Relativistic Generalization Research Articles

Relativistic Interacting Integrable Elliptic Tops

We propose relativistic generalization of integrable systems describing $M$ interacting elliptic ${\rm gl}(N)$ tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic ${\rm GL}(M)$ Ruijsenaars-Schneider model for $N=1$ case, while in the $M=1$ case they turn into relativistic integrable ${\rm GL}(N)$ elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed.

Reduction of a family of metric gravities

A recent proposal by Shuler regarding a postulate-based derivation of a family of metrics describing the gravitational field outside a static spherically symmetric mass distribution is reviewed. All of Shuler's gravities agree with the Schwarzschild solution in the weak-field limit, but they differ in the strong-field domain, i.e., close enough to a sufficiently compact source of the field. It is found that the evoked postulates of i) momentum conservation and ii) consistency of field strength measurement are satisfied in all metric theories of gravity compatible with the Einstein equivalence principle, no matter what the form of the metric. Therefore, they cannot be used, within any correct deduction, to derive a particular metric. Shuler's derivations are based on an inconsistent set of correspondences between local and distant quantities. Furthermore, it is shown here that out of the family of possible metrics given by Shuler only one member, the Schwarzschild metric, satisfies a standard relativistic generalization of Newton's law of gravitation, suggesting the others to be unphysical.

Quantum Hamilton equations for multidimensional systems

Within a stochastic picture quantum systems can be described in terms of kinematic and dynamic equations where the particles follow a conservative diffusion process. Pavon has shown that these equations and the Schrödinger equation can be derived from a quantum Hamilton principle, generalizing the Hamilton principle from classical mechanics. As shown recently by Köppe et al for the one-dimensional case, the reformulation of the quantum Hamilton principle as a stochastic optimal control problem leads to quantum Hamilton equations of motion which can be seen as a non relativistic generalization of Hamilton’s equations of motion to the quantum world. In this report we will extend this to higher dimensional problems and derive, by incorporating the concept of supersymmetric quantum mechanics, a general scheme that allows the determination of all bound states within the stochastic framework. Physical principles from classical mechanics like the decoupling of the center-of-mass motion in multi-particle systems or the solution of the Kepler problem, as a special case of the two-body problem, using space-time symmetries, are translated to this formulation of quantum mechanics. We will present numerical results for the ground state and lower excited states of the hydrogen atom in Cartesian coordinates, and, additionally, we will show that for spherically symmetric potentials the solution of the two-body problem can be reduced to a system of equations concerning only the radial part.

Free fall in curved spacetime—how to visualise gravity in general relativity

The first direct observation of gravitational waves in 2015 has led to an increased public interest in topics of general relativity (GR) and astronomy. Physics teachers and educators respond to this interest by introducing modern ideas of gravity and spacetime to high school students. Doing so, they face the challenge of finding suitable models that visualise gravity as the geometry of curved spacetime. Most models of GR, such as the popular rubber sheet model, only address spatial curvature. Yet, according to Albert Einstein, gravitational phenomena stem from deformations both in space and time. This paper presents a new model that builds on a relativistic generalisation of Newton’s first law. We use Einstein’s free fall thought experiment and a classical height-time diagram to explain how warped time gives rise to gravity. Our warped-time model acts as a convenient supplement to the rubber sheet model. To support teachers in integrating the model into their classroom practice, we have implemented the model as an interactive simulation that is freely accessible. The model is the result of a three-year period of developing and trialling digital learning resources in Norwegian high schools. Based on these trials, we suggest specific instructional strategies on how to use the warped-time model successfully in science classrooms.

Quantum walk hydrodynamics

A simple Discrete-Time Quantum Walk (DTQW) on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results show that suitable initial conditions indeed produce hydrodynamical shocks and that the coherence achieved in current experiments is robust enough to simulate quantum hydrodynamical phenomena through DTQWs. An analytical computation of the asymptotic quantum shock structure is presented. The non-relativistic limit is explored in the Supplementary Material (SM).

On The Contribution of the P and D Partial-Wave States to the Binding Energy of the Triton in the Bethe-Salpeter-Faddeev Approach

The influence of the partial-wave states with nonzero orbital moment of the nucleon pair on the binding energy of the triton in the relativistic case is considered. The relativistic generalization of the Faddeev equation in the Bethe-Salpeter formalism is applied. Two-nucleon t matrix is obtained from the Bethe-Salpeter equation with separable kernel of nucleon-nucleon interaction of the rank one. The kernel form factors are the relativistic type of the Yamaguchi functions. The following two-nucleon partial-wave states are considered: $^1S_0$, $^3S_1$, $^3D_1$, $^3P_0$, $^1P_1$, $^3P_1$. The system of the integral equations are solved by using the iteration method. The binding energy of the triton and three-nucleon amplitudes are found. The contribution of the P and D states to the binding energy of triton is given.

