Inphysics,Maxwell's equations in curved spacetime govern the dynamics of theelectromagnetic field incurvedspacetime (where themetric may deviate from theMinkowski metric) or where one uses an arbitrary (not necessarilyCartesian) coordinate system. These equations can be viewed as a generalization of thevacuum Maxwell's equations which are normally formulated in thelocal coordinates offlat spacetime. But becausegeneral relativity dictates that the presence of electromagnetic fields (orenergy/matter in general) induce curvature in spacetime,^[1] Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

When working in the presence of bulk matter, distinguishing between free and bound electric charges may facilitate analysis. When the distinction is made, they are called themacroscopic Maxwell's equations. Without this distinction, they are sometimes called the "microscopic" Maxwell's equations for contrast.

The electromagnetic field admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flatMinkowski space when using local coordinates that are not rectilinear. For example, the equations in this article can be used to write Maxwell's equations inspherical coordinates. For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as aspecial case of the general formulation.

Ingeneral relativity, themetric tensor${\displaystyle g_{\alpha \beta }}$ is no longer a constant (like${\displaystyle {\eta }_{\alpha \beta }}$ as inExamples of metric tensor) but can vary in space and time, and the equations of electromagnetism in vacuum become^{[citation needed]}

where${\displaystyle f_{\mu }}$ is the density of theLorentz force,${\displaystyle g^{\alpha \beta }}$ is the inverse of themetric tensor${\displaystyle g_{\alpha \beta }}$, and${\displaystyle g}$ is thedeterminant of the metric tensor. Notice that${\displaystyle A_{\alpha }}$ and${\displaystyle F_{\alpha \beta }}$ are (ordinary) tensors, while⁠${\displaystyle {\mathcal{D}}^{\mu \nu }}$⁠,⁠${\displaystyle J^{\nu }}$⁠, and${\displaystyle f_{\mu }}$ aretensordensities of weight +1. Despite the use ofpartial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus, if one replaced the partial derivatives withcovariant derivatives, the extra terms thereby introduced would cancel out (seeManifest covariance § Example).

Theelectromagnetic potential is a covariant vectorA_α, which is the undefined primitive of electromagnetism. Being a covariant vector, its components transform from one coordinate system to another according to

${\displaystyle {\overline{A}}_{\beta }(\overline{x})=\frac{\mathrm{\partial }{x}^{\gamma }}{\mathrm{\partial }{\overline{x}}^{\beta }}{A}_{\gamma }(x).}$

Theelectromagnetic field is a covariantantisymmetric tensor of degree 2, which can be defined in terms of the electromagnetic potential by$${\displaystyle F_{\alpha \beta }={\mathrm{\partial }}_{\alpha }{A}_{\beta }-{\mathrm{\partial }}_{\beta }{A}_{\alpha }.}$$

To see that this equation is invariant, we transform the coordinates as described in theclassical treatment of tensors:

This definition implies that the electromagnetic field satisfies$${\displaystyle {\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }=0,}$$which incorporatesFaraday's law of induction andGauss's law for magnetism. This is seen from$${\displaystyle {\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }={\mathrm{\partial }}_{\lambda }{\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\lambda }{\mathrm{\partial }}_{\nu }{A}_{\mu }+{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }{A}_{\lambda }-{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\lambda }{A}_{\nu }+{\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\lambda }{A}_{\mu }-{\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\mu }{A}_{\lambda }=0.}$$

Thus, the right-hand side of that Maxwell law is zero identically, meaning that the classic EM field theory leaves no room for magnetic monopoles or currents of such to act as sources of the field.

Although there appear to be 64 equations in Faraday–Gauss, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field, one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with {λ,μ,ν} being either {1, 2, 3}, {2, 3, 0}, {3, 0, 1}, or {0, 1, 2}.

