HOME

TheInfoList



OR:

A classical field theory is a
physical theory Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experime ...
that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
. In most contexts, 'classical field theory' is specifically intended to describe
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and gravitation, two of the
fundamental force In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electro ...
s of nature. A physical field can be thought of as the assignment of a physical quantity at each point of
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
and
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change. The first field theories,
Newtonian gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
and
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
of electromagnetic fields were developed in classical physics before the advent of
relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as ''non-relativistic'' and ''relativistic''. Modern field theories are usually expressed using the mathematics of
tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
. A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s.


Non-relativistic field theories

Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with Faraday's
lines of force A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction. According to J.J. Thomson, Faraday usually discusses ''lines of force'' as chains of polarized particles in a dielectric, yet sometimes Faraday discusses ...
when describing the electric field. The gravitational field was then similarly described.


Newtonian gravitation

The first field theory of gravity was
Newton's theory of gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
in which the mutual interaction between two
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
es obeys an inverse square law. This was very useful for predicting the motion of planets around the Sun. Any massive body ''M'' has a gravitational field g which describes its influence on other massive bodies. The gravitational field of ''M'' at a point r in space is found by determining the force F that ''M'' exerts on a small
test mass In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insu ...
''m'' located at r, and then dividing by ''m'': \mathbf(\mathbf) = \frac. Stipulating that ''m'' is much smaller than ''M'' ensures that the presence of ''m'' has a negligible influence on the behavior of ''M''. According to
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
, F(r) is given by \mathbf(\mathbf) = -\frac\hat, where \hat is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
pointing along the line from ''M'' to ''m'', and ''G'' is Newton's gravitational constant. Therefore, the gravitational field of ''M'' is \mathbf(\mathbf) = \frac = -\frac\hat. 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 General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. For a discrete collection of masses, ''Mi'', located at points, r''i'', the gravitational field at a point r due to the masses is \mathbf(\mathbf)=-G\sum_i \frac \,, If we have a continuous mass distribution ''ρ'' instead, the sum is replaced by an integral, \mathbf(\mathbf)=-G \iiint_V \frac \, , Note that the direction of the field points from the position r to the position of the masses r''i''; this is ensured by the minus sign. In a nutshell, this means all masses attract. In the integral form
Gauss's law for gravity In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux ( surface inte ...
is \iint\mathbf\cdot d \mathbf = -4\pi G M while in differential form it is \nabla \cdot\mathbf = -4\pi G\rho_m Therefore, the gravitational field g can be written in terms of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
: \mathbf(\mathbf) = -\nabla \phi(\mathbf). This is a consequence of the gravitational force F being
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
.


Electromagnetism


Electrostatics

A charged test particle with charge ''q'' experiences a force F based solely on its charge. We can similarly describe the electric field E generated by the source charge ''Q'' so that : \mathbf(\mathbf) = \frac. Using this and
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
the electric field due to a single charged particle is \mathbf = \frac \frac \hat \,. The electric field is
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
, and hence is given by the gradient of a scalar potential, \mathbf(\mathbf) = -\nabla V(\mathbf) \, .
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
for electricity is in integral form \iint\mathbf\cdot d\mathbf = \frac while in differential form \nabla \cdot\mathbf = \frac \,.


Magnetostatics

A steady current ''I'' flowing along a path ''ℓ'' will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by ''I'' on a nearby charge ''q'' with velocity v is \mathbf(\mathbf) = q\mathbf \times \mathbf(\mathbf), where B(r) is the magnetic field, which is determined from ''I'' by the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
: \mathbf(\mathbf) = \frac \int \frac. The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
, A(r): \mathbf(\mathbf) = \nabla \times \mathbf(\mathbf)
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
for magnetism in integral form is \iint\mathbf\cdot d\mathbf = 0, while in differential form it is \nabla \cdot\mathbf = 0. The physical interpretation is that there are no
magnetic monopole In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
s.


Electrodynamics

In general, in the presence of both a charge density ''ρ''(r, ''t'') and current density J(r, ''t''), there will be both an electric and a magnetic field, and both will vary in time. They are determined by
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, a set of differential equations which directly relate E and B to the electric charge density (charge per unit volume) ''ρ'' and current density (electric current per unit area) J. Alternatively, one can describe the system in terms of its scalar and vector potentials ''V'' and A. A set of integral equations known as ''
retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so th ...
s'' allow one to calculate ''V'' and A from ρ and J, and from there the electric and magnetic fields are determined via the relations \mathbf = -\nabla V - \frac \mathbf = \nabla \times \mathbf.


