Solutions of the Einstein field equations Wikipedia Laplace’s field equation. lineareccentricity Thedistance from thecenter of an ellipsoid to either of its foci. mean Earth ellipsoid Normal ellipsoid with para- meters closest to actual corresponding parameters for the Earth. mean tide geoid Geoid with all time-varying tidal effects removed. multipoles Stokes coefficients. Newtonian potential Harmonic function that approaches the potential of

(PDF) Gauge theory of Gravitation researchgate.net

Scrutiny of Einstein's Geodesic and Field Equations viXra. conclusively that the Einstein's geodesic and field equations cannot be reduced to Newton's equation of motion and Poisson's gravitational potential equation, respectively, and that the geometrization of gravity is unnecessary for physics., John Nash has proposed a new theory of gravity. We define a Nash-tensor equal to the curvature tensor appearing in the Nash field equations for empty space, and calculate its components for two cases..

22/08/2013 · This translation has moved. The text of this page has been moved. If you found this page by following a link, please go back and update it, or notify the webmaster. confused with equations (3) and (6) that give the individual rates of acceleration, a for the secondary mass under Newton’s theory, and A for the secondary mass in …

Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations actually gives a Lorentz manifold. Solutions are broadly classed as exact or non-exact. The Einstein field equations are = or more generally, if one allows a nonzero cosmological constant, + = where is a constant, and the Professor Susskind derives the Einstein field equations of general relativity. Beginning with Newtonian gravitational fields, an analogy with the four-current, and the continuity equation, he develops the stress-energy tensor (also known as the energy momentum tensor).

Chapter 5. Gravitation Notes: Equations (5.2) and (5.3) taken together show that gravitation is an attractive force, i.e., the two masses are drawn to one another. - 107 - 5.1.1 Gravitational Interactions with Spherically Symmetric Bodies Although the related calculations are beyond the scope of our studies, it can be shown that gravitational interactions involving spherically symmetric confused with equations (3) and (6) that give the individual rates of acceleration, a for the secondary mass under Newton’s theory, and A for the secondary mass in …

General Relativity and Solutions to Einstein’s Field Equations Abhishek Kumar Department of Physics and Astronomy, Bates College, Lewiston, ME 04240 General Relativity and Solutions to Einstein’s Field Equations A Senior Thesis Presented to the Department of Physics The field can be determined using Newton's law of universal gravitation. Determined in this way, the gravitational field around a single particle is a vector field consisting at every point of a vector pointing directly towards the particle.

Abstract. This is an English translation of a paper by Pascual Jordan, Jürgen Ehlers and Wolfgang Kundt, first published in 1960. The original paper was part 1 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. General Relativity and Solutions to Einstein’s Field Equations Abhishek Kumar Department of Physics and Astronomy, Bates College, Lewiston, ME 04240 General Relativity and Solutions to Einstein’s Field Equations A Senior Thesis Presented to the Department of Physics

This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out The field can be determined using Newton's law of universal gravitation. Determined in this way, the gravitational field around a single particle is a vector field consisting at every point of a vector pointing directly towards the particle.

Field Equations and Equations of Motion (General Relativity) The observer in modern physics: A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File: On Expansion of the Universe - PDF File : Field Equations & Equations of Motion (General Relativity) Velocity is a vector (tensor) or vector (tensor) field. In familiar notation, the velocity v is represented by. v GRAVITATIONAL FIELD THEORY Introduction. Itisby“instantaneousactionatadistance”—morespeciﬁcally: because not Lorentz covariant, and Newton’s Law of Universal Gravitation to be untenable because it drew upon a concept—“distant simultaneity”— which relativity had rendered obsolete. It placed upon physicists the burden of devising a “relativistic theory of gravitation

LibraryThing Review User Review - peridotite - LibraryThing "Gravitation" is a thorough introduction to Einstein's general relativity. Assuming basic calculus and mechanics, it introduces the mathematics, notation, and physics required to understand general Isotropic Statis Solutions of the Field Equations in Einstein's Theory of Gravitation Author(s): P. Y. Chou Source: American Journal of Mathematics, Vol. 59, No. 4 (Oct.,…

This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out Einstein's equation relates the matter content (right side of the equation) to the geometry (the left side) of the system. It can be summed up with "mass creates geometry, and geometry acts like mass".

Scalar theories of gravitation Note: This article focuses on relativistic classical field theories of gravitation. The best known relativistic classical field theory of gravitation, general relativity , is a tensor theory, in which the gravitational interaction is described using a tensor field. Abstract. Weyl and Eddington suggested three alternative general relativistic theories of gravitation with fourth-order field equations which in empty space admit the Schwarzschild metric as a solution.

