Doubly special relativity (DSR), with both an invariant velocity and an invariant length scale, elegantly preserves the principle of relativity between moving observers, and appears as a promising candidate of the quantum theory of gravity. We study the modifications of photon gas thermodynamics in the framework of DSR with an invariant length $|λ|$, after properly taking into account the effects of modified dispersion relation, upper bounded energy-momentum space, and deformed integration measure. We show that with a positive $λ$, the grand partition function, the energy density, the specific heat, the entropy, and the pressure are smaller than those of special relativity (SR), while the velocity of photons and the ratio of pressure to energy are larger. In contrast, with a negative $λ$, the quantum gravity effects show up in the opposite direction. However, these effects only manifest themselves significantly when the temperature is larger than $10^{-3} E_{\rm P}$. Thus, DSR can have considerable influence on the early universe in cosmological study.
We show that some of the recent results reported in gr-qc/0308049 are based on assumptions which are in contrast with general properties of ``Doubly Special Relativity'' and/or with Planck-scale physics models.
Using a BRST treatment, we show that the equivalence of General Relativity and Shape Dynamics can be extended to a theory that respects the BRST-symmetries of General Relativity as well as the ones of an extended version of Shape Dynamics. This version of Shape Dynamics implements local spatial Weyl transformations as well as a local and abstract analogue of special conformal transformations. Standard effective field theory arguments suggest that the definition of a gravity theory should implement this duality between General Relativity and Shape Dynamics, thus the name "Doubly General Relativ…
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.
We present commutation relations depending on the rainbow functions which are slightly different from the well-known results. However, the advantage of these new commutation relations are compatible with the calculation of the Hawking temperature in the rainbow Schwarzschild black hole.
We discuss the definition of particle velocity in doubly relativity theories. The general formula relating velocity and four-momentum of particle is given.
In the framework of the most-studied doubly special relativity models the use of the naive formula $v=dE/dp$ has been argued to lead to inconsistencies connected to different rules of transformation, under boosts, of particles with the same energy but with different masses. In this paper we show that, at least in 1+1 dimensions, doubly special relativity can be formulated in such a way that the formula $v=dE/dp$ is fully consistent with the invariance of the relative rest, easily fitting to the relativity principle. It is also argued that, always in 1+1d, is not necessary to renounce to the us…
In this paper we discuss how the Magueijo-Smolin Doubly Special Relativity proposal may obtained from a singular Lagrangian action. The deformed energy-momentum dispersion relation rises as a particular gauge, whose covariance imposes the non-linear Lorentz group action. Moreover, the additional invariant scale is present from the beginning as a coupling constant to a gauge auxiliary variable. The geometrical meaning of the gauge fixing procedure and its connection to the free relativistic particle are also described.
Based on the doubly special relativity we find a new type of generalized uncertainty principle (GUP) where the coordinate remain unaltered at the high energy while the momentum is deformed at the high energy so that it may be bounded from the above. For this GUP, we discuss some quantum mechanical problems in one dimension such as box problem, momentum wave function, and harmonic oscillator problem.
Magueijo and Smolin have introduced a modification of the Lorentz group for the momentum-space transformations in Doubly-Special Relativity. As presented the group is non-linear, but we show that it is a group of fraction-linear transformations in 4 dimensional real projective space. We pass to the associated 5 dimensional linear space and identify the subgroup as a conjugate of the ordinary Lorentz group, giving the conjugating matrix. Taking the dual of the 5-space, we identify a subgroup as the Lorentz transformations about a point different to the origin.
The current status of Doubly Special Relativity research program is shortly presented. I dedicate this paper to my teacher and friend Professor Jerzy Lukierski on occasion of his seventieth birthday.
I review the conceptual, algebraical, and geometrical structure of Doubly Special Relativity. I also speculate about the possible relevance of DSR for quantum gravity phenomenology.
We discuss the definition of velocity as dE/dp, where E,p are the energy and momentum of a particle, in Doubly Special Relativity (DSR). If this definition matches dx/dt appropriate for the space-time sector, then space-time can in principle be built consistently with the existence of an invariant length scale. We show that, within different possible velocity definitions, a space-time compatible with momentum-space DSR principles can not be derived.