The Faraday–Gauss equation is sometimes written$${\displaystyle F_{[\mu \nu ;\lambda ]}=F_{[\mu \nu ,\lambda ]}=\frac{1}{6}({\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }-{\mathrm{\partial }}_{\lambda }{F}_{\nu \mu }-{\mathrm{\partial }}_{\mu }{F}_{\lambda \nu }-{\mathrm{\partial }}_{\nu }{F}_{\mu \lambda })=\frac{1}{3}({\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu })=0,}$$where a semicolon indicates a covariant derivative, a comma indicates a partial derivative, and square brackets indicate anti-symmetrization (seeRicci calculus for the notation). The covariant derivative of the electromagnetic field is$${\displaystyle F_{\alpha \beta ;\gamma }=F_{\alpha \beta ,\gamma }-{{\mathrm{\Gamma }}^{\mu }}_{\alpha \gamma }{F}_{\mu \beta }-{{\mathrm{\Gamma }}^{\mu }}_{\beta \gamma }{F}_{\alpha \mu },}$$where Γ^α_βγ is theChristoffel symbol, which is symmetric in its lower indices.

Theelectric displacement fieldD and theauxiliary magnetic fieldH form an antisymmetric contravariant rank-2tensor density of weight +1. In vacuum, this is given by

${\displaystyle {\mathcal{D}}^{\mu \nu }=\frac{1}{{\mu }_0}{g}^{\mu \alpha }{F}_{\alpha \beta }{g}^{\beta \nu }\frac{\sqrt{-g}}{c}.}$

This equation is the only place where the metric (and thus gravity) enters into the theory of electromagnetism. Furthermore, the equation is invariant under a change of scale, that is, multiplying the metric by a constant has no effect on this equation. Consequently, gravity can only affect electromagnetism by changing thespeed of light relative to the global coordinate system being used. Light is only deflected by gravity because it is slower near massive bodies. So it is as if gravity increased the index of refraction of space near massive bodies.

More generally, in materials where themagnetization–polarization tensor is non-zero, we have

${\displaystyle {\mathcal{D}}^{\mu \nu }=\frac{1}{{\mu }_0}{g}^{\mu \alpha }{F}_{\alpha \beta }{g}^{\beta \nu }\frac{\sqrt{-g}}{c}-{\mathcal{M}}^{\mu \nu }.}$

The transformation law for electromagnetic displacement is

${\displaystyle {\overline{\mathcal{D}}}^{\mu \nu }=\frac{\mathrm{\partial }{\overline{x}}^{\mu }}{\mathrm{\partial }{x}^{\alpha }}\frac{\mathrm{\partial }{\overline{x}}^{\nu }}{\mathrm{\partial }{x}^{\beta }}{\mathcal{D}}^{\alpha \beta }det\left[\frac{\mathrm{\partial }{x}^{\sigma }}{\mathrm{\partial }{\overline{x}}^{\rho }}\right],}$

where theJacobian determinant is used. If the magnetization-polarization tensor is used, it has the same transformation law as the electromagnetic displacement.

The electric current is the divergence of the electromagnetic displacement. In vacuum,

${\displaystyle J^{\mu }={\mathrm{\partial }}_{\nu }{\mathcal{D}}^{\mu \nu }.}$

If magnetization–polarization is used, then this just gives the free portion of the current

${\displaystyle J_{\text{free}}^{\mu }={\mathrm{\partial }}_{\nu }{\mathcal{D}}^{\mu \nu }.}$

This incorporatesAmpere's law andGauss's law.

In either case, the fact that the electromagnetic displacement is antisymmetric implies that the electric current is automatically conserved:

${\displaystyle {\mathrm{\partial }}_{\mu }{J}^{\mu }={\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }{\mathcal{D}}^{\mu \nu }=0,}$

because the partial derivativescommute.

The Ampere–Gauss definition of the electric current is not sufficient to determine its value because the electromagnetic potential (from which it was ultimately derived) has not been given a value. Instead, the usual procedure is to equate the electric current to some expression in terms of other fields, mainly the electron and proton, and then solve for the electromagnetic displacement, electromagnetic field, and electromagnetic potential.

