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{{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''', or '''box operator''' 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 (using the box symbol notation)
: <math>
\begin{align}
\Box & = \partial^\mu \partial_\mu = g^{\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|''g<sup>μν</sup>''}}  is the inverse [[Minkowski metric]] with 
:<math>g_{00} = 1</math>, <math>g_{11} = g_{22} = g_{33} = -1</math>, <math>g_{\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]]. We have assumed units such that the speed of light {{mvar|c}} = 1.

(Some authors  alternatively use the negative [[metric signature]] of {{nowrap|(− + + +)}}, with <math>g_{00} = -1,\; g_{11} = g_{22} = g_{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 (<math>\Box</math>) and alternate notations==
There are a variety of notations for the d'Alembertian. The most common is the ''box'' symbol <math>\Box</math> ([[Unicode]]: {{unichar|2610|ballot box}}): the four sides of the box representing 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]]). This symbol is sometimes called the '''quabla''' (''cf''. [[nabla symbol]]). 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 <math>\Box</math> 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 (using box symbol notation)
:<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]].

The [[Klein–Gordon equation]] has the form
:<math>\left(\Box + m^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 (using 'box' symbol notation)
:<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|accessdate=2 January 2013}}</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==
*[[4-gradient]]
*[[d'Alembert's formula]]
*[[Klein–Gordon equation]]
*[[Relativistic heat conduction]]
*[[Ricci calculus]]

==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]]