Doubly Special Relativity Research Articles

Unruh effect using Doppler shift method in DSR framework

We study the Unruh effect in doubly special relativity (DSR) framework by generalising the Doppler-shift method to DSR. For both the scalar and Dirac particles, we observe a deviation in the power spectrum of Unruh radiation from the standard Bose–Einstein and Fermi–Dirac distributions, respectively, due to the presence of the frame independent length scale of DSR. We further show that this deviation results in the modification of Unruh temperature which then depends non-linearly on the proper acceleration in DSR.

Ideal gas thermodynamics with an invariant energy scale

A viable approach towards Quantum Gravity is the Doubly Special Relativity (DSR) framework in which an observer-independent finite energy upper bound (or a finite smallest length scale) appears quite naturally. In this work, we have studied the thermodynamic properties of an ideal gas in a specific DSR framework, known as the Magueijo-Smolin (MS) Model. We use the fact that DSR can be considered as nonlinear representation of Lorentz Group. Subsequently, various thermodynamic parameters of ideal gas have been derived in this modified framework to compare the corresponding deviations from the usual scenario due to the presence of the invariant energy (length) scale.

Time delays, choice of energy-momentum variables, and relative locality in doubly special relativity

Doubly Special Relativity (DSR) theories consider (quantum-gravity motivated) deformations of the symmetries of special relativity compatible with a relativity principle. The existence of time delays for massless particles, one of their proposed phenomenological consequences, is a delicate question, since, contrary to what happens with Lorentz Invariance Violation (LIV) scenarios, they are not simply determined by the modification in the particle dispersion relation. While some studies of DSR assert the existence of photon time delays, in this paper we generalize a recently proposed model for time delay studies in DSR and show that the existence of photon time delays does not necessarily follow from a DSR scenario, determining in which cases this is so. Moreover, we clarify long-standing questions about the arbitrariness in the choice of the energy-momentum labels and the independence of the time delay on this choice, as well as on the consistency of its calculation with the relative locality paradigm of DSR theories. Finally, we show that the result for time delays is reproduced in models that consider propagation in a noncommutative spacetime.

Deformed special relativity with an invariant minimum speed as an explanation of the cosmological constant

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-time (SSR) reveals a connection between $V$ and a preferred reference frame $S_V$ of background field that leads to the cosmological constant $\Lambda$, which can be associated with a cosmological anti-gravity. We will also investigate the effect of the number of baryons $N$ (Eddington number) of the observable universe on the Hydrogen atom. Finally, we will show that SSR-metric plays the role of a de-Sitter (dS)-metric with a positive cosmological constant, which could assume a tiny value.

Effect of non-commutativity of space-time on thermodynamics of photon gas

Doubly special relativity(DSR) introduce a minimal length scale i.e. the Planck length scale, which is independent length scale in addition to the speed of light in the normal special theory of relativity(STR). Doubly special relativity leads to study the $\kappa$-Minkowski space-time. In this paper, we present the result of our investigation on the thermodynamics of photon gas in the $\kappa$-Minkowski space-time. For studying this, we start with the $\kappa$-deformed dispersion relation and keep terms upto first order in the deformation parameter $a$ and we study that how does $\kappa$-deformed dispersion relation affect the thermodynamics of photon gas. In the limit, deformation parameter $a \rightarrow 0$, we get back all the results in the special theory of relativity(STR)\cite{partition}.

Maxwell’s equations and Lorentz force in doubly special relativity

On the basis of all commutation relations of the k-deformed phase space incorporating the k-Minkowski space-time, we have derived in this paper an extended first approximation of both Maxwell's equations and Lorentz force in doubly (or deformed) special relativity (DSR). For this purpose, we have used our approach of the special relativistic version of Feynman's proof by which we have established the explicit formulations of electric and magnetic fields. As in Fock's nonlinear relativity (FNLR), the laws of electrodynamics depend on the particle mass which therefore constitutes a common point between the two extended forms of special relativity. As one consequence, the corresponding equation of motion contains two different types of contributions. In addition to the usual type, another one emerges as a consequence of the coexistence of mass and charge which are coupled with the k-deformation and electromagnetic field. This new effect completely induced by the k-deformed phase space is interpreted as the gravitational-type Lorentz force. Unlike FNLR, the corrective terms all depend on the electromagnetic field in DSR.

