D'Alembert operator

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

${\displaystyle \begin{array}{cc}◻& ={\mathrm{\partial }}^{\mu }{\mathrm{\partial }}_{\mu }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\mu }=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}\\ & =\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-\mathrm{\Delta }.\end{array}}$

Here ${\displaystyle {\mathrm{\nabla }}^2:=\mathrm{\Delta }}$ is the 3-dimensional Laplacian and η^μν is the inverse Minkowski metric with

${\displaystyle {\eta }_{00}=1}$, ${\displaystyle {\eta }_{11}={\eta }_{22}={\eta }_{33}=-1}$, ${\displaystyle {\eta }_{\mu \nu }=0}$ for ${\displaystyle \mu \ne \nu }$.

Note that the μ and ν summation indices range from 0 to 3: see Einstein notation.

(Some authors alternatively use the negative metric signature of (− + + +), with ${\displaystyle {\eta }_{00}=-1,{\eta }_{11}={\eta }_{22}={\eta }_{33}=1}$.)

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

The box symbol and alternate notations

There are a variety of notations for the d'Alembertian. The most common are the box symbol ${\displaystyle ◻}$ (Unicode: U+2610 ☐ BALLOT BOX) whose four sides represent the four dimensions of space-time and the box-squared symbol ${\displaystyle {◻}^2}$ which emphasizes the scalar property through the squared term (much like the Laplacian). In keeping with the triangular notation for the Laplacian, sometimes ${\displaystyle {\mathrm{\Delta }}_M}$ is used.

Another way to write the d'Alembertian in flat standard coordinates is ${\displaystyle {\mathrm{\partial }}^2}$. This notation is used extensively in quantum field theory, where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian.

Sometimes the box symbol is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol ${\displaystyle \mathrm{\nabla }}$ is then used to represent the space derivatives, but this is coordinate-chart-dependent.