Editing D'Alembert operator

{{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.

==The box symbol and alternate notations==
There are a variety of notations for the d'Alembertian. The most common are the ''box'' symbol <math>\Box</math> ([[Unicode]]: {{unichar|2610|ballot box}}) whose four sides represent the four dimensions of space-time and the ''box-squared'' symbol <math>\Box^2</math> which emphasizes the scalar property through the squared term (much like the [[Laplacian]]). In keeping with the triangular notation for the [[Laplacian]], sometimes  <math>\Delta_M</math> is used.

Another way to write the d'Alembertian in flat standard coordinates is  <math>\partial^2</math>. 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 <math>\nabla</math> is then used to represent the space derivatives, but this is [[coordinate chart]] dependent.

==Applications==
The [[wave equation]] for small vibrations is of the form 
:<math> \Box_{c} u\left(x,t\right) \equiv u_{tt} - c^2u_{xx} = 0~,</math>
where {{math| ''u''(''x'', ''t'')}} is the displacement.

The [[wave equation]] for the electromagnetic field in vacuum is
:<math> \Box A^{\mu} = 0 </math>
where {{math|''A<sup>μ</sup>''}}  is the [[electromagnetic four-potential]] in [[Lorenz gauge condition|Lorenz gauge]].

The [[Klein–Gordon equation]] has the form
:<math>\left(\Box + \frac{m^2c^2}{\hbar^2}\right) \psi = 0~.</math>

==Green's function==
The [[Green's function]], <math>G\left(\tilde{x} - \tilde{x}'\right)</math>, for the d'Alembertian is defined by the equation 
:<math> \Box G\left(\tilde{x} - \tilde{x}'\right) = \delta\left(\tilde{x} - \tilde{x}'\right)</math>

where <math>\delta\left(\tilde{x} - \tilde{x}'\right)</math> is the multidimensional [[Dirac delta function]] and <math>\tilde{x}</math> and  <math>\tilde{x}'</math> are two points in Minkowski space.

A special solution is given by the ''retarded Green's function'' which corresponds to signal [[propagator|propagation]] only forward in time<ref>{{cite web|author=S. Siklos|title=The causal Green's function for the wave equation|url=http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/wave_equation.pdf|access-date=2 January 2013|archive-date=30 November 2016|archive-url=https://web.archive.org/web/20161130174612/http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/wave_equation.pdf|url-status=dead}}</ref>
:<math>G\left(\vec{r}, t\right) = \frac{1}{4\pi r} \Theta(t) \delta\left(t - \frac{r}{c}\right)</math>

where <math>\Theta</math> is the [[Heaviside step function]].

==See also==
*[[Four-gradient]]
*[[d'Alembert's formula]]
*[[Klein–Gordon equation]]
*[[Relativistic heat conduction]]
*[[Ricci calculus]]
*[[Wave equation]]

==References==
{{Reflist}}

==External links==
* {{springer|title=D'Alembert operator|id=p/d030080}}
* {{cite wikisource 
|title=Translation:On the Dynamics of the Electron (July)
|last=Poincaré
|first=Henri
|year=1906
}}, originally printed in [[Rendiconti del Circolo Matematico di Palermo]].
* {{MathWorld | urlname=dAlembertian | title=d'Alembertian}}

{{physics operators}}

[[Category:Differential operators]]
[[Category:Hyperbolic partial differential equations]]