(PDF) Gauge theory of Gravitation researchgate.net

The Theory of Space Time and Gravitation 2nd Edition. This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out, paper, “The General Form of the Field Equations of Gravitation”: “For if we consider a complete system (e.g. the solar system), the total mass of the system, ….

Dyon field equations in the presence of gravitation

Solutions Of The ECE Field Equations Aias. 26/04/2017 · The Field Equations of Gravitation By A. Einstein. I have shown in two recently published reports, how one can arrive at field equations of gravitation, that are in agreement with the postulate of general relativity, i.e. which in their general form are covariant in respect to arbitrary substitutions of space-time variables. https://simple.wikipedia.org/wiki/General_theory_of_relativity In general the Birkhoff theorem is violated in non-Einsteinian theories of gravitation. We show for theories in which the dynamical equations do not follow from the field equations that time-dependent vacuum solutions are needed in order to join nonstatic spherically symmetric incoherent matter ….

A GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM Myron W. Evans Institute for Advanced Study Alpha Foundation E-mail: emyrone@aol.com Einstein's 1916 derivation of the Field Equations Galina Weinstein 24/10/2013 Abstract: In his first November 4, 1915 paper Einstein wrote the Lagrangian form of his field equations. In the fourth November 25, 1915 paper, Einstein added a trace term of the energy-momentum tensor on the right-hand side of the generally covariant field equations. The main purpose of the present work is to show

The Theory of Space, Time, and Gravitation, 2nd Revised Edition focuses on Relativity Theory and Einstein's Theory of Gravitation and correction of the misinterpretation of the Einsteinian Gravitation … 22/08/2013 · This translation has moved. The text of this page has been moved. If you found this page by following a link, please go back and update it, or notify the webmaster.

The field can be determined using Newton's law of universal gravitation. Determined in this way, the gravitational field around a single particle is a vector field consisting at every point of a vector pointing directly towards the particle. Field Equations and Equations of Motion (General Relativity) The observer in modern physics: A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File: On Expansion of the Universe - PDF File : Field Equations & Equations of Motion (General Relativity) Velocity is a vector (tensor) or vector (tensor) field. In familiar notation, the velocity v is represented by. v

Isotropic Statis Solutions of the Field Equations in Einstein's Theory of Gravitation Author(s): P. Y. Chou Source: American Journal of Mathematics, Vol. 59, No. 4 (Oct.,… Jump to Einstein's equations - General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in and the current description of gravitation in modern physics. The relation is specified by the Einstein field equations, a system of partial differential equations.‎From classical mechanics · ‎Consequences of

Einstein's equation relates the matter content (right side of the equation) to the geometry (the left side) of the system. It can be summed up with "mass creates geometry, and geometry acts like mass". Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations actually gives a Lorentz manifold. Solutions are broadly classed as exact or non-exact. The Einstein field equations are = or more generally, if one allows a nonzero cosmological constant, + = where is a constant, and the

21/08/2013 · Einstein field equations. From Wikipedia, the free encyclopedia The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.[1] Laplace’s field equation. lineareccentricity Thedistance from thecenter of an ellipsoid to either of its foci. mean Earth ellipsoid Normal ellipsoid with para- meters closest to actual corresponding parameters for the Earth. mean tide geoid Geoid with all time-varying tidal effects removed. multipoles Stokes coefficients. Newtonian potential Harmonic function that approaches the potential of

Abstract. This is an English translation of a paper by Pascual Jordan, Jürgen Ehlers and Wolfgang Kundt, first published in 1960. The original paper was part 1 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. Chapter 5. Gravitation Notes: Equations (5.2) and (5.3) taken together show that gravitation is an attractive force, i.e., the two masses are drawn to one another. - 107 - 5.1.1 Gravitational Interactions with Spherically Symmetric Bodies Although the related calculations are beyond the scope of our studies, it can be shown that gravitational interactions involving spherically symmetric

confused with equations (3) and (6) that give the individual rates of acceleration, a for the secondary mass under Newton’s theory, and A for the secondary mass in … This equation is as elementary as the definition of Ampere that links mechanical forces to electromagnetism. In line with tradition we can represent the force using average spin current fields ˆ˙ ˆ˙ as follows: ˝ ˛˝ ˛ ˚ ˆ˙ ˆ˙ ˜˘ Force exists when both ˆ˙ ˆ˙ exist and interact. Spin current field potential Consider a field potential S due to spin current of an elementary

