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In this work, perfect fluid spacetimes obeying Gray’s decompositions have been studied. In the first part, after a brief summary about Gray’s decompositions used in this study, we work on properties of perfect fluid spacetimes obeying these decompositions. The results are obtained for different subspaces of a perfect fluid spacetime. Thereafter, we present the characterizations of these spacetimes by proving some theorems. [ABSTRACT FROM AUTHOR]

Within the scope of a Bianchi type-I (BI) cosmological model, we study the interacting system of spinor and electromagnetic fields and its role in the evolution of the Universe. In some earlier studies, it was found that in the case of a pure spinor field, the presence of nontrivial nondiagonal components of energy–momentum tensor (EMT) leads to some severe restrictions both on the space–time geometry and/or spinor field itself, whereas in the case of electromagnetic field with induced nonlinearity, such components impose severe restrictions on metric functions and the components of the vector potential. It is shown that in the case of interacting spinor and electromagnetic fields, the restrictions are not as severe as in the other cases and in this case, a nonlinear and massive spinor field with different components of vector potential can survive in a general Bianchi type-I space–time. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study geometrical aspects of an almost Ricci soliton on (wRs)4 perfect fluid spacetime obeying Einstein’s field equation. Among others, we first find the soliton constant and cosmological constant in terms of scalar curvature and potential vector field in (wRs)4 perfect fluid spacetime. Next, we discuss some physical phenomena related to dust fluid, dark fluid, and radiation era in (wRs)4 perfect fluid spacetime admitting an almost Ricci soliton with potential vector field as solenoidal vector field and basic vector field ρ under matter collineation condition. Further, we find an inequality for soliton constant when (wRs)4 perfect fluid spacetime obeys the timelike convergence condition. Finally, we obtain some results in a (wRs)4 perfect fluid spacetime whose metric represents an almost Ricci soliton when basic vector field and potential vector field both are torseforming vector field ρ. Also, for such spacetime we find the soliton constant in terms of cosmological constant, gravitational constant, energy density, and isotropic pressure. We also provide an example of (wRs)4 spacetime whose metric represents an almost Ricci soliton. [ABSTRACT FROM AUTHOR]

In this work, generalized weakly ℋ -symmetric space-times (GWHS) n are investigated, where ℋ is any symmetric (0 , 2) tensor. It is proved that, in a nontrivial (GWHS) n space-time, the tensor ℋ has a perfect fluid form. Accordingly, sufficient conditions for a nontrivial generalized weakly Ricci symmetric space-time (GWRS) n to be either an Einstein space-time or a perfect fluid space-time are obtained. Also, conditions for space-times admitting either a generalized weakly symmetric energy-momentum tensor or a generalized weakly symmetric tensor to be Einstein or perfect fluid space-times are provided. [ABSTRACT FROM AUTHOR]

We examine the basic matrix equation for a spin-2 field in spherical coordinates and a tetrad of Minkowski spacetime. The technique of Wigner D-functions is used to separate the variables. The operators of energy, the square and the third projection of the total angular momentum, and the space reflection operators are then diagonalized on the solutions. The separation of the variables is done, the problem reduces to 11 radial second-order differential equations for 11 variables related to the scalar and symmetric tensors. We then consider the case of a massless field. The four gauge solutions for the massless spin-2 field have been found in explicit form. These gauge solutions do not contribute to physically observable quantities such as the energy–momentum tensor. They are derived according to Pauli–Fierz theory from four independent spherical solutions for a spin-1 massless particle. We have also constructed two types of solutions, related to opposite parities, which are not the gauge solutions. These solutions describe physically observable states with spherical symmetry for the spin-2 massless field. [ABSTRACT FROM AUTHOR]

This article explores the differences between frame and coordinate transformations in relativistic theories. We highlight the key role of tetrad fields in connecting spacetime and frame indices. Using Maxwell’s electrodynamics as an example, we show that Maxwell’s equations are invariant under coordinate transformations but exhibit covariant behavior under frame transformations. We also analyze the energy-momentum of an electromagnetic field in different frames, providing deeper insights into the implications of different frames of reference and coordinate systems.

