Lagrangian Density Research Articles

Reformulation Complex Scalar Field Interacting With the Electromagnetic Lagrangian Density by Riemann- Liouville Factional Derivative

Reformulation Complex Scalar Field Interacting With the Electromagnetic Lagrangian Density by Riemann- Liouville Factional Derivative

Widened R<sup><i>n</i></sup> General Relativity

An extension of General Relativity is presented based on a scalar Lagrangian density which is a general function of all the independent invariant scalars that one is able to build by the powers of the Ricci tensor. It is shown how the new terms arising in the generalized Einstein filed equations may be interpreted as dark matter and dark energy contributions. Metricity condition fulfilled by a new tensor different than the usual metric tensor is also obtained. Moreover, it is shown that Schwarzschild-De Sitter, Robertson-Walker-De Sitter and Kerr-De Sitter metrics are exact solutions to the new field equations. Remarkably, the form of the equation of the geodesic trajectories of particle motions across space-time remains the same as in Einstein General Relativity unless the cosmological constant Lambda is no longer a constant becoming a function of the space-time co-ordinates.

The Lagrangian density of the disjoint union of a $3$-uniform tight path and a matching and the Turán number of its extension

The Lagrangian density of the disjoint union of a $3$-uniform tight path and a matching and the Turán number of its extension

FDTD Computation of Space/Time Integrated Electromagnetic Lagrangian: New Insights Into Design of Mutually Coupled Antennas

In this paper, we develop a formalism based on electromagnetic Lagrangian which provides new insights about the near-field reactive energy density around generic antennas for arbitrary spatio-temporal excitation signals. Using electric and magnetic fields calculated via FDTD technique and interpolation routines,we compute the EM Lagrangian density around antennas (thin-wire dipoles, Yagi-Uda Arrays and planar UWB Monopoles). Spatial maps of time-integrated EM Lagrangian density sheds light onto the capacitive/inductive nature of reactive energy density distribution around antennas and highlight its inherent connection with inter-element mutual coupling. Furthermore, we demonstrate that by minimizing the temporal fluctuations of spatially integrated EM Lagrangian density, it is possible to design MIMO antennas with wide impedance bandwidth and low mutual coupling.

Proving the Lorentz Invariance of the Entropy and the Covariance of Thermodynamics

The standard argument for the Lorentz invariance of the thermodynamic entropy in equilibrium is based on the assumption that it is possible to perform an adiabatic transformation whose only outcome is to accelerate a macroscopic body, keeping its rest mass unchanged. The validity of this assumption constitutes the very foundation of relativistic thermodynamics and needs to be tested in greater detail. We show that, indeed, such a transformation is always possible, at least in principle. The only two assumptions invoked in the proof are that there is at least one inertial reference frame in which the second law of thermodynamics is valid and that the microscopic theory describing the internal dynamics of the body is a field theory, with Lorentz invariant Lagrangian density. The proof makes no reference to the connection between entropy and probabilities and is valid both within classical and quantum physics. To avoid any risk of circular reasoning, we do not postulate that the laws of thermodynamics are the same in every reference frame, but we obtain this fact as a direct consequence of the Lorentz invariance of the entropy.

Statistics of thawing k -essence dark energy models

K-essence is a minimally-coupled scalar field whose Lagrangian density $\mathcal{L}$ is a function of the field value $\phi$ and the kinetic energy $X=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi$. In the thawing scenario, the scalar field is frozen by the large Hubble friction in the early universe, and therefore initial conditions are specified. We construct thawing k-essence models by generating Taylor expansion coefficients of $\mathcal{L}(\phi, X)$ from random matrices. From the ensemble of randomly generated thawing k-essence models, we select dark energy candidates by assuming negative pressure and non-growth of sub-horizon inhomogeneities. For each candidate model the dark energy equation of state function is fit to the Chevallier-Polarski-Linder parameterization $w(a) \approx w_0+w_a(1-a)$, where $a$ is the scale factor. The thawing k-essence dark models distribute very non-uniformly in the $(w_0, w_a)$ space. About 90\% models cluster in a narrow band in the proximity of a slow-roll line $w_a\approx -1.42 \left(\frac{\Omega_m}{0.3}\right)^{0.64}(1+w_0)$, where $\Omega_m$ is the present matter density fraction. This work is a proof of concept that for a certain class of models very non-uniform theoretical prior on $(w_0, w_a)$ can be obtained to improve the statistics of model selection.

