Editing D'Alembert operator (section)

{{Short description|Second-order differential operator}}
{{distinguish|d'Alembert's principle|d'Alembert's equation}}
{{DISPLAYTITLE:d'Alembert operator}}
In [[special relativity]], [[electromagnetism]] and [[Wave|wave theory]], the '''d'Alembert operator''' (denoted by a box: <math>\Box</math>), also called the '''d'Alembertian''', '''wave operator''', '''box operator''' or sometimes '''quabla operator'''<ref>{{Cite book |url=https://www.worldcat.org/oclc/899608232 |title=Theoretische Physik |date=2015 |isbn=978-3-642-54618-1 |edition=Aufl. 2015 |location=Berlin, Heidelberg |oclc=899608232 |last1=Bartelmann |first1=Matthias |last2=Feuerbacher |first2=Björn |last3=Krüger |first3=Timm |last4=Lüst |first4=Dieter |last5=Rebhan |first5=Anton |last6=Wipf |first6=Andreas }}</ref> (''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 {{math|(''t'', ''x'', ''y'', ''z'')}}, it has the form
: <math>
\begin{align}
\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{align}
</math>

Here  <math> \nabla^2 := \Delta </math> is the 3-dimensional [[Laplace operator|Laplacian]] and {{math|''η<sup>μν</sup>''}}  is the inverse [[Minkowski metric]] with 
:<math>\eta_{00} = 1</math>, <math>\eta_{11} = \eta_{22} = \eta_{33} = -1</math>, <math>\eta_{\mu\nu} = 0</math> for <math>\mu \neq \nu</math>.
Note that the {{math|''μ''}} and {{math|''ν''}} summation indices range from 0 to 3: see [[Einstein notation]]. 

(Some authors  alternatively use the negative [[metric signature]] of {{nowrap|(− + + +)}}, with <math>\eta_{00} = -1,\; \eta_{11} = \eta_{22} = \eta_{33} = 1</math>.)

[[Lorentz transformation]]s 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.