Energy-momentum Density Research Articles

Spacetime geometry of acoustics and electromagnetism

Both acoustics and electromagnetism represent measurable fields in terms of dynamical potential fields. Electromagnetic force-fields form a spacetime bivector that is represented by a dynamical energy–momentum 4-vector potential field. Acoustic pressure and velocity fields form an energy–momentum density 4-vector field that is represented by a dynamical action scalar potential field. Surprisingly, standard field theory analyses of spin angular momentum based on these traditional potential representations contradict recent experiments, which motivates a careful reassessment of both theories. We analyze extensions of both theories that use the full geometric structure of spacetime to respect essential symmetries enforced by vacuum wave propagation. The resulting extensions are geometrically complete and phase-invariant (i.e., dual-symmetric) formulations that span all five grades of spacetime, with dynamical potentials and measurable fields spanning complementary grades that are related by a spacetime vector derivative (i.e., the quantum Dirac operator). These complete representations correct the equations of motion, energy–momentum tensors, forces experienced by probes, Lagrangian densities, and allowed gauge freedoms, while making manifest the deep structural connections to relativistic quantum field theories. Finally, we discuss the implications of these corrections to experimental tests.

Monte Carlo Stellar Dynamics near Massive Black Holes: Two-dimensional Fokker–Planck Solutions of Multiple Mass Components

In this study we present a novel Monte Carlo code, referred to as GNC, which enables the investigation of dynamical relaxation in clusters comprising multiple mass components in the vicinity of supermassive black holes at the centers of galaxies. Our method is based on two-dimensional Fokker–Planck equations in the energy and angular momentum space, and allows the evolution of multiple mass components, including stars and compact objects. The code demonstrates remarkable flexibility in incorporating additional complex dynamics. By employing a weighting method, we effectively enhance the statistical accuracy of rare particle results. In this initial publication, we present the fundamental version of our method, focusing on two-body relaxations and loss cone effects. Through comparisons with previous studies, we establish consistent outcomes in terms of relaxation processes, energy and angular momentum distributions, density profiles, and loss cone consumption rates. We consistently observe the development of tangential anisotropy within the cluster, while the outer regions tend to retain near-isotropic characteristics. GNC holds great promise for exploring a wide range of intriguing phenomena within galactic nuclei, including relativistic stellar dynamics, providing detailed and insightful outcomes.

Exploring modifications to FLRW cosmology with general entropy and thermodynamics: A new approach

The investigation of modifications to the FLRW cosmology resulting from the consideration of a general entropy for the cosmological apparent horizon is the subject of this study. Building upon the work of Nojiri and colleagues in 2022, who introduced a class of generalized entropies with four parameters capable of converging to familiar entropies and addressing specific cosmological issues, our research explores the impact of correcting the entropy on the energy-momentum tensor of the cosmic fluid from the outset. Our calculations demonstrate that, by employing a correction function f(ρ) to modify the energy-momentum density tensor, the entropic area law (Bekenstein-Hawking entropy) can still be regarded as a general entropy. The construction of the function f(ρ) is facilitated through considerations of the thermodynamics associated with the apparent horizon. Additionally, we investigate the first and second laws of thermodynamics within this framework and illustrate how the limitations imposed on the equation of state of the cosmic fluid can be resolved through the incorporation of this correction function. Finally, we compute cosmography parameters to analyze the kinematics of the universe, with particular attention given to the notable influence of the correction function f(ρ) on these parameters. This paper provides valuable insights into the application of general entropies to the apparent horizon of the universe.