Two models unifying warm inflation with dark matter and dark energy

Two models that unify warm inflation with dark matter and dark energy are proposed. In the models, a single scalar field is responsible for the early expansion of the universe through the process of dissipative warm inflation and then acts as both dark matter and dark energy in subsequent stages. The first model is based on a noncanonical field with the Lagrangian density $\mathcal{L}=F(X)-V(\phi)$, where the potential is dominant at the slow-roll inflationary epoch and negligible in subsequent stages. The second model takes advantage of a $k$-essence Lagrangian density having the coupled form $\mathcal{L}=F(X)V(\phi)$. For both models, equations of the evolution for the fields and observational constraints are presented, and an evolution law describing how the energy density $\rho$ and state parameter $w$ scale with the scale factor $a$ is obtained.

Derivation of the Symmetric Stress-Energy-Momentum Tensor in Exterior Algebra

We present a derivation of a manifestly symmetric form of the stress-energy-momentum using the mathematical tools of exterior algebra and exterior calculus, bypassing the standard symmetrizations of the canonical tensor. In a generalized flat space-time with arbitrary time and space dimensions, the tensor is found by evaluating the invariance of the action to infinitesimal space-time translations, using Lagrangian densities that are linear combinations of dot products of multivector fields. An interesting coordinate-free expression is provided for the divergence of the tensor, in terms of the interior and exterior derivatives of the multivector fields that form the Lagrangian density. A generalized Leibniz rule, applied to the variation of action, allows to obtain a conservation law for the derived stress-energy-momentum tensor. We finally show an application to the generalized theory of electromagnetism.

Neutron stars in f(mathtt {R,L_m}) gravity with realistic equations of state: joint-constrains with GW170817, massive pulsars, and the PSR J0030+0451 mass-radius from NICER data

In this work, we investigate neutron stars (NS) in f(mathtt {R,L_m}) theory of gravity for the case f(mathtt {R,L_m})= mathtt {R}+ mathtt {L_m}+ sigma mathtt {R}mathtt {L_m}, where mathtt {R} is the Ricci scalar and mathtt {L_m} the Lagrangian matter density. In the term sigma mathtt {R}mathtt {L_m}, sigma represents the coupling between the gravitational and particles fields. For the first time the hydrostatic equilibrium equations in the theory are solved considering realistic equations of state and NS masses and radii obtained are subject to joint constrains from massive pulsars, the gravitational wave event GW170817 and from the PSR J0030+0451 mass-radius from NASA’s Neutron Star Interior Composition Explorer (NICER) data. We show that in this theory of gravity, the mass-radius results can accommodate massive pulsars, while the general theory of relativity can hardly do it. The theory also can explain the observed NS within the radius region constrained by the GW170817 and PSR J0030+0451 observations for masses around 1.4~M_{odot }.

