d'Alembert operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: ${\displaystyle \Box }$), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator[1] (cf. nabla symbol) 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

${\displaystyle {\begin{aligned}\Box &=\partial ^{\mu }\partial _{\mu }=\eta ^{\mu \nu }\partial _{\nu }\partial _{\mu }={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}\\&={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-\nabla ^{2}={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-\Delta ~~.\end{aligned}}}$

Here ${\displaystyle \nabla ^{2}:=\Delta }$ 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 ${\displaystyle \mu \neq \nu }$.

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 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.