Wave operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box: $\Box$ ), also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.

In Minkowski space, in standard coordinates $(t, x, y, z)$ , it has the form

Here ∇² is the 3-dimensional Laplacian and $g μν$ is the inverse Minkowski metric with

Note that the $μ$ and $ν$ summation indices range from 0 to 3: see Einstein notation. We have assumed units such that the speed of light $c$ = 1.

Some authors also use the negative metric signature of (− + + +), with $g_{00}=-1,\;g_{11}=g_{22}=g_{33}=1$ .