# Field equation

In physics, a field equation is a mathematical statement describing how the fundamental forces interact with matter and energy. The four fundamental forces are gravitation, electromagnetism, the strong interaction and the weak interaction.

Before the theory of quantum mechanics was fully developed, there were two existing field theories, namely gravitation and electromagnetism (both sometimes referred to as classical field theories, as they were formulated before the advent of quantum mechanics, and hence do not take into account quantum phenomena).

In 1839 James MacCullagh presented field equations to describe reflection and refraction in "An essay toward a dynamical theory of crystalline reflection and refraction".[1]

Modern field equations tend to be tensor equations.

## Potential theory

The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from scalar potentials which satisfied Laplace's equation. Poisson addressed the question of the stability of the planetary orbits, which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the equation (named Poisson's equation after him):

${\nabla}^2 \Phi = - 4\pi G \rho_g \,, \quad {\nabla}^2 \Phi = - {\rho_e \over \epsilon_0}$

where ρg and ρe represent the mass density and charge densities for gravitational and electrostatic fields, respectively.

To understand where this equation comes from, we need to examine the form and source of the force fields. We recognise that 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 for gravity is

$\iint\bold{g}\cdot{\rm d}\bold{S} = -4\pi G m \Rightarrow \bold{\nabla}\cdot\bold{g}=-4\pi G\rho_m$

and for electric fields:

$\iint\bold{E}\cdot {\rm d}\bold{S} = \frac{q_e}{\epsilon_0} \Rightarrow \bold{\nabla}\cdot\bold{E}=\frac{\rho_e}{\epsilon_0}$

Incidentally, this similarity arises from the similarity between Newton's law of gravitation and Coulomb's law. Since the force fields are related to their potentials by the gradient:

$\bold{g}=-\bold{\nabla}\phi_g\,\quad \bold{E}=-\bold{\nabla}\phi_e\quad$

we can substitute the potential for the field to get the Poisson's equations above.

In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation:

$\nabla^2 \phi = 0$

## Relativistic fields

Because of Lorentz covariance, in contemporary research, it is often desirable to model everything as a relativistic field. This is convenient under the formalism of relativistic (or covariant) classical field theory.

This works by finding a Lorentz scalar, the Lagrangian density, from which the field equations and symmetries can be readily derived.

## Gravitation

### Newtonian (classical) gravitation

The first field theory of gravity was Newton's theory of gravitation, which described gravity as obeying an inverse square law. This was very useful in describing the motion of planets around the Sun.

The gravitational field at the point r due to several masses, Mi, located at points, ri, is given by

$\bold{g}=-G\sum_i \frac{M_i(\bold{r}-\bold{r_i})}{|\bold{r}-\bold{r}_i|^3},$

where G is Newton's gravitational constant. Note that the direction of the field points from the position r to the position of the masses ri; this is ensured by the minus sign. In a nutshell, this means all masses attract.

### General relativity

Newtonian gravitation is now superseded by Einstein's theory of general relativity, in which gravitation is thought of as being due to a curved spacetime, caused by masses. The Einstein field equation describes how this curvature is produced by masses:

$G_{ab} = \kappa T_{ab}.$

where κ = 8πG/c4 is a constant (which appears in the Einstein field equations, and not the action).

The vacuum solution can be obtained by varying the following Einstein–Hilbert action with respect to the metric

$S[g]=k \int R \sqrt{-g} \, d^4x$

Vacuum field equations are the field equations written without matter (including sources). Solutions of the vacuum field equations are called vacuum solutions.

## Electromagnetism

### Maxwell's equations

Main article: Maxwell's equations

The electromagnetic force is best described by Maxwell's theory of electromagnetism. The field equations of classical electromagnetism are Maxwell's equations which describe how electromagnetic fields are produced from charged particles and are written in the framework of special relativity (which was devised to consistently describe electromagnetism and classical mechanics) as:

$F^{ab}{}_{,a} \, =k J^b$

This arises from the following Lagrangian

$\mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab} + j^aA_a.$

## Unification attempts

The Kaluza–Klein theory attempts to unify gravitation and electromagnetism, in a five-dimensional spacetime.