In special relativity, electromagnetism and wave theory, the d'Alembert operator \Box, 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:

\Box = \partial^\mu \partial_\mu = \eta^{\mu\nu} \partial_\nu \partial_\mu = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} - \frac{\partial^2}{\partial t^2}\right) = \left(\nabla_{\mathbf{R}^3}^2 - {\partial^2 \over \partial t^2} \right)

Here

• \nabla_{\mathbf{R}^3}^2 is the three dimensional Laplacian.
• \eta^{\mu\nu} is the Minkowski metric with \eta^{00} = -1, \eta^{11}=\eta^{22}=\eta^{33}=1 .

Note that the ? and ? summation indices range from 0 to 3: see Einstein notation. We have assumed units such that the speed of light c=1. Many authors are also using the negative metric signature of [+---] with \eta^{00} = 1, \eta^{11}=\eta^{22}=\eta^{33}=-1 , in that case, one can put a minus sign in the above definition.

Lorentz transformations 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 \Box or \Box^2: the four sides of the box representing the four dimensions of space-time. In keeping with the triangular notation for the Laplacian sometimes \Delta_M is used.

Another way to write the d'Alembertian in flat standard coordinates is \partial^2. 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 \Box is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol \nabla is then used to represent the space derivatives, but this is coordinate chart dependent. In such case, the three sides of the triangular nabla may be taken to represent the three dimensions of space.

Applications

The continuity equation for the four-current J = (?c, j)

\nabla^2_{\mathbf{R}^3} \cdot \mathbf{j} = -\frac{\partial \rho}{\partial t}

can be written

\Delta^{1/2} \cdot J = 0.

The Klein-Gordon equation is given by

(\Box - m^2) \psi = 0 .

The sign of the m^2 term depends on the metric signature we use to define \Box.

The wave equation for the electromagnetic field is

\Box \mathbf{A} = 0

where A is the vector potential.

An alternate wave equation for small vibrations is:

\Box_{c} u\left(x,t\right) = 0 = u_{tt} - c^2u_{xx}.

where u\left(x,t\right) is the displacement.

Green's function

The Green's function G(x-x') for the d'Alembertian is defined by the equation

\Box G(x-x') = \delta(x-x')

where \delta(x-x') is the Dirac delta function and x and x' are two points in Minkowski space.

Explicitly we have

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

where \,\Theta is the Heaviside step function.

External links

ca:Operador de d'Alembert cs:D'Alembert?v operátor de:D?Alembertoperator es:D'Alambertiano fr:D'alembertien it:Operatore di d'Alembert pl:Operator d'Alemberta pt:Operador de d'Alembert ru:???????? ?????????? sk:D'Alembertov operátor sr:?'????????? ???????? zh:??????

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