where ρg is the mass density, ρe the charge density, 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 and Coulomb's law. In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation: For a distribution of mass (or charge), the potential can be expanded in a series of spherical harmonics, and the nth term in the series can be viewed as a potential arising from the 2n-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 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. The action 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 , a scalar called the Lagrangian densitycan be constructed from and its derivatives. From this density, the action functional can be constructed by integrating over spacetime, Where is the volume form in curved spacetime. Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space. Then by enforcing the action principle, the Euler–Lagrange equations are obtained 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 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 more complete 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 four-potential is defined to be , and