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. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime, 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 the macroscopic 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 flat Minkowski space when using local coordinates that are not rectilinear. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates. For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case of the general formulation.
Summary
In general relativity, the metric tensor is no longer a constant (like as in Examples of metric tensor) but can vary in space and time, and the equations of electromagnetism in a vacuum become
where is the density of the Lorentz force, is the inverse of the metric tensor , and is the determinant of the metric tensor. Notice that and are (ordinary) tensors, while , , and are tensor densities of weight +1. Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus, if one replaced the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out (see ).
The electromagnetic potential
The electromagnetic potential is a covariant vector Aα, which is the undefined primitive of electromagnetism. Being a covariant vector, it transforms from one coordinate system to another as
Electromagnetic field
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:
This definition implies that the electromagnetic field satisfies
which incorporates Faraday's law of induction and Gauss's law for magnetism. This is seen from
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
where a semicolon indicates a covariant derivative, a comma indicates a partial derivative, and square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is
where Γαβγ is the Christoffel symbol, which is symmetric in its lower indices.
Electromagnetic displacement
The electric displacement field D and the auxiliary magnetic field H form an antisymmetric contravariant rank-2 tensor density of weight +1. In a vacuum, this is given by
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 the speed 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 the magnetization–polarization tensor is non-zero, we have