In the weak-field and slow motion limit of general relativity, the phenomenon of gravitoelectromagnetism (in short "GEM") occurs, creating a parallel between gravitation and electromagnetism. The gravitational field is the analogue of the electric field , while the gravitomagnetic field , which results from circulations of masses due to their angular momentum , is the analogue of the magnetic In this paper, the Maxwell-Proca type field equations of linear gravity are formulated in terms of hyperbolic octonions (split octonions). A hyperbolic octonionic gravitational wave equation with massive gravitons and gravitomagnetic monopoles is proposed. The real gravitoelectromagnetic field

The field equations of gravitation (EFE) violates the basic of modern astronomy Einstein’s general relativity has never been right. Einstein has no idea on the basic of astronomy. Albert Einstein. Nov 1915 - 4 pages Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.) 1915 (1915) 844-847

29 March—Einstein’s discovery of the field equation èDerivation of the Field Equations èCosmological constant èEinstein’s toy "Derivation" of Einstein's field equation E’s plan was to write measure of curvature = source of gravity. üEinstein’s happiest thought, Nov 1907 üMathematics of curvature 1913 Einstein & Grossman, Z. Math. Physik, 62, 225, (1913) the mathematics of Abstract. Weyl and Eddington suggested three alternative general relativistic theories of gravitation with fourth-order field equations which in empty space admit the Schwarzschild metric as a solution.

CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

On theories of gravitation with higher-order field equations. solutions of the ece field equations and this fundamental property is known conventionally as the tetrad postulate. However, nothing is postulated (i.e. nothing is needed to derive Eq.6.20) other, Physical Review & Research International, 3(4): 270-292, 2013 271 1. INTRODUCTION In 1916, Karl Schwarzschild published an exact solution to Einstein’s field equations for the.

Dyon field equations in the presence of gravitation Il

Scalar theories of gravitation Wikipedia. In this paper, the Maxwell-Proca type field equations of linear gravity are formulated in terms of hyperbolic octonions (split octonions). A hyperbolic octonionic gravitational wave equation with massive gravitons and gravitomagnetic monopoles is proposed. The real gravitoelectromagnetic field, In this paper, the Maxwell-Proca type field equations of linear gravity are formulated in terms of hyperbolic octonions (split octonions). A hyperbolic octonionic gravitational wave equation with massive gravitons and gravitomagnetic monopoles is proposed. The real gravitoelectromagnetic field.

This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out Jump to Einstein's equations - General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in and the current description of gravitation in modern physics. The relation is specified by the Einstein field equations, a system of partial differential equations.‎From classical mechanics · ‎Consequences of

General Relativity and Solutions to Einstein’s Field Equations Abhishek Kumar Department of Physics and Astronomy, Bates College, Lewiston, ME 04240 General Relativity and Solutions to Einstein’s Field Equations A Senior Thesis Presented to the Department of Physics A gauge theory of gravitation having the de-Sitter group SO (1, 4) as local sym-metry is presented. The strength tensor field of the gravitational gauge potentials is obtained and then the field

confused with equations (3) and (6) that give the individual rates of acceleration, a for the secondary mass under Newton’s theory, and A for the secondary mass in … In general the Birkhoff theorem is violated in non-Einsteinian theories of gravitation. We show for theories in which the dynamical equations do not follow from the field equations that time-dependent vacuum solutions are needed in order to join nonstatic spherically symmetric incoherent matter …

John Nash has proposed a new theory of gravity. We define a Nash-tensor equal to the curvature tensor appearing in the Nash field equations for empty space, and calculate its components for two cases. The gravitational field strength and the gravitational torsion field describe gravitational field in the Lorentz-invariant theory of gravitation and obey the Maxwell-like gravitational equations. In general relativity , the gravitational field strength is called the strength of gravitoelectric field, and the torsion field corresponds to the gravitomagnetic field.