Unsettled Problems of Second-Order Quantum Theories of Elementary Particles

The physical community agrees that the variational principle is a cornerstone of a quantum fields theory (QFT) of an elementary particle. This approach examines the variation of the action of a Lagrangian density whose form is \(S = \int d^4 x \mathcal {L}(\psi,\psi_{,\mu}).\) The dimension of the action \(S\) and \(d^4x\) prove that the quantum function \(\psi\) of any specific Lagrangian density \(\mathcal {L}(\psi,\psi_{,\mu})\) has a definite dimension. This evidence determines the results of new consistency tests of QFTs. This work applies these tests to several kinds of quantum functions of a QFT of elementary particles. It proves that coherent results are derived from the standard form of quantum electrodynamics which depends on the Dirac linear equation of a massive charged particle and Maxwell theory of the electromagnetic fields. In contrast, contradictions stem from second-order quantum theories of an elementary particle, such as the Klein-Gordon equation and the electroweak theory of the \(W^\pm\) boson. An observation of the literature that discusses the latter theories indicates that they do not settle the above-mentioned crucial problems. This issue supports the main results of this work.

Nilpotent (anti-)BRST and (anti-)co-BRST symmetries in 2D non-Abelian gauge theory: Some novel observations

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-) co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two [Formula: see text]-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

Impact of gauge fixing on angular momentum operators of the covariantly quantized electromagnetic field

Covariant quantization of the electromagnetic field imposes the so-called gauge-fixing modification on the Lagrangian density. As a result of that, the total angular momentum operator receives at least one gauge-fixing-originated contribution, whose presence causes some confusion in the literature. The goal of this work is to discuss in detail why such a contribution, having no classical interpretation, is actually indispensable. For this purpose, we divide canonical and Belinfante-Rosenfeld total angular momentum operators into different components and study their commutation relations, their role in the generation of rotations of quantum fields, and their action on states from the physical sector of the theory. Then, we examine physical matrix elements of operators having gauge-fixing-related contributions, illustrating problems that one may encounter due to careless employment of the resolution of identity during their evaluation. The resolution of identity, in the indefinite-metric space of the covariantly quantized electromagnetic field, is extensively discussed because it takes a not-so-intuitive form if one insists on explicit projection onto states from the physical sector of the theory. Our studies are carried out in the framework of the Gupta-Bleuler theory of the free electromagnetic field. Relevant remarks about interacting systems, described by covariantly quantized electrodynamics, are given.

ModMax meets Susy

We give a prescription for mathcal{N} = 1 supersymmetrization of any (four-dimensional) nonlinear electrodynamics theory with a Lagrangian density satisfying a convexity condition that we relate to semi-classical unitarity. We apply it to the one-parameter ModMax extension of Maxwell electrodynamics that preserves both electromagnetic duality and conformal invariance, and its Born-Infeld-like generalization, proving that duality invariance is preserved. We also establish superconformal invariance of the superModMax theory by showing that its coupling to supergravity is super-Weyl invariant. The higher-derivative photino-field interactions that appear in any supersymmetric nonlinear electrodynamics theory are removed by an invertible nonlinear superfield redefinition.

Rotating black holes in general relativity coupled to nonlinear electrodynamics

We find an exact spherically symmetric magnetically charged black hole solution to general relativity (GR) coupled to nonlinear electrodynamics (NED) with an appropriate Lagrangian density. In turn, starting with this spherical black hole as a seed metric, we construct a rotating spacetime, a modification of Kerr black hole, using the revised Newman–Janis algorithm that depends on mass, spin, and a NED parameter g. We find an exact expression for thermodynamic quantities of the black holes like the mass, Hawking temperature, entropy, heat capacity, and free energy expressed in terms of horizon radius, and they show significant deviations from the Kerr case owing to NED. We also calculate analytical expressions for effective Komar mass and angular momentum for the rotating black hole and demonstrate that the Komar conserved quantity corresponding to the null Killing vector at horizon obeys Kχ=2S+T+. The radiating counterpart renders a generalization of Carmeli’s spacetime as well as Vaidya’s spacetime in the appropriate limits.

Soliton dynamics in the ABS nonlinear spinor model with external fields

We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva et al (Ann. Phys., NY 403 198) (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a collective coordinates (CCs) theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the CCs. This Lagrangian differs only in the charge and mass with the Lagrangian of a CCs theory for the Gross–Neveu equation. Thus the soliton dynamics in the Alexeeva–Barashenkov–Saxena (ABS) spinor model is qualitatively the same as in the Gross–Neveu equation, but quantitatively it differs. These results of the CCs theory are confirmed by simulations, i.e. by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials.