Photon gas at the Planck scale within the doubly special relativity

Within the approach to doubly special relativity (DSR) suggested by Magueijo and Smolin, a new algebraically justified rule of so-called κ-addition for the energies of identical particles is proposed. This rule permits to introduce the nonlinear κ-dependent Hamiltonian for one-mode multi-photon (sub)system. On its base, with different modes treated as independent, the thermodynamics of black-body radiation is explored within DSR, and main thermodynamic quantities are obtained. In their derivation, we use both the analytical tools within mean field approximation (MFA) and numerical evaluations based on exact formulas. The entropy of one-mode subsystem turns out to be finite (bounded). Another unusual result is the existence of threshold temperature above which radiation is present. Specific features of the obtained results are explained and illustrated with a number of plots. Comparison with some works of relevance is given.

A naturally regularized path integral formalism

A minimal uncertainty in position measurement comes into play if, for instance, one incorporates gravitational interaction of photon and electron in the Heisenberg’s Electron Microscope Gedanken Experiment. This interaction modifies the Heisenberg’s standard uncertainty principle (HUP) to the so-called Generalized (Gravitational) Uncertainty Principle (GUP). The finite resolution of spacetime structure (through a minimal uncertainty in position measurement) nontrivially reflects the existence of a minimal measurable length of the order of the Planck length. The existence of such a minimal length is addressed in several approaches to quantum gravity proposal. On the other hand, Doubly Special Relativity (DSR) theories have revealed that due to the existence of a minimal measurable length, there would be also a maximal momentum for a test particle. These are actually two faces of one fact, that is, a natural ultraviolet cutoff which plays the role of a regulator in quantum field theory. A GUP consisting these cutoffs has some important impacts on the foundation of standard quantum mechanics; one of which is the discreteness of the spacetime manifold. In this paper we study Kernel and Feynman Path Integrals in the framework of a GUP that admits a minimal length and a maximal momentum. We work in the quasi-space representation by adopting maximally localized states in position space. We perform our analysis for both the particle-like and wave-like scenarios. We show that unlike the ordinary quantum mechanics, it is possible to construct a path integral even in the wave-like approach due to the presence of natural cutoffs. As an important application, by using the connection between the path integral formalism and statistical mechanics we investigate the modifications imposed on some thermodynamical properties of an ideal gas in this setup.

Testing Lorentz symmetry violation with an invariant minimum speed

This work presents an experimental test of Lorentz invariance violation in the infrared (IR) regime by means of an invariant minimum speed in spacetime and its effects on the time when an atomic clock given by a certain radioactive single-atom (e.g. isotope Na[Formula: see text]) is a thermometer for an ultracold gas like the dipolar gas Na[Formula: see text]K[Formula: see text]. So, according to a Deformed Special Relativity (DSR) so-called Symmetrical Special Relativity (SSR), where there emerges an invariant minimum speed V in the subatomic world, one expects that the proper time of such a clock moving close to V in thermal equilibrium with the ultracold gas is dilated with respect to the improper time given in lab, i.e. the proper time at ultracold systems elapses faster than the improper one for an observer in the lab, thus leading to the so-called proper time dilation so that the atomic decay rate of an ultracold radioactive sample (e.g. Na[Formula: see text]) becomes larger than the decay rate of the same sample at room temperature. This means a suppression of the half-life time of a radioactive sample thermalized with an ultracold cloud of dipolar gas to be investigated by NASA in the Cold Atom Lab (CAL).

Salecker-Wigner inequalities and the black hole thermodynamics in doubly special relativity

Salecker-Wigner (SW) inequalities are investigated within doubly special relativity (DSR) to take into account the quantum gravitational effects. Based on the modified SW inequalities in DSR, the black hole lifetime and the number of bits required to specify the information content of the black holes are calculated. On the other hand, the black hole is studied from the viewpoint of thermodynamics. The temperature, entropy, specific heat and the emission rate are modified in the presence of DSR effects. The lifetime obtained from the emission rate can be compared with the lifetime coming from the modified SW inequalities. It is shown that they can behave functionally the same. In DSR, the existence of the black hole remnant is predicted from the viewpoint of the SW approach as well as the thermodynamics approach.

