Set Of Field Equations Research Articles

Scattering theory with matter fields of classical gravitons in the null surface formulation

Using a set of field equations in the null surface formulation we obtain the linearized coupling between the gravitational and matter fields. We first derive a formula for the metric of the space time and then we use this formula to study the scattering of incoming gravitational waves when matter is present, obtaining explicit formulae relating the radiation modes at past and future null infinity for a general asymptotically flat spacetime. An example application is made at the end of this work when the matter field is a massless real scalar field. The relevance of this result for a perturbation procedure is discussed.

STABILITY ANALYSIS OF BIOCONVECTIVE HEAT TRANSFER IN A SUSPENSION OF GYROTACTIC MICROORGANISMS UNDER VERTICAL VIBRATION

Vertical vibration is an important technique to reduce or increase the bioconvective heat transfer. It has great applications in biotechnology, biomedicine, enhancing oil recovery, mixing in microvolumes, analysis of whole body vibration (WBV), and industry. The present study aims to investigate the influence of vertical vibration on the initiation of thermo-bioconvection in a suspension of randomly swimming gyrotactic microorganisms in a shallow thermally stratified horizontal fluid layer. The time-averaged method, linear stability analysis, and single-term Galerkin technique with different boundary layers are employed on the set of field equations. A secular equation is obtained in terms of vibrational and bioconvectional parameters for rigid-rigid and rigid-free boundary conditions. The analysis reported that vertical vibration destabilizes the suspension due to randomly swimming of gyrotactic microorganisms, i.e., enhancing the rate of bioconvective heat transfer. The authors also found that pertinent controlling parameters influence the size of the cell of vibrational thermo-bioconvection. It is observed that the diffusion and average volume of randomly swimming microorganisms affect the bioconvective heat transfer of the vibrational system inversely.

Class I stars in the background of [formula omitted] cosmological model

The present paper is designed to study the structure of some compact objects such as PSRJ1614−2230,LMXB4U1608−52,CenX−3,EXO1785−248 and SMCX−1 in the framework of f(T,τ) gravity, where τ and T represent the trace of energy–momentum tensor and torsion scalar, respectively. For this work, we use anisotropic fluid distribution filled in spherical symmetric geometry providing the interior of star model and utilize the off-diagonal tetrad components for deriving the corresponding set of field equations. We construct stellar structure by using a well-known model of f(T,τ) gravity, given by f(T,τ)=αT(r)2+βτ(r)+ϕ, where α, β and ϕ are arbitrary constants. This model has interesting cosmological properties, in addition to providing an effective dark energy sector explanation (Harko et al., 2014). For checking viability of the constructed model, we set α<0, β<0 and discuss physical and stability features like anisotropy, energy conditions, EoS parameters, TOV equation, redshift, compactness and mass function of strange stars. It is found that these features comprehensively match with the realistic nature of compact stars in the realm of f(T,τ) gravity.

Study of anisotropic strange stars in Krori Barua metric under [formula omitted] gravity

In this paper, we focused on the study of anisotropic strange stars with anisotropic matter distribution in the framework of f(T,T)=αT(r)2+βT(r) model, where α, β are arbitrary constants, T and T represent the torsion scalar and trace of energy–momentum tensor, respectively. This model has interesting cosmological properties, in addition to providing an effective dark energy sector explanation (Harko et al., 2014). In order to achieve our goal, we considered the Krori–Barua space–time and utilize the off-diagonal tetrad components for deriving the corresponding set of field equations. We calculate the relation for the evaluation of effective density, effective radial and tangential pressure using the EoS of strange star i.e., MIT bag model pr(r)=13[ρ(r)−4Bg]. The unknown constants appearing in the solution have been evaluated by using the conventional matching of interior and exterior space time. To check the feasibility of our proposed model we discuss the anisotropy, energy condition, TOV forces, abreu condition, EoS parameters, adiabatic index, mass function, compactness and redshift for different compact stars models namely, PSRJ1416−2230, 4U1608−52, CenX−3, EXO1785−248 and SMCX−1.

Killing superalgebras for lorentzian five-manifolds

We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an mathfrak{sp} (1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds. We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations. The maximally supersymmetric backgrounds are characterised and their Killing superalgebras are explicitly described as filtered deformations of the Poincaré superalgebra.

