In 3+1 dimensional flat spacetime, electromagnetic waves travel strictly on the null cone. How does one quantify this statement? It is through the retarded Green’s function of its wave operator.

In Lorentz covariant notation, the electromagnetic fields are encoded with the antisymmetric tensor , which in turn is built out of the vector potential through

(1)

If is the electromagnetic current, the electromagnetic fields themselves are sourced through the equation

(1)

Throughout this post, we shall assume the time-space coordinates and parametrizes some global Lorentzian inertial frames; i.e., they describe the invariant intervals

(1′) and

Suppose we have solved the massless scalar Green’s function , which obeys

(2)

where is the dimensional Dirac delta function; and suppose further we’ve managed to specify the initial spatial components of the vector potential and the electric field . It is then possible to express at any later time via the following Kirchhoff integral representation:

(3)

where the un-primed indices denote derivatives with respect to the observer location at and the primed ones the source location at .

Note that the magnetic field can be defined as ; hence, eq. (3) may be viewed a variant of the statement:

To determine the electric and magnetic fields at some later time , it suffices to specify them at the initial time .

**Huygens’ principle in 3+1D** Suppose we were dealing with free electromagnetic waves — i.e., homogeneous solutions to the wave equation, with — then we have

(3′)

In dimensions, the Green’s function is non-zero strictly on the light cone.

(3.1′)

When eq. (3.1′) is inserted into eq. (3′), we obtain the quantitative form of Huygens’ principle: the electromagnetic field at each point in space at the initial time will spread out in an infinitesimally thin spherical shell at the speed of light.

If we focus instead on the inhomogeneous solution — i.e., attribute entirely to , set all initial fields to zero, and send .

(3”)

What this statement says is: the electromagnetic field generated from the electric current at each spacetime point again propagates outwards in an infinitesimally thin spherical shell at the speed of light: .

**Other dimensions** In higher dimensions, the situation is a tad more complicated; splitting into even versus odd cases.

For even dimensions, the Green’s function again propagates signals strictly on the null cone. Hence, Huygens’ principle continues to hold; except the structure of the Green’s function does become more involved — see eq. (12) of this post.

In odd dimensions, the Green’s function — see eq. (12′) of the same post — now has a non-zero piece inside the light cone (). This inside-the-light-cone portion is known as the tail. Equations (3′), which really holds in all , tells us the homogeneous signal now receives contributions from inside the past light cone of the observer. And equation (3”) tells us the signal produced by the electric current at now travels inside the forward light cone.

Huygens’ principle is violated in odd dimensional flat spacetime; but respected in even dimensional ones (higher than 2).