The Effect of f(R, T) Modified Gravity on the Mass and Radius of Pulsar HerX1

Millisecond pulsars are the perfect testable to examine potential matter-geometry coupling and its physical consequences in the context of the recent Neutron Star Interior Composition Explorer discoveries. We apply the field equations of modified gravity, f(R, T) = R + α T, to a spherically symmetric spacetime, where R is the Ricci scalar, α is a dimensional parameter, and T is the matter of the geometry. Five unknown functions are present in the output system of differential equations, which consists of three equations. To close the system, we make explicit assumptions about the anisotropy and the radial metric potential, g rr . We then solve the output differential equations and derive the explicit forms of the components of the energy-momentum tensor, i.e., density, radial, and tangential pressures. We look into the possibility that all of the physical parameters in the star can be reexpressed in terms of α and the compactness parameters, C = 2 GM Rc−2. We show that, for a given mass, the size permitted by Einstein’s general relativity is less due to the matter-geometry coupling in f(R, T). The validity of the hypothesis was validated by observations from an extra 21 pulsars. To achieve a surface density that is compatible with a neutron core at nuclear saturation density, the mass–radius curve enables masses up to 3.35M ⊙. We emphasize that although there is no assumption of an equation of state, the model fits well with a linear behavior. When comparing the surface densities of these 20 pulsars, we divided them into three groups. We show that these three groups are compatible with neutron cores.

Relativistic probability densities for location

Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ μ ( x , t) = (ℏ/2i)[ψ +( x , t)∂ μ ψ( x , t) − ∂ μ ψ +( x , t) · ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields ϕ( x , t), and therefore the canonical local energy-momentum densities and differ from ℘ μ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density , the canonical charge density ϱ( x , t), the energy pseudo-density , and the Born density ∣ψ( x , t)∣2 for the massless free Klein–Gordon field. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case: in spite of their nonlocal mathematical relations, they are mutually local in the sense that their maxima do not deviate beyond a common position uncertainty Δx. Indeed, they are practically indistinguishable in cases where we would expect a normalized quantum state to produce particle-like position signals, viz. if we are observing quanta with momenta p ≫ Δp ≥ ℏ/2Δx. We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density provides a suitable measure for positions of bosons, whereas normalized charge density ϱ( x , t)/q provides a suitable measure for fermions.

Relativistic Fermion and Boson Fields: Bose-Einstein Condensate as a Time Crystal

In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated.

Spherically symmetric wormholes in minimally coupled f(R,T) gravity

In this manuscript, we construct a spherical thin shell wormhole in the background of the minimally coupled f(R, T) model. In this gravitational theory, f is an arbitrary functional form that depends on the Ricci scalar (R) as well as the trace of stress-energy tensor (T). To continue our systematic analysis, we use a cut and paste approach to link two surfaces, i.e., interior and exterior. We find energy-momentum density along with surface pressure for f(R, T) gravitational f(R, T) = R + α R 2 + λ T model, by using the Lanczos equation. We utilize the polytropic equation of state to check the dynamical behavior of the wormhole. A standard potential approach is applied to check the stability of constructed wormhole with throat radius a 0. Graphical analysis shows that the stability regions of wormholes rely on the specific values mass M to charge Q ratio.

Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories (II): Furstenberg's Theorem and Exceptions to Heating Phases

In this sequel (to [Phys. Rev. Res. 3, 023044(2021)], arXiv:2006.10072), we study randomly driven (1+1)(1+1) dimensional conformal field theories (CFTs), a family of quantum many-body systems with soluble non-equilibrium quantum dynamics. The sequence of driving Hamiltonians is drawn from an independent and identically distributed random ensemble. At each driving step, the deformed Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength and therefore induces a Möbius transformation on the complex coordinates. The non-equilibrium dynamics is then determined by the corresponding sequence of Möbius transformations, from which the Lyapunov exponent \lambda_LλL is defined. We use Furstenberg’s theorem to classify the dynamical phases and show that except for a few exceptional points that do not satisfy Furstenberg’s criteria, the random drivings always lead to a heating phase with the total energy growing exponentially in the number of driving steps nn and the subsystem entanglement entropy growing linearly in nn with a slope proportional to central charge cc and the Lyapunov exponent \lambda_LλL. On the contrary, the subsystem entanglement entropy at an exceptional point could grow as \sqrt{n}n while the total energy remains to grow exponentially. In addition, we show that the distributions of the operator evolution and the energy density peaks are also useful characterizations to distinguish the heating phase from the exceptional points: the heating phase has both distributions to be continuous, while the exceptional points could support finite convex combinations of Dirac measures depending on their specific type. In the end, we compare the field theory results with the lattice model calculations for both the entanglement and energy evolution and find remarkably good agreement.

Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3‑dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth ‘spacetime harmonic functions’ in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.

Comment on the paper of A. K. Singal ‘Contribution of pressure to the energy-momentum density in a moving perfect fluid: A physical perspective.’

Comment on the paper of A. K. Singal ‘Contribution of pressure to the energy-momentum density in a moving perfect fluid: A physical perspective.’

Metric inertia for eddy densities of nonlocal matter-space

ABSTRACT Monistic thermomechanics of eddy mater-space with metric stresses of Maxwellian type and inelastic metric waves can be developed for purely kinetic densities of the nonlocal mass-energy integral. Pseudo-Riemannian 4-geometry uses locally warped time to preserve the Euclidean sublight transport of nonlocally correlated densities with instantaneous metric connections and coherent conservation of local 4-currents everywhere. The kinematic viscosity of geometrised energy-momentum densities and their geodesic self-pushes by correlated metric stresses quantitatively clarify the adaptive all-unity of continuous kinetic energy, including material ether or variable rest-energy. The metric self-organisation of matter-space+time implies a nonlocal feedback of correlated eddy densities and their tensor response to the vector density of external forces. Such tensor inertial responses of correlated metric stresses reveal the nonlocal nature of turbulence and predict adaptive auto-modes or thermal waves of Euclidean material space due to the modified geodesic equation with viscose autowaves and the inverse Cavendish constant. The proposed nonlocal alternative to Euler/Navier-Stokes hydrodynamics can distinguish between Cartesian and Newtonian worldviews in conceptual laboratory probes of new macroscopic mechanics for metric self-organisations of thermokinetic energies of viscose material space without negative gravitational energies.

Gravitational energy in Van Stockum space-time

The purpose of this paper is to illustrate the problem of energy and momentum distributions of Van Stockum space-time within the framework of two different theories of gravity, general relativity and teleparallel gravity. We have shown that for all homogeneous space-times with metric components $$g_{\mu \nu }$$ being functions of time variable, t, alone and independent of space variables the total gravitational energy for any finite volume is identically zero. By working with general relativity, we have calculated the energy-momentum density for Van Stockum space-time using double index complexes and in the framework teleparallel gravity, we used the energy-momentum complexes of Einstein, Bergmann–Thomson and Landau–Lifshitz. In our analysis, we sustained that general relativity and teleparallel gravity are equivalent theories of space-time under consideration. For space-time under consideration, we have shown that different complexes of energy-momentum density do not provide the same results neither in general relativity nor in teleparallel gravity.

Periodically, quasiperiodically, and randomly driven conformal field theories

In this paper and its sequel, we study non-equilibrium dynamics in driven 1+1D conformal field theories (CFTs) with periodic, quasi-periodic, and random driving. We study a soluble family of drives in which the Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength. The resulting time evolution is then captured by a M\"obius coordinate transformation. In this Part I, we establish the general framework and focus on the first two classes. In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement/energy evolution in different phases, i.e. the heating, non-heating phases and the phase transition between them. In quasi-periodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence. We find that (i) the non-heating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the non-heating fixed point, where the entanglement entropy/energy oscillate at the Fibonacci numbers, but grow logarithmically/polynomially at the non-Fibonacci numbers; (v) for certain choices of driving Hamiltonians, the non-heating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasi-crystal. In addition, another quasi-periodically driven CFT with an Aubry-Andr\'e like sequence is also studied. We compare the CFT results to lattice calculations and find remarkable agreement.