The electric current is a contravariant vector density, and as such it transforms as follows:

${\displaystyle {\overline{J}}^{\mu }=\frac{\mathrm{\partial }{\overline{x}}^{\mu }}{\mathrm{\partial }{x}^{\alpha }}{J}^{\alpha }det\left[\frac{\mathrm{\partial }{x}^{\sigma }}{\mathrm{\partial }{\overline{x}}^{\rho }}\right].}$

Verification of this transformation law:

So all that remains is to show that

${\displaystyle \frac{{\mathrm{\partial }}^2{\overline{x}}^{\nu }}{\mathrm{\partial }{\overline{x}}^{\nu }\mathrm{\partial }{x}^{\beta }}+\frac{\mathrm{\partial }{\overline{x}}^{\rho }}{\mathrm{\partial }{x}^{\sigma }}\frac{{\mathrm{\partial }}^2{x}^{\sigma }}{\mathrm{\partial }{x}^{\beta }\mathrm{\partial }{\overline{x}}^{\rho }}=0,}$

which is a version of a known theorem (seeInverse functions and differentiation § Higher derivatives).

The density of theLorentz force is a covariant vector density given by$${\displaystyle f_{\mu }=F_{\mu \nu }{J}^{\nu }.}$$

The force on a test particle subject only to gravity and electromagnetism is$${\displaystyle \frac{d{p}_{\alpha }}{dt}={\mathrm{\Gamma }}_{\alpha \gamma }^{\beta }{p}_{\beta }\frac{d{x}^{\gamma }}{dt}+q{F}_{\alpha \gamma }\frac{d{x}^{\gamma }}{dt},}$$wherep_α is the linear 4-momentum of the particle,t is any time coordinate parameterizing the world line of the particle, Γ^β_αγ is theChristoffel symbol (gravitational force field), andq is the electric charge of the particle.

This equation is invariant under a change in the time coordinate; just multiply by${\displaystyle dt/d\overline{t}}$ and use thechain rule. It is also invariant under a change in thex coordinate system.

Using the transformation law for the Christoffel symbol,$${\displaystyle {\overline{\mathrm{\Gamma }}}_{\alpha \gamma }^{\beta }=\frac{\mathrm{\partial }{\overline{x}}^{\beta }}{\mathrm{\partial }{x}^{ϵ}}\frac{\mathrm{\partial }{x}^{\delta }}{\mathrm{\partial }{\overline{x}}^{\alpha }}\frac{\mathrm{\partial }{x}^{\zeta }}{\mathrm{\partial }{\overline{x}}^{\gamma }}{\mathrm{\Gamma }}_{\delta \zeta }^{ϵ}+\frac{\mathrm{\partial }{\overline{x}}^{\beta }}{\mathrm{\partial }{x}^{\eta }}\frac{{\mathrm{\partial }}^2{x}^{\eta }}{\mathrm{\partial }{\overline{x}}^{\alpha }\mathrm{\partial }{\overline{x}}^{\gamma }},}$$we get

In vacuum, theLagrangian density for classical electrodynamics (in joules per cubic meter) is a scalardensity$${\displaystyle \mathcal{L}=-\frac{1}{4{\mu }_0}{F}_{\alpha \beta }{F}^{\alpha \beta }\frac{\sqrt{-g}}{c}+A_{\alpha }{J}^{\alpha },}$$where$${\displaystyle F^{\alpha \beta }=g^{\alpha \gamma }{F}_{\gamma \delta }{g}^{\delta \beta }.}$$

The 4-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables.

If we separate free currents from bound currents, the Lagrangian becomes$${\displaystyle \mathcal{L}=-\frac{1}{4{\mu }_0}{F}_{\alpha \beta }{F}^{\alpha \beta }\frac{\sqrt{-g}}{c}+A_{\alpha }{J}_{\text{free}}^{\alpha }+\frac{1}{2}{F}_{\alpha \beta }{\mathcal{M}}^{\alpha \beta }.}$$

As part of the source term in theEinstein field equations, the electromagneticstress–energy tensor is a covariant symmetric tensor$${\displaystyle T_{\mu \nu }=-\frac{1}{{\mu }_0}\left(F_{\mu \alpha }{g}^{\alpha \beta }{F}_{\beta \nu }-\frac{1}{4}{g}_{\mu \nu }{F}_{\sigma \alpha }{g}^{\alpha \beta }{F}_{\beta \rho }{g}^{\rho \sigma }\right),}$$using a metric of signature(−, +, +, +). If using the metric with signature(+, −, −, −), the expression for${\displaystyle T_{\mu \nu }}$ will have opposite sign. The stress–energy tensor is trace-free:$${\displaystyle T_{\mu \nu }{g}^{\mu \nu }=0}$$because electromagnetism propagates at the localinvariant speed, and is conformally invariant.^{[citation needed]}