An Exterior Algebraic Derivation of the Euler–Lagrange Equations from the Principle of Stationary Action

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh–Plesset Cavitation Dynamics

The following article extends a decomposition to the Navier–Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on the existence of solutions relative to the NSEs and to identify a potential manner of investigation into the classical Millennial Problem encompassed in Fefferman’s presentation. The article also presents the utilization of a recently developed versatile variational framework by both authors in order to study a related fluid-mechanics phenomena, namely the Raleigh–Plesset equations, which are ultimately obtained from the NSEs. The article develops, for the first time, a Lagrangian density functional for a closed surface which when minimized produced the Raleigh–Plesset equations. The article then proceeds with the demonstration that the Raleigh–Plesset equations may be obtained from this energy functional and identifies the energy dissipation predicted by the proposed Lagrangian density. The importance of the novel Raleigh–Plesset functional in the greater scheme of fluid mechanics is commented upon.

On the Derivation of Multisymplectic Variational Integrators for Hyperbolic PDEs Using Exponential Functions

We investigated the derivation of numerical methods for solving partial differential equations, focusing on those that preserve physical properties of Hamiltonian systems. The formulation of these properties via symplectic forms gives rise to multisymplectic variational schemes. By using analogy with the smooth case, we defined a discrete Lagrangian density through the use of exponential functions, and derived its Hamiltonian by Legendre transform. This led to a discrete Hamiltonian system, the symplectic forms of which obey the conservation laws. The integration schemes derived in this work were tested on hyperbolic-type PDEs, such as the linear wave equations and the non-linear seismic wave equations, and were assessed for their accuracy and the effectiveness by comparing them with those of standard multisymplectic ones. Our error analysis and the convergence plots show significant improvements over the standard schemes.

Searches for beyond-Riemann gravity

Many effective field theories describing gravity cannot arise from an underlying theory based on Riemann geometry or its extensions to include torsion and nonmetricity but may instead emerge from another geometry or may have a nongeometric basis. The Lagrange density for a broad class of such theories is investigated. The action for fermions coupled to gravity is linearized about a Minkowski background and is found to include terms describing small deviations from Lorentz invariance and gravitational gauge invariance. The corresponding nonrelativistic hamiltonian is derived at second order in the fermion momentum. The implications for laboratory experiments and astrophysical observations with fermions are studied, with primary focus on anomalous spin-gravity couplings. First constraints on some coefficients are extracted from existing data obtained via measurements at different potentials, comparisons of gravitational accelerations, interferometric methods, and investigations of gravitational bound states. Some prospects for future experimental studies are discussed.

Three-body inertia tensor

We derive a general formula for the inertia tensor of a rigid body consisting of three particles with which students can learn basic properties of the inertia tensor without calculus. The inertia-tensor operator is constructed by employing the Dirac’s bra-ket notation to obtain the inertia tensor in an arbitrary frame of reference covariantly. The principal axes and moments of inertia are computed when the axis of rotation passes the center of mass. The formulas are expressed in terms of the relative displacements of particles that are determined by introducing Lagrange’s undetermined multipliers. This is a heuristic example analogous to the addition of a gauge-fixing term to the Lagrangian density in gauge field theories. We confirm that the principal moments satisfy the perpendicular-axis theorem of planar lamina. Two special cases are considered as pedagogical examples. One is a water-molecule-like system in which a particle is placed on the vertical bisector of two identical particles. The other is the case in which the center of mass coincides with the incenter of the triangle whose vertices are placed at the particles. The principal moment of the latter example about the normal axis is remarkably simple and proportional to the product ‘abc’ of the three relative distances. We expect that this new formula can be used in actual laboratory classes for general physics or undergraduate classical mechanics.