Maxwell's equations in curved spacetime

In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system.

But because general relativity dictates that the presence of electromagnetic fields (or energy/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.

Without this distinction, they are sometimes called the "microscopic" Maxwell's equations for contrast.

Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not rectilinear.

For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case of the general formulation.

as in Examples of metric tensor) but can vary in space and time, and the equations of electromagnetism in vacuum become [citation needed] where

Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations.

Being a covariant vector, its components transform from one coordinate system to another according to The electromagnetic field is a covariant antisymmetric tensor of degree 2, which can be defined in terms of the electromagnetic potential by

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

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.

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}.

In vacuum, this is given by This equation is the only place where the metric (and thus gravity) enters into the theory of electromagnetism.

Consequently, gravity can only affect electromagnetism by changing the speed of light relative to the global coordinate system being used.

So it is as if gravity increased the index of refraction of space near massive bodies.

More generally, in materials where the magnetization–polarization tensor is non-zero, we have The transformation law for electromagnetic displacement is where the Jacobian determinant is used.

In either case, the fact that the electromagnetic displacement is antisymmetric implies that the electric current is automatically conserved: because the partial derivatives commute.

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.

So all that remains is to show that which is a version of a known theorem (see Inverse functions and differentiation § Higher derivatives).

because electromagnetism propagates at the local invariant speed, and is conformal-invariant.

The nonhomogeneous electromagnetic wave equation in terms of the field tensor is modified from the special-relativity form to[2] where Racbd is the covariant form of the Riemann tensor, and

Using Maxwell's source equations can be written in terms of the 4-potential [ref.

2[clarification needed], p. 569] as or, assuming the generalization of the Lorenz gauge in curved spacetime, where

The wave equation in this form also bears some resemblance to the Lorentz force in curved spacetime, where Aa 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.

This equation is completely coordinate- and metric-independent and says that the electromagnetic flux through a closed two-dimensional surface in space–time is topological, more precisely, depends only on its homology 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 the Poincaré lemma, this equation implies (at least locally) that there exists a 1-form

The dependence of Maxwell's equation on the metric of spacetime lies in the Hodge star operator

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 the Aharonov–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.