Energy-momentum Vectors Research Articles

On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

Ultrafast visualization of an electric field under the Lorentz transformation

In special relativity, four-vectors, such as space–time, energy–momentum and electromagnetic potential vectors, follow the Lorentz transformation. The transformations of space–time and energy–momentum vectors have been tested by time dilation and rest mass energy experiments, respectively. However, the Lorentz transformation of electromagnetic potentials has never been directly demonstrated in experiments due to the limitations of the diagnostics. Here we show the spatiotemporal electric-field profiles around a highly energetic electron beam with a subpicosecond temporal resolution obtained by a terahertz technique based on electro-optic sampling. We observe the Coulomb field contraction in the propagation direction of the beam under the Lorentz transformation. After passing the beam through a metallic plate, we investigate the birth of the Coulomb field around it and follow the spatiotemporal evolution of the spherical wavefront, which is derived by the Liénard–Wiechert potentials. The measured electric-field profile in the far propagation distance is experimentally confirmed to coincide with the calculation of the Liénard–Wiechert potentials and the Lorentz transformation. Our ultrafast measurement of a spatiotemporal relativistic electric field provides both additional experimental evidence of special relativity and a platform for detailed experimental research of energetic charged particle beams and electromagnetic radiation physics.

On the Significance of the Stress–Energy Tensor in Finsler Spacetimes

We revisit the physical arguments that led to the definition of the stress–energy tensor T in the Lorentz–Finsler setting (M,L) starting with classical relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes T an anisotropic 2-tensor (i.e., a tensor for each L-timelike direction), in contrast with the energy-momentum vectors defined on M. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is revised from a geometric viewpoint, and, then, the conservation laws of T for each observer field are revisited. We introduce a natural anisotropic Lie bracket derivation, which leads to a divergence obtained from the volume element and the non-linear connection associated with L alone. The computation of this divergence selects the Chern anisotropic connection, thus giving a geometric interpretation to previous choices in the literature.

Energy–momentum density and pressure relations for a relativistic ideal gas with a bulk motion

We derive here, from first principles, the energy–momentum density of a perfect fluid in the form of an ideal molecular gas, in an inertial frame where the fluid possesses a bulk motion. We begin from the simple expressions for the energy density and pressure of a perfect fluid in the rest frame of the fluid, where the fluid constituents (gas molecules) may possess a random motion, but no bulk motion. From a Lorentz transformation of the velocity vectors of the molecules moving in different directions in the rest frame of the fluid, we compute their energy–momentum vectors and number densities in an inertial frame moving with respect to the rest frame of the liquid. From that, we arrive at the energy–momentum density of the fluid in a frame where it has a bulk motion. This way, we explicitly demonstrate how a couple of curious pressure-dependent terms make an appearance in the energy–momentum density of a perfect fluid having a bulk motion. In addition to an ideal molecular gas, we compute the energy–momentum density for a photon gas also, which of course matches the energy–momentum density expression obtained for a molecular gas having an ultra-relativistic random motion.

Polarization tensor for tilted Dirac fermion materials: Covariance in deformed Minkowski spacetime

The rich structure of solid state physics provides us with Dirac materials the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic lattices, the Lorentz symmetry will be deformed in a way that the null energy-momentum vectors will correspond to on-shell condition for tilted Dirac cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as solid state systems where the effective spacetime is non-Minkowski. In this work, we show that the polarization tensor for tilted Dirac cone systems acquires a covariant from only when the spacetime is considered to be an appropriate deformation of the Minkowski spacetime. As a unique consequence of the deformation of the geometry of the spacetime felt by the electrons in tilted Dirac cone materials, the Coulomb density-density interactions will generate corrections in both longitudinal and transverse channels. Therefore the transverse photons also participate in mediating the Coulomb forces, implying emergent Amperean forces associated with the tilt of the spacetime.

Towards Asymptotic Completeness of Two-Particle Scattering in Local Relativistic QFT

We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Let $\ti p_1$ and $\ti p_2$ be two energy-momentum vectors of a massive particle and let $\De$ be a small neighbourhood of $\ti p_1+\ti p_2$. We construct asymptotic observables (two-particle Araki-Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods of $\ti p_1$ and $\ti p_2$. We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of $\De$. Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag-Ruelle scattering states with total energy-momenta in $\De$. The result holds under very general conditions which are satisfied, for example, in $\la \phi^4_2$. The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.