IN EINSTEIN’S FIELD EQUATIONS AND GRAND UNIFICATION THEORY is directed towards the center of gravity. If the total force, F 0 then p 0 and linear momentum is conserved. Angular frequency, (6) 2 2 2 1 dr d V m in the generalized case where E V T where E is the total energy, V is the potential energy, T is kinetic energy and m is the mass of the system. A revolving of a particle has The Gravity Field The law of gravitational attraction was formulated by Isaac Newton (1642-1727) and published in 1687, that is, about three generations after Galileo had deter-

conclusively that the Einstein's geodesic and field equations cannot be reduced to Newton's equation of motion and Poisson's gravitational potential equation, respectively, and that the geometrization of gravity is unnecessary for physics. LibraryThing Review User Review - peridotite - LibraryThing "Gravitation" is a thorough introduction to Einstein's general relativity. Assuming basic calculus and mechanics, it introduces the mathematics, notation, and physics required to understand general

GRAVITATIONAL FIELD THEORY Introduction. Itisby“instantaneousactionatadistance”—morespeciﬁcally: because not Lorentz covariant, and Newton’s Law of Universal Gravitation to be untenable because it drew upon a concept—“distant simultaneity”— which relativity had rendered obsolete. It placed upon physicists the burden of devising a “relativistic theory of gravitation Professor Susskind derives the Einstein field equations of general relativity. Beginning with Newtonian gravitational fields, an analogy with the four-current, and the continuity equation, he develops the stress-energy tensor (also known as the energy momentum tensor).

LibraryThing Review User Review - peridotite - LibraryThing "Gravitation" is a thorough introduction to Einstein's general relativity. Assuming basic calculus and mechanics, it introduces the mathematics, notation, and physics required to understand general gravity (but not in general) the Riemann tensor is gauge-invariant. Thus one way to form Thus one way to form gauge-invariant quantities is to replace equation (4) by the geodesic deviation equation,

Field Equations and Equations of Motion (General Relativity) The observer in modern physics: A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File: On Expansion of the Universe - PDF File : Field Equations & Equations of Motion (General Relativity) Velocity is a vector (tensor) or vector (tensor) field. In familiar notation, the velocity v is represented by. v Abstract. Weyl and Eddington suggested three alternative general relativistic theories of gravitation with fourth-order field equations which in empty space admit the Schwarzschild metric as a solution.

DYON FIELD EQUATIONS IN THE PRESENCE OF GRAVITATION 247 Electromagnetic field tensorsE mnandH mnare associated with electric and magnetic four-potentials,Arepresents the dual part and the other symbols have their usual gravity (but not in general) the Riemann tensor is gauge-invariant. Thus one way to form Thus one way to form gauge-invariant quantities is to replace equation (4) by the geodesic deviation equation,

The Field Equations of Gravitation INSPIRE-HEP. 26/04/2017 · The Field Equations of Gravitation By A. Einstein. I have shown in two recently published reports, how one can arrive at field equations of gravitation, that are in agreement with the postulate of general relativity, i.e. which in their general form are covariant in respect to arbitrary substitutions of space-time variables., confused with equations (3) and (6) that give the individual rates of acceleration, a for the secondary mass under Newton’s theory, and A for the secondary mass in ….

The Einstein Field Equations Miles Mathis

3.02 Potential Theory and Static Gravity Field of the Earth. Dyon field equations in the presence of gravitation Rawat, A.; Negi, O. 2016-01-09 00:00:00 Maxwell’s-Dirac field equations of dyons are formulated in the presence of gravitation and the corresponding Lorentz force equation of motion, energy-momentum tensor and Einstein’s field equations are derived consistently from an invariant action principle., 26/04/2017 · The Field Equations of Gravitation By A. Einstein. I have shown in two recently published reports, how one can arrive at field equations of gravitation, that are in agreement with the postulate of general relativity, i.e. which in their general form are covariant in respect to arbitrary substitutions of space-time variables..

Quantum Gravity and the Holographic Mass

3.02 Potential Theory and Static Gravity Field of the Earth. Physical Review & Research International, 3(4): 270-292, 2013 271 1. INTRODUCTION In 1916, Karl Schwarzschild published an exact solution to Einstein’s field equations for the https://en.wikipedia.org/wiki/Gravitoelectromagnetism Albert Einstein. Nov 1915 - 4 pages Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.) 1915 (1915) 844-847.

• Quantum Theory of Gravitation General Formulation and
• (PDF) Exact Solutions of the Field Equations for Empty

• On this basis we surmise that the field equations coupled to the source term can be written in the form G μν = κT μν where κ is a constant which must equal −8π (remembering that Newton's gravitational constant is 1 in our units) in order for the field equations to reduce to Newton's law in the weak field limit. Thus we have the complete expression of Einstein's metrical law of general Field Equations and Equations of Motion (General Relativity) The observer in modern physics: A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File: On Expansion of the Universe - PDF File : Field Equations & Equations of Motion (General Relativity) Velocity is a vector (tensor) or vector (tensor) field. In familiar notation, the velocity v is represented by. v