Continuum mechanics


Fluid dynamics

Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation is a continuity equation, representing the conservation of mass \frac + \nabla \cdot (\rho \mathbf u) = 0 and the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid, \frac (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u + p \mathbf I) = \nabla \cdot \boldsymbol \tau + \rho \mathbf b if the density , pressure ,
deviatoric stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
of the fluid, as well as external body forces b, are all given. The
velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
u is the vector field to solve for.


Other examples

In 1839,
James MacCullagh James MacCullagh (1809 – 24 October 1847) was an Irish mathematician. Early Life MacCullagh was born in Landahaussy, near Plumbridge, County Tyrone, Ireland, but the family moved to Curly Hill, Strabane when James was about 10. He was the e ...
presented field equations to describe
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
and
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
in "An essay toward a dynamical theory of crystalline reflection and refraction".


Potential theory

The term "
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
s which satisfied Laplace's equation. Poisson addressed the question of the stability of the planetary
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
s, which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
, named after him. The general form of this equation is \nabla^2 \phi = \sigma where ''σ'' is a source function (as a density, a quantity per unit volume) and φ the scalar potential to solve for. In Newtonian gravitation; masses are the sources of the field so that field lines terminate at objects that have mass. Similarly, charges are the sources and sinks of electrostatic fields: positive charges emanate electric field lines, and field lines terminate at negative charges. These field concepts are also illustrated in the general divergence theorem, specifically Gauss's law's for gravity and electricity. For the cases of time-independent gravity and electromagnetism, the fields are gradients of corresponding potentials \mathbf = - \nabla \phi_g \,,\quad \mathbf = - \nabla \phi_e so substituting these into Gauss' law for each case obtains \nabla^2 \phi_g = 4\pi G \rho_g \,, \quad \nabla^2 \phi_e = 4\pi k_e \rho_e = - where ''ρg'' is the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, ''ρe'' the
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
, ''G'' the gravitational constant and ''ke = 1/4πε0'' the electric force constant. Incidentally, this similarity arises from the similarity between
Newton's law of gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
and
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
. In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation: \nabla^2 \phi = 0. For a distribution of mass (or charge), the potential can be expanded in a series of
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
, and the ''n''th term in the series can be viewed as a potential arising from the 2''n''-moments (see multipole expansion). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations.


Relativistic field theory

Modern formulations of classical field theories generally require
Lorentz covariance In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
s. This is a function that, when subjected to an
action principle In physics, action is a scalar quantity describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple cas ...
, gives rise to the
field equations A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
and a conservation law for the theory. The
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
is a Lorentz scalar, from which the field equations and symmetries can be readily derived. Throughout we use units such that the speed of light in vacuum is 1, i.e. ''c'' = 1.


Lagrangian dynamics

Given a field tensor \phi, a scalar called the
Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
\mathcal(\phi,\partial\phi,\partial\partial\phi, \ldots ,x)can be constructed from \phi and its derivatives. From this density, the action functional can be constructed by integrating over spacetime, \mathcal = \int. Where \sqrt \, \mathrm^4x is the volume form in curved spacetime. (g\equiv \det(g_)) Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space. Then by enforcing the
action principle In physics, action is a scalar quantity describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple cas ...
, the Euler–Lagrange equations are obtained \frac = \frac -\partial_\mu \left(\frac\right)+ \cdots +(-1)^m\partial_ \partial_ \cdots \partial_ \partial_ \left(\frac\right) = 0.


Relativistic fields

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


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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and magnetic fields. With the advent of special relativity, a more complete formulation using
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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 four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
is defined to be , and the electromagnetic four-current . The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor F_ = \partial_a A_b - \partial_b A_a.


The Lagrangian

To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have \mathcal = -\fracF^F_\,. We can use
gauge field theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
to get the interaction term, and this gives us \mathcal = -\fracF^F_ - j^aA_a\,.


The equations

To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potential ''A'', and it's this potential which enters the Euler-Lagrange equations. The EM field ''F'' is not varied in the EL equations. Therefore, \partial_b\left(\frac\right)=\frac \,. Evaluating the derivative of the Lagrangian density with respect to the field components \frac = \mu_0 j^a \,, and the derivatives of the field components \frac = F^ \,, obtains
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are \partial_b F^=\mu_0 j^a \, . while the other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that ''F'' is the 4-curl of ''A'', or, in other words, from the fact that the
Bianchi identity In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie alge ...
holds for the electromagnetic field tensor. 6F_ \, = F_ + F_ + F_ = 0. where the comma indicates a partial derivative.


