In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.
For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space.
Modern field theories are usually expressed using the mathematics of tensor calculus.
Michael Faraday coined the term "field" and lines of forces to explain electric and magnetic phenomena.
Lord Kelvin in 1851 formalized the concept of field in different areas of physics.
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 m located at r, and then dividing by m:[1]
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, F(r) is given by[1]
is a unit vector pointing along the line from M to m, and G is Newton's gravitational constant.
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.
We can similarly describe the electric field E generated by the source charge Q so that F = qE:
Using this and Coulomb's law the electric field due to a single charged particle is
The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential.
They are determined by Maxwell's equations, 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.
A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J,[note 1] and from there the electric and magnetic fields are determined via the relations[3]
Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum.
if the density ρ, pressure p, deviatoric stress tensor τ of the fluid, as well as external body forces b, are all given.
where σ is a source function (as a density, a quantity per unit volume) and ø the scalar potential to solve for.
These field concepts are also illustrated in the general divergence theorem, specifically Gauss's law's for gravity and electricity.
In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation:
Modern formulations of classical field theories generally require Lorentz covariance as this is now recognised as a fundamental aspect of nature.
This is a function that, when subjected to an action principle, gives rise to the field equations and a conservation law for the theory.
The action is a Lorentz scalar, from which the field equations and symmetries can be readily derived.
With the advent of special relativity, a more complete formulation using tensor fields was found.
describe how this curvature is produced by matter and radiation, where Gab is the Einstein tensor,
During the years between the two World Wars, the idea of unification of gravity with electromagnetism was actively pursued by several mathematicians and physicists like Albert Einstein, Theodor Kaluza,[6] Hermann Weyl,[7] Arthur Eddington,[8] Gustav Mie[9] and Ernst Reichenbacher.
[10] Early attempts to create such theory were based on incorporation of electromagnetic fields into the geometry of general relativity.
There are several ways of extending the representational framework for a unified field theory which have been considered by Einstein and other researchers.
[11] An example of the first option is relaxing the restrictions to four-dimensional space-time by considering higher-dimensional representations.
For the second, the most prominent example arises from the concept of the affine connection that was introduced into the theory of general relativity mainly through the work of Tullio Levi-Civita and Hermann Weyl.