In the expression for the conservation of energy and linear momentum, the electromagnetic stress–energy tensor is best represented as a mixed tensor density$${\displaystyle {\mathfrak{T}}_{\mu }^{\nu }=T_{\mu \gamma }{g}^{\gamma \nu }\frac{\sqrt{-g}}{c}.}$$

From the equations above, one can show that$${\displaystyle {{\mathfrak{T}}_{\mu }^{\nu }}_{;\nu }+f_{\mu }=0,}$$where the semicolon indicates acovariant derivative.

This can be rewritten as$${\displaystyle -{{\mathfrak{T}}_{\mu }^{\nu }}_{,\nu }=-{\mathrm{\Gamma }}_{\mu \nu }^{\sigma }{\mathfrak{T}}_{\sigma }^{\nu }+f_{\mu },}$$which says that the decrease in the electromagnetic energy is the same as the work done by the electromagnetic field on the gravitational field plus the work done on matter (via the Lorentz force), and similarly the rate of decrease in the electromagnetic linear momentum is the electromagnetic force exerted on the gravitational field plus the Lorentz force exerted on matter.

Derivation of conservation law:

which is zero because it is the negative of itself (see four lines above).

Thenonhomogeneous electromagnetic wave equation in terms of the field tensor is modified from thespecial-relativity form to^[2]

${\displaystyle ◻F_{ab}\stackrel{\text{def}}{=}{F}_{ab;}{}^d{}_d=-2R_{acbd}{F}^{cd}+R_{ae}{F}^e{}_b-R_{be}{F}^e{}_a+J_{a;b}-J_{b;a},}$

whereR_acbd is the covariant form of theRiemann tensor, and${\displaystyle ◻}$ is a generalization of thed'Alembertian operator for covariant derivatives. Using

${\displaystyle ◻A^a={{A^{a;}}^b}_b.}$

Maxwell's source equations can be written in terms of the4-potential [ref. 2^{[clarification needed]}, p. 569] as

${\displaystyle ◻A^a-{A^{b;a}}_b=-{\mu }_0{J}^a}$

or, assuming the generalization of theLorenz gauge in curved spacetime,

where${\displaystyle R_{ab}\stackrel{\text{def}}{=}{R^s}_{asb}}$ is theRicci curvature tensor.

This is the same form of the wave equation as in flat spacetime, except that the derivatives are replaced by covariant derivatives and there is an additional term proportional to the curvature. The wave equation in this form also bears some resemblance to the Lorentz force in curved spacetime, whereA^a plays the role of the 4-position.

For the case of a metric signature in the form(+, −, −, −), the derivation of the wave equation in curved spacetime is carried out in the article.^{[citation needed]}

When Maxwell's equations are treated in abackground-independent manner, that is, when the spacetime metric is taken to be a dynamical variable that depends on the electromagnetic field, then the electromagnetic wave equation and Maxwell's equations are nonlinear. This can be seen through the dependence of the curvature tensor on the stress–energy tensor through theEinstein field equation

${\displaystyle G_{ab}=\kappa T_{ab},}$

where

${\displaystyle G_{ab}\stackrel{\text{def}}{=}{R}_{ab}-\frac{1}{2}R{g}_{ab}}$

is theEinstein tensor,κ is theEinstein gravitational constant,g_ab is themetric tensor, andR is thescalar curvature, equal to the trace of the Ricci curvature tensor. The stress–energy tensor is composed of the stress–energy of all matter and fields, including the electromagnetic field. This mutual dependency is the origin of the nonlinearity.

