# D'Alembert operator

In special relativity, electromagnetism and wave theory, the **d'Alembert operator** (represented by a box: ), also called the **d'Alembertian** or the **wave 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 (*t*, *x*, *y*, *z*) it has the form:

Here is the 3-dimensional Laplacian and is the inverse Minkowski metric with , , for . 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 . Some authors also use the negative metric signature of [− + + +] with .

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 : the four sides of the box representing the four dimensions of space-time and the 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 is used.

Another way to write the d'Alembertian in flat standard coordinates is . 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 is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol 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

The wave equation for the electromagnetic field in vacuum is

- where is the electromagnetic four-potential.

The Klein–Gordon equation has the form

## Green's function

The Green's function for the d'Alembertian is defined by the equation

where is the multidimensional Dirac delta function and and are two points in Minkowski space.

A special solution is given by the *retarded Green's function* which corresponds
to signal propagation only forward in time

where is the Heaviside step function.

## See also

## References

## External links

- {{#invoke:citation/CS1|citation

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- Template:Cite wikisource, originally printed in Rendiconti del Circolo Matematico di Palermo.
- Weisstein, Eric W., "d'Alembertian",
*MathWorld*.