The field equations of gravitation (EFE) violates the basic of modern astronomy Einstein’s general relativity has never been right. Einstein has no idea on the basic of astronomy. Einstein's 1916 derivation of the Field Equations Galina Weinstein 24/10/2013 Abstract: In his first November 4, 1915 paper Einstein wrote the Lagrangian form of his field equations. In the fourth November 25, 1915 paper, Einstein added a trace term of the energy-momentum tensor on the right-hand side of the generally covariant field equations. The main purpose of the present work is to show

LibraryThing Review User Review - peridotite - LibraryThing "Gravitation" is a thorough introduction to Einstein's general relativity. Assuming basic calculus and mechanics, it introduces the mathematics, notation, and physics required to understand general Einstein's equation relates the matter content (right side of the equation) to the geometry (the left side) of the system. It can be summed up with "mass creates geometry, and geometry acts like mass".

The review of gauge theory, namely for the field equations of gravitation (Newton, weak field-slow motion approximation, SU1-gauge) is followed by a spacetime definition of the deforming geoid as well as by a computational scheme for the four-gravitational potential inside and outside a … Laplace’s field equation. lineareccentricity Thedistance from thecenter of an ellipsoid to either of its foci. mean Earth ellipsoid Normal ellipsoid with para- meters closest to actual corresponding parameters for the Earth. mean tide geoid Geoid with all time-varying tidal effects removed. multipoles Stokes coefficients. Newtonian potential Harmonic function that approaches the potential of

This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out Albert Einstein. Nov 1915 - 4 pages Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.) 1915 (1915) 844-847

Physical consequences from gravitation equations based on Poincar\'{e} ideas of relativity of space and time in respect of measuring instruments are considered. A gauge theory of gravitation having the de-Sitter group SO (1, 4) as local sym-metry is presented. The strength tensor field of the gravitational gauge potentials is obtained and then the field

History . According to McDonald, the first scientist who used Maxwell equations to describe gravitation was Oliver Heaviside. A conclusion was reached that stated, in weak gravitational fields the standard theory of gravitation could be written in the form of Maxwell equations with two gravitational constants. Scalar theories of gravitation Note: This article focuses on relativistic classical field theories of gravitation. The best known relativistic classical field theory of gravitation, general relativity , is a tensor theory, in which the gravitational interaction is described using a tensor field.

Volume 58A, number 3 PHYSICS LETTERS 23 August 1976 THE FIELD EQUATIONS OF THE POINCARE GAUGE THEORY OF GRAVITATION P. Von der HEYDE Institut far Theoretisehe Physik, Unfversitdt zu KOln, D-5000 Kn 41, Fed. Rep. Germany Received 9 June 1976 We define local active Poincar~ transfmations of matter fields and the corresponding invariance. This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out

29 March—Einstein’s discovery of the field equation èDerivation of the Field Equations èCosmological constant èEinstein’s toy "Derivation" of Einstein's field equation E’s plan was to write measure of curvature = source of gravity. üEinstein’s happiest thought, Nov 1907 üMathematics of curvature 1913 Einstein & Grossman, Z. Math. Physik, 62, 225, (1913) the mathematics of IN EINSTEIN’S FIELD EQUATIONS AND GRAND UNIFICATION THEORY is directed towards the center of gravity. If the total force, F 0 then p 0 and linear momentum is conserved. Angular frequency, (6) 2 2 2 1 dr d V m in the generalized case where E V T where E is the total energy, V is the potential energy, T is kinetic energy and m is the mass of the system. A revolving of a particle has

In this paper, the Maxwell-Proca type field equations of linear gravity are formulated in terms of hyperbolic octonions (split octonions). A hyperbolic octonionic gravitational wave equation with massive gravitons and gravitomagnetic monopoles is proposed. The real gravitoelectromagnetic field Physical Review & Research International, 3(4): 270-292, 2013 271 1. INTRODUCTION In 1916, Karl Schwarzschild published an exact solution to Einstein’s field equations for the

Laplace’s field equation. lineareccentricity Thedistance from thecenter of an ellipsoid to either of its foci. mean Earth ellipsoid Normal ellipsoid with para- meters closest to actual corresponding parameters for the Earth. mean tide geoid Geoid with all time-varying tidal effects removed. multipoles Stokes coefficients. Newtonian potential Harmonic function that approaches the potential of Scalar theories of gravitation Note: This article focuses on relativistic classical field theories of gravitation. The best known relativistic classical field theory of gravitation, general relativity , is a tensor theory, in which the gravitational interaction is described using a tensor field.