D'Alembert operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box: $\scriptstyle\Box$), 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:

$\begin{align} \Box & = \partial_\mu \partial^\mu = g_{\mu\nu} \partial^\nu \partial^\mu = \frac{1}{c} \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} \\ & = {\partial^2 \over \partial t^2} - \nabla^2 = {\partial^2 \over \partial t^2} - \Delta \end{align}$

Here $\scriptstyle g_{\mu\nu}$ is the Minkowski metric with $\scriptstyle g_{00} \,=\, 1$, $\scriptstyle g_{11} \,=\, g_{22} \,=\, g_{33} \,=\, -1$, $\scriptstyle g_{\mu\nu} \,=\, 0$ for $\scriptstyle\mu \,\neq\, \nu$. 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 $\scriptstyle c \,=\, 1$. Some authors also use the negative metric signature of [− + + +] with $\scriptstyle\eta_{00} \,=\, -1,\; \eta_{11} \,=\, \eta_{22} \,=\, \eta_{33} \,=\, 1$.

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 are a variety of notations for the d'Alembertian. The most common is the symbol $\scriptstyle\Box$: the four sides of the box representing the four dimensions of space-time and the $\scriptstyle\Box^2$ 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 $\scriptstyle\Delta_M$ is used.

Another way to write the d'Alembertian in flat standard coordinates is $\scriptstyle\partial^2$. 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 $\scriptstyle\Box$ is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol $\scriptstyle\nabla$ is then used to represent the space derivatives, but this is coordinate chart dependent.

Applications

The Klein–Gordon equation has the form

$(\Box + m^2) \psi = 0. \,$

The wave equation for the electromagnetic field in vacuum is

$\Box A^{\mu} = 0$

where A^μ is the electromagnetic four-potential.

The wave equation for small vibrations is of the form

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

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

Green's function

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

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

where $\scriptstyle\delta(x-x')$ is the Dirac delta function and $\scriptstyle x$ and $\scriptstyle x'$ 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 $\scriptstyle\,\Theta$ is the Heaviside step function.

See also

• Klein-Gordon equation
• Relativistic heat conduction

External links

• Weisstein, Eric W., "d'Alembertian" from MathWorld.