scholarly journals Karmarkar scalar condition

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
J. Ospino ◽  
L. A. Núñez
Keyword(s):  
Differential Equation ◽  
Energy Density ◽  
Density Profile ◽  
Riemann Tensor ◽  
Orthogonal Decomposition ◽  
Algebraic Relation ◽  
Structure Scalars ◽  
Physical Variables ◽  
Adiabatic Case ◽  
Dissipative Dynamic

AbstractIn this work we present the Karmarkar condition in terms of the structure scalars obtained from the orthogonal decomposition of the Riemann tensor. This new expression becomes an algebraic relation among the physical variables, and not a differential equation between the metric coefficients. By using the Karmarkar scalar condition we implement a method to obtain all possible embedding class I static spherical solutions, provided the energy density profile is given. We also analyse the dynamic adiabatic case and show the incompatibility of the Karmarkar condition with several commonly assumed simplifications to the study of gravitational collapse. Finally, we consider the dissipative dynamic Karmarkar collapse and find a new solution family.

Complexity factors for static axial system in self-interacting Brans–Dicke gravity

2019 ◽  
Vol 16 (11) ◽  
pp. 1950174 ◽  
Author(s):  
M. Sharif ◽  
Amal Majid
Keyword(s):  
Scalar Field ◽  
Evolution Equations ◽  
Riemann Tensor ◽  
Additional Source ◽  
Bianchi Identities ◽  
Structure Scalars ◽  
Physical Attributes ◽  
Physical Features ◽  
Axial System

This paper explores the physical attributes of a static axial source that induce complexity within the fluid in the background of self-interacting Brans–Dicke theory. Bel’s approach is used to split the Riemann tensor and construct structure scalars that involve physical features of the fluid in the presence of scalar field. Using the evolution equations derived from Bianchi identities as well as structure scalars, five complexity factors are identified which include constraints on the scalar field. Finally, the conditions of vanishing complexity are used to present solutions for an anisotropic inhomogeneous spheroid. It is concluded that scalar field is an additional source of complexity.

scholarly journals Complexity of dynamical sphere in self-interacting Brans–Dicke gravity

2020 ◽  
Vol 80 (12) ◽  
Author(s):  
M. Sharif ◽  
Amal Majid
Keyword(s):  
Scalar Field ◽  
Riemann Tensor ◽  
Dissipative Systems ◽  
Anisotropic Pressure ◽  
Massive Scalar ◽  
Initial State ◽  
Spherical System ◽  
Massive Scalar Field ◽  
Structure Scalars ◽  
Definition Of

AbstractThis paper aims to derive a definition of complexity for a dynamic spherical system in the background of self-interacting Brans–Dicke gravity. We measure complexity of the structure in terms of inhomogeneous energy density, anisotropic pressure and massive scalar field. For this purpose, we formulate structure scalars by orthogonally splitting the Riemann tensor. We show that self-gravitating models collapsing homologously follow the simplest mode of evolution. Furthermore, we demonstrate the effect of scalar field on the complexity and evolution of non-dissipative as well as dissipative systems. The criteria under which the system deviates from the initial state of zero complexity is also discussed. It is concluded that complexity of the sphere increases in self-interacting Brans–Dicke gravity because the homologous model is not shear-free.

A Proper Orthogonal Decomposition Analysis Method for Multimedia Heat Conduction Problems

2016 ◽  
Vol 138 (7) ◽  
Author(s):  
Xiaowei Gao ◽  
Jinxiu Hu ◽  
Shizhang Huang
Keyword(s):  
Heat Conduction ◽  
Proper Orthogonal Decomposition ◽  
Decomposition Analysis ◽  
Orthogonal Decomposition ◽  
Analysis Method ◽  
Medium Level ◽  
Conduction Behavior ◽  
Physical Variables ◽  
The Individual ◽  
Proper Orthogonal

In this paper, a new proper orthogonal decomposition (POD) analysis method is proposed for numerical analysis of thermal mechanical engineering problems consisting of multiple media. After the creation of a heat conduction solution database for each medium, the “snapshot” approach of the POD technique is applied to facilitate reduced-order modeling (ROM) of the unsteady heat conduction behavior. The snapshot matrix is constructed medium by medium by collecting individual medium solutions at different instances in time through a columnwise manner. By means of expressing physical variables in terms of reduced modes at the individual medium level, a system of differential equations with respect to time is formed by utilizing the consistency conditions of the physical variables on interface boundaries. The solutions of the problem can be obtained by solving the system of equations at different time stops. Two numerical examples are given to demonstrate the efficiency of the proposed method.