Sensitivity of elastic electron scattering off the 3He to the nucleon form factors

Elastic electron-3He scattering is studied in the relativistic impulse approximation. The amplitudes for the three-nucleon system – 3He – are obtained by solving the relativistic generalization of the Faddeev equation. The charge and magnetic form factor are calculated and compared with the experimental data for the momentum transfer squared up to 100 fm-2. The influence of the various nucleon form factors is investigated.

Evaluation of relativistic transport coefficients and the generalized Spitzer function in devices with 3D geometry and finite collisionality

The relativistic generalization of the linearized drift kinetic equation solver NEO-2 is presented which is used for computation of neoclassical transport coefficients and the generalized Spitzer function in 3D toroidal fusion devices (tokamaks and stellarators). This upgrade allows computations of the Spitzer function playing the role of current drive efficiency in the whole experimentally relevant temperature range, from mild temperatures where finite plasma collisionality effects are important to high temperatures where relativistic effects should be taken into account. Within the Galerkin method used for problem discretization over energy, relativistic effects are included into a set of matrices constant on a flux surface. Those matrices determine coefficients of a coupled set of integro-differential equations with a reduced dimension which is of the same form as in the non-relativistic case. For energy discretization of the linearized relativistic Coulomb collision operator, it is presented in spherical momentum space variables in the symmetric integral form derived directly from Beliaev-Budker expressions. The cancellation problem pertinent to the fully analytical representation of Braams and Karney [Phys. Fluids B 1, 1355 (1989)] does not appear in this form. Examples of evaluation of relativistic transport coefficients and the Spitzer function are presented.

Solitonic metrics and harmonic maps

We investigate the relationship between solitonic metrics $$g_u = - (\sin ^2 \frac{u}{2}) c^2 d t^2 + (\cos ^2 \frac{u}{2}) \sum _{i=1}^n (d x^i )^2$$ with $$u \in C^\infty ({{\mathbb {R}}}^{n+1})$$ and stationary points of the functional $$E_\Omega (\Phi ) = \frac{1}{2} \int _\Omega \Vert d \Phi \Vert ^2 d^{n+1} \mathbf{x}$$ with $$\Phi \in C^\infty ({{\mathbb {R}}}^{n+1}, S^2 )$$. Building on work by F.L. Williams (cf. Williams, in: Milton (ed) Quantum field theory under the influence of external conditions, Rinton Press, Princeton, pp 370–372, 2004; in: 4th international winter conference on mathematical methods in physics, Rio de Janeiro, http://pos.sissa.it/archive/conferences/013/003/wc2004_003.pdf, 2004; in: Chen (ed) Trends in soliton research, Nova Science Publications, Hauppauge, pp 1–14, 2006; in: Maraver, Kevrekidis, Williams (eds) The sine-gordon modeland its applications, Springer, Berlin, pp 177–205, 2014) we show that a map $$\Phi = (\cos \beta \sin \frac{u}{2}, \sin \beta \sin \frac{u}{2}, \cos \frac{u}{2} )$$ with $$\beta (\mathbf{x}, t) = m \big ( 1 + | \mathbf{v} |^2 \big )^{-1/2} (t + \mathbf{v} \cdot \mathbf{x})$$ is harmonic if and only if $$u_{tt} + \Delta u = m^2 \sin u$$ (the sine-Gordon equation) and $$u_t + \mathbf{v} \cdot \nabla u = 0$$ (the convection equation). In the spirit of work by B. Solomon (cf. Solomon in J Differ Geom 21:151–162, 1985) we build a 1-parameter variation of $$\Phi $$ which singles out [from the full Euler–Lagrange system of the variational principle $$\delta E_\Omega (\Phi ) = 0$$] the convection equation. Williams’ harmonic maps $$\Phi ^\pm = (1 + v^2 )^{-1/2} (\tau \cos \beta , \tau \sin \beta , \pm (1 + v^2 )^{1/2}\tanh \rho )$$ are shown to be harmonic morphisms of dilation $$\lambda = m(1 + v^2 )^{-1/2}\tau $$, further explaining the relationship between Jackiw–Teitelboim 2-dimensional dilation-gravity theory (cf. Jackiw, in: Christensen (ed) Quantum theory of gravity, MIT, Cambridge, pp 403–420, 1982; Teitelboim, in: Christensen (ed) Quantum theory of gravity, Adam Hilger Ltd, Bristol, pp 403–420, 1984) and harmonic map theory (cf. Baird and Wood Harmonic morphisms between Riemannian manifolds, London mathematical society monographs, new series, 29, The Clarendon Press, Oxford University Press, Oxford, ISBN 0-19-850362-8, 2003). We show that geodesic motion in a weak solitonic gravitational field $$g_{u_\epsilon }$$ [$$u_\epsilon = u_0 + 2 \epsilon \rho $$ with $$\rho \in C^\infty ({{\mathbb {R}}}^{n+1})$$ bounded and $$\epsilon<< 1$$] in the Newtonian velocity limit ($$\Vert \mathbf{u}\Vert /c<< 1$$) obeys to the law of motion $$d^2 \mathbf{r}/d t^2 = - \nabla \phi $$ in a central force field of potential $$\phi \equiv \epsilon c^2 \tan \frac{u_0}{2} \, \rho $$. To justify the use of geodesic equations as equations of motion we address a relativistic mechanics problem i.e. the rotation of a disc in a solitonic gravitational field and exhibit a class of metrics $$g_u$$ one of whose geodesic equations yields a relativistic generalization $$d^2 r/d s^2 = (r w^2 )/c^2$$ of centrifugal accelerations in classical mechanics.