Variational Principles in Teleparallel Gravity Theories

We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that, in addition to the tetrad or metric, they employ a flat connection as additional field variable, but dthey iffer by the presence of absence of torsion and nonmetricity for this independent connection. Besides the different underlying geometric formulation using a tetrad or metric as fundamental field variable, one has different choices to introduce the conditions of vanishing curvature, torsion, and nonmetricity, either by imposing them a priori and correspondingly restricting the variation of the action when the field equations are derived, or by using Lagrange multipliers. Special care must be taken, since these conditions form non-holonomic constraints. Here, we explicitly show that all of the aforementioned approaches are equivalent, and that the same set of field equations is obtained, independently of the choice of the geometric formulation and variation procedure. We further discuss the consequences arising from the diffeomorphism invariance of the gravitational action, and show how they establish relations between the gravitational field equations.

Non-relativistic limit of gravity theories in the first order formalism

We consider the non-relativistic limit of gravity in four dimensions in the first order formalism. First, we revisit the case of the Einstein-Hilbert action and formally discuss some geometrical configurations in vacuum and in the presence of matter at leading order. Second, we consider the more general Mardones-Zanelli action and its non-relativistic limit. The field equations and some interesting geometries, in vacuum and in the presence of matter, are formally obtained. Remarkably, in contrast to the Einstein-Hilbert limit, the set of field equations is fully determined because the boost connection appears in the action and field equations. It is found that the cosmological constant must disappear in the non-relativistic Mardones-Zanelli action at leading order. The conditions for Newtonian absolute time be acceptable are also discussed. It turns out that Newtonian absolute time can be safely implemented with reasonable conditions.

An anisotropic version of Tolman VII solution in f(R, T) gravity via gravitational decoupling MGD approach

In this work, we have adopted gravitational decoupling by Minimal Geometric Deformation (MGD) approach and have developed an anisotropic version of well-known Tolman VII isotropic solution in the framework of $f(R, T)$ gravity, where $R$ is Ricci scalar and $T$ is trace of energy momentum tensor. The set of field equations has been developed with respect to total energy momentum tensor, which combines effective energy momentum tensor in $f(R,T)$ gravity and additional source $\phi_{ij}$. Following MGD approach, the set of field equations has been separated into two sections. One section represents $f(R, T)$ field equations, while the other is related to the source $\phi_{ij}$. The matching conditions for inner and outer geometry have also been discussed and an anisotropic solution has been developed using mimic constraint for radial pressure. In order to check viability of the solution, we have considered observation data of three different compact star models, named PSR J1614-2230, PSR 1937+21 and SAX J1808.4-3658 and have discussed thermodynamical properties analytically and graphically. The energy conditions are found to be satisfied for the three compact stars. The stability analysis has been presented through causality condition and Herrera's cracking concept, which ensures physical acceptability of the solution.

Some Bianchi I dark energy models in Brans–Dicke theory

The present article deals with the study of interacting and non-interacting dark energy (DE) and dark matter (DM) in the spatially homogeneous and anisotropic Bianchi I space–time within the framework of Brans–Dicke (BD) scalar–tensor theory of gravitation. As the set of field equations is not closed, exact solutions are obtained using power-law relation and assuming a linearly varying deceleration parameter. The physical acceptability and stability of the obtained model are scrutinised using energy conditions and squared speed of sound. The statefinder diagnostic method is adopted to discuss and measure the deviation of the considered model from the $$\Lambda $$ cold dark matter ( $$\Lambda $$ CDM) model.

Matrix Algorithm of Approximate Solution of Wave Equations in Inhomogeneous Magnetoactive Plasma

The problem of propagation of an electromagnetic wave in plane-stratified magnetoactive plasma is analyzed. A matrix algorithm of approximate solution of a set of wave equations is proposed. The algorithm consists in successive finding of the medium-inhomogeneity-induced corrections to the local roots of the dispersion relation and local polarization vectors. The set of field equations is reduced to a set of algebraic equations. The proposed algorithm is convenient for numerical calculations. In contrast to the classical geometrical-optics approximation, the proposed algorithm allows one to take into account the weak effect of linear mode interaction. Examples of numerical calculations of the power reflection coefficient of whistler waves incident on the ionosphere from above are presented. The proposed matrix algorithm can be useful to find the coefficients of reflection and linear transformation of waves in a smoothly inhomogeneous ionosphere.

