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Four dimensional field formulation

Equations in Gaussian units convention

Name	Integral equations	Differential equations	Meaning
Gauss's law	${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} =4\pi \iiint _{\Omega }\rho \,\mathrm {d} V$	$\nabla \cdot \mathbf {E} =4\pi \rho$	There are no magnetic monopoles; the total magnetic flux through a closed surface is zero.
Gauss's law for magnetism	${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$	There are no magnetic monopoles; the total magnetic flux through a closed surface is zero.
Maxwell–Faraday equation (Faraday's law of induction)	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } =-{\frac {1}{c}}{\frac {d}{dt}}\iint \mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$	The voltage induced in a closed circuit is proportional to the rate of change of the magnetic flux it encloses.
Ampère's law (with Maxwell's extension)	$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {\ell } ={\frac {1}{c}}\left(4\pi \iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\frac {d}{dt}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)$	$\nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$	Electric currents and changes in electric fields are proportional to the magnetic fields circulating about the areas where they accumulate.

Equations in Gaussian units convention

Name	Microscopic equations	Macroscopic equations
Gauss's law	$\nabla \cdot \mathbf {E} =4\pi \rho$	$\nabla \cdot \mathbf {D} =4\pi \rho _{\text{f}}$
Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$
Ampère's law (with Maxwell's extension)	$\nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$	$\nabla \times \mathbf {H} ={\frac {1}{c}}\left(4\pi \mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}\right)$

Four dimensional field formulation

With the benefit of hindsight, the split between electric and magnetic field and their intertwinedness in the Maxwell equations is a consequence of a a split in space and time. Consider spacetime and choose coordinates $x^{\mu }=(x^{0},\ldots ,x^{3})=(ct,x,y,z)$. We can assume that the metric tensor i.e. such that the metric tensor has o We can assume that the

Alternative formulations of Maxwell's equations

Main articles: Mathematical descriptions of the electromagnetic field and classical electromagnetism and special relativity

Following is a summary of the numerous other ways to write the "microscopic" equations. See the main articles for the details of each formulation. SI units and conventions are used all over.

Four dimensional field formulation

The four dimensional formulation is formulated on spacetime rather than space and time seperately and is manifestly Lorentz invariant. Properly formulated they are even general coordinate invariant. The four dimensional formulation of the Maxwell equations was first given by Minkowski in 19XXXX shortly after the introduction of special relativity by Einstein. Historically, Lorentz invariance was discovered by studying the full invariance properties of the conventional Maxwell equations and was a starting point of Einsteins theory of special relativity. With the benefit of hindsight, however, the conventional equations can be considered as a specialisation of the four dimensional formulation in a frame of reference that singles out the time direction, obscuring their natural invariance.

The basic quantity in the four dimensional formulation is the Faraday tensor $F_{\alpha \beta }$. This is an antisymmetic covariant tensor that supersedes both the electric field E and the magnetic field B. Spacetime implicitly has a Lorentzian metric $g$ or $\eta$, that may or may not depend depend on the coordinates respectively. For orthonormal coordinates, $x^{\alpha }=(x^{0},\ldots ,x^{3})=(ct,x,y,z)$ with a constant metric tensor $\eta _{\alpha ,\beta }=diag(1,-1,-1,-1)$ we have $E_{i}/c=F_{0i}$ and $B_{i}=\epsilon ^{ijk}F_{jk}$ i.e.

$F_{\alpha \beta }=\left({\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{matrix}}\right)\,$

The four-current $J^{\alpha }$ is a contravariant four-vector density which supersedes electric current density J and electric charge density ρ. In orthonormal coordinates it is given by

$J^{\alpha }=\,(c\rho ,-\mathbf {J} )\,$

A modern coordinate free notation for the Faraday tensor is the Faraday twoform $F=F_{\alpha \beta }dx^{\alpha }\wedge dx^{\beta }$. In orthogonal coordinates

$F=E_{x}dt\wedge dx+E_{y}dt\wedge dy+E_{z}dt\wedge dz+B_{x}dy\wedge dz+B_{y}dz\wedge dx+B_{z}dx\wedge dy$

The four current is conveniently expressed as the three form<footnote> Conventions can differ as to where to put the factor c. If we define the Coulomb as an essentially dimensionless of electron charges (as it soon will be) the vectorfield $J^{\alpha }$ has the dimension of velocity while the three form j is dimensionless</footnote>

$\mathbf {j} ={\frac {1}{c}}J^{\alpha }{\sqrt {-g}}\epsilon _{\alpha \beta \gamma \delta }dx^{\beta }\wedge dx^{\gamma }\wedge dx^{\delta }$

i.e. in conventional coordinates

$\mathbf {j} =\rho dx\wedge dy\wedge dz+J_{x}dt\wedge dy\wedge dz+J_{y}dt\wedge dz\wedge dx+J_{z}dt\wedge dx\wedge dy$

A three form can be integrated over a three dimensional region in spacetime e.g. to get an amount of charge in a spatial region or the amount of charge flowing through a two dimensional surface during some period of time.