In this paper, the relation between the modified Lorenz boosts, proposed in the doubly relativity theories and a linear combination of Conformal Group generators in $R^{1,d-1}$ is investigated. The introduction of a new generator is proposed in order to deform the Conformal Group to achieve the connection conjectured. The new generator is obtained trough a formal dimensional reduction from a free massless particle living in a $R^{2,d}$ space. Due this treatment it is possible to say that even DSR theories modify light cone structure in $R^{1,d-1}$, it could remains, in some cases, untouched in…
The modification of the quantum mechanical commutators in a relativistic theory with an invariant length scale (DSR) is identified. Two examples are discussed where a classical behavior is approached in one case when the energy approaches the inverse of the invariant length which appears as a cutoff in the energy and in the second case when the mass is much larger than the inverse of the invariant length.
There are two major alternatives for violating the (usual) Lorentz invariance at large (Planckian) energies or momenta - either not all inertial frames (in the Planck regime) are equivalent (e.g., there is an effectively preferred frame) or the transformations from one frame to another are (non-linearly) deformed (``doubly special relativity''). We demonstrate that the natural (and reasonable) assumption of an energy-dependent speed of light in the latter method goes along with violations of locality/separability (and even translational invariance) on macroscopic scales. PACS: 03.30.+p, 11.30.…
We show that depending on the direction of deformation of $κ$-Poincaré algebra (time-like, space-like, or light-like) the associated phase spaces of single particle in Doubly Special Relativity theories have the energy-momentum spaces of the form of de Sitter, anti-de Sitter, and flat space, respectively.
Double Special Relativity theories are the relativistic theories in which the transformations between inertial observers are characterized by two observer-independent scales of the light speed and the Planck length. We study two main examples of these theories and want to show that these theories are not the new theories of relativity, but only are re-descriptions of Einstein's special relativity in the non-conventional coordinates.
An action principle of singular hypersurfaces in general relativity and scalar-tensor type theories of gravity in the Einstein frame is presented without assuming any symmetry. The action principle is manifestly doubly covariant in the sense that coordinate systems on and off a hypersurface are disentangled and can be independently specified. It is shown that, including variation of the metric, the position of the hypersurface and matter fields, the variational principle gives the correct set of equations of motion: the Einstein equation off the hypersurface, Israel's junction condition in a d…
In these notes, based on the lectures given at 40th Winter School on Theoretical Physics, I review some aspects of Doubly Special Relativity (DSR). In particular, I discuss relation between DSR and quantum gravity, the formal structure of DSR proposal based on $κ$-Poincaré algebra and non-commutative $κ$-Minkowski space-time, as well us some results and puzzles related to DSR phenomenology.
We argue that existing doubly special relativities may not be operationally distinguishable from the special relativity. In the process we point out that some of the phenomenologically motivated modifications of dispersion relations, and arrived conclusions, must be reconsidered. Finally, we reflect on the possible conceptual issues that arise in quest for a theory of spacetime with two invariant scales.
A kind of doubly special relativity theory proposed by J. Magueijo and L. Smolin [Phys. Rev. Lett. 88, 190403 (2002)] is analysed. It is shown that this theory leads to serious physical difficulties in interpretation of kinematical quantities. Moreover, it is argued that statistical mechanics and thermodynamics cannot be resonably formulate within the model proposed in the mentioned paper.
We consider the Doubly Special Relativity (DSR) generalization of Dirac equation in an external potential in the Magueijo-Smolin base. The particles obey a modified energy-momentum dispersion relation. The semiclassical diagonalization of the Dirac Hamiltonian reveals the intrinsic Berry phase effects in the particle dynamics.
This is a write-up of two lectures delivered at COST CA18108 First Training School. They cover the motivations and some basic technical results in the field of Doubly Special Relativity and Relative Locality. The energy-dependent speed of light is a recurring theme here. The soccer ball problem is also briefly described. The video recording of these and other lectures can be found at https://www.youtube.com/channel/UChRfzgjTkO9sOkwq358kHZw/videos.
