A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics (quantum field theories). Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Quantum field theory (QFT) is the quantum theory of fields. ...

A physical field can be thought of as the assignment of a physical quantity at each point of space and time (usually in a continuous manner). For example, on weather forecasts, the wind velocity during a day over a country is described by assigning a vector at each point of space (with moving arrows representing the change in wind velocity during the day). The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature. A physical quantity is either a quantity within physics that can be measured (e. ... Space has been an interest for philosophers and scientists for much of human history. ... This article is about the measurement concept. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... For other uses, see Gravitation (disambiguation). ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ...

Descriptions of physical fields were given before the advent of relativity theory and then revised in light of this theory. Consequently, classical field theories are usually categorised as non-relativistic and relativistic. Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ...

Non-relativistic field theories

Some of the simplest physical fields are vector force fields. Historically, the first time fields were taken seriously was with Faraday's lines of force when describing the electric field. The gravitational field was then similarly described. In physics, the faraday (not to be confused with the farad) is a unit of electrical charge; one faraday is equal to the charge of 6. ... Line of force or line of flux , usually taken in the context of electromagnetism, is the curve whose tangent gives the direction of the field at that point. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... A gravitational field is a model used within physics to explain how gravity exists in the universe. ...

Newtonian gravitation

A classical field theory describing gravity was Newtonian gravitation, which describes the gravitational force as a mutual interaction between two masses. Gravity is a force of attraction that acts between bodies that have mass. ... This article or section is in need of attention from an expert on the subject. ...

In a gravitational field, a test particle of gravitational mass m experiences a force, F. The gravitational field strength is then defined by g=F/m, whereit is required that the test mass, m, is so small that its presence effectively does not disturb the gravitational field. Newton's law of gravitation says that two masses separated by a distance, r, experience a force In metric theories of gravitation, particularly general relativity, a test particle is an idealized model of a small object whose mass is so small that it does not appreciably disturb the ambient graviational field. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ...

Using Newton's 2nd law (for constant inertial mass), F=ma leads to a definition of the gravitational field strength due to a mass m as Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle, which leads to general relativity. In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

Electrostatics

Main article: Electrostatics

A charged test particle, charge q, experiences a force, F, based solely on its charge. We can similarly describe the electric field, E, so that F=qE. Using this and Coulomb's law tells us that, we define the electric field due to a single charged particle as A student demonstrating the effects of electrostatics. ... Look up charge in Wiktionary, the free dictionary. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ...

Magnetism

Main article: Magnetism

For other senses of this word, see magnetism (disambiguation). ...

Hydrodynamics

Main article: Hydrodynamics

Hydrodynamics is fluid dynamics applied to liquids, such as water, alcohol, oil, and blood. ...

Relativistic field theory

Main article: Covariant classical field theory

Modern formulations of classical field theories generally require Lorentz covariance as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using Lagrangians. This is a function that, when subjected to an action principle, gives rise to the field equations and a conservation law for the theory. In recent years, there has been renewed interest in the covariant formalism of classical field theory. ... In physics, Lorentz covariance is a key property of spacetime that follows from the special theory of relativity, where it applies globally. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... A field equation is an equation in a physical theory that describes how a fundamental force (or a combination of such forces) interacts with matter. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

We use units where c=1 throughout.

Lagrangian dynamics

Main article: Lagrangian

Given a field tensor - which could be a tensor of any rank but to simplify matters, a scalar will be used - φ, a scalar called the Lagrangian density can be constructed from φ and its derivatives. A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

From this density, the functional action can be constructed by integrating over spacetime

Then by enforcing the action principle, the Euler-Lagrange equations are obtained In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

Relativistic fields

Two of the most well-known Lorentz covariant classical field theories are now described.

Electromagnetism

Main articles: Electromagnetic field and Electromagnetism

Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the electromagnetic field. Maxwell's theory of electromagnetism describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the electric and magnetic fields. With the advent of special relativity, a better (and more consistent with mechanics) formulation using tensor fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together is used. The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Maxwell is a common Scottish, English, or Irish name that may refer to: // Anna Maxwell (1851–1929) Augustus Maxwell (1820–1903) Blakey Harris James 2006 Colt Telecom Brian Maxwell (1953–2004) Carmen Maxwell Cedric Maxwell (born 1955) Charlie Maxwell (born 1927) David Maxwell (academic) David Maxwell Fyfe, 1st Earl of... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The article on electrical energy is located elsewhere. ... In physics, magnetism is a phenomenon by which materials exert an attractive or repulsive force on other materials. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

We have the electromagnetic potential, , and the electromagnetic four-current . The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor In theoretical physics, the electromagnetic potential is a physical quantity that unifies the electric potential and the vector potential (see also magnetic potential) into a single quantity with four components (four is the dimension of the spacetime). ... In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ... In electromagnetism, the electromagnetic tensor, or electromagnetic field tensor, F, is defined as: where Ai is the vector potential. ...

The Lagrangian

To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have We can use gauge field theory to get the interaction term, and this gives us Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

The Equations

This, coupled with the Euler-Lagrange equations, gives us the desired result, since the E-L equations say that

After some enlightening algebra, this yields

This gives us a vector equation, which are Maxwell's equations in vacuum. The other two are obtained from the fact that F is the 4-curl of A: In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...

where the comma indicates a partial derivative. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

Gravitation

Main articles: Gravitation and General Relativity

Newtonian gravitation being found to be inconsistent with special relativity, a new theory of gravitation called general relativity was formulated by Albert Einstein. This treats gravitation as a geometric phenomena ('curved spacetime') caused by masses and the gravitational field is represented mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. The field equations may be derived by using the Einstein-Hilbert action. The Lagrangian For other uses, see Gravitation (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... “Einstein” redirects here. ... For other uses, see Gravitation (disambiguation). ... For other uses of this term, see Spacetime (disambiguation). ... A gravitational field is a model used within physics to explain how gravity exists in the universe. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen...

where is the Ricci scalar written in terms of the Ricci tensor and the metric tensor , will yield the vacuum EFE: In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...

where is the Einstein tensor. Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ...

See also

• Covariant classical field theory
• Electromagnetism
• Field (physics)
• General relativity
• Quantum field theory
• Variational methods in general relativity

In recent years, there has been renewed interest in the covariant formalism of classical field theory. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Quantum field theory (QFT) is the quantum theory of fields. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

External links

• Thidé, Bo. Electromagnetic Field Theory. Retrieved on February 14, 2006.
• Carroll, Sean M. Lecture Notes on General Relativity. Retrieved on February 14, 2006.
• Binney, Prof. James J. Lecture Notes on Classical Fields. Retrieved on April 30, 2007.