On the coherent states for a relativistic scalar particle

The three approaches to relativistic generalization of coherent states are discussed in the simplest case of a spinless particle: the standard, canonical coherent states, the Lorentzian states and the coherent states introduced by Kaiser and independently by Twareque Ali, Antoine and Gazeau. All treatments utilize the Newton–Wigner localization and dynamics described by the Salpeter equation. The behavior of expectation values of relativistic observables in the coherent states is analyzed in detail and the Heisenberg uncertainty relations are investigated.

Intershell-correlation-induced time delay in atomic photoionization

We predict an observable Wigner time delay in outer atomic shell photoionization near inner shell thresholds. The near-threshold increase of time delay is caused by intershell correlation and serves as a sensitive probe of this effect. The time delay increase is present even when the inner and outer shell thresholds are hundreds of electron volts apart. We illustrate this observation by several prototypical examples in noble gas atoms from Ne to Kr. In our study, we employ the random phase approximation with exchange and its relativistic generalization. We also support our findings by a simplified, yet quite insightful, treatment within the lowest-order perturbation theory.

Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing

We study the ergodicity properties of weak solutions to a relativistic generalization of Burgers equation posed on a curved background and, specifically, a Schwarzschild black hole. We investigate the interplay between the dynamics of shocks, a curved geometric background, and a random boundary forcing, and solve three problems of independent interest. First of all, we consider the standard Burgers equation on a half-line and establish a ‘one-force-one-solution’ principle when the random forcing at the boundary is sufficiently “strong” in comparison with the velocity of the solutions at infinity. Secondly, we consider the Burgers–Schwarzschild model and establish a generalization of the Hopf–Lax–Oleinik formula. This novel formula takes the curved geometry into account and allows us to establish the existence of bounded variation solutions. Thirdly, under a random boundary forcing in the vicinity of the horizon of the black hole, we prove the existence of a random global attractor and we again validate the ‘one-force-one-solution’ principle. Finally, we extend our main results to the pressureless Euler system which includes a transport equation satisfied by the integrated fluid density.

Ruijsenaars-Schneider three-body models with N = 2 supersymmetry

The Ruijsenaars-Schneider models are conventionally regarded as relativistic generalizations of the Calogero integrable systems. Surprisingly enough, their supersymmetric generalizations escaped attention. In this work, N = 2 supersymmetric extensions of the rational and hyperbolic Ruijsenaars-Schneider three-body models are constructed within the framework of the Hamiltonian formalism. It is also known that the rational model can be described by the geodesic equations associated with a metric connection. We demonstrate that the hyperbolic systems are linked to non-metric connections.

Spacetime from locality of interactions in deformations of special relativity: The example of κ -Poincaré Hopf algebra

A new proposal for the notion of spacetime in a relativistic generalization of special relativity based on a modification of the composition law of momenta is presented. Locality of interactions is the principle which defines the spacetime structure for a system of particles. The formulation based on $\kappa$-Poincar\'e Hopf algebra is shown to be contained in this framework as a particular example.