Could LIV/noncommutativity alleviate the cosmological constant problem?

We test the hypothesis that in models with an explicit breaking of Lorentz invariance violation (LIV), the problem with the cosmological constant may, if not already be solved, then at least be partially mitigated by improving its UV behavior. Certain class of noncommutative models that have a well-defined dual momentum space geometry and in addition, setting up a suitable frame for developing the appropriate doubly (or deformed) special relativity (DSR) models are also subject to the same test.

Gravity's rainbow: A bridge between LQC and DSR

The doubly special relativity (DSR) theories are investigated in order to take into account an observer-independent length scale in special relativity framework. It is widely believed that any quantum theory of gravity would reduce to a DSR model at the flat limit when purely gravitational and quantum mechanical effects are negligible. Gravity's rainbow is a simple generalization of DSR theories to incorporate gravity. In this paper, we show that the effective Friedmann equations that are suggested by loop quantum cosmology (LQC) can be exactly reobtained in rainbow cosmology setup. The deformed geometry of LQC then completely fixes the modified dispersion relation and results in unique DSR model. In comparison with standard LQC scenario where only the geometry is modified, both of the geometry and matter parts get modifications in our setup. In this respect, we find that the total number of microstates for the universe is finite which suggests the statistical origin for the energy and entropy density bounds. These results explicitly show that the DSR theories are appropriate candidates for the flat limit of loop quantum gravity.

Planck scale effects on the thermodynamics of photon gases

A particular framework for quantum gravity is the doubly special relativity (DSR) formalism that introduces a new observer independent scale (the Planck scale). We resort to the methods of statistical mechanics in this framework to determine how the deformed dispersion relation affects the thermodynamics of a photon gas. The ensuing modifications to the density of states, partition function, pressure, internal energy, entropy, free energy and specific heat are calculated. These results are compared with the outcome obtained in the Lorentz violating model of Camacho and Marcias (Gen. Relativ. Gravit. 39: 1175-1183, 2007). The two types of models predict different results due to different spacetime structure near the Planck scale. The resulting modifications can be interpreted as a consequence of the deformed Lorentz symmetry present in the particular model we have considered. In the low energy limit, our calculation coincides with usual results of photon thermodynamics in special relativity (SR) theory, in contrast to the study presented in (Phys. Rev. D81, 085039 (2010)).

Free-fall frame black hole in gravity’s rainbow

Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use "Gravity's rainbow", where the spacetime background felt by a test particle would depend on its energy. In this scenario, one could rewrite the rainbow metric $g_{\mu\nu}\left( E\right) $ in terms of some orthonormal frame fields and use the modified equivalence principle to determine the energy dependence of $g_{\mu\nu}\left( E\right) $. Obviously, the form of $g_{\mu\nu}\left( E\right) $ depends on the choice of the orthonormal frame. For a static black hole, there are two natural orthonormal frames, the static one hovering above it and freely falling one along geodesics. The cases with the static orthonormal frame have been extensively studied by many authors. The aim of this paper is to investigate properties of rainbow black holes in the scenario with the free-fall orthonormal frame. We first derive the metric of rainbow black holes and their Hawking temperatures in this free-fall scenario. Then, the thermodynamics of a rainbow Schwarzschild black hole is studied. Finally, we use the brick wall model to compute the thermal entropy of a massless scalar field near the horizon of a Schwarzschild rainbow black hole in this free-fall scenario.

Some features of scattering problem in a κ‐Deformed minkowski spacetime

The doubly special relativity (DSR) theories are suggested in order to incorporate an observer‐independent length scale in special theory of relativity. The Magueijo‐Smolin proposal of DSR is realizable through a particular form of the noncommutative (NC) spacetime (known as κ‐Minkowski spacetime) in which the Lorentz symmetry is preserved. In this framework, the NC parameter κ provides the origin of natural cutoff energy scale. Using a nonlinear deformed relativistic dispersion relation along with the Lorentz transformations, we investigate some phenomenological facets of two‐body collision problem (without creation of new particles) in a κ‐Minkowski spacetime. By treating an elastic scattering problem, we study effects of the Planck scale energy cutoff on some relativistic kinematical properties of this scattering problem. The results are challenging in the sense that as soon as one turns on the κ‐spacetime extension, the nature of the two‐body collision alters from elastic to inelastic one. It is shown also that a significant kinematical variable involving in heavy ion collisions, the rapidity, is not essentially an additive quantity under a sequence of the nonlinear representation of the Lorentz transformations.image