The primary goal of this paper is to examine spacetimes admitting semiconformal curvature tensor in f (ℛ) modify gravity. The semiconformal flatness of general spacetime and spacetime in f (ℛ) gravity with perfect fluid, has been analyzed. For this consideration, we generate the forms of isotropic pressure p and energy density σ. After that, a few energy conditions are taken into account. Finally, we study the divergence-free semiconformal curvature tensor in f (ℛ) gravity in presence of perfect fluid. We emphasize that for recurrent or bi-recurrent energy–momentum tensor, Ricci tensor of this spacetime is semi-symmetric and consequently, the resulting spacetimes either accomplish inflation or possess fixed isotropic pressure and energy density. [ABSTRACT FROM AUTHOR]

We prove a theorem on scalar‐valued functions of tensors, where "scalar" refers to absolute scalars as well as relative scalars of weight w. The present work thereby generalizes an identity referred to earlier by Rosenfeld in his publication "On the energy–momentum tensor". The theorem provides a (1,1)‐tensor identity which can be regarded as the tensor analogue of the identity following from Euler's theorem on homogeneous functions. The remarkably simple identity is independent of any internal symmetries of the constituent tensors, providing a powerful tool for deriving relations between field‐theoretical expressions and physical quantities. We apply the identity especially for analyzing the metric and canonical energy–momentum tensors of matter and gravity and the relation between them. Moreover, we present a generalized Einstein field equation for the arbitrary version of vacuum space–time dynamics—including torsion and non‐metricity. The identity allows to formulate an equivalent representation of this equation. Thereby the conjecture of a zero‐energy universe is confirmed. [ABSTRACT FROM AUTHOR]

We have investigated a model of colliding plain gravitational waves, proposed by Szekeres, whose structure of singularities is determined. We have evaluated a total energy of matter as a volume integral of the energy–momentum tensor (EMT), whose contributions arise only at these singularities. The total matter energy is conserved before a collision of two plane gravitational waves but decreases during the collision and becomes zero at the end of the collision. We thus interpret that this model of colliding plane gravitational waves is a space–time describing a pair annihilation of plan gravitational waves. We have also calculated a matter conserved charge proposed by the present author and his collaborators. The matter charge is indeed conserved but is zero due to a cancelation between two plain gravitational waves. This seems natural since nothing remains after a pair annihilation, and gives a hint on a physical interpretation of the conserved charge, which we call the gravitational charge. By modifying the space–time for the pair annihilation, we newly construct two types of a scattering plane gravitational wave and a pair creation of plane gravitational waves, and combining all, a Minkowski vacuum bottle, a Minkowski space–time surrounded by two moving plane gravitational waves. [ABSTRACT FROM AUTHOR]

The semiconformal curvature tensor has been studied for the spacetime of general relativity. It is shown that the energy-momentum tensor with divergence-free semiconformal curvature tensor is of Codazzi type, as well as the energy-momentum tensor of a spacetime having semi-symmetric semiconformal curvature tensor is also semi-symmetric. The semiconformal curvature tensor has also been expressed in terms of different tensors already known in the literature, and the relationship between their divergences has been established. [ABSTRACT FROM AUTHOR]

Within the scope of a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model we study the role of a nonlinear spinor field in the evolution of the universe. In doing so, we exploit the FLRW models given in both Cartesian and spherical coordinates. It is found that if the FLRW model is given in the spherical coordinates the energy-momentum tensor (EMT) of the spinor field possesses nontrivial non-diagonal components, which is not the case for Cartesian coordinates. These non-diagonal components do not depend on either the spinor field nonlinearity or the parameter k that defines the type of curvature of the FLRW model. The presence of such components imposes some restrictions on the spinor field. The problem is studied for open, flat and close geometries and the spinor field is used to simulate different types of sources including dark energies. Some qualitative numerical solutions are given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We reconsider four-dimensional scalar field theory in presence of Robin boundary conditions on two parallel plates. These boundary conditions are directly imposed in the path integral definition of the theory via auxiliary fields living on the plates. We discuss how this leads to boundary corrections to the standard energy momentum tensor operator. Via a dimensional reduction to an effective three-dimensional boundary theory, we compute the Casimir energy in terms of the plate separation and the two Robin parameters, as well as the scalar field propagator in the presence of the plates. Coincidentally, the boundary contribution vanishes in the expectation value for the vacuum energy, thereby giving results in full accordance with other energy expressions in the literature for the same setup. We also discuss for which values of the Robin parameters this energy is real-valued. [ABSTRACT FROM AUTHOR]

In the present article, some geometric and physical properties of M G (Q E) n were investigated. Moreover, general relativistic viscous fluid M G (Q E) 4 spacetimes with some physical applications were studied. Finally, through a non-trivial example of M G (Q E) 4 spacetime, we proved its existence. [ABSTRACT FROM AUTHOR]

