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 (cf. nabla symbol) is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert, with the box notation being introduced by French mathematician Henri Poincaré in his lectures on electromagnetism. In Minkowski space, in standard coordinates (t, x, y, z), it has the form ◻ = ∂ μ ∂ μ = η μ ν ∂ ν ∂ μ = 1 c 2 ∂ 2 ∂ t 2 − ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 − ∂ 2 ∂ z 2 = 1 c 2 ∂ 2 ∂ t 2 − ∇ 2 = 1 c 2 ∂ 2 ∂ t 2 − Δ . {\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 ∇ 2 := Δ {\displaystyle \nabla ^{2}:=\Delta } is the 3-dimensional Laplacian and ημν is the inverse Minkowski metric with η 00 = 1 {\displaystyle \eta _{00}=1} , η 11 = η 22 = η 33 = − 1 {\displaystyle \eta _{11}=\eta _{22}=\eta _{33}=-1} , η μ ν = 0 {\displaystyle \eta _{\mu \nu }=0} for μ ≠ ν {\displaystyle \mu \neq \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 η 00 = − 1 , η 11 = η 22 = η 33 = 1...