On the use of a SEBS polymer gel block as a new ballistic target to assess blunt ballistic impacts: Application to a wide range of LLKE projectiles

The objective of the present paper is to determine the ability of a transparent polymeric gel, used as a reference material with a mechanical response comparable to that of a human body, to assess less-lethal kinetic energy (LLKE) projectiles. To do this, a consistent measurement method of the dynamic deformation of the gel impacted by a projectile is first of all presented. It relies on high-speed images and image processing routines. Several metrics from the measured gel wall displacement profile are used to investigate experiments. Secondly, repeated impact conditions on various gel blocks confirm that, the proposed material meets the requirements of mechanical consistency, batch-to-batch reproducibility and easy handling. Thirdly, 138 ballistic experiments covering kinetic energies from 10 J up to 300 J are performed, involving commercially available LLKE projectiles of various stiffness. The important dataset is used to perform a statistical analysis through Spearman’s correlation matrix between metrics and projectile parameters. It mainly reveals the need to use a ballistic target to determine the wounding potential of a given projectile rather than kinetic energy density or momentum density. Robust power law mathematical fits describe the relations between three metrics (maximum gel displacement, displacement - rate of displacement-based metric and maximum displaced volume) and kinetic energy density depending on LLKE projectiles’ stiffness. Fourthly, three case reports available in literature and involving LLKE projectiles have been replicated on a gel block. An experimental procedure is proposed based on numerous case reports to give design guidelines in terms of projectile mass, velocity and stiffness. Finally, this original study demonstrates the gel material capabilities to assess LLKE weapons before their field deployment. However, further investigations need to be pursued to propose a relevant injury predictive tool and associated risk curves to law enforcement agencies and regulatory bodies.

Floquet conformal field theories with generally deformed Hamiltonians

In this work, we study non-equilibrium dynamics in Floquet conformal field theories (CFTs) in 1+1D, in which the driving Hamiltonian involves the energy-momentum density spatially modulated by an arbitrary smooth function. This generalizes earlier work which was restricted to the sine-square deformed type of Floquet Hamiltonians, operating within a \mathfrak{sl}_2𝔰𝔩2 sub-algebra. Here we show remarkably that the problem remains soluble in this generalized case which involves the full Virasoro algebra, based on a geometrical approach. It is found that the phase diagram is determined by the stroboscopic trajectories of operator evolution. The presence/absence of spatial fixed points in the operator evolution indicates that the driven CFT is in a heating/non-heating phase, in which the entanglement entropy grows/oscillates in time. Additionally, the heating regime is further subdivided into a multitude of phases, with different entanglement patterns and spatial distribution of energy-momentum density, which are characterized by the number of spatial fixed points. Phase transitions between these different heating phases can be achieved simply by changing the duration of application of the driving Hamiltonian. %In general, there are rich internal structures in the heating phase characterized by different numbers of spatial fixed points, which result in different entanglement patterns and distribution of energy-momentum density in space. %Interestingly, after each driving cycle, these spatial fixed points will shuffle to each other in the array, and come back to the original locations after pp (p\ge 1p≥1) driving cycles. We demonstrate the general features with concrete CFT examples and compare the results to lattice calculations and find remarkable agreement.

Contribution of Pressure to the Energy–Momentum Density in a Moving Perfect Fluid: A Physical Perspective

In the energy–momentum density expressions for a relativistic perfect fluid with a bulk motion, one comes across a couple of pressure-dependent terms, which though well known, are to an extent, lacking in their conceptual basis and the ensuing physical interpretation. In the expression for the energy density, the rest mass density along with the kinetic energy density of the fluid constituents due to their random motion, which contributes to the pressure as well, are already included. However, in a fluid with a bulk motion, there are, in addition, a couple of explicit, pressure-dependent terms in the energy–momentum density, whose presence to an extent, is shrouded in mystery, especially from a physical perspective. We show here that one such pressure-dependent term appearing in the energy density, represents the work done by the fluid pressure against the Lorentz contraction during transition from the rest frame of the fluid to another frame in which the fluid has a bulk motion. This applies equally to the electromagnetic energy density of electrically charged systems in motion and explains in a natural manner an apparently paradoxical result that the field energy of a charged capacitor system decreases with an increase in the system velocity. The momentum density includes another pressure-dependent term, that represents an energy flow across the system, due to the opposite signs of work being done by pressure on two opposite sides of the moving fluid. From Maxwell’s stress tensor we demonstrate that in the expression for electromagnetic momentum of an electric charged particle, it is the presence of a similar pressure term, arising from electrical self-repulsion forces in the charged sphere, that yields a natural solution for the notorious, more than a century old but thought by many as still unresolved, 4/3 problem in the electromagnetic mass.