We show that Doubly Special Relativity (DSR) can be viewed as a theory with energy-momentum space being the four dimensional de Sitter space. Different formulations (bases) of the DSR theory considered so far can be therefore understood as different coordinate systems on this space. The emerging geometrical picture makes it possible to understand the universality of the non-commutative structure of space-time of Doubly Special Relativity. Moreover, it suggests how to construct the most natural DSR bases and makes it possible to address the long standing problem of total momentum of many partic…
Doubly Special Relativity (DSR) is a class of theories of relativistic motion with two observer-independent scales. We investigate the velocity of particles in DSR, defining velocity as the Poisson bracket of position with the appropriate hamiltonian, taking care of the non-trivial structure of the DSR phase space. We find the general expression for four-velocity, and we show further that the three-velocity of massless particles equals 1 for all DSR theories. The relation between the boost parameter and velocity is also clarified.
In a quantum gravity theory, spacetime at mesoscopic scales can acquire a novel structure very different from the classical concept of general relativity. A way to effectively characterize the quantum nature of spacetime is through a momentum dependent space-time metric. There is a vast literature showing that this geometry is related to deformed relativistic kinematics, which is precisely a way to capture residual effects of a quantum gravity theory. In this work, we study the notion of surface gravity in a momentum dependent Schwarzschild black hole geometry. We show that using the two main …
In this paper we recall the construction of Doubly Special Relativity (DSR) as a theory with energy-momentum space being the four dimensional de Sitter space. Then the bases of the DSR theory can be understood as different coordinate systems on this space. We investigate the emerging geometrical picture of Doubly Special Relativity by presenting the basis independent features of DSR that include the non-commutative structure of space-time and the phase space algebra. Next we investigate the relation between our geometric formulation and the one based on quantum $κ$-deformations of the Poincaré…
Building on our previous work [Phys.Rev.D82,085016(2010)], we show in this paper how a Brownian motion on a short scale can originate a relativistic motion on scales that are larger than particle's Compton wavelength. This can be described in terms of polycrystalline vacuum. Viewed in this way, special relativity is not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength is taken. By analyzing the robustness of such a special relativity under small variations in the polycrystalline grain-size dis…
We argue that recently proposed by Amelino-Camelia et all [1,2] so-called doubly special relativity (DSR), with deformed boost transformations identical with the formulae for $κ$-deformed kinematics in bicrossproduct basis is a classical special relativity in nonlinear disguise. The choice of symmetric composition law for deformed fourmomenta as advocated in [1, 2] implies that DSR is obtained by considering nonlinear fourmomenta basis of classical Poincaré algebra and it does not lead to noncommutative space-time. We also show how to construct large two classes of doubly special relativity th…
In this work we discuss the deformed relativistic wave equations, namely the Klein--Gordon and Dirac equations in a Doubly Special Relativity scenario. We employ what we call a geometric approach, based on the geometry of a curved momentum space, which should be seen as complementary to the more spread algebraic one. In this frame we are able to rederive well-known algebraic expressions, as well as to treat yet unresolved issues, to wit, the explicit relation between both equations, the discrete symmetries for Dirac particles, the fate of covariance, and the formal definition of a Hilbert spac…
We present and discuss two different possibilities to construct position space version for Magueijo-Smolin (MS) doubly special relativity proposal. The first possibility is to start from ordinary special relativity and then to define conserved momentum in special way. It generates MS invariant as well as nonlinear MS transformations on the momentum space, leading to consistent picture for one-particle sector of the theory. The second possibility is based on the following observation. Besides the nonlinear MS transformations, the MS energy-momentum relation is invariant also under some inhomoge…
In an attempt to clarify what is the velocity of a particle in doubly special relativity, we solve Maxwell's equations invariant under the position-space nonlinear Lorentz transformation proposed by Kimberly, Magueijo, and Medeiros. We show that only the amplitude of the Maxwell wave, not the phase, is affected by the nonlinearity of the transformation. Thus, although the Maxwell wave appears to have infinitely large energy near the Planck time, the wave velocity is the same as the conventional light velocity. Surprisingly, the velocity of the Maxwell wave is not the same as the maximum signal…