Kraftfreie Magnetfelder II

1957 ◽  
Vol 12 (10) ◽  
pp. 855-859 ◽  
Author(s):  
A. Schlüter
Keyword(s):  
Differential Equation ◽  
Angular Momentum ◽  
Energy Density ◽  
Magnetic Fields ◽  
General Solution ◽  
Current Density ◽  
Cylindrical Symmetry ◽  
Axis Of Symmetry ◽  
Free Fields ◽  
Electric Flux

Die allgemeine Lösung der Differentialgleichung der kraftfreien Magnetfelder wird für den zylindersymmetrischen Fall angegeben. Die Energiedichte des Magnetfeldes kann als Funktion des Abstandes von der Symmetrieachse vorgegeben werden, sie muß nur zwei Ungleichungen befriedigen. Die Komponenten des magnetischen Feldvektors und der elektrischen Stromdichte folgen dann durch Differentiation. Durch Konstruktion eines Beispiels wird gezeigt, daß ein solches Feld Impuls und Drehimpuls in Richtung der Achse transportieren kann in einem Ausmaß, das durch den Gesamtstrom und den gesamten magnetischen Fluß nicht bestimmt ist.The general solution of the differential equation of force-free magnetic fields is given in the case of cylindrical symmetry. The energy density has to fulfil two inequalities; apart from this, it can be freely chosen as function of the distance from the axis of symmetry. The components of the field as well as the electric current density follow from it purely by differentiation. By explicitly constructing a three-parameter family of such force-free fields it is shown that the amount of momentum and of angular momentum transported by the field in the direction of the axis is not determined by the total magnetic and electric flux of the field.

Complexity factors for static anisotropic axially symmetric fluid distributions in f(R) gravity

2020 ◽  
Vol 17 (03) ◽  
pp. 2050043
Author(s):  
G. Abbas ◽  
H. Nazar
Keyword(s):  
Energy Density ◽  
General Theory ◽  
Graphical Analysis ◽  
Fluid Distribution ◽  
Anisotropic Fluid ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Axially Symmetric ◽  
Structure Scalars ◽  
Isotropic Pressure

In this paper, we have analyzed the complexity factor for the most general axially symmetric static anisotropic fluid distributions in context of [Formula: see text] theory of gravity. For this purpose, we have studied three distinct complexity factors that are organized in terms of three scalar variables (structure scalars) comes from the orthogonal splitting of the curvature tensor. The vanishing of all complexity factors condition for what we choose the simplest fluid distribution that in which system having energy density is homogeneous with isotropic pressure. Although, it has been found that the complexity factors condition can also vanish when inhomogeneous energy density and anisotropy of the pressure cancel each other. Next, we express a class of exact solutions and their graphical analysis as compatible to our models that satisfies the vanishing condition of complexity factors. Finally, it is worth mentioning that these results can reproduce the results of General theory of Relativity under some constraints.

Role of structure scalars on viscous and heat conducting spherical systems in f(R,T) gravity

2017 ◽  
Vol 26 (14) ◽  
pp. 1750155 ◽  
Author(s):  
T. Hussain ◽  
M. Khurshudyan ◽  
S. Ahmed ◽  
As. Khurshudyan
Keyword(s):  
Evolution Equations ◽  
Energy Momentum Tensor ◽  
Gravitational Theory ◽  
Riemann Tensor ◽  
Momentum Tensor ◽  
Compact Objects ◽  
Heat Conducting ◽  
Structure Scalars ◽  
Dynamical Features

In this paper, we analyze some dynamical features of spherical celestial objects through structure scalars in [Formula: see text] gravitational theory, where [Formula: see text] and [Formula: see text] are the Ricci scalar and the trace of energy–momentum tensor, respectively. In this framework, we consider our relativistic geometry to be spherical in shape filled with radiating viscous and shearing fluid content. We formulate extended version of structure scalars by orthogonal decomposition of the Riemann tensor with and without constant [Formula: see text] and [Formula: see text] backgrounds. We discuss the effects of dark source corrections on the construction of expansion and shear evolution equations via scalar variables. It is inferred that like general relativity, one can investigate the evolutionary stages of stellar compact objects with the help of extended scalar parameters.

Role of modified scalar variables in the modeling of spherical fluids

2018 ◽  
Vol 15 (08) ◽  
pp. 1850140 ◽  
Author(s):  
A. Akram ◽  
A. Rehman Jami ◽  
S. Ahmad ◽  
M. Sufyan ◽  
R. Munir
Keyword(s):  
Degrees Of Freedom ◽  
Dust Cloud ◽  
Mass Function ◽  
Riemann Tensor ◽  
Ricci Scalar ◽  
Structure Scalars ◽  
Specific Distribution ◽  
Scalar Matter ◽  
Symmetric Geometry

The aim of this work is to analyze the role of shear evolution equation in the modeling of relativistic spheres in [Formula: see text] gravity. We assume that non-static diagonally symmetric geometry is coupled with dissipative anisotropic viscous fluid distributions in the presence of [Formula: see text] dark source terms. A specific distribution of [Formula: see text] cosmic model has been assumed and the spherical mass function through generic formula introduced by Misner-Sharp has been formulated. Some very important relations regarding Weyl scalar, matter variables and mass functions are being computed. After decomposing orthogonally the Riemann tensor, some scalar variables in the presence of [Formula: see text] extra degrees of freedom are calculated. The effects of the polynomial modified structure scalars in the modeling of through Weyl, shear, expansion scalar differential equations are investigated. The energy density irregularity factor has been calculated for both anisotropic radiating viscous with varying Ricci scalar and for dust cloud with present Ricci scalar corrections.

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