Energy-Momentum Density of Non-Diagonal Bianchi Type Space-Time in General Relativity and Teleparallel Gravity

We calculated the energy-momentum density of non-diagonal Bianchi type space-time in two different theories of gravity, General relativity (GR) and the theory of Teleparallel gravity (TG). Firstly, by applying Einstein, Landau-Lifshitz, Bergmann-Thomson and Moller prescriptions, using double index complexes in GR. Secondly, in the frame work of TG, we used the energy momentum complexes of Einstein, Bergmann-Thomson and Landau-Lifshitz. We also study the spacial cases of non-diagonal Bianchi type space-time BII, BVIII and BIX. We obtained the same energy-momentum density components for Einstein and Bergmann-Thomson prescriptions for the above four mentioned space-times that we considered in our work. Also, we found that the energy density component in Moller prescription is zero for all Bianchi types space-times in GR. Furthermore, we show that if the metric components are functions of time t alone, then the total gravitational energy is identically zero.

Classical and quantal aspects of Minkowski's four-momentum in analog gravity

The electrodynamic theory of continuous media is probably the most convenient platform when trying to construct analog gravity theories. Quite naturally, this topic has gained considerable interest. One peculiar but not so very known feature in this context is the unconventional behavior of radiation energy and momentum in cases where superluminal fluid velocities are encountered, what, as known, is a major ingredient in analog gravity theories. These peculiar features are intimately connected with the spacelike character of Minkowski's four-momentum in electrodynamics. Here, we first consider an artificial model in which a Kerr-induced superluminal region is created in the right-hand region ($z>0$) in a left-moving, originally subluminal, fluid. We analyze the behavior of energy density, Poynting vector, and momentum density, and calculate the force on the artificial black hole horizon. Also, we delve into quantal aspects, looking for eventual production of particles associated with the sudden creation of the horizon, finding, however, that no particles are predicted to occur. The present paper continues a previous investigation by the author on the same topic, in Phys. Rev. A {\bf 100}, 032109 (2019). The subject as such is closely related to the famous Abraham-Minkowski problem.

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.

Strong spin–orbit interaction of photonic skyrmions at the general optical interface

Abstract Photonic skyrmions have applications in many areas, including the vectorial and chiral optics, optical manipulation, deep-subwavelength imaging and nanometrology. Much effort has been focused on the experimental characterization of photonic skyrmions. Here, we give an insight into the spin and orbital features of photonic skyrmions constructed by the p-polarized and s-polarized surface waves at an interface with various electric and magnetic properties by analyzing the continuity of chirality, energy flow and momentum densities through the electric and magnetic interface. The continuity of chirality density indicates that the photonic skyrmion has a property of the optical transverse spin. Most importantly, the continuity of energy flow and momentum densities results in four spin–orbit interaction quantities, which indicate the gradient of electric polarizability or permeability governs the spin–orbit interaction of photonic skyrmions and leads to the discontinuity and even the reversal of spin orientation through the optical interface. Our investigations on the spin–orbit properties of photonic skyrmions, which can give rise to the spin-dependent force and topological unidirectional transportation, is thorough and can be extended to other classical wave, such as acoustic and fluid waves. The findings help in understanding the spin–orbit feature of photonic topological texture and in constructing further optical manipulation, sensing, quantum and topological techniques.