D'Alembert Operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box: ), also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named for French mathematician and physicist Jean le Rond d'Alembert. In Minkowski space in standard coordinates (t, x, y, z) it has the form:

$\begin{align} \Box & = \partial^\mu \partial_\mu = g^{\mu\nu} \partial_\nu \partial_\mu = \frac{1}{c^{2}} \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} \\ & = \frac{1}{c^2} {\partial^2 \over \partial t^2} - \nabla^2 = \frac{1}{c^2}{\partial^2 \over \partial t^2} - \Delta. \end{align}$

Here is the inverse Minkowski metric with, for . 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 . Some authors also use the negative metric signature of with .

Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian is a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.

Read more about D'Alembert Operator: Alternate Notations, Applications, Green's Function