Einstein-Maxwell equations

Equations best understood by starting with the Maxwell equations and then adding to them the flavour of Einstein's theory of relativity (cf. also Relativity theory).

The connection between electricity and magnetism (and hence the term "electromagnetism" ; cf. also Electromagnetism) was first observed by H.C. Oersted in 1820, when he showed that an electric current in a wire can affect a compass needle. The Ampere law is a mathematical formulation of the magnetic effect of a changing electric current. Later, in 1831, M. Faraday proved the law of induction: A changing magnetic field has an electrical effect. J.C. Maxwell (1831–1879) connected the equations of his predecessors and formulated a set of the following four equations, called the Maxwell equations:

a) $\nabla \cdot E = \rho$ (Gauss' law);

b) $\nabla \times H = \frac { 1 } { c } \left( \frac { \partial E } { \partial t } + J \right)$ (Ampere's law);

c) $\nabla \times E = - \frac { 1 } { c } \frac { \partial H } { \partial t }$ (Faraday's law);

d) $\nabla \cdot H = 0$ ( "no magnetic monopole" ).

The symbols used in these equations have their usual meanings ($\rho$ is the charge density; $J$ is the current density; $E$ is the electric field vector; $H$ is the magnetic field vector; $c$ is the speed of light; $\nabla\cdot$ and $\nabla \times$ are the divergence and curl operators.)

Items a)–d) describe electromagnetic waves and their propagation in various media. Their application includes devices such a radio, TV, microwave radar, microscope, telescope, etc. Maxwell's equations play the same role in electromagnetism that Newton's laws of motion do in mechanics. See [a1], [a2], [a7], [a8] and also Newton laws of mechanics.

Einstein's special theory of relativity owes its origin principally to Maxwell's theory of electromagnetic fields, [a5], [a6]. The basic characteristic of Einstein's theory is the Lorentz transformation, which relates scientific measurements in one frame of reference to another. The theory establishes a similarity between time and space, and has been applied to electromagnetism. In fact, a magnetic field in one coordinate system appears simultaneously in another coordinate system (in relative motion) as an electric field and vice versa. Maxwell's equations, together with the transformations of the special theory of relativity, are referred to as the Einstein–Maxwell equations.

Einstein's equations, describing constraints on the possible curvature of space-time, are:

\begin{equation*} R _ { \mu \nu } - \frac { 1 } { 2 } R g _ { \mu \nu } - \Lambda g _ { \mu \nu } = \chi T _ { \mu \nu }. \end{equation*}

These equations are invariant under any continuous and differentiable transformations of coordinate systems [a3]. Maxwell's equations can also be transformed accordingly.

One defines a contravariant vector $S ^ { \sigma }$ as $S ^ { \sigma } = ( \rho , J / c )$ for $\sigma = 0,1,2,3$, and a contravariant tensor $F^{\mu \nu}$ as

\begin{equation*} F ^ { \mu \nu } = \left( \begin{array} { c c c c } { 0 } & { E _ { x } } & { E _ { y } } & { E _ { z } } \\ { - E _ { x } } & { 0 } & { H _ { z } } & { - H _ { y } } \\ { - E _ { y } } & { - H _ { z } } & { 0 } & { H _ { x } } \\ { - E _ { z } } & { H _ { y } } & { - H _ { x } } & { 0 } \end{array} \right), \end{equation*}

where $E = ( E _ { x } , E _ { y } , E _ { z } )$ and $H = ( H _ { x } , H _ { y } , H _ { z } )$. Since the distance in four-dimensional space satisfies the equation

\begin{equation*} ( d \sigma ) ^ { 2 } = g _ { \mu \nu } d x ^ { \mu } d x ^ { \nu }, \end{equation*}

one obtains the metric tensor

\begin{equation*} g _ { \mu \nu } = \left( \begin{array} { c c c c } { 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } \end{array} \right). \end{equation*}

Now, if one sets $F _ { \mu \nu } = g _ { \mu \alpha } g _ { \nu \beta } F ^ { \alpha \beta }$, then it can be seen that Maxwell's equations a)–b) take the from $F _ { ,\nu } ^ { \mu \nu } = S ^ { \mu }$ for $\mu = 0,1,2,3$. Here, the symbol ",n" stands for covariant tensor differentiation. If the symbol ",,n" denotes the usual partial differentiation, then in the case of special relativity,

\begin{equation} \tag{a1} F ^ { \mu \nu_{ , \nu} } = F ^ { \mu \nu_{ , , \nu}} = S ^ { \mu }. \end{equation}

Finally the internal equations c)–d) take the form:

\begin{equation} \tag{a2} \frac { \partial F _ { \mu \nu } } { \partial x ^ { \sigma } } + \frac { \partial F _ { \nu \sigma } \sigma } { \partial x ^ { \mu } } + \frac { \partial F _ { \sigma \mu } } { \partial x ^ { \nu } } = 0. \end{equation}

These equations together are the Einstein–Maxwell equations. In a nutshell, they are the Maxwell equations of electromagnetism, viewed from a different frame of reference. A detailed derivation can be found in [a4], [a5], [a6].

References

[a1]	D. Halliday, R. Resnick, "Physics" , 2 , Wiley (1978) (Edition: Third)
[a2]	Lucas, Hodgson, "Spacetime and electromagnetism" , Oxford Univ. Press (1990)
[a3]	L. Nottale, "Fractal space-time and microphysics" , World Sci. (1993)
[a4]	C.W. Kilmister, "General theory of relativity" , Pergamon (1973)
[a5]	G. Tauber, "Albert Einstein's theory of general relativity" , Crown Publ. (1979)
[a6]	E. Zahar, "Einstein's revolution" , Open Court (1989)
[a7]	A. Kyrala, "Theoretical physics: Applications of vectors, matrices, tensors and quaternions" , Saunders (1967)
[a8]	Marshall, D. Zohar, "Who's afraid of Schrodinger's cat?" , William Morrow (1997)