Definitions

# Field equation

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. The four fundamental forces are the gravitational force, the electromagnetic force, the strong force and the weak force.

Before the theory of quantum mechanics was fully developed, there were two known field theories, namely gravitation and electromagnetism (these two are 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).

Modern field equations tend to be tensor equations.

## Newton's theory of universal 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

$vec\left\{g\right\}=-G_csum_i frac\left\{M_i\left(vec\left\{r\right\}-vec\left\{r_i\right\}\right)\right\}\left\{|vec\left\{r\right\}-vec\left\{r\right\}_i|^3\right\},$

where Gc 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.

## Poisson's equation

The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from 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 for the disturbing forces and created Poisson's equation:

$\left\{nabla\right\}^2 Phi = - \left\{rho over epsilon_0\right\}$

To understand where this equation comes from, we need to examine the form and source of the force fields. We recognise that charges are the sources and sinks of electrostatic fields: positive charges emanate electric field lines, and field lines terminate at negative charges. Similarly, in Newton's gravitation masses are the sources of the field so that field lines terminate at objects that have mass. Formalised Gauss' Law for electric fields (using the more general divergence theorem):

$iintvec\left\{E\right\}cdotvec\left\{dS\right\} = frac\left\{q_e\right\}\left\{epsilon_0\right\} Rightarrow vec\left\{nabla\right\}cdotvec\left\{E\right\}=frac\left\{rho_e\right\}\left\{epsilon_0\right\}$

and for masses

$iintvec\left\{g\right\}cdotvec\left\{dS\right\} = -4pi G_c m Rightarrow vec\left\{nabla\right\}cdotvec\left\{g\right\}=-4pi G_crho_m$

where ρe and ρm represents the charge and mass densities respectively. Incidentally, this similarity arises from the similarity of the form of Newton's law of gravitation and Coulomb's law.

Since the force fields are related to their potentials by the gradient:

$vec\left\{E\right\}=-vec\left\{nabla\right\}phi_e,,,,vec\left\{g\right\}=-vec\left\{nabla\right\}phi_g,$

we can substitute the potential for the field to get Poisson's equation:

$nabla^2phi_e = -frac\left\{rho\right\}\left\{epsilon_0\right\},$

$nabla^2phi_g = 4pi G_c rho.$

### Laplace's equation

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

When it was realised that Lorentz invariance is an essential feature of nature, it became desirable to model everything as a relativistic field. This could be conveniently done 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.

## A scalar particle with non-zero mass

$hbar^2\left(frac\left\{1\right\}\left\{tau^2\right\}frac\left\{partial^2\right\}\left\{partial v^2\right\} - nabla^2\right) Psi = M^2 c^2 Psi$

This is the familiar Klein–Gordon equation in the four dimensions of space and velocity.

## 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:

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

This arises from the following Lagrangian

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

## Einstein's field equation

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 - which describes how this curvature is produced by masses - is:

$G_\left\{ab\right\} , = kappa T_\left\{ab\right\}.$

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

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

## Vacuum field equations

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