Gravitation

After Newtonian gravitation was found to be inconsistent with
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
formulated a new theory of gravitation called
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. This treats gravitation as a geometric phenomenon ('curved
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
') caused by masses and represents the gravitational field mathematically by a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
called the metric tensor. The
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
describe how this curvature is produced.
Newtonian gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
is now superseded by Einstein's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, in which gravitation is thought of as being due to a curved
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, caused by masses. The Einstein field equations, G_ = \kappa T_ describe how this curvature is produced by matter and radiation, where ''Gab'' is the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field ...
, G_ \, = R_-\frac R g_ written in terms of the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
''Rab'' and
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
, is the stress–energy tensor and is a constant. In the absence of matter and radiation (including sources) the ' vacuum field equations'', G_ = 0 can be derived by varying the
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
, S = \int R \sqrt \, d^4x with respect to the metric, where ''g'' is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the metric tensor ''gab''. Solutions of the vacuum field equations are called
vacuum solution In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or no ...
s. An alternative interpretation, due to Arthur Eddington, is that R is fundamental, T is merely one aspect of R, and \kappa is forced by the choice of units.


Further examples

Further examples of Lorentz-covariant classical field theories are * Klein-Gordon theory for real or complex scalar fields *
Dirac Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
theory for a Dirac spinor field *
Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
for a non-abelian gauge field


Unification attempts

Attempts to create a unified field theory based on classical physics are classical unified field theories. During the years between the two World Wars, the idea of unification of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
with
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
was actively pursued by several mathematicians and physicists like
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
,
Theodor Kaluza Theodor Franz Eduard Kaluza (; 9 November 1885 – 19 January 1954) was a German mathematician and physicist known for the Kaluza–Klein theory, involving field equations in five-dimensional space-time. His idea that fundamental forces can be ...
, Hermann Weyl, Arthur Eddington,
Gustav Mie Gustav Adolf Feodor Wilhelm Ludwig Mie (; 29 September 1868 – 13 February 1957) was a German physicist. Life Mie was born in Rostock, Mecklenburg-Schwerin, Germany in 1868. From 1886 he studied mathematics and physics at the University of ...
and Ernst Reichenbacher. Early attempts to create such theory were based on incorporation of electromagnetic fields into the geometry of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. In 1918, the case for the first geometrization of the electromagnetic field was proposed in 1918 by Hermann Weyl. In 1919, the idea of a five-dimensional approach was suggested by
Theodor Kaluza Theodor Franz Eduard Kaluza (; 9 November 1885 – 19 January 1954) was a German mathematician and physicist known for the Kaluza–Klein theory, involving field equations in five-dimensional space-time. His idea that fundamental forces can be ...
. From that, a theory called Kaluza-Klein Theory was developed. It attempts to unify gravitation and
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, in a five-dimensional space-time. There are several ways of extending the representational framework for a unified field theory which have been considered by Einstein and other researchers. These extensions in general are based in two options. The first option is based in relaxing the conditions imposed on the original formulation, and the second is based in introducing other mathematical objects into the theory. An example of the first option is relaxing the restrictions to four-dimensional space-time by considering higher-dimensional representations. That is used in Kaluza-Klein Theory. For the second, the most prominent example arises from the concept of the
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
that was introduced into the theory of general relativity mainly through the work of
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
and Hermann Weyl. Further development of quantum field theory changed the focus of searching for unified field theory from classical to quantum description. Because of that, many theoretical physicists gave up looking for a classical unified field theory. Quantum field theory would include unification of two other fundamental forces of nature, the strong and
weak nuclear force In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
which act on the subatomic level.


See also

*
Relativistic wave equations In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
* Quantum field theory *
Classical unified field theories Since the 19th century, some physicists, notably Albert Einstein, have attempted to develop a single theoretical framework that can account for all the fundamental forces of nature – a unified field theory. Classical unified field theories are at ...
* Variational methods in general relativity *
Higgs field (classical) Spontaneous symmetry breaking, a vacuum Higgs field, and its associated fundamental particle the Higgs boson are quantum phenomena. A vacuum Higgs field is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interact ...
*
Lagrangian (field theory) Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
*
Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory ...
*
Covariant Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory ...


Notes


References


Citations


Sources

*


External links

* * * * {{DEFAULTSORT:Classical field theory Mathematical physics Theoretical physics Lagrangian mechanics Equations