Formulation	Homogeneous equations		Nonhomogeneous equations
Tensor calculus	Failed to parse (syntax error): {\displaystyle \partial_{[\alpha F_{\beta\gamma]} = 0 }		$\nabla _{\alpha }F^{\beta \alpha }=\mu _{0}J^{\beta }$
Differential forms	$\mathrm {d} F=0$		$\mathrm {d} \,{\star F}={\frac {1}{\epsilon _{0}}}\mathbf {j}$

Here $[\ ]$ means antisymmetrisation $d$ is the exterior derivative, $\nabla$ is the covariant derivative (Levi Civita connection) and $\star$ is the Hodge star of the Lorentzian metric. If the metric Failed to parse (syntax error): {\displaystyle g = \eta <math> is coordinate independant, we can replace the covariant derivative with an ordinary derivative. Note that the homogeneous equations are metric independent, and the form version is conformally invariant. ===Potential formulation=== In this formulation the homogeneous equations are solved automatically by the introduction of potentials, with the fields being gradients or rotations in three dimensions or an antisymmetrised/exterior derivative in four dimensions. :{|class="wikitable" style="text-align: center;" |- !scope="column" width="160px"|Formulation !colspan="2"| Homogeneous equations !colspan="2"| Nonhomogeneous equations |- ![[Vector calculus]] (fields) || <math>\nabla\cdot\mathbf{B}=0} || $\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0$ || $\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$|| $\nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {J}$ |- !Vector calculus (potentials, any gauge) |colspan="2"| identities || $\nabla ^{2}\varphi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}$ || $\Box \mathbf {A} +\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=\mu _{0}\mathbf {J}$ |- !Tensor calculus (potentials, Lorenz gauge) |colspan="2"| identities |colspan="2"| $\Box A^{\mu }=\mu _{0}J^{\mu }$ }

! QED, vector calculus (potentials, Lorenz gauge) |colspan="2"| identities || $\Box \varphi =-{\frac {1}{\varepsilon _{0}}}e\psi ^{\dagger }\psi$|| $\Box \mathbf {A} =-\mu _{0}e\psi ^{\dagger }{\boldsymbol {\alpha }}\psi$ |-

where

$\Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}$

is the D'Alembert operator. Following are the reasons for using such formulations:

• Potential formulation approach: In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a potential formulation involving the electric potential (also called scalar potential) φ, and the magnetic potential A, (also called vector potential). These are defined such that:

$\mathbf {E} =-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .$

Many different choices of A and φ are consistent with a given E and B, making these choices physically equivalent – a flexibility known as gauge freedom. Suitable choice of A and φ can simplify these equations, or can adapt them to suit a particular situation.

• Manifestly covariant (tensor) approach: Maxwell's equations are exactly consistent with special relativity—i.e., if they are valid in one inertial reference frame, then they are automatically valid in every other inertial reference frame. In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation that involves a seemingly miraculous cancellation of different terms.

For example, consider a conductor moving in the field of a magnet. In the frame of the magnet, that conductor experiences a magnetic force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an electric field. The motion is exactly consistent in these two different reference frames, but it mathematically arises in quite different ways.

For this reason and others, it is often useful to rewrite Maxwell's equations in a way that is "manifestly covariant"—i.e. obviously consistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors. This can be done using the EM tensor F, or the 4-potential A, with the 4-current J – see covariant formulation of classical electromagnetism.

• Differential forms approach: Gauss's law for magnetism and the Faraday–Maxwell law can be grouped together since the equations are homogeneous, and be seen as geometric identities expressing the field F (a 2-form), which can be derived from the 4-potential A. Gauss's law for electricity and the Ampere–Maxwell law could be seen as the dynamical equations of motion of the fields, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal action, see electromagnetic tensor.

Often, the time derivative in the Faraday–Maxwell equation motivates calling this equation "dynamical", which is somewhat misleading in the sense of the preceding analysis. This is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F *F for A; and take into account the non-physical degrees of freedom which can be removed by gauge transformation A → A' = A − dα. See also gauge fixing and Faddeev–Popov ghosts.

• Geometric algebra summarizes the entire content of Maxwell's equations into a single equation, using the Riemann–Silberstein multivector F and the four-current J.