We propose a new Doubly Special Relativity theory based on the generalization of the $κ$-deformation of the Poincaré algebra acting along one of the null directions. We recall the quantum Hopf structure of such deformed Poincaré algebra and use it to derive the phase space commutation relations. As in the DSR based on the standard quantum $κ$-Poincaré algebra we find that the space time is non-commutative. We investigate the fate of the properties of Special Relativity in the null basis: the split of the algebra of Lorentz and momentum generators into kinematical and dynamical parts, the actio…
We generate non-linear representations of the Lorentz Group by unitary transformation over the Lorentz generators. To do that we use deformed scale transformations by introducing momentum-depending parameters. The momentum operator transformation is found to be equivalent to a particle momentum transformation. The configuration space transformation is found to depend on the old momentum operator and we show that this transformation generates models with two scales, one for the velocity ($c$) and another one for the energy. A Lagrangian formalism is proposed for these models and an effective me…
Several issues concerning quantum kappa-Poincare algebra are discussed and reconsidered here. We propose two different formulations of kappa-Poincare quantum algebra. Firstly we present a complete Hopf algebra formulae of kappa-Poincare in classical Poincare basis. Further by adding one extra generator, which modifies the classical structure of Poincare algebra, we eliminate non polynomial functions in the kappa-parameter. Hilbert space representations of such algebras make Doubly Special Relativity (DSR) similar to the Stueckelberg's version of (proper-time) relativistic Quantum Mechanics.
We investigate the thermodynamic stability of neutral black objects with (at least) two angular momenta. We use the quasilocal formalism to compute the grand canonical potential and show that the doubly spinning black ring is thermodynamically unstable. We consider the thermodynamic instabilities of ultra-spinning black objects and point out a subtle relation between the microcanonical and grand canonical ensembles. We also find the location of the black string/membrane phases of doubly spinning black objects.
It is shown that gravity in 2+1 dimensions coupled to point particles provides a nontrivial example of Doubly Special Relativity (DSR). This result is obtained by interpretation of previous results in the field and by exhibiting an explicit transformation between the phase space algebra for one particle in 2+1 gravity found by Matschull and Welling and the corresponding DSR algebra. The identification of 2+1 gravity as a $DSR$ system answers a number of questions concerning the latter, and resolves the ambiguity of the basis of the algebra of observables. Based on this observation a heuristic …
It is shown how a Doubly-Special Relativity model can emerge from a quantum cellular automaton description of the evolution of countably many interacting quantum systems. We consider a one-dimensional automaton that spawns the Dirac evolution in the relativistic limit of small wave-vectors and masses (in Planck units). The assumption of invariance of dispersion relations for boosted observers leads to a non-linear representation of the Lorentz group on the $(ω,k)$ space, with an additional invariant given by the wave-vector $k=π/2$. The space-time reconstructed from the $(ω,k)$ space is intrin…
Doubly Special Relativity (DSR) theory is a theory with two observer-independent scales, of velocity and mass (or length). Such a theory has been proposed by Amelino--Camelia as a kinematic structure which may underline quantum theory of relativity. Recently another theory of this kind has been proposed by Magueijo and Smolin. In this paper we show that both these theories can be understood as particular bases of the $κ$--Poincaré theory based on quantum (Hopf) algebra. This observation makes it possible to construct the space-time sector of Magueijo and Smolin DSR. We also show how this const…
In the present paper, dynamics of generalized charged particles are studied in the presence of external electromagnetic interactions. This particular extension of the free relativistic particle model lives in Non-Commutative $κ$-Minkowski space-time, compatible with Doubly Special Relativity, that is motivated to describe Quantum Gravity effects. Furthermore we have also considered the electromagnetic field to be dynamical and have derived the modified forms of Lienard-Wiechert like potentials for these extended charged particle models. In all the above cases we exploit the new and extended fo…