Thermodynamics and luminosities of rainbow black holes

Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. As result of the nonlinearity of the Lorentz transformation, the energy-momentum dispersion relation is modified. One simple way to import DSR to curved spacetime is ``Gravity's rainbow'', where the spacetime background felt by a test particle would depend on its energy. Focusing on the ``Amelino-Camelia dispersion relation'' which is E2 = m2+p2[1−η(E/mp)n] with n > 0, we investigate the thermodynamical properties of a Schwarzschild black hole and a static uncharged black string for all possible values of η and n in the framework of rainbow gravity. It shows that there are non-vanishing minimum masses for these two black holes in the cases with η < 0 and n ⩾ 2. Considering effects of rainbow gravity on both the Hawking temperature and radius of the event horizon, we use the geometric optics approximation to compute luminosities of a 2D black hole, a Schwarzschild one and a static uncharged black string. It is found that the luminosities can be significantly suppressed or boosted depending on the values of η and n.

Extended Lorentz Transformations in Clifford Space Relativity Theory

Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constant ħ, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.

Effects of a Maximal Energy Scale in Thermodynamics for Photon Gas and Construction of Path Integral

In this article, we discuss some well-known theoretical models where an observer-independent energy scale or a length scale is present. The presence of this invariant scale necessarily deforms the Lorentz symmetry. We study different aspects and features of such theories about how modifications arise due to this cutoff scale. First we study the formulation of energy-momentum tensor for a perfect fluid in doubly special relativity (DSR), where an energy scale is present. Then we go on to study modifications in thermodynamic properties of photon gas in DSR. Finally we discuss some models with generalized uncertainty principle (GUP).

Novel Physical Consequences of the Extended Relativity in Clifford Spaces

Novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) are explored. The latter theory provides a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. Furthermore, an elegant nonlinear momentum-addition law is derived in order to tackle the “soccer-ball” problem in DSR. Neither derivation in C-spaces requires a curved momentum space nor a deformation of the Lorentz algebra. While the constant (energy-independent) speed of photon propagation is always compatible with the generalized photon dispersion relations in C-spaces, another important consequence is that these generalized photon dispersion relations allow also for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, while breaking the extended Lorentz symmetry in C-spaces. This does not occur in DSR nor in other approaches, like the presence of quantum spacetime foam. We conclude with some comments on the quantization program and the key role that quantum Clifford-Hopf algebras might have in the future developments since the latter q-Clifford algebras naturally contain the κ-deformed Poincare algebras which are essential ingredients in the formulation of DSR.

Geometrical Lorentz Violation and Quantum Mechanical Physics

On the basis of the results of some experiments dealing with the violation of Local Lorentz Invariance (LLI) and on the formalism of the Deformed Special Relativity (DSR), we examine the connections between the local geometrical structure of space-time and the foundation of Quantum Mechanics. We show that Quantum Mechanics, beside being an axiomatic theory, can be considered also a deductive physical theory, deducted from the primary physical principle of Relativistic Correlation. This principle is synonym of LLI and of a rigid and at minkowskian space-time. The results of the experiments mentioned above show the breakdown of LLI and hence the violation of the principle of Relativistic Correlation. The formalism of DSR allows to highlight the deep meaning of LLI breakdown in terms of the geometrical structure of local space-time which, far from being rigid and at, is deformed by the energy of the physical phenomena that take place and in this sense it has an active part in the dynamics of the whole physical process. This perspective has a far reaching physical meaning that extends its consequences to the foundations of Quantum Mechanics according to the interpretation of Copenhagen. It provides a 'real' explanation and description of quantum phenomena enriching, by the concept of deformed space-time, the realistic interpretation in terms of pilot wave and hence it uncovers the reality hidden below the probabilistic interpretation and dualistic nature of quantum objects.