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In the differential geometric formulation of the electromagnetic field, the antisymmetric Faraday tensor can be considered as theFaraday 2-form⁠${\displaystyle \mathbf{F}}$⁠. In this view, one of Maxwell's two equations is

${\displaystyle \mathrm{d}\mathbf{F}=0,}$

where${\displaystyle \mathrm{d}}$ is theexterior derivative operator. This equation is coordinate- and metric-independent and says that the electromagnetic flux through a closed two-dimensional surface in spacetime is topological, more precisely, constrained by itshomology class (a generalization of the integral form of Gauss law and Maxwell–Faraday equation, as the homology class in Minkowski space is automatically 0). By thePoincaré lemma, this equation implies that there exists locally a 1-form${\displaystyle \mathbf{A}}$ satisfying

${\displaystyle \mathbf{F}=\mathrm{d}\mathbf{A}.}$

The other equation is

${\displaystyle \mathrm{d}\star \mathbf{F}=\mathbf{J}.}$

In this context,${\displaystyle \mathbf{J}}$ is thecurrent 3-form, and the star${\displaystyle \star }$ denotes theHodge star operator. The dependence of Maxwell's equation on the metric of spacetime lies in the Hodge star operator${\displaystyle \star }$ on 2-forms, which isconformally invariant. Written this way, Maxwell's equation is the same in any spacetime, manifestly coordinate-invariant, and convenient to use (even in Minkowski space or Euclidean space and time, especially with curvilinear coordinates).

An alternative geometric interpretation is that the Faraday 2-form${\displaystyle \mathbf{F}}$ is (up to a factor⁠${\displaystyle i}$⁠) thecurvature 2-form${\displaystyle F(\mathrm{\nabla })}$ of a U(1)-connection${\displaystyle \mathrm{\nabla }}$ on aprincipal U(1)-bundle whose sections represent charged fields. The connection is much like the vector potential, since every connection can be written as${\displaystyle \mathrm{\nabla }={\mathrm{\nabla }}_0+iA}$ for a "base" connection⁠${\displaystyle {\mathrm{\nabla }}_0}$⁠, and

${\displaystyle \mathbf{F}={\mathbf{F}}_0+\mathrm{d}\mathbf{A}.}$

In this view, the Maxwell "equation"${\displaystyle \mathrm{d}\mathbf{F}=0}$ is a mathematical identity known as theBianchi identity. The equation${\displaystyle \mathrm{d}\star \mathbf{F}=\mathbf{J}}$ is the only equation with any physical content in this formulation.^{[citation needed]} This point of view is particularly natural when considering charged fields or quantum mechanics. It can be interpreted as saying that, much like gravity can be understood as being the result of the necessity of a connection to parallel transport vectors at different points, electromagnetic phenomena, or more subtle quantum effects like theAharonov–Bohm effect, can be understood as a result from the necessity of a connection to parallel transport charged fields or wave sections at different points. In fact, just as the Riemann tensor is theholonomy of theLevi-Civita connection along an infinitesimal closed curve, the curvature of the connection is the holonomy of the U(1)-connection.

• Electromagnetic wave equation
• Inhomogeneous electromagnetic wave equation
• Mathematical descriptions of the electromagnetic field
• Covariant formulation of classical electromagnetism
• Theoretical motivation for general relativity
• Introduction to the mathematics of general relativity
• Electrovacuum solution
• Paradox of radiation of charged particles in a gravitational field

• Einstein, A. (1961).Relativity: The Special and General Theory. New York: Crown.ISBN 0-517-02961-8.{{cite book}}:ISBN / Date incompatibility (help)
• Misner, Charles W.;Thorne, Kip S.;Wheeler, John Archibald (1973).Gravitation. San Francisco:W. H. Freeman.ISBN 0-7167-0344-0.
• Landau, L. D.;Lifshitz, E. M. (1975).Classical Theory of Fields (Fourth Revised English ed.). Oxford: Pergamon.ISBN 0-08-018176-7.
• Feynman, R. P.; Moringo, F. B.; Wagner, W. G. (1995).Feynman Lectures on Gravitation.Addison-Wesley.ISBN 0-201-62734-5.

• Electromagnetic fields in curved spacetimes

• General relativity
• Partial differential equations
• Maxwell's equations

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