In this paper we recall the construction of scalar field action on $κ$-Minkowski space-time and investigate its properties. In particular we show how the co-product of $κ$-Poincaré algebra of symmetries arises from the analysis of the symmetries of the action, expressed in terms of Fourier transformed fields. We also derive the action on commuting space-time, equivalent to the original one. Adding the self-interaction $Φ^4$ term we investigate the modified conservation laws. We show that the local interactions on $κ$-Minkowski space-time give rise to 6 inequivalent ways in which energy and mom…
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 evaluati…
We describe firstly the basic features of quantum $κ$-Poincaré symmetries with their Hopf algebra structure. The quantum $κ$-Poincaré framework in any basis relates rigidly the quantum $κ$-Poincaré algebra with quantum $κ$-Poincaré group, noncommutative space-time and $κ$-deformed phase space. Further we present the approach of Doubly Special Relativity (DSR) theories, which introduce(in the version DSR1) kinematically the frame - independent fundamental mass parameter as described by maximal three-momentum $|\overrightarrow{p}|=κc$. We argue why the DSR theories in one-particle sector can be …
We propose version of doubly special relativity theory starting from position space. The version is based on deformation of ordinary Lorentz transformations due to the special conformal transformation. There is unique deformation which does not modify rotations. In contrast to the Fock-Lorentz realization (as well as to recent position-space proposals), maximum signal velocity is position (and observer) independent scale in our formulation by construction. The formulation admits one more invariant scale identified with radius of three-dimensional space-like hypersection of space-time. We prese…
A mass-like quantum Weyl-Poincare algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, a new relativistic theory with two observer-independent scales (or DSR theory). Deformed momentum representation, finite boost transformations, range of rapidity, energy and momentum, as well as position and velocity operators are explicitly studied and compared with those of previous DSR theories based on kappa-Poincare algebra. The main novelties of the DSR theory here presented are the new features of momentum saturation and a new type of deformed …
Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there is infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the $κ$-Poincaré quantum (Hopf) algebra. Then we use the co-product of this algebra and the known construction of $κ$-deformed phase space via Heisenberg double in order to derive the non-commutative space-time structure and description of the whole of the DSR …
The theories known as doubly special relativity are introduced in order to take into account an observer-independent length scale and the speed of light in the framework of special relativity. These theories can be generally formulated on the de Sitter and also recently proposed anti-de Sitter momentum spaces. In the context of these theories, we study the statistical mechanics and to do this, we consider the natural measure on the corresponding extended phase space. The invariant measure on the space of distinct microstates is obtained by restriction of the natural measure of the extended pha…
The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed. The main building block of the proper relativistic extension of the classical thermodynamics laws is the four-temperature vector β, which plays a major role in the quantum framework and defines a very convenient hydrodynamic frame. The general relativistic thermodynamic equilibrium condition demands βto be a Killing vector field. We show that a remarkable consequence is that all Lie derivatives of all physical observables along the four-temperature flow must then vanish.
We study the doubly charged Callan-Giddings-Harvey-Strominger (CGHS) model, which has black hole solutions that were found to be U-dual to the D1-D5-KK black holes of the IIB supergravity. We derive the action of the model via a spontaneous compactification on S^3 of the IIB supergravity on S^1*T^4 and obtain the general static solutions including black holes corresponding to certain non-asymptotically flat black holes in the IIB supergravity. Thermodynamics of them is established by computing the entropy, temperature, chemical potentials, and mass in the two-dimensional setup, and the first l…
In this Ph.D. thesis several topics in doubly special relativity are explored. The starting point of this theory is very different from other perspectives: it is not a fundamental theory, but it is considered a low energy limit of a quantum gravity theory that tries to study its possible residual elements. In particular, in doubly special relativity the Einstenian relativity principle is generalized, adding to the speed of light $c$ another relativistic invariant, the Planck length $l_p$. This idea can really have possible experimental observations, giving place to what is known as quantum gra…
