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In [[special relativity]], [[electromagnetism]] and [[Wave|wave theory]], the '''d'Alembert operator''' <math>\Box^2</math>, also called the '''d'Alembertian''' or the '''wave operator''',  is the [[Laplace operator]] of [[Minkowski space]]. The operator is named for French mathematician and physicist [[Jean le Rond d'Alembert]]. In Minkowski space in standard coordinates (t, x, y, z) it has the form:

<math>\Box^2 = \partial^\mu \partial_\mu = g^{\mu\nu} \partial_\nu \partial_\mu = \left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}\right) = {\partial^2 \over \partial t^2} - \left(\nabla_{\mathbf{R}^3}^2 \right)    </math>

Here
* <math>\nabla_{\mathbf{R}^3}^2</math> is the three dimensional [[Laplacian]].
* <math> g^{\mu\nu}</math> is the [[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\not=\nu</math>.
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 <math>c=1</math>. Few authors also use the negative [[metric signature]] of [-+++] 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 is a [[Lorentz scalar]]. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.

==Alternate notations==

There is a variety of notations for the d'Alembertian. The most common is the symbol <math>\Box^2</math>: the four sides of the box representing the four dimensions of space-time. 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 <math>\Box^2</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.   In such case, the three sides of the triangular [[del|nabla]] may be taken to represent the three dimensions of space.  

==Applications==

The [[continuity equation]] for the [[four-current]] ''J'' = (''ρc'', '''j''')
:<math>\nabla_{\mathbf{R}^3} \cdot \mathbf{j} = -\frac{\partial \rho}{\partial t} </math>.

The [[Klein-Gordon equation]] is given by
:<math> (\Box^2 + m^2) \psi = 0 </math>.

The [[wave equation]] for the electromagnetic field is
:<math> \Box^2 \mathbf{A} = 0 </math>
where '''A''' is the [[vector potential]].

An alternate [[wave equation]] for small vibrations is:
:<math> \Box^2_{c} u\left(x,t\right) = 0 = u_{tt} - c^2u_{xx}</math>.

where <math>u\left(x,t\right)</math> is the displacement.

==Green's function==

The [[Green's function]] <math>G(x-x')</math> for the d'Alembertian is defined by the equation

:<math> \Box^2 G(x-x') = \delta(x-x')</math>

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

Explicitly we have

:<math>G(t,x,y,z) = \frac{1}{2\pi} \Theta(t) \delta(t^2 - x^2 - y^2 - z^2)</math>

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

==See Also==
*[[Relativistic heat conduction]]

==External links==

{{MathWorld | urlname=dAlembertian | title=d'Alembertian}}

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

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