Theory of Electro Relativity

A unified field theory is created by using a simple dipole antenna to model the Earth’s electrical fields. From Maxwell’s equation a set of standard radio field equations are presented for the Electrostatic, Inductive and Radiative field. By inserting three gravity field equations (1/r, 1/r2 and 1/r3) a new set of field equations are created that can be used to define space, time, gravity and the electromagnetic field at any point P in the field.

Heat transfer across a nanoscale pressurized air gap and its application in magnetic recording

In this study, we investigated how a thermally actuated air bearing slider heats up a fast-spinning storage disk through a highly pressurized nanoscale air gap in a magnetic recording system. A Eulerian-description-based computational approach is developed considering heat conduction through a pressurized air film and near-field radiation across the gap. A set of field equations that govern the air bearing dynamics, slider thermo-mechanics and disk heat dissipation are solved simultaneously through an iterative approach. A temperature field on the same order as the hot slider surface itself is found to be established in the disk. The effective local heat transfer coefficient is found to vary substantially with disk materials and linear speeds. This approach quantifies the magnitude of different thermal transport schemes and the accuracy is verified by an excellent agreement with our experiment, which measures the local slider temperature rise with a resistance temperature sensor. It also demonstrates an effective computational approach to treat transient thermal processes in a system of components with fast relative speed and different length scales. Finally, the investigated thermal transport mechanism leads to a substantial spacing change that has a significant impact on the spacing margin of today’s magnetic storage systems.

Modified Saez–Ballester scalar–tensor theory from 5D space-time

In this paper, we bring together the five-dimensional Saez–Ballester (SB) scalar–tensor theory (Saez and Ballester 1986 Phys. Lett. 113A 9) and the induced-matter-theory (IMT) setting (Wesson and Ponce de Leon 1992 J. Math. Phys. 33 3883), to obtain a modified SB theory (MSBT) in four dimensions. Specifically, by using an intrinsic dimensional reduction procedure into the SB field equations in five-dimensions, a MSBT is obtained onto a hypersurface orthogonal to the extra dimension. This four-dimensional MSBT is shown to bear distinctive new features in contrast to the usual corresponding SB theory as well as to IMT and the modified Brans–Dicke theory (MBDT) (Rasouli et al 2014 Class. Quantum Grav. 31 115002). In more detail, besides the usual induced matter terms retrieved through the IMT, the MSBT scalar field is provided with additional physically distinct (namely, SB induced) terms as well as an intrinsic self-interacting potential (interpreted as a consequence of the IMT process and the concrete geometry associated with the extra dimension). Moreover, our MSBT has four sets of field equations, with two sets having no analog in the standard SB scalar–tensor theory. It should be emphasized that the herein appealing solutions can emerge solely from the geometrical reductional process, from the presence also of extra dimension(s) and not from any ad-hoc matter either in the bulk or on the hypersurface. Subsequently, we apply the herein MSBT to cosmology and consider an extended spatially flat FLRW geometry in a five-dimensional vacuum space-time. After obtaining the exact solutions in the bulk, we proceed to construct, by means of the MSBT setting, the corresponding dynamic, on the four-dimensional hypersurface. More precisely, we obtain the (SB) components of the induced matter, including the induced scalar potential terms. We retrieve two different classes of solutions. Concerning the first class, we show that the MSBT yields a barotropic equation of state for the induced perfect fluid. We then investigate vacuum, dust, radiation, stiff fluid and false vacuum cosmologies for this scenario and contrast the results with those obtained in the standard SB theory, IMT and BD theory. Regarding the second class solutions, we show that the scale factor behaves in a similar way to a de Sitter (DeS) model. However, in our MSBT setting, this behavior is assisted by non-vanishing induced matter instead, without any a priori cosmological constant. Moreover, for all these solutions, we show that the extra dimension contracts with the cosmic time.

Exact solutions for Weyl fermions with gravity

We consider the single-handed spinor field in interaction with its own gravitational field described by the set of field equations given by Weyl field equations written in terms of derivatives that are covariant with respect to the gravitational connection plus Einstein field equations soured with the energy tensor of the spinor: for the Weyl spinor and the ensuing spacetime of Weyl-Lewis-Papapetrou structure, we will find all exact solutions. The obtained solution for the metric tensor is that of a PP-wave spacetime while the spinor field is a flag-dipole.

Is the Bianchi identity always hyperbolic?