We consider an extended theory of gravity with Lagrangian ℒ = f (R , T (n)) , with T (n) being a 2 n th-order invariant made of contractions of the energy–momentum tensor. When n = 1 , this theory reduces to f (R , T) gravity, where T accounts for the trace of the energy–momentum tensor. We study the gravitational wave polarization modes, from which it results that when the matter Lagrangian contains dynamical scalar fields minimally coupled to the geometry, further polarization modes arise with respect to General Relativity. Finally, we show that the motion for test particles is nongeodesic and we explicitly obtain the extra force. [ABSTRACT FROM AUTHOR]

In quantum theory, for a system with macroscopic wavefunction, the charge density and current density are represented by non-commuting operators. It follows that the anomaly I=∂tρ+∇·j, being essentially a linear combination of these two operators in the frequency-momentum domain, does not admit eigenstates and has a minimum uncertainty fixed by the Heisenberg relation ΔNΔϕ≃1, which involves the occupation number and the phase of the wavefunction. We give an estimate of the minimum uncertainty in the case of a tunnel Josephson junction made of Nb. Due to this violation of the local conservation of charge, for the evaluation of the e.m. field generated by the system it is necessary to use the extended Aharonov–Bohm electrodynamics. After recalling its field equations, we compute in general form the energy–momentum tensor and the radiation power flux generated by a localized oscillating source. The physical requirements that the total flux be positive, negative or zero yield some conditions on the dipole moment of the anomaly I.

An important issue in phenomenological macroscopic electrodynamics of moving media is the definition of the energy and momentum of the electromagnetic field in matter. Rather surprisingly, this topic has demonstrated a remarkable longevity, and the problem of the electromagnetic energy and momentum in matter remained open, despite numerous theoretical and experimental investigations. We overview the definition of the momentum of light in matter and demonstrate that, for the correct understanding of the problem, one needs to carefully distinguish situations when the material medium is modeled either as a background for light or as a dynamical part of the total system. The status of Minkowski and Abraham energy-momentum tensors of the electromagnetic field is clarified for the two particular types of complex matter, the spinning fluid and the liquid crystal medium, and summarized for the case of general anisotropic moving material media with a linear constitutive law. [ABSTRACT FROM AUTHOR]

The present paper aims to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field ξ in connection with conformal Ricci–Yamabe metric and conformal η -Ricci–Yamabe metric. We delineate the conditions for conformal Ricci–Yamabe soliton to be expanding, steady or shrinking. We also discuss conformal Ricci–Yamabe soliton on some special types of perfect fluid spacetime such as dust fluid, dark fluid and radiation era. Furthermore, we design conformal η -Ricci–Yamabe soliton to find its characteristics in a perfect fluid spacetime and lastly acquired Laplace equation from conformal η -Ricci–Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type. Overall, the main novelty of the paper is to study the geometrical phenomena and characteristics of our newly introduced conformal Ricci–Yamabe and conformal η -Ricci–Yamabe solitons to apply their existence in a perfect fluid spacetime. [ABSTRACT FROM AUTHOR]

The main object of this paper is to investigate spacetimes admitting concircular curvature tensor in f(R) gravity theory. At first, concircularly flat and concircularly flat perfect fluid spacetimes in fR gravity are studied. In this case, the forms of the isotropic pressure p and the energy density σ are obtained. Next, some energy conditions are considered. Finally, perfect fluid spacetimes with divergence free concircular curvature tensor in f(R) gravity are studied; amongst many results, it is proved that if the energy-momentum tensor of such spacetimes is recurrent or bi-recurrent, then the Ricci tensor is semi-symmetric and hence these spacetimes either represent inflation or their isotropic pressure and energy density are constants.

In this paper, we study M-projective curvature tensors on an (LCS) 2 ⁢ n + 1 {(\mathrm{LCS})_{2n+1}} -manifold. Here we study M-projectively Ricci symmetric and M-projectively flat admitting spacetime. [ABSTRACT FROM AUTHOR]

In quantum theory, for a system with macroscopic wavefunction, the charge density and current density are represented by non-commuting operators. It follows that the anomaly I = tr + r - j, being essentially a linear combination of these two operators in the frequencymomentum domain, does not admit eigenstates and has a minimum uncertainty fixed by the Heisenberg relation DNDf '1, which involves the occupation number and the phase of the wavefunction. We give an estimate of the minimum uncertainty in the case of a tunnel Josephson junction made of Nb. Due to this violation of the local conservation of charge, for the evaluation of the e.m. field generated by the system it is necessary to use the extended Aharonov-Bohm electrodynamics. After recalling its field equations, we compute in general form the energy-momentum tensor and the radiation power flux generated by a localized oscillating source. The physical requirements that the total flux be positive, negative or zero yield some conditions on the dipole moment of the anomaly I. [ABSTRACT FROM AUTHOR]