A family of asymptotically hyperbolic manifolds with arbitrary energy-momentum vectors

A family of non-radial solutions to the Yamabe equation, modeled on the hyperbolic space, is constructed using power series. As a result, we obtain a family of asymptotically hyperbolic metrics, with spherical conformal infinity, with scalar curvature greater than or equal to −n(n − 1), but which are a priorinot complete. Moreover, any vector of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{n+1}$\end{document}Rn+1 is performed by an energy-momentun vector of one suitable metric of this family. They can in particular provide counter-examples to the positive energy-momentum theorem when one removes the completeness assumption.

Gravitomagnetic Field and Penrose Scattering Processes

In this paper we present theoretical model calculations involving Monte Carlo computer simulations of Compton scattering and electron-positron (e-e+) pair production processes in the ergosphere of a supermassive rotating black hole. Particles from an accretion disk surrounding the rotating black hole fall into the ergosphere and are scattered by particles that are confined in equatorial and nonequatorial orbits. The energy-momentum vectors are calculated for the scattered escaping particles. Particles escape with energies of about 3 GeV or greater. Importantly, these model calculations show that the Lense-Thirring effect, that is, the dragging of local inertial frames into rotation, inside the ergosphere, caused by the angular momentum of the rotating black hole, results in a gravitomagnetic force being exerted on the scattered escaping particles. Effects of this force on the Penrose scattered particles are analyzed and discussed.

The Lund fragmentation process for a multi-gluon string according to the area law

The Lund Area Law describes the probability for the production of a set of colourless hadrons from an initial set of partons, in the Lund string fragmentation model. In this paper we will present a general method to implement the Area Law for a multi-gluon string state. The partonic states are in general given by a perturbative QCD cascade and are consequently defined only down to a cutoff in the energy momentum fluctuations. We will show that our method defines the states down to the hadronic mass scale inside an analytically calculable scenario. We will then show that there is a differential version of our process which is closely related to the generalised rapidity range \lambda, which has been used as a measure on the partonic states. We identify \lambda as the area spanned between the directrix curve (the curve given by the parton energy momentum vectors laid out in colour order, which determines the string surface) and the average curve (to be called the P-curve) of the stochastic X-curves (curves obtained when the hadronic energy-momentum vectors are laid out in rank order). Finally we show that from the X-curve corresponding to a particular stochastic fragmentation situation it is possible to reproduce the directrix curve (up to one starting vector and a set of sign choices, one for each hadron).

Gravitational waves, black holes and cosmic strings in cylindrical symmetry

Gravitational waves in cylindrically symmetric Einstein gravity are described by an effective energy tensor with the same form as that of a massless Klein-Gordon field, in terms of a gravitational potential generalizing the Newtonian potential. Energy-momentum vectors for the gravitational waves and matter are defined with respect to a canonical flow of time. The combined energy-momentum is covariantly conserved, the corresponding charge being the modified Thorne energy. Energy conservation is formulated as the first law expressing the gradient of the energy as work and energy-supply terms, including the energy flux of the gravitational waves. Projecting this equation along a trapping horizon yields a first law of black-hole dynamics containing the expected term involving area and surface gravity, where the dynamic surface gravity is defined with respect to the canonical flow of time. A first law for dynamic cosmic strings also follows. The Einstein equation is written as three wave equations plus the first law, each with sources determined by the combined energy tensor of the matter and gravitational waves.

An example of a field theory with indefinite metric in Hilbert space II

A Lagrangian formalism is set up for the field equations proposed in ref. 1). Energy-momentum tensors and current vectors are derived for boson and fermion fields and the corresponding conservation laws are considered. In space-time regions close to the light cone and hyperboloid the energy density of the boson field and charge density of the fermion field are not positive definite. The integrals of these quantities over the whole volume (energy and charge) are definite. The appearance of negative densities is interpreted in accord with the modification of the concept of a particle coordinate proposed in ref. 1).

Mass Centre in Relativity

UNDER this title, in NATURE of April 13, p. 587, Prof. M. Born and K. Fuchs gave some relations between the total and a relative momentum vector of a system of two free particles. They only define the magnitude, not the direction of their relative momentum vector. I believe things become clearer by the following statement. Let Va, Wa (a = 0, 1, 2, 3) be (1 + 3)-dimensional velocity time-space vectors of the particles (V2 = W2 = c2), and m1, m2 their scalar masses. Their individual energy-momentum vectors being ia = m1Va and ja = m2Wa we may define the total energy-momentum vector by the relative energy-momentum vector by These definitions entail two identities which are equivalent to equations (4), (5), (6) of Born and Fuchs.