We consider n + 1 dimensional smooth Riemannian and Lorentzian spaces satisfying Einsteinʼs equations. The base manifold is assumed to be smoothly foliated by a one-parameter family of hypersurfaces. In both cases—likewise it is usually done in the Lorentzian case—Einsteinʼs equations may be split into ‘Hamiltonian’ and ‘momentum’ constraints and a ‘reduced’ set of field equations. It is shown that regardless of whether the primary space is Riemannian or Lorentzian, whenever the foliating hypersurfaces are Riemannian the ‘Hamiltonian’ and ‘momentum’ type expressions are subject to a subsidiary first order symmetric hyperbolic system. Since this subsidiary system is linear and homogeneous in the ‘Hamiltonian’ and ‘momentum’ type expressions, the hyperbolicity of the system implies that in both cases the solutions to the ‘reduced’ set of field equations are also solutions to the full set of equations provided that the constraints hold on one of the hypersurfaces foliating the base manifold.

GENERALIZED EINSTEIN–MAXWELL FIELD EQUATIONS IN THE PALATINI FORMALISM

We derive a new set of field equations within the framework of the Palatini formalism. These equations are a natural generalization of the Einstein–Maxwell equations which arise by adding a function [Formula: see text], with [Formula: see text] to the Palatini Lagrangian f(R, Q). The result we obtain can be viewed as the coupling of gravity with a nonlinear extension of the electromagnetic field. In addition, a new method is introduced to solve the algebraic equation associated to the Ricci tensor.

A bio-chemo-hydro-mechanical model for microbially induced calcite precipitation in soils

Microbially Induced Calcite Precipitation (MICP) is an innovative technique for soil grouting involving a bacterial reactive grout. This comprehensive research study is carried out to better understand and describe the coupled phenomena of multispecies reactive biogrout transport in a saturated, deformable soil. A unique predictive model of the behaviour of the porous media during biogrout injection is presented. The general field equations describing the system are derived from the macroscopical balance equations and constitutive equations. The set of field equations is numerically discretised. Finally, numerical examples are provided as a first step to validate the capabilities of the proposed model.

BEM/FDM conjugate heat transfer analysis of a two-dimensional air-cooled turbine blade boundary layer

A coupled boundary element method (BEM) and finite difference method (FDM) are applied to solve conjugate heat transfer problem of a two-dimensional air-cooled turbine blade boundary layer. A loosely coupled strategy is adopted, in which each set of field equations is solved to provide boundary conditions for the other. The Navier-Stokes equations are solved by HIT-NS code. In this code, the FDM is adopted and is used to resolve the convective heat transfer in the fluid region. The BEM code is used to resolve the conduction heat transfer in the solid region. An iterated convergence criterion is the continuity of temperature and heat flux at the fluid-solid interface. The numerical results from the BEM adopted in this paper are in good agreement with the results of analytical solution and the results of commercial code, such as Fluent 6.2. The BEM avoids the complicated mesh needed in other computation method and saves the computation time. The results prove that the BEM adopted in this paper can give the same precision in numerical results with less boundary points. Comparing the conjugate results with the numerical results of an adiabatic wall flow solution, it reveals a significant difference in the distribution of metal temperatures. The results from conjugate heat transfer analysis are more accurate and they are closer to realistic thermal environment of turbines.

Microstructure Sensitive Design and Quantitative Prediction of Effective Conductivity in Fuel Cell Design

Statistical continuum approach is used to predict effective conductivity of anisotropic random porous heterogeneous media using two-point correlation functions. Probability functions play a critical role in describing the statistical distribution of different constituents in a heterogeneous media. In this study a 3-dimensional two-point correlation function is utilized to characterize the anisotropic porous media of a Cathode materials to incorporate all the details of the microstructure. These correlation functions are then linked to the effective properties using homogenization relations. An anisotropioc Green’s function solution is used to solve the set of field equations. Examples in this study demonstrated how the model captured the anisotropy in effective conductivity of the random heterogeneous media. Predicted results showed the influence of microstructure on the effective conductivity tensor.

Modelling of turbulent flow and multi-phase heat transfer under electromagnetic force

In this paper, we develop a mathematical model and a numerical technique to study the coupled turbulent flow and heat transfer process in continuous steel casting under an electromagnetic force. The complete set of field equations are established and solved numerically. The influences of the electromagnetic field on a flow pattern of molten steel and the temperature field as well as steel solidification are presented in the paper.