The electron-graviton interaction can be described in terms of the gravitational form factors of the QED energy-momentum tensor. Here we focus on the form factor D(t), and we examine its properties and its interpretation in terms of internal forces at one-loop accuracy in QED. We perform the calculation by keeping separate the contributions due to the electron and the photon parts of the energy-momentum tensor. We also study the case of a nonzero photon mass. Furthermore, we discuss similarities with and differences to the form factor D(t) of hadronic bound states.

The first order variation of the matter energy–momentum tensor T μ ν with respect to the metric tensor g α β plays an important role in modified gravity theories with geometry-matter coupling, and in particular in the f (R , T) modified gravity theory. We obtain the expression of the variation δ T μ ν / δ g α β for the baryonic matter described by an equation given in a parametric form, with the basic thermodynamic variables represented by the particle number density, and by the specific entropy, respectively. The first variation of the matter energy–momentum tensor turns out to be independent on the matter Lagrangian, and can be expressed in terms of the pressure, the energy–momentum tensor itself, and the matter fluid four-velocity. We apply the obtained results for the case of the f (R , T) gravity theory, where R is the Ricci scalar, and T is the trace of the matter energy–momentum tensor, which thus becomes a unique theory, also independent on the choice of the matter Lagrangian. A simple cosmological model, in which the Hilbert–Einstein Lagrangian is generalized through the addition of a term proportional to T n is considered in detail, and it is shown that it gives a very good description of the observational values of the Hubble parameter up to a redshift of z ≈ 2. 5. [ABSTRACT FROM AUTHOR]

In this paper, we consider conformally symmetric generalized Riccirecurrent manifolds. We prove that such a manifold is a quasi-Einstein manifold and study its geometric properties. Also, we obtain several interesting results. Among others, the universal cover of this manifold splits geometrically as L¹xNn−1 , where L is a line, (Nn−1, g Nn−1 ) is Einstein, ϕ = − 1/n-r. Moreover, we demonstrate the applications of the conformally symmetric generalized Ricci-recurrent spacetime with non-zero constant scalar curvature in the theory of general relativity. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study space-time admitting generalized projective curvature tensor. [ABSTRACT FROM AUTHOR]

We present a precise definition of a conserved quantity from an arbitrary covariantly conserved current available in a general curved space–time with Killing vectors. This definition enables us to define energy and momentum for matter by the volume integral. As a result we can compute charges of Schwarzschild and BTZ black holes by the volume integration of a delta function singularity. Employing the definition we also compute the total energy of a static compact star. It contains both the gravitational mass known as the Misner–Sharp mass in the Oppenheimer–Volkoff equation and the gravitational binding energy. We show that the gravitational binding energy has the negative contribution at maximum by 68% of the gravitational mass in the case of a constant density. We finally comment on a definition of generators associated with a vector field on a general curved manifold. [ABSTRACT FROM AUTHOR]

In this paper, the solutions of the Gauss-Bonnet theory of gravity was proposed in the presence of a perfect fluid with thermodynamic pressure P and energy gravity ρ in n-dimension. Accordingly, perfect fluid tensor was regarded as energy-momentum tensor and the static and radiating solutions for the linear equation of state p = wρ was calculated. This solution contains all the solutions already being proposed for Gauss-Bonnet gravity, such as asymptotically flat or asymptotically uncharged (anti)-de Sitter, radiating asymptotically flat and (anti)-de Sitter solutions as well as some new static and radiating solutions for different w values. We go through the properties of the new static and radiating solutions. We find solutions for w = 0 and w = (n-2)-2 in static and radiating states. We also show that the solutions satisfy the dominant and weak energy conditions and they present black holes with one or two horizons or naked singularities provided that the parameters of the solutions are chosen suitable.

