation with the gravitational degrees of freedom is suggested. In the second part, some original empirical tests are proposed based on the observation that torsion induces new interactions between different spin-polarized particles.
In the general case, torsion couples to the spin current of the Dirac field. In General Relativity, the apparent torsion field to which the spin current of the Dirac field couples is a mere manifestation of the tetrad anholonomy. Seen from the tetrad frame itself, it has for components the anholonomy coefficients. The latter represent mechanical characteristics of the frame. In the teleparallel equivalent of General Relativity, this coefficient turns out to be the only torsion present.
Near a spinning point particle in (2+1)-dimensional gravity (or near an infinitely thin, straight, spinning string in 3+1 dimensions) there is a region of space-time with closed timelike curves. Exact solutions for extended sources with apparently physically acceptable energy-momentum tensors, have produced the same exterior space-time structure. Here it is pointed out that in the case with torsion, closed timelike curves appear only for spin densities so high that the spin energy density is higher than the net effective energy density. In models without torsion, the presence of closed time-li…
Starting from the Siklos waves in general relativity with a cosmological constant, interpreted as gravitational waves on the anti-de Sitter background, a new class of exact torsion waves is constructed in the framework of three-dimensional gravity with propagating torsion. In the asymptotic limit, the geometry of torsion waves takes the anti-de Sitter form. In the sector with massless torsion modes, we found a set of asymptotic conditions that leads to the conformal asymptotic symmetry.
This paper shows the need of the emergence of a universal minimum speed in the space-time by means of a more thorough investigation of Dirac's large number hypothesis (LNH). We will realize that there should be a minimum speed $V$ with the same status of the invariance of the speed of light $c$, however $V$ has gravitational origin. Hence, such a minimum speed forms a new kinematic basis in the space-time, thus leading to a new Deformed Special Relativity (DSR) for the quantum world so-called Symmetrical Special Relativity (SSR). Furthermore, we will show that such a new structure of space-tim…