Disparities of Larmor’s/Liénard’s formulations with special relativity and energy-momentum conservation

It is shown that the familiar Larmor’s formula or its relativistic generalization, Liénard’s formula, widely believed to represent the instantaneous radiative losses from an accelerated charge, are not compatible with the special theory of relativity (STR). It is also shown that energy-momentum conservation is violated in all inertial frames when we compute the instantaneous rate of loss of kinetic energy and momentum of the radiating charge in accordance with Larmor’s/Liénard’s formulation. It is emphasized that one should clearly distinguish between the electromagnetic power going as radiation into space and the instantaneous loss of mechanical power by the charge. In literature both powers are treated as not only equal but almost synonymous; the two need not be the same, however. It is pointed out that a mathematical subtlety in the applicability of Poynting’s theorem seems to have been overlooked in the text-book derivation of Larmor’s formula, where a proper distinction between ‘real’ and ‘retarded’ times was not maintained. This has led to some century-old apparent anomalies, including the mysterious Schott energy, which happens to be nothing but merely a change in self-field energy of the charge between retarded and real times.

Effects of general relativity on glitch amplitudes and pulsar mass upper bounds

Pinning of vortex lines in the inner crust of a spinning neutron star may be the mechanism that enhances the differential rotation of the internal neutron superfluid, making it possible to freeze some amount of angular momentum which eventually can be released, thus causing a pulsar glitch. We investigate the general relativistic corrections to pulsar glitch amplitudes in the slow-rotation approximation, consistently with the stratified structure of the star. We thus provide a relativistic generalization of a previous Newtonian model that was recently used to estimate upper bounds on the masses of glitching pulsars. We find that the effect of general relativity on the glitch amplitudes obtained by emptying the whole angular momentum reservoir is less than 30\%. Moreover we show that the Newtonian upper bounds on the masses of large glitchers obtained from observations of their largest recorded event differ by less than a few percent from those calculated within the relativistic framework. This work can also serve as a basis to construct more sophisticated models of angular momentum reservoir in a relativistic context. In particular, we present two alternative scenarios for "rigid" and "slack" vortex lines, and we generalize the Feynman-Onsager relation to the case when both entrainment coupling between the fluids and a strong gravitational field are present.

The general setting for the zero-flux condition: The lagrangian and zero-flux conditions that give the heisenberg equation of motion.

Generalizing our recent work on relativistic generalizations of the quantum theory of atoms in molecules, we present the general setting under which the principle of stationary action for a region leads to open quantum subsystems. The approach presented here is general and works for any Hamiltonian, and when a reasonable Lagrangian is selected, it often leads to the integral of the Laplacian of the electron density on the region vanishing as a necessary condition for the zero-flux surface. Alternatively, with this method, one can design a Lagrangian that leads to a surface of interest (though this Lagrangian may not be, and indeed probably will not be, "reasonable"). For any reasonable Lagrangian for the electronic wave function and any two-component method (related by integration by parts to the Hamiltonian) considered, the Bader definition of an atom is recaptured. © 2018 Wiley Periodicals, Inc.

Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the perturbation and solution schemes that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given to any order of the perturbations, together with extensions and applications for accessing the nonperturbative regime. We here discuss more in detail the general first-order scheme within the Cartan formalism including and concentrating on the gravitational wave propagation in matter. We provide master equations for all parts of Lagrangian-linearized perturbations propagating in the perturbed spacetime, and we outline the solution procedure that allows one to find general solutions. Particular emphasis is given to global properties of the Lagrangian perturbation fields by employing results of Hodge-de Rham theory. We here discuss how the Hodge decomposition relates to the standard scalar-vector-tensor decomposition. Finally, we demonstrate that we obtain the known linear perturbation solutions of the standard relativistic perturbation scheme by performing two steps: first, by restricting our solutions to perturbations that propagate on a flat unperturbed background spacetime and, second, by transforming to Eulerian background coordinates with truncation of nonlinear terms.

Does a deformation of special relativity imply energy dependent photon time delays?

Theoretical arguments in favor of energy dependent photon time delays from a modification of special relativity (SR) have met with recent gamma ray observations that put severe constraints on the scale of such deviations. We review the case of the generality of this theoretical prediction in the case of a deformation of SR and find that, at least in the simple model based on the analysis of photon worldlines which is commonly considered, there are many scenarios compatible with a relativity principle which do not contain a photon time delay. This will be the situation for any modified dispersion relation which reduces to for photons, independently of the quantum structure of spacetime. This fact opens up the possibility of a phenomenologically consistent relativistic generalization of SR with a new mass scale many orders of magnitude below the Planck mass.

Dirac Green function for δ potentials

The Green function for a singular one-dimensional potential is explicitly obtained in the relativistic context of the Dirac equation, using Dyson's equation. From it, two bound states are easily found, one for the particle and another for the antiparticle, both depending on the δ potential intensity. When a second δ perturbation is introduced at a different point, the problem can also be solved analytically, for the possible bound states. A transcendental equation is obtained, which can be considered as a relativistic generalization of the well-known Lambert equation.