The flexoelectricity, which is a new electromechanical coupling phenomenon between strain gradients and electric polarization, has a great influence on the fracture analysis of flexoelectric solids due to the large gradients near the cracks. On the other hand, although the flexoelectricity has been extensively investigated in recent decades, the study on flexoelectricity in nonhomogeneous materials is still rare, especially the fracture problems. Therefore, in this manuscript, the conservation integrals for nonhomogeneous flexoelectric materials are obtained to solve the fracture problem. Application of operators such as grad, div, and curl to electric Gibbs free energy and internal energy, the energy-momentum tensor, angular momentum tensor, and dilatation flux can also be derived. We examine the correctness of the conservation integrals by comparing with the previous work and discuss the operator method here and Noether theorem in the previous work. Finally, considering the flexoelectric effect, a nonhomogeneous beam problem with crack is solved to show the application of the conservation integrals. [ABSTRACT FROM AUTHOR]

In this paper, we have studied space-time with W4 -curvature tensor and proved that a 4-dimensional relativistic W4 -flat space-time satisfying Einstein's field equation with cosmological constant, the energy-momentum tensor is covariant constant. It is also observed that in a 4-dimensional relativistic space-time M has conservative W4 - curvature tensor if and only if the energy momentum tensor is Codazzi tensor provided that the scalar curvature is constant in both the cases. [ABSTRACT FROM AUTHOR]

Here, we have investigated the interaction of Bianchi type-I anisotropic cloud string cosmological model universe with electromagnetic field in the context of general relativity. In this paper, the energy-momentum tensor is assumed to be the sum of the rest energy density and string tension density with an electromagnetic field. To obtain exact solution of Einstein's field equations, we take the average scale factor as an integrating function of time. Also, the dynamics and significance of various physical parameters of model are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the effect of backreaction on the scale factor of the Friedmann–Lemaître–Robertson–Walker (FLRW) and de Sitter spaces. We consider a non-minimally coupled massive scalar field to the curvature scalar. For our purpose, we use the results of vacuum expectation values of energy–momentum tensor, which have been obtained previously. By substituting the quantum energy density into the Friedmann equation, we obtain the linear order perturbation of the scale factor. So, the effect of backreaction leads to the new scale factor. [ABSTRACT FROM AUTHOR]

By analyzing the energy-momentum tensor and equations of state of ideal gas, scalar, spinor and vector potential in detail, we find that the total mass density of all matter is always positive, and the initial total pressure is negative. Under these conditions, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: (i) K=1, namely, the global spatial structure of the universe should be a three-dimensional sphere S3; (ii) 0≤Λ<10−24ly−2, the cosmological constant should be zero or an infinitesimal; (iii) a(t)>0, the initial singularity of the universe is unreachable, and the evolution of the universe should be cyclic in time. Since the matter components considered are quite complete and the proof is very elementary and strict, these conclusions are quite reliable in logic and compatible with all observational data. Obviously, these conclusions will be very helpful to correct some popular misconceptions and bring great convenience to further research other problems in cosmology such as the properties of dark matter and dark energy. In addition, the macroscopic Lagrangian of fluid model is derived.

We discuss in detail the spatial distribution of angular momentum inside the nucleon. We show that the discrepancies between different definitions originate from terms that integrate to zero. Even though these terms can safely be dropped at the integrated level, they have to be taken into account when discussing distributions. Using the scalar diquark model, we illustrate our results and, for the first time, check explicitly that the equivalence between kinetic and canonical orbital angular momentum persists at the level of distributions, as expected in a system without gauge degrees of freedom.

By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of the Dirac equation, the energy-momentum tensor of spinors and so on. To split the spinor connection into the Keller connection Υμ∈Λ1 and the pseudo-vector potential Ωμ∈Λ3 not only makes the calculation simpler, but also highlights their different physical meanings. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton’s second law of particles. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields; however, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. This uncertainty increases the difficulty of deriving rigorous expression. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table Sabμν, which plays an important role in the interaction between spinor and gravity. From this paper we find that Clifford algebra has irreplaceable advantages in the study of geometry and physics.

The structure of the matrix elements of the energy–momentum tensor play an important role in determining the properties of the form factors A(q2), B(q2) and C(q2) which appear in the Lorentz covariant decomposition of the matrix elements. In this paper we apply a rigorous frame-independent distributional-matching approach to the matrix elements of the Poincaré generators in order to derive constraints on these form factors as q→0. In contrast to the literature, we explicitly demonstrate that the vanishing of the anomalous gravitomagnetic moment B(0) and the condition A(0)=1 are independent of one another, and that these constraints are not related to the specific properties or conservation of the individual Poincaré generators themselves, but are in fact a consequence of the physical on-shell requirement of the states in the matrix elements and the manner in which these states transform under Poincaré transformations.